Advances in neural networks - ISNN 2008 5th International Symposium on Neural Networks, ISNN 2008, Beijing, China, September 24-28, 2008: proceedings

Lecture Notes in Computer Science Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Alfred Kobsa University of California, Irvine, CA, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen University of Dortmund, Germany Madhu Sudan Massachusetts Institute of Technology, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany

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Fuchun Sun Jianwei Zhang Ying Tan Jinde Cao Wen Yu (Eds.)

Advances in Neural Networks – ISNN 2008 5th International Symposium on Neural Networks, ISNN 2008 Beijing, China, September 24-28, 2008 Proceedings, Part I

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Volume Editors Fuchun Sun Tsinghua University, Dept. of Computer Science and Technology Beijing 100084, China E-mail: [email protected] Jianwei Zhang University of Hamburg, Institute TAMS 22527 Hamburg, Germany E-mail: [email protected] Ying Tan Peking University, Department of Machine Intelligence Beijing 100871, China E-mail: [email protected] Jinde Cao Southeast University, Department of Mathematics Nanjing 210096, China E-mail: [email protected] Wen Yu Departamento de Control Automático, CINVESTAV-IPN México D.F., 07360, México E-mail: [email protected]

Library of Congress Control Number: 2008934862 CR Subject Classification (1998): F.1.1, I.2.6, I.5.1, H.2.8, G.1.6 LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues ISSN ISBN-10 ISBN-13

0302-9743 3-540-87731-2 Springer Berlin Heidelberg New York 978-3-540-87731-8 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2008 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper SPIN: 12529735 06/3180 543210

Preface

This book and its companion volume, LNCS vols. 5263 and 5264, constitute the proceedings of the 5th International Symposium on Neural Networks (ISNN 2008) held in Beijing, the capital of China, during September 24–28, 2008. ISNN is a prestigious annual symposium on neural networks with past events held in Dalian (2004), Chongqing (2005), Chengdu (2006), and Nanjing (2007). Over the past few years, ISNN has matured into a well-established series of international symposiums on neural networks and related fields. Following the tradition, ISNN 2008 provided an academic forum for the participants to disseminate their new research findings and discuss emerging areas of research. It also created a stimulating environment for participants to interact with each other and exchange information on future challenges and opportunities of neural network research. ISNN 2008 received 522 submissions from about 1,306 authors in 34 countries and regions (Australia, Bangladesh, Belgium, Brazil, Canada, China, Czech Republic, Egypt, Finland, France, Germany, Hong Kong, India, Iran, Italy, Japan, South Korea, Malaysia, Mexico, The Netherlands, New Zealand, Poland, Qatar, Romania, Russia, Singapore, South Africa, Spain, Switzerland, Taiwan, Turkey, UK, USA, Virgin Islands (UK)) across six continents (Asia, Europe, North America, South America, Africa, and Oceania). Based on rigorous reviews by the Program Committee members and reviewers, 192 high-quality papers were selected for publication in the proceedings with an acceptance rate of 36.7%. These papers were organized in 18 cohesive sections covering all major topics of neural network research and development. In addition to the contributed papers, the ISNN 2008 technical program included four plenary speeches by Dimitri P. Bertsekas (Massachusetts Institute of Technology, USA), Helge Ritter (Bayreuth University, Germany), Jennie Si (Arizona State University, USA), and Hang Li (Microsoft Research Asia, China). Besides the regular sessions and panels, ISNN 2008 also featured four special sessions focusing on some emerging topics. As organizers of ISNN 2008, we would like to express our sincere thanks to Tsinghua University, Peking University, The Chinese University of Hong Kong, and Institute of Automation at the Chinese Academy of Sciences for their sponsorship, to the IEEE Computational Intelligence Society, International Neural Network Society, European Neural Network Society, Asia Pacific Neural Network Assembly, the China Neural Networks Council, and the National Natural Science Foundation of China for their technical co-sponsorship. We thank the National Natural Science Foundation of China and Microsoft Research Asia for their financial and logistic support. We would also like to thank the members of the Advisory Committee for their guidance, the members of the International Program Committee and additional reviewers for reviewing the papers, and members of the Publications Committee for checking the accepted papers in a short period of time. In particular, we would

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Preface

like to thank Springer for publishing the proceedings in the prestigious series of Lecture Notes in Computer Science. Meanwhile, we wish to express our heartfelt appreciation to the plenary and panel speakers, special session organizers, session Chairs, and student helpers. In addition, there are still many more colleagues, associates, friends, and supporters who helped us in immeasurable ways; we express our sincere gratitude to them all. Last but not the least, we would like to thank all the speakers, authors, and participants for their great contributions that made ISNN 2008 successful and all the hard work worthwhile.

September 2008

Fuchun Sun Jianwei Zhang Ying Tan Jinde Cao Wen Yu

Organization

General Chair Bo Zhang, China

General Co-chair Jianwei Zhang, Germany

Advisory Committee Chairs Xingui He, China Yanda Li, China Shoujue Wang, China

Advisory Committee Members Hojjat Adeli, USA Shun-ichi Amari, Japan Zheng Bao, China Tianyou Chai, China Guoliang Chen, China Ruwei Dai, China Wlodzislaw Duch, Poland Chunbo Feng, China Walter J. Freeman, USA Kunihiko Fukushima, Japan Aike Guo, China Zhenya He, China Frank L. Lewis, USA Ruqian Lu, China Robert J. Marks II, USA Erkki Oja, Finland Nikhil R. Pal, India Marios M. Polycarpou, USA Leszek Rutkowski, Poland DeLiang Wang, USA Paul J. Werbos, USA Youshou Wu, China Donald C. Wunsch II, USA Youlun Xiong, China

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Organization

Lei Xu, Hong Kong Shuzi Yang, China Xin Yao, UK Gary G. Yen, USA Bo Zhang, China Nanning Zheng, China Jacek M. Zurada, USA

Program Committee Chairs Ying Tan, China Jinde Cao, China Wen Yu, Mexico

Steering Committee Chairs Zengqi Sun, China Jun Wang, China

Organizing Committee Chairs Fuchun Sun, China Zengguang Hou, China

Plenary Sessions Chair Derong Liu, USA

Special Sessions Chairs Xiaoou Li, Mexico Changyin Sun, China Cong Wang, China

Publications Chairs Zhigang Zeng, China Yunong Zhang, China

Publicity Chairs Andrzej Cichocki, Japan Alois Knoll, Germany Yi Shen, China

Organization

Finance Chair Yujie Ding, China Huaping Liu, China

Registration Chair Fengge Wu, China

Local Arrangements Chairs Lei Guo, China Minsheng Zhao, China

Electronic Review Chair Xiaofeng Liao, China

Steering Committee Members Shumin Fei, China Chengan Guo, China Min Han, China Xiaofeng Liao, China Baoliang Lu, China Zongben Xu, China Zhang Yi, China Hujun Yin, UK Huaguang Zhang, China Ling Zhang, China Chunguang Zhou, China

Program Committee Members Ah-Hwee Tan, Singapore Alan Liew, Australia Amir Hussain, UK Andreas Stafylopatis, Greece Andries Engelbrecht, South Africa Andrzej Cichocki, Japan Bruno Apolloni, Italy Cheng Xiang, Singapore Chengan Guo, China Christos Tjortjis, UK

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Organization

Chuandong Li, China Dacheng Tao, Hong Kong Daming Shi, Singapore Danchi Jiang, Australia Dewen Hu, China Dianhui Wang, Australia Erol Gelenbe, UK Fengli Ren, China Fuchun Sun, China Gerald Schaefer, UK Guangbin Huang, Singapore Haibo He, USA Haijun Jiang, China He Huang, Hong Kong Hon Keung Kwan, Canada Hongtao Lu, China Hongyong Zhao, China Hualou Liang, USA Huosheng Hu, UK James Lam, Hong Kong Jianquan Lu, China Jie Zhang, UK Jinde Cao, China Jinglu Hu, Japan Jinling Liang, China Jinwen Ma, China John Qiang Gan, UK Jonathan H. Chan, Thailand Jos´ e Alfredo F. Costa, Brazil Ju Liu, China K. Vijayan Asari, USA Kang Li, UK Khurshid Ahmad, UK Kun Yuan, China Liqing Zhang, China Luonan Chen, Japan Malik Ismail, USA Marco Gilli, Italy Martin Middendorf, Germany Matthew Casey, UK Meiqin Liu, China Michael Li, Australia Michel Verleysen, Belgium Mingcong Deng, Japan Nian Zhang, USA

Organization

Nikola Kasabov, New Zealand Norikazu Takahashi, Japan Okyay Kaynak, Turkey Paul S. Pang, New Zealand ´ P´eter Erdi, USA Peter Tino, UK Ping Guo, China Ping Li, Hong Kong Qiankun Song, China Qing Ma, Japan Qing Tao, China Qinglong Han, Australia Qingshan Liu, China Quanmin Zhu, UK Rhee Man Kil, Korea Rubin Wang, China Sabri Arik, Turkey Seiichi Ozawa, Japan Sheng Chen, UK Shunshoku Kanae, Japan Shuxue Ding, Japan Stanislaw Osowski, Poland Stefan Wermter, UK Sungshin Kim, Korea Tingwen Huang, Qatar Wai Keung Fung, Canada Wei Wu, China Wen Yu, Mexico Wenjia Wang, UK Wenlian Lu, China Wenwu Yu, Hong Kong Xiaochun Cheng, UK Xiaoli Li, UK Xiaoqin Zeng, China Yan Liu, USA Yanchun Liang, China Yangmin Li, Macao Yangquan Chen, USA Yanqing Zhang, USA Yi Shen, China Ying Tan, China Yingjie Yang, UK Zheru Chi, Hong Kong

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Organization

Reviewers Dario Aloise Ricardo de A. Araujo Swarna Arniker Mohammadreza Asghari Oskoei Haibo Bao simone Bassis Shuhui Bi Rongfang Bie Liu Bo Ni Bu Heloisa Camargo Liting Cao Jinde Cao Lin Chai Fangyue Chen Yangquan Chen Xiaofeng Chen Benhui Chen Sheng Chen Xinyu Chen Songcan Chen Long Cheng Xiaochun Cheng Zunshui Cheng Jungik Cho Chuandong Li Antonio J. Conejo Yaping Dai Jayanta Kumar Debnath Jianguo Du Mark Elshaw Christos Emmanouilidis Tolga Ensari Yulei Fan Mauricio Figueiredo Carlos H. Q. Foster Sabrina Gaito Xinbo Gao Zaiwu Gong Adilson Gonzaga Shenshen Gu Dongbing Gu Suicheng Gu Qianjin Guo

Jun Guo Chengan Guo Hong He Fengqing Han Wangli He Xiangnan He Yunzhang Hou Wei Hu Jin Hu Jun Hu Jinglu Hu Yichung Hu Xi Huang Chuangxia Huang Chi Huang Gan Huang He Huang Chihli Hung Amir Hussain Lei Jia Qiang Jia Danchi Jiang Minghui Jiang Lihua Jiang Changan Jinag Chi-Hyuck Jun Shunshoku Kanae Deok-Hwan Kim Tomoaki Kobayashi Darong Lai James Lam Bing Li Liping Li Chuandong Li Yueheng Li Xiaolin Li Kelin Li Dayou Li Jianwu Li Ping Li Wei Li Xiaoli Li Yongmin Li Yan Li

Organization

Rong Li Guanjun Li Jiguo Li Lulu Li Xuechen Li Jinling Liang Clodoaldo Aparecido de Moraes Lima Yurong Liu Li Liu Maoxing Liu Nan Liu Chao Liu Honghai Liu Xiangyang Liu Fei Liu Lixiong Liu Xiwei Liu Xiaoyang Liu Yang Liu Gabriele Lombardo Xuyang Lou Jianquan Lu Wenlian Lu Xiaojun Lu Wei Lu Ying Luo Lili Ma Shingo Mabu Xiangyu Meng Zhaohui Meng Cristian Mesiano Xiaobing Nie Yoshihiro Okada Zeynep Orman Stanislaw Osowski Tsuyoshi Otake Seiichi Ozawa Neyir Ozcan Zhifang Pan Yunpeng Pan Zhifang Pang Federico Pedersini Gang Peng Ling Ping Chenkun Qi

Jianlong Qiu Jianbin Qiu Dummy Reviewer Zhihai Rong Guangchen Ruan Hossein Sahoolizadeh Ruya Samli Sibel Senan Zhan Shu Qiankun Song Wei Su Yonghui Sun Junfeng Sun Yuan Tan Lorenzo Valerio Li Wan Lili Wang Xiaofeng Wang Jinlian Wang Min Wang Lan Wang Qiuping Wang Guanjun Wang Duan Wang Weiwei Wang Bin Wang Zhengxia Wang Haikun Wei Shengjun Wen Stefan Wermter Xiangjun Wu Wei Wu Mianhong Wu Weiguo Xia Yonghui Xia Tao Xiang Min Xiao Huaitie Xiao Dan Xiao Wenjun Xiong Junlin Xiong Weijun Xu Yan Xu Rui Xu Jianhua Xu

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Organization

Gang Yan Zijiang Yang Taicheng Yang Zaiyue Yang Yongqing Yang Bo Yang Kun Yang Qian Yin Xiuxia Yang Xu Yiqiong Simin Yu Wenwu Yu Kun Yuan Zhiyong Yuan Eylem Yucel Yong Yue Jianfang Zeng Junyong Zhai Yunong Zhang Ping Zhang Libao Zhang Baoyong Zhang

Houxiang Zhang Jun Zhang Qingfu Zhang Daoqiang Zhang Jiacai Zhang Yuanbin Zhang Kanjian Zhang Leina Zhao Yan Zhao Cong Zheng Chunhou Zheng Shuiming Zhong Jin Zhou Bin Zhou Qingbao Zhu Wei Zhu Antonio Zippo Yanli Zou Yang Zou Yuanyuan Zou Zhenjiang Zhao

Table of Contents – Part I

Computational Neuroscience Single Trial Evoked Potentials Study during an Emotional Processing Based on Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ling Zou, Renlai Zhou, Senqi Hu, Jing Zhang, and Yansong Li

1

Robust Speaker Modeling Based on Constrained Nonnegative Tensor Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qiang Wu, Liqing Zhang, and Guangchuan Shi

11

A Hypothesis on How the Neocortex Extracts Information for Prediction in Sequence Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weiyu Wang

21

MENN Method Applications for Stock Market Forecasting . . . . . . . . . . . . Guangfeng Jia, Yuehui Chen, and Peng Wu

30

New Chaos Produced from Synchronization of Chaotic Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zunshui Cheng

40

A Two Stage Energy Model Exhibiting Selectivity to Changing Disparity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaojiang Guo and Bertram E. Shi

47

A Feature Extraction Method Based on Wavelet Transform and NMFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suwen Zhang, Wanyin Deng, and Dandan Miao

55

Cognitive Science Similarity Measures between Connection Numbers of Set Pair Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Junjie Yang, Jianzhong Zhou, Li Liu, Yinghai Li, and Zhengjia Wu

63

Temporal Properties of Illusory-Surface Perception Probed with Poggendorff Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qin Wang and Marsanori Idesawa

69

Interval Self-Organizing Map for Nonlinear System Identification and Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luzhou Liu, Jian Xiao, and Long Yu

78

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Table of Contents – Part I

A Dual-Mode Learning Mechanism Combining Knowledge-Education and Machine-Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yichang Chen and Anpin Chen The Effect of Task Relevance on Electrophysiological Response to Emotional Stimuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baolin Liu, Shuai Xin, Zhixing Jin, Xiaorong Gao, Shangkai Gao, Renxin Chu, Yongfeng Huang, and Beixing Deng A Detailed Study on the Modulation of Emotion Processing by Spatial Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baolin Liu, Shuai Xin, Zhixing Jin, Xiaorong Gao, Shangkai Gao, Renxin Chu, Beixing Deng, and Yongfeng Huang

87

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107

Mathematical Modeling of Neural Systems MATLAB Simulation and Comparison of Zhang Neural Network and Gradient Neural Network for Time-Varying Lyapunov Equation Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yunong Zhang, Shuai Yue, Ke Chen, and Chenfu Yi

117

Improved Global Exponential Stability Criterion for BAM Neural Networks with Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yonggang Chen and Tiheng Qin

128

Global Exponential Stability and Periodicity of CNNs with Time-Varying Discrete and Distributed Delays . . . . . . . . . . . . . . . . . . . . . . . Shengle Fang, Minghui Jiang, and Wenfang Fu

138

Estimation of Value-at-Risk for Exchange Risk Via Kernel Based Nonlinear Ensembled Multi Scale Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kaijian He, Chi Xie, and Kinkeung Lai

148

Delay-Dependent Global Asymptotic Stability in Neutral-Type Delayed Neural Networks with Reaction-Diffusion Terms . . . . . . . . . . . . . . . . . . . . . Jianlong Qiu, Yinlai Jin, and Qingyu Zheng

158

Discrimination of Reconstructed Milk in Raw Milk by Combining Near Infrared Spectroscopy with Biomimetic Pattern Recognition . . . . . . . . . . . Ming Sun, Qigao Feng, Dong An, Yaoguang Wei, Jibo Si, and Longsheng Fu Data Fusion Based on Neural Networks and Particle Swarm Algorithm and Its Application in Sugar Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yanmei Meng, Sijie Yan, Zhihong Tang, Yuanling Chen, and Jingneng Liu

168

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Table of Contents – Part I

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Asymptotic Law of Likelihood Ratio for Multilayer Perceptron Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joseph Rynkiewicz

186

An On-Line Learning Radial Basis Function Network and Its Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nini Wang, Xiaodong Liu, and Jianchuan Yin

196

A Hybrid Model of Partial Least Squares and RBF Neural Networks for System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nini Wang, Xiaodong Liu, and Jianchuan Yin

204

Nonlinear Complex Neural Circuits Analysis and Design by q-Value Weighted Bounded Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hong Hu and Zhongzhi Shi

212

Fuzzy Hyperbolic Neural Network Model and Its Application in H∞ Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shuxian Lun, Zhaozheng Guo, and Huaguang Zhang

222

On the Domain Attraction of Fuzzy Neural Networks . . . . . . . . . . . . . . . . . Tingwen Huang, Xiaofeng Liao, and Hui Huang

231

CG-M-FOCUSS and Its Application to Distributed Compressed Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhaoshui He, Andrzej Cichocki, Rafal Zdunek, and Jianting Cao

237

Dynamic of Cohen-Grossberg Neural Networks with Variable Coefficients and Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xuehui Mei and Haijun Jiang

246

Permutation Free Encoding Technique for Evolving Neural Networks . . . Anupam Das, Md. Shohrab Hossain, Saeed Muhammad Abdullah, and Rashed Ul Islam

255

Six-Element Linguistic Truth-Valued Intuitionistic Reasoning in Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Li Zou, Wenjiang Li, and Yang Xu

266

A Sequential Learning Algorithm for RBF Networks with Application to Ship Inverse Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gexin Bi and Fang Dong

275

Stability and Nonlinear Analysis Implementation of Neural Network Learning with Minimum L1 -Norm Criteria in Fractional Order Non-gaussian Impulsive Noise Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daifeng Zha

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Table of Contents – Part I

Stability of Neural Networks with Parameters Disturbed by White Noises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wuyi Zhang and Wudai Liao

291

Neural Control of Uncertain Nonlinear Systems with Minimum Control Effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dingguo Chen, Jiaben Yang, and Ronald R. Mohler

299

Three Global Exponential Convergence Results of the GPNN for Solving Generalized Linear Variational Inequalities . . . . . . . . . . . . . . . . . . . Xiaolin Hu, Zhigang Zeng, and Bo Zhang

309

Disturbance Attenuating Controller Design for a Class of Nonlinear Systems with Unknown Time-Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geng Ji

319

Stability Criteria with Less Variables for Neural Networks with Time-Varying Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tao Li, Xiaoling Ye, and Yingchao Zhang

330

Robust Stability of Uncertain Neural Networks with Time-Varying Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wei Feng, Haixia Wu, and Wei Zhang

338

Novel Coupled Map Lattice Model for Prediction of EEG Signal . . . . . . . Minfen Shen, Lanxin Lin, and Guoliang Chang

347

Adaptive Synchronization of Delayed Chaotic Systems . . . . . . . . . . . . . . . . Lidan Wang and Shukai Duan

357

Feedforward and Fuzzy Neural Networks Research on Fish Intelligence for Fish Trajectory Prediction Based on Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yanmin Xue, Hongzhao Liu, Xiaohui Zhang, and Mamoru Minami

364

A Hybrid MCDM Method for Route Selection of Multimodal Transportation Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lili Qu and Yan Chen

374

Function Approximation by Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . Fengjun Li

384

Robot Navigation Based on Fuzzy RL Algorithm . . . . . . . . . . . . . . . . . . . . . Yong Duan, Baoxia Cui, and Huaiqing Yang

391

Table of Contents – Part I

Nuclear Reactor Reactivity Prediction Using Feed Forward Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shan Jiang, Christopher C. Pain, Jonathan N. Carter, Ahmet K. Ziver, Matthew D. Eaton, Anthony J.H. Goddard, Simon J. Franklin, and Heather J. Phillips Active Noise Control Using a Feedforward Network with Online Sequential Extreme Learning Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qizhi Zhang and Yali Zhou

XIX

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410

Probabilistic Methods A Probabilistic Method to Estimate Life Expectancy of Application Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shengzhong Yuan and Hong He

417

Particle Filter with Improved Proposal Distribution for Vehicle Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huaping Liu and Fuchun Sun

422

Cluster Selection Based on Coupling for Gaussian Mean Fields . . . . . . . . Yarui Chen and Shizhong Liao Multiresolution Image Fusion Algorithm Based on Block Modeling and Probabilistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chenglin Wen and Jingli Gao An Evolutionary Approach for Vector Quantization Codebook Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carlos R.B. Azevedo, Esdras L. Bispo Junior, Tiago A.E. Ferreira, Francisco Madeiro, and Marcelo S. Alencar Kernel-Based Text Classification on Statistical Manifold . . . . . . . . . . . . . . Shibin Zhou, Shidong Feng, and Yushu Liu A Boost Voting Strategy for Knowledge Integration and Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Haibo He, Yuan Cao, Jinyu Wen, and Shijie Cheng

432

442

452

462

472

Supervised Learning A New Strategy for Pridicting Eukaryotic Promoter Based on Feature Boosting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shuanhu Wu, Qingshang Zeng, Yinbin Song, Lihong Wang, and Yanjie Zhang Searching for Interacting Features for Spam Filtering . . . . . . . . . . . . . . . . . Chuanliang Chen, Yunchao Gong, Rongfang Bie, and Xiaozhi Gao

482

491

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Table of Contents – Part I

Structural Support Vector Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hui Xue, Songcan Chen, and Qiang Yang

501

The Turning Points on MLP’s Error Surface . . . . . . . . . . . . . . . . . . . . . . . . . Hung-Han Chen

512

Parallel Fuzzy Reasoning Models with Ensemble Learning . . . . . . . . . . . . . Hiromi Miyajima, Noritaka Shigei, Shinya Fukumoto, and Toshiaki Miike

521

Classification and Dimension Reduction in Bank Credit Scoring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bohan Liu, Bo Yuan, and Wenhuang Liu Polynomial Nonlinear Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . JinFeng Wang, KwongSak Leung, KinHong Lee, and Zhenyuan Wang Testing Error Estimates for Regularization and Radial Function Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Petra Vidnerov´ a and Roman Neruda

531 539

549

Unsupervised Learning A Practical Clustering Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wei Li, Haohao Li, and Jianye Chen

555

Concise Coupled Neural Network Algorithm for Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lijun Liu, Jun Tie, and Tianshuang Qiu

561

Spatial Clustering with Obstacles Constraints by Hybrid Particle Swarm Optimization with GA Mutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xueping Zhang, Hui Yin, Hongmei Zhang, and Zhongshan Fan

569

Analysis of the Kurtosis-Sum Objective Function for ICA . . . . . . . . . . . . . Fei Ge and Jinwen Ma

579

BYY Harmony Learning on Weibull Mixture with Automated Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhijie Ren and Jinwen Ma

589

A BYY Split-and-Merge EM Algorithm for Gaussian Mixture Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lei Li and Jinwen Ma

600

A Comparative Study on Clustering Algorithms for Multispectral Remote Sensing Image Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lintao Wen, Xinyu Chen, and Ping Guo

610

Table of Contents – Part I

A Gradient BYY Harmony Learning Algorithm for Straight Line Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gang Chen, Lei Li, and Jinwen Ma

XXI

618

Support Vector Machine and Kernel Methods An Estimation of the Optimal Gaussian Kernel Parameter for Support Vector Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wenjian Wang and Liang Ma

627

Imbalanced SVM Learning with Margin Compensation . . . . . . . . . . . . . . . Chan-Yun Yang, Jianjun Wang, Jr-Syu Yang, and Guo-Ding Yu

636

Path Algorithms for One-Class SVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liang Zhou, Fuxin Li, and Yanwu Yang

645

Simulations for American Option Pricing Under a Jump-Diffusion Model: Comparison Study between Kernel-Based and Regression-based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyun-Joo Lee, Seung-Ho Yang, Gyu-Sik Han, and Jaewook Lee

655

Global Convergence Analysis of Decomposition Methods for Support Vector Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jun Guo and Norikazu Takahashi

663

Rotating Fault Diagnosis Based on Wavelet Kernel Principal Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Guo, G.M. Dong, J. Chen, Y. Zhu, and Y.N. Pan

674

Inverse System Identification of Nonlinear Systems Using LSSVM Based on Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Changyin Sun, Chaoxu Mu, and Hua Liang

682

A New Approach to Division of Attribute Space for SVR Based Classification Rule Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dexian Zhang, Ailing Duan, Yanfeng Fan, and Ziqiang Wang

691

Chattering-Free LS-SVM Sliding Mode Control . . . . . . . . . . . . . . . . . . . . . . Jianning Li, Yibo Zhang, and Haipeng Pan

701

Selection of Gaussian Kernel Parameter for SVM Based on Convex Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Changqian Men and Wenjian Wang

709

Multiple Sources Data Fusion Strategies Based on Multi-class Support Vector Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luo Zhong, Zhe Li, Zichun Ding, Cuicui Guo, and Huazhu Song

715

XXII

Table of Contents – Part I

A Generic Diffusion Kernel for Semi-supervised Learning . . . . . . . . . . . . . . Lei Jia and Shizhong Liao

723

Weighted Hyper-sphere SVM for Hypertext Classification . . . . . . . . . . . . . Shuang Liu and Guoyou Shi

733

Theoretical Analysis of a Rigid Coreset Minimum Enclosing Ball Algorithm for Kernel Regression Estimation . . . . . . . . . . . . . . . . . . . . . . . . . Xunkai Wei and Yinghong Li

741

Kernel Matrix Learning for One-Class Classification . . . . . . . . . . . . . . . . . . Chengqun Wang, Jiangang Lu, Chonghai Hu, and Youxian Sun

753

Structure Automatic Change in Neural Network . . . . . . . . . . . . . . . . . . . . . Han Honggui, Qiao Junfei, and Li Xinyuan

762

Hybrid Optimisation Algorithms Particle Swarm Optimization for Two-Stage FLA Problem with Fuzzy Random Demands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yankui Liu, Siyuan Shen, and Rui Qin T-S Fuzzy Model Identification Based on Chaos Optimization . . . . . . . . . Chaoshun Li, Jianzhong Zhou, Xueli An, Yaoyao He, and Hui He ADHDP for the pH Value Control in the Clarifying Process of Sugar Cane Juice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiaofeng Lin, Shengyong Lei, Chunning Song, Shaojian Song, and Derong Liu Dynamic PSO-Neural Network: A Case Study for Urban Microcosmic Mobile Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chaozhong Wu, Chengwei Xu, Xinping Yan, and Jing Gong An Improvement to Ant Colony Optimization Heuristic . . . . . . . . . . . . . . . Youmei Li, Zongben Xu, and Feilong Cao Extension of a Polynomial Time Mehrotra-Type Predictor-Corrector Safeguarded Algorithm to Monotone Linear Complementarity Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mingwang Zhang and Yanli Lv QoS Route Discovery of Ad Hoc Networks Based on Intelligence Computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cong Jin and Shu-Wei Jin Memetic Algorithm-Based Image Watermarking Scheme . . . . . . . . . . . . . . Qingzhou Zhang, Ziqiang Wang, and Dexian Zhang

776 786

796

806 816

826

836 845

Table of Contents – Part I

A Genetic Algorithm Using a Mixed Crossover Strategy . . . . . . . . . . . . . . Li-yan Zhuang, Hong-bin Dong, Jing-qing Jiang, and Chu-yi Song Condition Prediction of Hydroelectric Generating Unit Based on Immune Optimized RBFNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhong Liu, Shuyun Zou, Shuangquan Liu, Fenghua Jin, and Xuxiang Lu

XXIII

854

864

Synthesis of a Hybrid Five-Bar Mechanism with Particle Swarm Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ke Zhang

873

Robust Model Predictive Control Using a Discrete-Time Recurrent Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yunpeng Pan and Jun Wang

883

A PSO-Based Method for Min-ε Approximation of Closed Contour Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bin Wang, Chaojian Shi, and Jing Li

893

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

903

Table of Contents – Part II

Machine Learning and Data Mining Rough Set Combine BP Neural Network in Next Day Load Curve Forecasting

1

Improved Fuzzy Clustering Method Based on Entropy Coefficient and Its Application

11

An Algorithm of Constrained Spatial Association Rules Based on Binary

21

Sequential Proximity-Based Clustering for Telecommunication Network Alarm Correlation

30

A Fast Parallel Association Rules Mining Algorithm Based on FP-Forest

40

Improved Algorithm for Image Processing in TCON of TFT-LCD

50

Clustering Using Normalized Path-Based Metric

57

Association Rule Mining Based on the Semantic Categories of Tourism Information

67

The Quality Monitoring Technology in the Process of the Pulping Papermaking Alkaline Steam Boiling Based on Neural Network

74

A New Self-adjusting Immune Genetic Algorithm

81

Calculation of Latent Semantic Weight Based on Fuzzy Membership

91

Research on Spatial Clustering Acetabuliform Model and Algorithm Based on Mathematical Morphology

100

Intelligent Control and Robotics Partner Selection and Evaluation in Virtual Research Center Based on Trapezoidal Fuzzy AHP

110

A Nonlinear Hierarchical Multiple Models Neural Network Decoupling Controller

119

Adaptive Dynamic Programming for a Class of Nonlinear Control Systems with General Separable Performance Index

128

A General Fuzzified CMAC Controller with Eligibility

138

Case-Based Decision Making Model for Supervisory Control of Ore Roasting Process

148

An Affective Model Applied in Playmate Robot for Children

158

The Application of Full Adaptive RBF NN to SMC Design of Missile Autopilot

165

Multi-Objective Optimal Trajectory Planning of Space Robot Using Particle Swarm Optimization

171

The Direct Neural Control Applied to the Position Control in Hydraulic Servo System

180

An Application of Wavelet Networks in the Carrying Robot Walking

190

TOPN Based Temporal Performance Evaluation Method of Neural Network Based Robot Controller

200

A Fuzzy Timed Object-Oriented Petri Net for Multi-Agent Systems

210

Fuzzy Reasoning Approach for Conceptual Design

220

Extension Robust Control of a Three-Level Converter for High-Speed Railway Tractions

227

Pattern Recognition Blind Image Watermark Analysis Using Feature Fusion and Neural Network Classifier

237

Gene Expression Data Classification Using Independent Variable Group Analysis

243

The Average Radius of Attraction Basin of Hopfield Neural Networks

253

A Fuzzy Cluster Algorithm Based on Mutative Scale Chaos Optimization

259

A Sparse Sampling Method for Classification Based on Likelihood Factor

268

Estimation of Nitrogen Removal Effect in Groundwater Using Artificial Neural Network

276

Sequential Fuzzy Diagnosis for Condition Monitoring of Rolling Bearing Based on Neural Network

284

Evolving Neural Network Using Genetic Simulated Annealing Algorithms for Multi-spectral Image Classification

294

Detecting Moving Targets in Ground Clutter Using RBF Neural Network

304

Application of Wavelet Neural Networks on Vibration Fault Diagnosis 313 321 331

341

Audio, Image Processing and Computer Vision Denoising Natural Images Using Sparse Coding Algorithm Based on the Kurtosis Measurement

351

A New Denoising Approach for Sound Signals Based on Non-negative Sparse Coding of Power Spectra

359

Building Extraction Using Fast Graph Search

367

376 Image Denoising Using Neighbouring Contourlet Coefficients

384

Robust Watermark Algorithm Based on the Wavelet Moment Modulation and Neural Network Detection

392

Manifold Training Technique to Reconstruct High Dynamic Range Image

402

Face Hallucination Based on CSGT and PCA

410

Complex Effects Simulation Based Large Particles System on GPU

419

A Selective Attention Computational Model for Perceiving Textures

429

Classifications of Liver Diseases from Medical Digital Images

439

A Global Contour-Grouping Algorithm Based on Spectral Clustering

449

Emotion Recognition in Chinese Natural Speech by Combining Prosody and Voice Quality Features

457

Fault Diagnosis On-Line Diagnosis of Faulty Insulators Based on Improved ART2 Neural Network

465

Diagnosis Method for Gear Equipment by Sequential Fuzzy Neural Network

473

Study of Punch Die Condition Discrimination Based on Wavelet Packet and Genetic Neural Network

483

Data Reconstruction Based on Factor Analysis

492

Synthetic Fault Diagnosis Method of Power Transformer Based on Rough Set Theory and Bayesian Network

498

Fuzzy Information Fusion Algorithm of Fault Diagnosis Based on Similarity Measure of Evidence

506

Other Applications and Implementations NN-Based Near Real Time Load Prediction for Optimal Generation Control

516

A Fuzzy Neural-Network-Driven Weighting System for Electric Shovel

526

Neural-Network-Based Maintenance Decision Model for Diesel Engine

533

Design of Intelligent PID Controller Based on Adaptive Genetic Algorithm and Implementation of FPGA

542

Fragile Watermarking Schemes for Tamperproof Web Pages

552

Real-Time Short-Term Traffic Flow Forecasting Based on Process Neural Network

560

Fuzzy Expert System to Estimate Ignition Timing for Hydrogen Car

570

Circuitry Analog and Synchronization of Hyperchaotic Neuron Model

580

A Genetic-Neural Method of Optimizing Cut-Off Grade and Grade of Crude Ore

588

A SPN-Based Delay Analysis of LEO Satellite Networks

598

Research on the Factors of the Urban System Influenced Post-development of the Olympics’ Venues

607

A Stock Portfolio Selection Method through Fuzzy Delphi

615

A Prediction Algorithm Based on Time Series Analysis

624

Applications of Neural Networks in Electronic Engineering An Estimating Traffic Scheme Based on Adaline

632

SVM Model Based on Particle Swarm Optimization for Short-Term Load Forecasting

642

A New BSS Method of Single-Channel Mixture Signal Based on ISBF and Wavelet

650

A Novel Pixel-Level and Feature-Level Combined Multisensor Image Fusion Scheme

658

Combining Multi Wavelet and Multi NN for Power Systems Load Forecasting

666

An Adaptive Algorithm Finding Multiple Roots of Polynomials

674

Cellular Neural Networks and Advanced Control with Neural Networks Robust Designs for Directed Edge Overstriking CNNs with Applications

682

Application of Local Activity Theory of Cellular Neural Network to the Chen’s System

692

Application of PID Controller Based on BP Neural Network Using Automatic Differentiation Method

702

Neuro-Identifier-Based Tracking Control of Uncertain Chaotic System

712

Robust Stability of Switched Recurrent Neural Networks with Discrete and Distributed Delays under Uncertainty

720

Nature Inspired Methods of High-dimensional Discrete Data Analysis WHFPMiner: Efficient Mining of Weighted Highly-Correlated Frequent Patterns Based on Weighted FP-Tree Approach

730

Towards a Categorical Matching Method to Process High-Dimensional Emergency Knowledge Structures

740

Identification and Extraction of Evoked Potentials Based on Borel Spectral Measure for Less Trial Mixtures

748

A Two-Step Blind Extraction Algorithm of Underdetermined Speech Mixtures

757

A Semi-blind Complex ICA Algorithm for Extracting a Desired Signal Based on Kurtosis Maximization

764

Fast and Efficient Algorithms for Nonnegative Tucker Decomposition

772

Pattern Recognition and Information Processing Using Neural Networks Neural Network Research Progress and Applications in Forecast

783

Adaptive Image Segmentation Using Modified Pulse Coupled Neural Network

794

Speech Emotion Recognition System Based on BP Neural Network in Matlab Environment

801

Broken Rotor Bars Fault Detection in Induction Motors Using Park’s Vector Modulus and FWNN Approach

809

Coal and Gas Outburst Prediction Combining a Neural Network with the Dempster-Shafter Evidence

822

Using the Tandem Approach for AF Classification in an AVSR System

830

Author Index

841

Single Trial Evoked Potentials Study during an Emotional Processing Based on Wavelet Transform Ling Zou1,2,3, Renlai Zhou2,3,*, Senqi Hu4, Jing Zhang2, and Yansong Li2 1

Faculty of Information Science & Engineering, Jiangsu Polytechnic University, Changzhou, Jiangsu, 213164, China 2 State Key Laboratory of Cognitive Neuroscience and Learning, Beijing Normal University, Beijing, 100875, China 3 Beijing Key Lab of Applied Experimental Psychology, Beijing, 100875, China 4 Department of Psychology, Humboldt State University {Ling Zou,Renlai Zhou,Senqi Hu,Jing Zhang,Yansong Li, rlzhou}@bnu.edu.cn

Abstract. The present study aimed at examining the event-related potentials (ERPs) single-trial extraction during an emotional processing by wavelet transform and analyzing the brain responses to emotional stimuli. ERPs were recorded from 64 electrodes in 10 healthy university students while three types of emotional pictures (pleasant, neural, and unpleasant) from the international affective picture system were presented. All the subjects showed significantly greater P300 and slow waves amplitudes at antero-inferior, medial-inferior and posterior electrode sites for pleasant and unpleasant pictures than for neural pictures and unpleasant pictures elicited more positive P300 and slow wave effects than pleasant pictures. The results indicated the effectiveness of the wavelet transform-based approach in ERP single-trial extraction and further supported the view that emotional stimuli are processed more intensely. Keywords: ERPs, wavelet transform, emotion, P300, slow wave.

1 Introduction In recent years there has been growing interest in understanding brain mechanisms subserving emotion, brain asymmetries related to emotion, and the influence of emotion on memory [1-3]. A number of neuroimaging studies have been published on investigating brain response to the passive viewing of affective pictures by using many methods [4-6]. Among these studies, event related potentials (ERPs) of the electroencephalogram (EEG) were widely used for its non-invasive and readily available to community clinics [5-6]. Investigations of evoked potentials to emotional visual stimuli have revealed higher cortical positivity in response to emotional compared with neutral stimuli. Radilovà studied evoked potentials in response to emotional pictures, and found that unpleasant, Compared to neutral, visual stimuli elicited more robust P300 effects [7]. Radilovà and his coworkers also reported that a *

Corresponding author.

F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 1–10, 2008. © Springer-Verlag Berlin Heidelberg 2008

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significantly greater P300 for erotic compared with non-erotic scenes leading to the suggestion that the arousing quality of emotional stimuli produces a heightened P300 component independently of the valence of the stimuli [8]. Similarly, other scientists reported significantly larger evoked potentials to arousing vs. neutral stimuli [6, 9]. Interestingly, the effects occurred primarily from frontal to parietal recording sites, also associated with P300 generation [10]. Keil et. Al. demonstrated that both the P300 and the late positive slow wave show an arousal-related signal enhancement with largest differences in late VEPs as a function of emotional arousal near PZ electrode [5-6]. The most common way to visualize the ERPs is to take an average over time locked single-trial measurements. The implicit assumption in the averaging is that the task-related cognitive process does not vary much in timing from trial to trial. However, it has been evident for a few decades that in many cases this assumption is not valid. The observation of variation in the parameters of the ERPs permits the dynamic assessment of changes in cognitive state. Thus, the goal in the analysis of ERPs is currently the estimation of the single potentials, that we call single-trial extraction. Several techniques have been proposed to improve the visualization of the ERPs from the background EEG with various successes [11-13]. Among these ways, wavelet transform (WT) was taken as the promising one for its optimal resolution both in the time and in the frequency domain. WT has been used to decompose the ERP signal onto a space with basis functions. With this technique the ERP is assumed to be the result of the superimposition of wave packets in various frequencies with varying degrees of frequency stabilization, enhancement and time locking within conventional frequency bands of the ongoing EEG activity such as delta, theta, alpha and gamma ranges. The wavelet analysis treats ERP responses in time frequency plane and has yielded to new knowledge about ERP components [14]. However, the selection of relevant frequency band and the interpretation of the results in frequency domain is a challenging task. In this paper, we first estimated single-trial ERPs during an emotional processing based on wavelet multiresolution analysis (MRA) [15], and then analyzed the brain responses to emotional stimuli by using the extracted ERPs. The results indicated the effectiveness of the wavelet transform-based approach in ERP single-trial extraction and further supported the view that emotional stimuli are processed more intensely.

2 Materials and Methods 2.1 Subjects Ten health undergraduate students (7men) from Beijing normal University with aged from 19 to 25 years old Participated in the experiment. All subjects were right-handed with normal or correct-normal vision. 2.2 Stimuli and Design 210 colorful pictures were selected from the International Affective Picture System (IAPS), consisting of 70 highly arousing pleasant, 70 neural, and 70 highly arousing unpleasant images. The pictures were chosen according to the normative ratings of the

Single Trial Evoked Potentials Study during an Emotional Processing

3

IAPS. The order of the pictures was arranged so that 70 neural, 70 pleasant and 70 unpleasant pictures were shown respectively in each block. Emotional pictures were presented on a 19-in. computer screen with a refresh rate of 60Hz. The screen was placed approximately 1.5m in front of the viewer. Each picture was presented for 1000 ms, with inter-trial intervals varying between 2500 and 3000 ms. After the EEG recordings of each type emotion stimuli, subjects were asked to rate the respective picture on a 11-point scale, among them, 0 means no sense while 100 means very pleasant or unpleasant. 2.3 Electrophysiological Recordings EEG activity was recorded continuously from 64 leads with a DC amplifier in AC mode (bandpass: 0.01-100Hz; SYNAMPS, Neuroscan) and digitized at a rate of 500 Hz. For each trial, 1.1s of data was saved on a hard disc (from 0.1 s pre- to 1 s poststimulation). Horizontal and vertical electrooculograms (EOG) were recorded by electrodes placed above and below the left eye (VEOG) and lateral to the outer canthus of each eye (HEOG). Offline, the EEG was re-referenced to linked mastoids. For the purpose of statistical analysis, the mean voltages of the averaged visually evoked potentials (VEPs) were obtained with horizontal plane (anterior, medial, posterior), and vertical plane (inferior, superior), based on recording sites of the international 1020 system [16]. The locations of these regions with respect to sites of the international 10-20 system are shown in Fig. 1. 2.3 Multiresolution Analysis: Discrete Wavelets Transform (DWT) DWT is a time-frequency analysis technique that is most suited for non-stationary signal such as the ERPs. DWT analyzes the signal at different frequency bands with different resolutions by decomposing the signal into a coarse approximation and detail information. DWT employs two sets of functions, called scaling functions and wavelet functions, which are associated with lowpass and highpass filters, respectively. The decomposition of the signal into different frequency bands is simply obtained by successive highpass and lowpass filtering of the time domain signal. The original signal x [n] is first passed through a half-band highpass filter g [n] and a lowpass filter h[n]. After the filtering, half of the samples can be eliminated according to Nyquist’s rule, since the signal now has a highest frequency of π/2 radians instead of π. The signal can therefore be subsampled by 2, simply by discarding every other sample. This constitutes one level of decomposition and can mathematically be expressed as follows:

y high [k ] = ∑ x[n] ⋅ g[2k − n] .

(1)

ylow [k ] = ∑ x[n] ⋅ h[2k − n] .

(2)

n

n

Where

yhigh [k ] and ylow [k ] are the outputs of the highpass and lowpass filters after the

subsampling, and are referred to as detail coefficients and approximation coefficients, respectively. This procedure is repeated by decomposing the approximation coefficients

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until further decomposition is not possible. The detail coefficients di at the level I then constitute Level i DWT coefficients. At each level, the successive filtering and subsampling result in half the time resolution and double the frequency resolutions, hence multiresolution analysis.

Fig. 1. Layout of the electrode array. 1, 2(left/right antero-superior); 3, 4(left/right anteroinferior); 5, 6(left/right medial-superior); 7, 8(left/right medial-inferior); 9, 10(left/right postero-inferior); 11, 12(left/right postero-superior).

In this study, we chose the Daubechies wavelets as the basic wavelet functions for their simplicity and general purpose applicability in a variety of time-frequency representation problems [17]. According to the sampling frequency of 500 Hz, a 6 level decomposition was used, thus having 6 scales of details (d1-d6) and a final approximation (a6).

3 Results Fig. 2 showed the seven signals obtained from 6-level decomposition of a sample VEP and the reconstruct single-trial signal (from a subject during unpleasant pictures stimuli process from PO3 electrode site). For a single-trial signal x[n] from P1 electrode and using Daubechies-5 wavelets, these levels correspond to the following frequency bands: d1:125-250Hz, d2:62.5-125Hz, d3:31.5-62.5Hz (Gamma), d4:15.2-31.3Hz (Beta),

Single Trial Evoked Potentials Study during an Emotional Processing

5

d5:7.8-15.6 Hz (Alpha), d6: 3.9-7.8 Hz (Theta), a6:0.1-3.9 Hz (Delta). The wavelet transform yielded 280 coefficients in d1, 144 in d2, 76 in d3, 42 in d4, 25 in d5, 17 in d6 and 17 in a6. Then, we calculated the wavelet energy for each frequency band introduced in [18] and we got the relative energy values in percentage to reflect the probability distribution of energy at different resolution levels. In this case, delta band preserved most of the signal energy (approx. 72% at PO3, which indicated that the waveform morphology is determined predominantly by this band. In order to capture an adequate proportion of the signal energy, the theta band was also included into the analysis. Combined, the delta and theta band preserve 77% of the signal energy at site PO3. In our study, the combination of delta and theta band preserve over 67% (i.e. at least two thirds) of the signal energy at all electrodes sites. The delta band corresponded to the approximation level (a6) of the MRA while the theta band corresponded to the highest detail level (d6). All activity from frequency bands higher than the theta band was suppressed by setting corresponding wavelet coefficients to zero and subsequent inverse transform to the time domain. Delta and theta frequencies had been proven very important in the generation of the P3 response to auditory stimuli [11, 14]. Fig.3 showed the time-frequency distributions for the above sample VEP and its wavelet-based VEP estimate, respectively. The unpleasant stimuli appeared at 0 s. The same axis range for the amplitude is used here. Visual-related activity is clearly

Fig. 2. Sample VEP decomposition and reconstruction. The original signal was shown at the uppermost panel in the left column. Left column showed the decomposed signal reflecting the time course of the signal in the respective frequency band. Right column showed the reconstructed single-trial signal by the sum of a6 and d6.

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Fig. 3. Sample results for the time-frequency plot of a single trial of VEP. Left column corresponding the original signal. Right column corresponding the reconstructed single-trial signal by wavelet transform.

noticeable in the time-frequency distribution of the wavelet-based VEP estimate, whereas such activity can hardly be seen from the raw signal. Therefore, we conclude that the wavelet-based method can recover the evoked potential. To investigate that the different brain areas are activated and compare the VEPs components of the same brain area during the processing of different emotions, the above described wavelet transform method was applied to the single trials of each subject at 62 electrodes described in Fig.1 (not included the VEOG and HEOG sites). For each subject the results of the wavelet decomposition of the 15 single trials were averaged, and then the grand mean visually evoked potentials (VEPs) under the three types of stimuli of the 10 subjects were obtained for 12 scalp areas. Mean voltages in these regions were assessed in the P300 (300-500 ms) and in the slow wave window (550-900 ms) [5-6, 16]. Fig. 4 showed the grand mean VEPs at PZ site were composed of five components: A N100, a P200, a N200, a P300 component and a late positive slow wave. Here, we focused on the P300 and slow wave time window indication the sustained and highlevel processing of salient visual stimuli [6, 16]. Grand average ERPs to unpleasant, pleasant, and neutral stimuli are presented in Fig.5. Here, we selected the P3/4, C3/4, CP3/4, F3/4, PO7/8, PZ, CZ, CPZ, FZ and OZ electrodes, which distributed in the antero-inferior, medial-inferior, postero-inferio and postero-superio scalp areas. Table.1 showed the mean and standard deviation (SD) of the P300 window (300500 ms) amplitude of the grand-average VEPs in response to the three types of emotional stimuli on the postero, medial-inferior and antero-inferior sites as shown in Fig.1. Table.2 showed the mean and standard deviation (SD) of the slow wave window (550-900 ms) under the same conditions.

Single Trial Evoked Potentials Study during an Emotional Processing

7

Fig. 4. Grand-average VEPs at the electrode PZ in response to three types of emotional stimuli

Fig. 5. Grand-average VEPs in response to three types of emotional stimuli at P3/4, C3/4, CP3/4, F3/4, PO7/8, PZ, CZ, CPZ, FZ and OZ electrode sites

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Table 1. Mean amplitude and standard deviation (SD) of the P300 window of the grandaverage VEPs in response to three types of stimuli at different electrode sites

Electrodes sites

unpleasant Mean SD ( v) ( v) 5.27 0.38 4.47 0.28 0.65 0.53

μ

Postero-inferio Postero-superio Medial-inferior (up) (C1,C3, C5, C2, C4,C6) Medial-inferior (down) 3.63 (CP1,CP3,CP5, CP2, CP4, CP6) Antero-inferior -1.61

Neural Mean ( v) 0.42 0.60 -0.01

μ

SD ( v) 0.39 0.25 0.55

pleasant Mean ( v) 3.49 3.47 -1.06

μ

SD ( v) 0.37 0.27 0.45

0.40

0.16

0.56

1.59

0.23

0.33

-0.18

0.50

-2.54

0.41

μ

μ

μ

Table 2. Mean amplitude and standard deviation (SD) of the Slow wave window of the grandaverage VEPs in response to three types of stimuli at different electrode sites

Electrodes sites

unpleasant Mean SD ( v) ( v) 2.51 0.79 2.11 0.67 1.50 0.17

μ

Postero-inferio Postero-superio Medial-inferior (up) (C1,C3, C5, C2, C4,C6) Medial-inferior (down) 2.57 (CP1,CP3,CP5, CP2, CP4, CP6) Antero-inferior 0.05

Neural Mean ( v) -0.00 0.12 0.22

μ

SD ( v) 0.55 0.08 0.12

pleasant Mean ( v) 1.45 1.53 0.26

μ

SD ( v) 0.07 0.52 0.70

0.50

-0.05

0.06

1.44

0.23

0.55

0.14

0.14

-0.02

0.68

μ

μ

μ

From Fig 5, Table.1 and Table.2, we can see the distribution of P300 and slow wave over the scalp areas. The positive P300 and slow wave were found greatest over posterior sites, both inferior and superior, as well as medial-inferior (down) sites. Negativity of the P300 and slow wave were over anterior-inferior and medial-inferior (up) sites. The distribution areas were the same as the described in [15]. The results demonstrated the voltages evoked by unpleasant images and pleasant pictures were greater than neural pictures. Statistical analyses showed unpleasant pictures evoked greater positive P300 voltages than pleasant pictures (e.g. at postero-infero sites under the unpleasant stimuli, MeanP300=5.27μv, SDP300=0.38μv, while under the pleasant stimuli, MeanP300 = 3.49μv, SDP300 = 0.37μv) ; Pleasant pictures evoked greater negative P300 voltages than unpleasant pictures(e.g. at antero-inferior sites under the unpleasant stimuli, MeanP300=-1.61μv, SDP300=0.33μv, while under the pleasant stimuli, MeanP300 = -2.54 μv, SDP300=0.41μv ); Unpleasant pictures evoked greater slow wave voltages than pleasant pictures (e.g. at the medial-inferior(up) sites under the unpleasant stimuli, Mean Slow=1.50μv, SDSlow=0.17μv, while under the pleasant stimuli, MeanSlow=0.26μv, SDSlow=0.70μv ), which were different from [16].

Single Trial Evoked Potentials Study during an Emotional Processing

9

4 Discussion In this paper we have pursued two complimentary goals: (1) to improve the visualization of the single-trial ERPs based on wavelet transform method and seek its application to cognitive VEPs; (2) to analyze the brain responses to emotional stimuli by using the extracted single-trial VEPs. Firstly, we used MRA method to estimate the single-trial VEPs by keeping the wavelet coefficients of low frequency bands (delta band and theta band) and then reconstructed the original experimental example signal. We then got the grand mean VEPs at scalp areas by applying the above wavelet-based approach to the 10 subjects, each subjects including 15 trials. The results showed that the VEPs obtained by wavelet method could be used as a reliable, sensitive, and high-resolution indicator for emotion study after only 15 trials of ensemble averaging. Secondly, our results showed greater P300 and slow wave amplitudes for unpleasant and pleasant pictures compared to neural stimuli, indicating that motivationally relevant stimuli automatically direct attentional resources, are processed more deeply and thus provoke an arousan-related enhancement of VEPs, which further support the view that emotional stimuli are processed more intensely. We observed greater P300 and greater slow wave amplitudes for unpleasant pictures compared to pleasant pictures, while in previous studies, pleasant pictures evoked the greatest P300 as well as slow wave amplitudes [6, 15].Our results based on the estimated single-trial VEPs also showed significantly greater P300 and slow waves amplitudes at antero-inferior, medial-inferior and posterior electrode sites for pleasant and unpleasant pictures than for neural pictures. These findings are in accordance with results demonstrating the largest differences in late VEPs as a function of emotional arousal for electrode sites near PZ [5, 6, 10]. The MRA of the wavelet transform method enables the latency and the amplitude of the VEP to be detected more accurately, while it is difficult to achieve this performance using other methods based solely on either a time-domain or frequencydomain approach [12, 13]. In addition, the WT method can significantly reduce the number of stimuli required for detection of small VEPs. The wavelet method suggested in this paper therefore has great potential for clinical cognitive practicability. In the future work, we’re going to explore how other factors (e.g. sex, hemisphere) influence emotion perception by using WT method or WT-based methods. Acknowledgments. This work was supported by the open project of State Key Laboratory of Cognitive Neuroscience and Learning, Beijing Normal University, and Jiangsu Education Nature Foundation (07KJD510038).

References 1. Renlai, Z., Senqi, H.: Effects of Viewing Pleasant and Unpleasant Photographs on Facial EMG Asymmetry. Perceptual and Motor Skills 99, 1157–1167 (2004) 2. Gasbarri, A., Arnone, B., Pompili, A., Marchetti, A., Pacitti, F., Saadcalil, S., Pacitti, C., Tavares, M.C., Tomaz, C.: Sex-related Lateralized Effect of Emotional Content on Declarative Memory: an Event Related Potential Study. Behav. Brain Res. 168, 177–184 (2006)

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3. Wiens, S.: Interoception in Emotional Experience. Curr. Opin. Neurol. 18, 442–447 (2005) 4. Phan, K.L., Wager, T., Taylor, S.F., Liberzon, I.: Functional Neuroanatomy of Emotion: a Meta-analysis of Emotion Activation Studies in PET and fMRI. NeuroImage 16, 331–348 (2002) 5. Keil, A., Müller, M.M., Gruber, T., Stolarova, M., Wienbruch, C., Elbert, T.: Effects of emotional arousal in the cerebral hemispheres: a study of oscillatory brain activity and event-related potentials. Clin. Neurophysiol. 112, 2057–2068 (2001) 6. Cuthberg, B., Schupp, H., Bradley, M., Birbaumer, N., Lang, P.: Brain Potentials in Affective Picture Processing: Covariation with Autonomic Arousal and Affective Report. Biol. Psychol. 52, 95–111 (2000) 7. Radilovà, J.: The Late Positive Components of Visual Evoked Responses Sensitive to Emotional Factors. Act. Nerv. Super. (suppl. 3), 334 (1982) 8. Radilovà, J.: P300 and the Emotional States Studied by Psycho Physiological Methods. Int. J. Psychophysiol. 7, 364–366 (1989) 9. Dolcos, F., Cabeza, R.: Event-related Potentials of Emotional Memory: Encoding Pleasant, Unpleasant, and Neutral Pictures. Cogn. Affect. Behav. Neurosci. 2, 252–263 (2002) 10. Polich, J., Kok, A.: Cognitive and biological determinants of P300: an integrative review. Biol. Psychol. 41, 103–146 (1995) 11. Roth, A., Roesch-Ely, D., Bender, S., Weisbrod, M., Kaiser, S.: Increased Event-related Potential Latency and Amplitude Variability in Schizophrenia Detected through Waveletbased Single Trial Analysis. Int. J. Psychophysiology 66, 244–254 (2007) 12. Vorobyov, S., Cichocki, A.: Blind noise reduction for multisensory signals using ICA and subspace filtering with application to EEG analysis. Biol. Cybern. 86, 293–303 (2002) 13. Yin, H.E., Zeng, Y.J., Zhang, J.H.: Application of adaptive noise cancellation with neuralnetwork-based fuzzy inference system for visual evoked potentials estimation. Med. Eng. Phys. 26, 87–92 (2004) 14. Demiralp, T., Ademoglu, A., Istefanopulos, Y., Basar-Eroglu, C., Basar, E.: Wavelet analysis of oddball P300. Int. J. Psychophysiology 39, 221–227 (2001) 15. Mallat, S.: A Theory for Multiresolution Signal Decomposition: the Wavelet Representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(7), 674–693 (1989) 16. Herbert, B.M., Pollatos, O., Schandre, R.: Interoceptive Sensitivity and Emotion Processing: An EEG study. Int. J. Psychophysiology 65, 214–227 (2007) 17. Polikar, R., Topalis, A., Green, D., Kounios, J., Clark, C.M.: Comparative Multiresolution Wavelet Analysis of ERP Spectral Bands Using an Ensemble of Classifiers Approach for Early Diagnosis of Alzheimer’s Disease. Computers in Biology and Medicine 37, 542–556 (2007) 18. Rosso, O.A., Blanco, S., Yordanova, J., Kolev, V., Figliola, A., Schürmann, M., Basar, E.: Wavelet entropy: a new tool for analysis of short duration brain electrical signals. J. Neuro. Met. 105, 65–75 (2001)

Robust Speaker Modeling Based on Constrained Nonnegative Tensor Factorization Qiang Wu, Liqing Zhang, and Guangchuan Shi Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China {johnnywu,lqzhang,sgc1984}@sjtu.edu.cn

Abstract. Nonnegative tensor factorization is an extension of nonnegative matrix factorization(NMF) to a multilinear case, where nonnegative constraints are imposed on the PARAFAC/Tucker model. In this paper, to identify speaker from a noisy environment, we propose a new method based on PARAFAC model called constrained Nonnegative Tensor Factorization (cNTF). Speech signal is encoded as a general higher order tensor in order to learn the basis functions from multiple interrelated feature subspaces. We simulate a cochlear-like peripheral auditory stage which is motivated by the auditory perception mechanism of human being. A sparse speech feature representation is extracted by cNTF which is used for robust speaker modeling. Orthogonal and nonsmooth sparse control constraints are further imposed on the PARAFAC model in order to preserve the useful information of each feature subspace in the higher order tensor. Alternating projection algorithm is applied to obtain a stable solution. Experiments results demonstrate that our method can improve the recognition accuracy specifically in noise environment.

1 Introduction Speaker recognition is the task of determining the identification of a person from one’s voice which has great potential applications in industry, business and security, etc. For a speaker recognition system, feature extraction is one of important tasks, which aims at finding succinct, robust, and discriminative features from acoustic data. Acoustic features such as linear predictive cepstral coefficients (LPCC)[1], mel-frequency cepstral coefficients (MFCC)[1], perceptual linear predictive coefficients (PLP) [2] are commonly used. The conventional speaker modeling methods such as Gaussian mixture models(GMM)[3] achieve very high performance for speaker identification and verification tasks on high-quality data when training and testing conditions are well controlled. However, in the real application such systems usually do not perform well for a large variety of speech signals corrupted by adverse conditions such as environmental noise and channel distortions. Feature compensation techniques [2,4] such as CMS, RASTA have been developed for robust speech recognition. Spectral subtraction [5] and subspacebased filtering[6] techniques assuming a priori knowledge of the noise spectrum have been widely used because of their simplicity. Recently the computational auditory nerve models and sparse coding attract much attention from both neuroscience and speech signal processing communities. Smith et al.[7] proposed an algorithm for learning efficient auditory codes using a theoretical model for coding sound in terms of spikes.Much research F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 11–20, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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about sparse coding and representation for sound and speech[8,9,10] is also proved to be useful for auditory modeling and speech separation which will be a potential way for robust speech feature extraction. As a powerful data modeling tool for pattern recognition, multilinear algebra of the higher order tensors has been proposed as a potent mathematical framework to manipulate the multiple factors underlying the observations. Currently common tensor decomposition methods include: (1) the CANDECOMP/PARAFAC model [11,12,13]; (2) the Tucker Model[14,15]; (3) Nonnegative Tensor Factorization (NTF) which imposes the nonnegative constraint on the CANDECOMP/PARAFAC model [16,17]. In computer vision applications, Multilinear ICA [18]and tensor discriminant analysis [19] are applied to image representation and recognition, which improve recognition performance. In this paper, we proposed a new feature extraction method for robust speaker recognition based on auditory periphery model and tensor factorization. A novel tensor factorization method called cNTF is derived by imposing orthogonal and nonnegative constraints on the tensor structure. The advantages of our feature extraction method include following: (1) simulation of the auditory perception mechanism of human being provides a higher frequency resolution at low frequencies which helps to obtain robust spectro-temporal feature; (2) a supervised feature extraction procedure via cNTF learns the basis functions of multi-related feature subspaces which preserve the individual, spectro-temporal information in the tensor structure; furthermore the orthogonal constraint ensures redundancy minimization between different basis functions; (3) sparse constraint on cNTF enhances energy concentration of speech signal which will preserve the useful feature during the noise reduction. The sparse tensor feature extracted by cNTF can be further processed into a representation called auditory-based nonnegative tensor feature(ANTF) via discrete cosine transform, which can be used as feature for speaker recognition.

2 Method 2.1 Multilinear Algebra and PARAFAC Model Multilinear algebra is the algebra of higher order tensors. A tensor is a higher order generalization of a matrix. Let X ∈ RN1 ×N2 ×...×NM denotes a tensor. The order of X is M . An element of X is denoted by xn1 ,n2 ,...,nM , where 1 ≤ nd ≤ Nd and 1 ≤ d ≤ M . The mode-d matricization or matrix unfolding of an M th-order tensor X ∈ RN1 ×N2 ×...×NM rearranges the elements of X to form the matrix X(d) ∈ RNd ×Nd+1 Nd+2 ···NM N1 ···Nd−1 , which is the ensemble of vectors in RNd obtained by keeping index nd fixed and varying the other indices. Matricizing a tensor is similar to vectoring a matrix. The PARAFAC model was suggested independently by Carroll and Chang[11] under the name CANDECOMP(canonical decomposition) and by Harshman[12] under the name PARAFAC(parallel factor analysis) which has gained increasing attention in the data mining field. This model has structural resemblance with many physical models of common real-world data and its uniqueness property implies that the data following the PARAFAC model can be uniquely decomposed into individual contributions.

Robust Speaker Modeling Based on Constrained Nonnegative Tensor Factorization

13

An M -way tensor X ∈ RN1 ×N2 ×...×NM can be decomposed into a sum of M rank-1 terms, i.e. represented by the outer product of M vectors: X = a(1) ◦ a(2) ◦ · · · ◦ a(M) ,

(1)

where ◦ is the outer product operator, a(d) ∈ RNd , for d = 1, 2, . . . , M . The rank of tensor X , denoted R = rank(X ), is the minimal number of rank-1 tensors that is required to yield X : X =

R 

(2) (M) A(1) :,r ◦ A:,r ◦ · · · ◦ A:,r ,

(2)

r=1

where A:,r represents the rth column vector of the mode matrix A(d) ∈ RNd ×R . The PARAFAC model aims to find a rank-R approximation of the tensor X , (d)

X ≈

R 

(2) (M) A(1) :,r ◦ A:,r ◦ · · · ◦ A:,r ,

(3)

r=1

The PARAFAC model can also be written in matrix notation by use of the Khatri-Rao product, which gives the equivalent expressions:  T X(d) ≈ A(d) A(d−1)  . . .  A(1)  A(M)  . . .  A(d+1) ,

(4)

where  is the Khatri-Rao product operator. 2.2 Constrained Nonnegative Tensor Factorization Given a nonnegative M -way tensor X ∈ RN1 ×N2 ×...×NM , nonnegative tensor factorization(NTF) seeks a factorization of X in the form: X ≈ Xˆ =

R 

(2) (M) A(1) :,r ◦ A:,r ◦ · · · ◦ A:,r ,

(5)

r=1

where the mode matrices A(d) ∈ RNd ×R for d = 1, . . . , M are restricted to have only nonnegative elements in the factorization. In order to find an approximate tensor factorization Xˆ , we can construct Least Square cost function JLS and KL-divergence cost function JKL based on the approximate factorization model (4). The cost functions with mode matrices A(d) are given by 1 X(d) − A(d) Z(d) 2F 2 M

JLS1 (A(d) ) =

d=1

=

Nd¯  M Nd  1 

2

d=1 p=1 q=1

[X(d) ]pq − [A(d) Z(d) ]pq

2 (6)

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Q. Wu, L. Zhang, and G. Shi

JKL1 (A(d) ) =

M 

D(X(d) A(d) Z(d) )

d=1

Nd¯  Nd  M   = [X(d) ]pq log d=1 p=1 q=1

[X(d) ]pq − [X(d) ]pq + [A(d) Z(d) ]pq [A(d) Z(d) ]pq



(7)

T M where Z(d) = A(d−1)  . . .  A(1)  A(M)  . . .  A(d+1) and Nd¯ = j=d Nj . These cost functions are quite similar to NMF[20], which performs matrix factorization in each mode and minimizes the error for all modes. By above model, we can add additional constraint which makes the basis functions be as orthogonal as possible, i.e. ensures redundancy minimization between different basis This orthogonal constraint can be imposed by minimizing the formula functions. (d)T (d) [A A ]pq . p=q For the traditional NMF methods, many approaches have been proposed to control the sparsenses by additional constraints or penalization terms. These constraints or penalizations can be applied to the basis vectors or both basis and encoding vectors. The nsNMF model[22] proposed a factorization model V = WSH, providing a smoothing matrix S ∈ Rq×q given by θ S = (1 − θ)I + 11T (8) q where I is the identify matrix, 1 is a vector of ones, and the parameter θ satisfies 0 ≤ θ ≤ 1. For θ = 0, the model(8) is equivalent to the original NMF. As θ → 1, stronger smoothness is imposed on S, leading to a strong sparseness on both W and H. By this nonsmooth approach, we can control the sparseness of basis vectors and encoding vectors and maintain the faithfulness of the model to the data. The same idea can be applied to the NTF. Then the corresponding cost functions with orthogonal and sparse control constraints can be given by ⎛ ⎞ Nd¯  Nd  M 2    1 ⎝ [X(d) ]pq − [A(d) SZ(d) ]pq + α JLS2 (A(d) ) = [A(d)T A(d) ]pq ⎠ 2 p=1 q=1 d=1

JKL2 (A(d) ) =

M  d=1

+α

p=q

(9)

⎛

Nd¯  Nd   ⎝ [X(d) ]pq log p=1 q=1



⎞

[A(d)T A(d) ]pq ⎠

[X(d) ]pq − [X(d) ]pq + [A(d) SZ(d) ]pq [A(d) SZ(d) ]pq



(10)

p=q

where α > 0 is a balancing parameter between reconstruction and orthogonality. We can derive multiplicative learning algorithms for mode matrices A(d) using the exponential gradient, which are similar to those in NMF. Updating algorithms in an element-wise manner for minimizing the cost function (9) and (2.2) are directly derived as done in [16,17]:

Robust Speaker Modeling Based on Constrained Nonnegative Tensor Factorization

15

– LS: (d)

(d)

Aij ← Aij

[X(d) Z(d)T ST ]ij [A(d) SZ(d) Z(d)T ST ]ij + α p=j [A(d)T ]pi

(11)

– KL:

(d) Aij

←

(d) Aij

[X

]

(d) ik [SZ(d) ]jk [A(d) SZ (d) ] ik (d) ] (d)T ] pi k [SZ jk + α p=j [A

k

(12)

3 Feature Extraction Based on Auditory Model and Tensor Representation As we know, human auditory system is of powerful capability in speech recognition and speaker recognition. Much of research on auditory model has already shown that the features based on simulation of auditory system are more robust than traditional features under noisy background. In our feature extraction framework, we calculate the frequency selectivity information by imitating the process performed in the auditory periphery and pathway. And the robust speech features are obtained by the projections of the extracted auditory information mapped into multiple interrelated feature subspace via cNTF. A diagram of feature extraction and speaker recognition framework is shown in Figure 1. 6SHHFK Pre-Emphasis

DCT

Recognition Result

GMM

A

Cochlear Filters

X

F17)

Nonlinearity &RFKOHDU)HDWXUH )HDWXUH7HQVRUE\ 'LIIHUHQW6SHDNHUV

6SHFWUR7HPSURDO %DVLV)XQFWLRQV

Fig. 1. Feature extraction and recognition framework

3.1 Feature Extraction Based on Auditory Model We extract the features by imitating the process occurred in the auditory periphery and pathway, such as outer ear, middle ear, basilar membrane, inner hair-cell, auditory nerves, and cochlear nucleus. We implement traditional pre-emphasis to model the combined outer and middle ear functions, which is xpre (t) = x(t)−0.97x(t−1), where x(t) is the discrete time speech signal, t = 1, 2, ..., and xpre (t) is the filtered output signal. The frequency selectivity of peripheral auditory system such as basilar membrane is simulated by a bank of cochlear filters, which have an impulse response in the following form: gi (t) = ai tn−1 e2πbi ERB(fi )t cos(2πfi t + φi ), (1 ≤ i ≤ N ),

(13)

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Q. Wu, L. Zhang, and G. Shi

where n is the order of the filters, N is the number of filterbanks. For the ith filter bank, fi is the center frequency, ERB(fi ) is the equivalent rectangular bandwidth (ERB) of the auditory filter, φi is the phase, and ai , bi ∈ R are constants where bi determines the rate of decay of the impulse response, which is related to bandwidth. In order to model nonlinearity of the inner hair-cells, we compute the power of each band in every frame k with a logarithmic nonlinearity:  {xig (t)}2 ), (14) P (i, k) = log(1 + γ t∈f rame k

where P (i, k) is the output power, γ is a scaling constant, and xig (t) = τ xpre (τ )gi (t− τ ) is the outputs of each gammatone filterbanks. This model can be considered as average firing rates in the inner hair-cells, which simulate the higher auditory pathway. The resulting power feature vector P (i, k) at frame k with component index of frequency fi , comprises the spectro-temporal power representation of the auditory response. Similar to Mel-scale processing in MFCC extraction, this power spectrum provides a much higher frequency resolution at low frequencies than at high frequencies. 3.2 Sparse Tensor Representation In order to extract robust features based on tensor structure, we model the cochlear power feature of different speakers as 3-order tensor X ∈ RNf ×Nt ×Ns . Each feature tensor is an array with three modals frequency × time × speaker identity which comprises the cochlear power feature matrix X ∈ RNf ×Nt of different speakers. Then we transform the auditory feature tensor into multiple interrelated subspaces by cNTF to learn the basis functions A(d) , (d = 1, 2, 3). Figure 2 shows the tensor model for the calculation of basis functions. Compared with traditional subspace learning methods, the extracted tensor features may characterize the differences of speakers and preserve the discriminative information for classification. As described in Section 3.1, the

cNTF

Basis Functions

Fig. 2. Tensor model for calculation of basis functions via cNTF

cochlear power feature can be considered as neurons response in the inner hair-cells. The hair-cells have receptive fields which refer to a coding of sound frequency. Here we employ the sparse localized basis function A ∈ RNf ×R in time-frequency subspace to transform the auditory feature into the sparse feature subspace, where R is the dimension of sparse feature subspace. The representation of auditory sparse feature Xs is obtained via the following transformation: ˆ Xs = AX

(15)

Robust Speaker Modeling Based on Constrained Nonnegative Tensor Factorization

10

2

20

1

30

0

40

2

50

0

20

40

60

80

0

20

40

60

80

0

20

40

60

80

17

1

60

0

70

2

80

1

90 100 20

40

60

(a) Basis functions

80

0

(b) Examples of encoding vector

Fig. 3. Results of cNTF applied to the clean speech data. (a) basis functions (100×80) in spectrotemproal domain. (b) Examples for encoding feature vector.

ˆ consists of the nonnegative elements of A−1 , i.e. A ˆ = [A−1 ]+ . Figure 3(a) where A shows an example of basis functions in spectro-temporal domain. From this result we can see that most elements of basis function are near to zero, which accords with the sparse constraint of cNTF. Figure 3(b) gives several examples for the encoding feature vector after transformation which also prove the sparse characteristic of feature. Our feature extraction model is based on the fact that in sparse coding the energy of the signal is concentrated on a few components only, while the energy of additive noise remains uniformly spreading on all the components. As a soft-threshold operation, the absolute values of pattern from the sparse coding components are compressed towards to zero. The noise is reduced while the signal is not strongly affected. We also impose orthogonal constraint to cNTF which helps to extract the helpful feature by minimizing the redundancy of different basis functions.

4 Experiments Results In this section we provide the evaluation results of a speaker identification system using ANTF. Aurora2 speech corpus is used to test the recognition performance, which is designed to evaluate speech recognition algorithms in noisy conditions. Different noise classes were considered to evaluate the performance of ANTF against MFCC, MelNMF, Mel-PCA feature and identification accuracy was assessed. In our experiments the sampling rate of speech signals was 8kHz. For the given speech signals, we employed time window of length 40000 samples (5s). For computational simplicity, we selected 36 cochlear filter banks and time duration 10 samples(1.25ms). Then the dimension of the speaker data is 36 × 10 = 360. We calculated the basis functions using cNTF after the calculation of cochlear power feature. For learning the basis functions in different subspaces, 550 sentences (5 sentences each person) were selected randomly as the training data and 200 dimension sparse tensor representation is extracted. In order to estimate the speaker model and test the efficiency of our method, we use 5500 sentences (50 sentences each person) as training data and 1320 sentences (12 sentences each person) mixed with different kinds of noise were used as testing data. The

18

Q. Wu, L. Zhang, and G. Shi

Table 1. Identification accuracy in four noisy conditions(subway, car noise, babble, exhibition hall) for Aurora2 noise testing dataset Noise SNR(dB) ANTF(%) Mel-NMF(%) Mel-PCA(%) MFCC(%)

5 24.5 15.5 3.6 2.7

Subway 10 15 58.2 82.7 40.9 67.3 12.7 50.9 16.4 44.6

20 86.4 88.2 88.2 76.4

5 24.6 23.6 21.8 16.4

Babble 10 15 60.0 83.6 41.8 61.8 51.8 79.1 51.8 79.1

20 89.1 82.7 96.4 93.6

5 23.6 3.6 2.7 5.5

Car noise 10 15 57.3 79.1 26.4 57.3 10.0 38.2 17.3 44.6

20 86.4 74.6 79.1 78.2

Exhibition hall 5 10 15 20 16.4 50.9 82.7 90.9 9.1 29.1 68.2 86.4 3.6 20.9 59.1 89.1 1.8 20.0 50.0 76.4

testing data were mixed with subway, babble, car noise, exhibition hall in SNR intensities of 20dB, 15dB, 10dB and 5dB. For the final feature set, 16 cepstral coefficients were extracted and used for speaker modeling. GMM was used to build the recognizer with 64 gaussian mixtures. For comparison, the performance of MFCC, Mel-NMF and Mel-PCA with 16-order cepstral coefficients are also tested. We use PCA and NMF to learn the part-based representation in the spectro-temporal domain after mel filtering, which is similar to [9]. The feature after PCA or NMF projection was further processed into the cesptral domain via discrete cosine transform. Table 1 presents the identification accuracy obtained by ANTF and baseline system in all testing conditions. We can observe from Table 1 that the performance degradation of ANTF is slower with increasing noise intensity that compared with other features. It performs better than other three features in the high noise conditions such as 5dB condition noise. Figure 4 describes the identification rate in four noisy conditions averaged over SNRs between 5-20 dB, and the overall average accuracy across all the conditions. The results suggest that this auditory-based tensor representation feature is robust against the additive noise, which indicates the potential of the new feature for dealing with a wider variety of noisy conditions. 100% ANTF Mel−NMF Mel−PCA MFCC

Identification rate

80% 60% 40% 20% 0

Subway

Babble

Car noise

Exhibition hall

Average

Fig. 4. Identification accuracy in four noisy conditions averaged over SNRs between 5-20dB, and the overall average accuracy across all the conditions, for ANTF and other three features using Aurora2 noise testing dataset

Robust Speaker Modeling Based on Constrained Nonnegative Tensor Factorization

19

5 Conclusion In this paper, we presented a novel speech feature extraction framework which is robust to noise with different SNR intensities, for evaluation with identification systems operating under a wide variety of conditions. This approach is primarily data-driven and effectively extracts robust feature of speech called ANTF that is invariant to noise types and interference with different intensities. We derived new feature extraction methods called cNTF for robust speaker identification. The research is mainly focused on the encoding of speech based on general higher order tensor structure to extract the robust auditory-based feature from interrelated feature subspace. The frequency selectivity features at basilar membrane and inner hair cells were used to represent the speech signals in the spectro-temporal domain, and then cNTF algorithm was employed to extract the sparse tensor representation for robust speaker modeling. The discriminative and robust information of different speakers may be preserved after the multi-related subspace projection. Experiment on Aurora2 has shown the improvement of the noise robustness by the new method, in comparison with baseline systems trained on the same amount of information.

Acknowledgment The work was supported by the National High-Tech Research Program of China (Grant No.2006AA01Z125) and the National Natural Science Foundation of China (Grant No. 60775007).

References 1. Rabiner, L.R., Juang, B.: Fundamentals on Speech Recognition. Prentice Hall, New Jersey (1996) 2. Hermansky, H., Morgan, N.: RASTA Processing of Speech. IEEE Trans. Speech Audio Process 2, 578–589 (1994) 3. Reynolds, D.A., Quatieri, T.F., Dunn, R.: Speaker Verification Using Adapted Gaussian Mixture Models. Digital Signal Processing 10, 19–41 (2000) 4. Reynolds, D.A.: Experimental Evaluation of Features for Robust Speaker Identification. IEEE Trans. Speech Audio Process 2, 639–643 (1994) 5. Berouti, M., Schwartz, R., Makhoul, J., Beranek, B., Newman, I., Cambridge, M.A.: Enhancement of Speech Corrupted by Acoustic Noise. Acoustics, Speech, and Signal Processing. In: IEEE International Conference on ICASSP 1979, vol. 4, pp. 208–211 (1979) 6. Hermus, K., Wambacq, P., Van hamme, H.: A Review of Signal Subspace Speech Enhancement and Its Application to Noise Robust Speech Recognition. EURASIP Journal on Applied Signal Processing 1, 195–209 (2007) 7. Smith, E., Lewicki, M.S.: Efficient Auditory Coding. Nature 439, 978–982 (2006) 8. Kim, T., Lee, S.Y.: Learning Self-organized Topology-preserving Complex Speech Features at Primary Auditory Cortex. Neurocomputing 65, 793–800 (2005) 9. Cho, Y.C., Choi, S.: Nonnegative Features of Spectro-temporal Sounds for Classification. Pattern Recognition Letters 26, 1327–1336 (2005) 10. Asari, H., Pearlmutter, B.A., Zador, A.M.: Sparse Representations for the Cocktail Party Problem. Journal of Neuroscience 26, 7477–7490 (2006)

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11. Carroll, J.D., Chang, J.J.: Analysis of Individual Differences in Multidimensional Scaling via An n-way Generalization of “Eckart-Young” Decomposition. Psychometrika 35, 283– 319 (1970) 12. Harshman, R.A.: Foundations of the PARAFAC Procedure: Models and Conditions for An “Explanatory” Multi-modal Factor Analysis. UCLA Working Papers in Phonetics 16, 1–84 (1970) 13. Bro, R.: PARAFAC: Tutorial and Applications. Chemometrics and Intelligent Laboratory Systems 38, 149–171 (1997) 14. De Lathauwer, L., De Moor, B., Van de walle, J.: A Multilinear Singular Value Decomposition. SIAM Journal on Matrix Analysis and Applications 21, 1253–1278 (2000) 15. Kim, Y.D., Choi, S.: Nonnegative Tucker Decomposition. In: IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2007, pp. 1–8 (2007) 16. Welling, M., Weber, M.: Positive Tensor Factorization. Pattern Recognition Letters 22, 1255– 1261 (2001) 17. Shashua, A., Hazan, T.: Non-negative Tensor Factorization with Applications to Statistics and Computer Vision. In: Proceedings of the International Conference on Machine Learning (ICML), pp. 792–799 (2005) 18. Vasilescu, M.A.O., Terzopoulos, D.: Multilinear independent components analysis, 2005. In: Proc. IEEE Conf. on Computer Vision and Pattern Recognition, vol. 1, pp. 547–553 (2005) 19. Tao, D.C., Li, X.L., Wu, X.D., Maybank, S.J.: General Tensor Discriminant Analysis and Gabor Feature for Gait Recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 29, 1700–1715 (2007) 20. Lee, D.D., Seung, H.S.: Algorithms for Non-negative Matrix Factorization. Advances in Neural Information Processing Systems 13, 556–562 (2001) 21. Li, S.Z., Hou, X.W., Zhang, H.J., Cheng, Q.S.: Learning Spatially Localized, Parts-based Representation. In: Proc. IEEE Conf. Computer Vision and Pattern Recognition, pp. 1–6 (2001) 22. Pascual-Montano, A., Carazo, J.M., Kochi, K., Lehmann, D., Pascual-Marqui, R.D.: Nonsmooth Nonnegative Matrix Factorization. IEEE Transactions on. Pattern Analysis and Machine Intelligence. 28, 403–415 (2006)

A Hypothesis on How the Neocortex Extracts Information for Prediction in Sequence Learning Weiyu Wang Department of Biology, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong [email protected]

Abstract. From the biological view, each component of a temporal sequence is represented by neural code in cortical areas of different orders. In whatever order areas, minicolumns divide a component into sub-components and parallel process them. Thus a minicolumn is a functional unit. Its layer IV neurons form a network where cell assemblies for sub-components form. Then layer III neurons are triggered and feed back to layer IV. Considering the delay, through Hebbian learning the connections from layer III to layer IV can associate a sub-component to the next. One sub-component may link multiple following sub-components plus itself, but the prediction is deterministic by a mechanism involving competition and threshold dynamic. So instead of learning the whole sequence, minicolumns selectively extract information. Information for complex concepts are distributed in multiple minicolumns, and long time thinking are in the form of integrated dynamics in the whole cortex, including recurrent activity. Keywords: Sequence prediction; Columnar architecture; Neocortex; Connectionism; Associative memory.

1 Introduction Most human and animal learning processes can be viewed as sequence learning. Sun and Giles summarize problems related to sequence learning into four categories: sequence prediction, sequence generation, sequence recognition, and sequential decision making [1]. The four categories are closely related [1], and sequence prediction is arguably the foundation of the other three. Sequence learning can be touched by various disciplines, while typically it deals with sequences of symbols and is applied to language processing. In this problem, a temporal pattern is defined as a temporal sequence and each static pattern constituting it is defined as a component (Wang and Arbib, [2]). Because of the intrinsic complexity of language, a component usually cannot be determined solely by the previous component, but by a previous sequence segment defined as context [2]. To learn complex sequences, a short-term memory (STM) at least of the maximum degree of these sequences is inevitable. And at least one context detector is assigned to each context. So according to the model proposed by Wang and Yuwono in 1995 [3][4], a neural network with 2m+(n+1)r neurons (m context sensors, m modulators, n terminals each with a STM of length r) can learn an arbitrary sequence at most of length m and degree r, with at most n symbols. Starzyk F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 21–29, 2008. © Springer-Verlag Berlin Heidelberg 2008

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and He proposed a more complex model with hierarchical structure in 2007 [5]. To learn a sequence of length l with n symbols, the primary level network requires 3nl+2n+2l+m neurons, where m is for the number of output neurons for the next hierarchical level network (equals the number of symbols in the next level), and the total number of neurons should include all hierarchical levels [5]. Such expensive cost makes the application of sequence learning undesirable. Another problem is if we expand the discipline from language to others, for example vision, the input is nearly a continuous time temporal sequence with continuous value components, as the time interval is in milliseconds and thousands of neurons are involved for the primary visual representation. This leads to an extremely large symbol set, and extremely long sequence to be learn even within a few minutes. So it is obviously impossible to take the traditional sequence learning method aiming at remembering the whole sequence and the relationships from each context to its corresponding component. So we have to think out other methods to solve these problems. And there is surely an answer, as the guarantee is just the existence of us ourselves. We do read piles of articles and indeed learn something from them. We receive tremendous amount of information from our sense organs throughout our lives, and even at the last moment of our life, we can recall some scenes in our earliest life stage. Obviously, what’s important is not only how to learn, but also what to learn. This article touches sequence learning from a different viewpoint—how to pick up useful information from the input sequences and store it in an organized way. This is defined as “information extraction”. Our idea is to solve this problem by studying the biological architecture of the nervous system. A mechanism for information extraction is hypothesized based on the hierarchical and columnar organization of the cerebral cortex in part 2. A neural network is built to simulate the function of a single minicolumn according to this hypothesis in part 3. Part 4 gives the conclusion and summarizes the significance of this model.

2 The Mechanism for Information Extraction 2.1 The Hierarchical Structure of Neocortex and Abstraction Take the visual pathway as example. Light enters the eyes and is transduced to electrical signal in retina. The neural signal is transferred to primary visual areas via thalamus. Then information is submitted to secondary visual areas. For forming declarative memory, further transfer is to medial temporal lobe memory system and back to higher order cortex areas [6][7][8]. Though the mechanism of declarative memory formation is not completely clear yet, it is widely accepted that forming abstract concepts requires high level integration of information. Along this pathway the integration level rises, so is the abstraction level. If we describe this pathway mathematically as a vector series V1, V2,…,Vn, where Vi is a Ni elements 0-1 vector, then a component of a temporal sequence is represented by a assignment to each vector in this vector series, instead of only one vector. Notice the higher footnote i is, the higher abstraction level Vi has. And Vi depends onVi-1 ( i=2, 3,…n). This structure is somewhat an analog of Starzyk and He’s hierarchical model [5], in the difference that

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it deals with real neural code instead of symbols, and much more complex integration (computation) is applied between two hierarchical levels. 2.2 The Columnar Organization of Neocortex and PDP The neocortex is horizontally divided into 6 layers. Layer IV contains different types of stellate and pyramidal cells, and is the main target of thalamocortical and intrahemispheric corticocortical afferents. Layers I through III are the main target of interhemispheric corticocortical afferents. Layer III contains predominantly pyramidal cells and is the principal source of corticocortical efferents. Layer V and VI are efferents to motor-related subcortical structures and thalamus separately [9]. Vertically neocortex is columnar organized with elementary module minicolumn. The minicolumn is a discrete module at the layers IV, II, and VI, but connected to others for most neurons of layer III [10]. Considering the vector series V1, V2,…,Vn, vector Vi is divided into sub-vectors in corresponding minicolumns for any i. Each sub-vector represents a sub-component, and is processed independently in its minicolumn. Minicolumns transmit processed information to the next hierarchical level minicolumns. This accords with the idea of “Parallel Distributed Processing” (PDP) proposed by Rumelhart and McClelland [11][12]. 2.3 Minicolumn Architecture A model for the structure of a minicolumn is shown in figure 1. In this model, all pyramidal cells and stellate cells in layer IV of the minicolumn form a symmetrical Hebbian network. As neurons involved are limited and closely packed, we can assume any neuron is connected to all other neurons through short axons, whose transmission delay can be omitted. If the pyramidal cells connect to pyramidal cells directly, the connections are excitatory. If the pyramidal cells connect to other pyramidal cells through stellate cells, the connections are inhibitory. Thus this network contains both excitatory and inhibitory connections. Typical Hebbian learning in this network will form cell assemblies [13]. Each cell assembly stands for a sub-component. Signals are transmitted from layer IV pyramidal cells to layer III pyramidal cells through long axons. As layer III contains predominantly pyramidal cells, the connections are mainly excitatory. Thus layer III is not an idea place for forming cell assemblies, as without inhibitory connections two cell assemblies will intermingle with each other and become one if only they have very small overlapping. The representations in layer III are just corresponding to the cell assemblies in layer IV, and we can assume no overlapping in layer III, as this can be automatically achieved through a winner-take-all (WTA) mechanism also used in Wang and Arbib’s model [2]. Signals are transmitted from layer III through long axons either to other minicolumns, or back to layer IV. 2.4 Association in Minicolumns during Learning What’s important is the transmission from layer III back to layer IV (the feedback). Typically the function of a feedback is thought for refinement or synchronization,

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Fig 1. Structure of a minicolumn. Focusing on layer IV (afferent) and layer III (efferent). Pyramidal cells and stellate cells in layer IV connect with each other through short axons, forming a network with both excitatory and inhibitory synapses. Layer IV pyramidal cells transmit signals to layer III pyramidal cells through long axons. Layer III pyramidal cells may transmit signals to layer IV pyramidal cells of other minicolumns through long axons, or transmit signals back to its own layer IV pyramidal cells through long axons, forming feedback loop (indicated by the thick lines).

for example in the model proposed by Korner etc. [14]. But in our view, the feedback loop along with the transmission delay is the base for associating a sub-component to the next sub-component. Notice the involved two sub-components are not input at the same time, but Hebbian learning based on the synapse plasticity requires the two involved neurons exciting at the same time [13][15-17]. This is solved by the transmission delay of this feedback loop. The synapse modification can only happen in the synaptic junction, by the changes of the amount of neurotransmitter released by the presynaptic neuron, or the number of postsynaptic receptors [15-17]. Suppose the delay from the excitation of layer IV pyramidal cell bodies (dendrites) to the excitation of layer III pyramidal cell axon terminals is Δt, and the lasting time of subcomponent A and sub-component B are t1 and t2 respectively (t1, t2>>Δt), B follows A. Then from time 0 to Δt, no Hebbian learning happens at the synaptic junctions between layer III pyramidal cell axons and layer IV pyramidal cell bodies (dendrites), for only the later is exciting. From Δt to t1, the Hebbian learning associates subcomponent A with itself, denoted as learning the ordered pair (A,A). From t1 to t1+Δt , the layer III pyramidal cell axon terminals still represent sub-component A, while the

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layer IV pyramidal cell bodies (dendrites) already code for sub-component B. Hence the association is (A,B). From t1+Δt to t2, the association will be (B,B). 2.5 Competition and Threshold Dynamic during Retrieval Suppose A, B, C, B, D, E, A, B, F, D, E, C, A, B denotes a sequence composed of sub-components of a temporal sequence in a minicolumn. Then after learning (A,B), (B,C), (B,D), (B,F), (C,B), (C,A), (D,E), (E,A), (E,C), (F,D) plus (A,A), (B,B), (C,C), (D,D), (E,E), (F,F) are learned. Now input A (lasting time t> Δt ). The cell assembly for A in layer IV is evoked. From Δt to t, the feedback from layer III try to evoke both A and B. But A is exciting, supported by the exterior input. It will inhibit the exciting of cell assembly for B. Until the exterior input ceases at time t, the only remaining stimulation is from layer III, and this stimulation will last exactly Δt. Because cell assembly for A has excited, the threshold of its neurons raises, thus it cannot be evoked again for quite a while (at least Δt). Thus cell assembly for B finally gets its chance to excite. After another Δt cell assembly for B ceases exciting and cannot be evoked again, and layer III feedback try to evoke three cell assemblies for C, D, F separately. They all want to excite and inhibit the other two, the competition leads to nothing excited (more accurately, the three may excite as a “flash” for inhibition is triggered by exciting, but this “flash” is so short compared with Δt and disappears without further effect). Hence from the exterior performance of the minicolumn, only (A,B), (D,E), (F,D) are learned. 2.6 Summary By the mechanism described above, an input temporal sequence is understood at different abstraction level in different hierarchical levels of the neocortex. In each hierarchical level, the components (temporal sequences) are divided into sub-components (sub-temporal sequences) by minicolumns. Each minicolumn only extracts the deterministic feature of the sub-temporal sequences: if sub-component A is always followed by sub-component B and no other sub-components, the minicolumn learns A predicts B.

3 The Neural Network Simulation of the Minicolumn We only built a small network containing 10 layer IV pyramidal cells and 10 layer III pyramidal cells for demonstration. Of course the network can be expanded to hundreds of neurons to simulate the real minicolumn. Let binary arrays F[10] and T[10] denote the layer IV neurons and layer III neurons separately. For simplicity, we let T[i] = F[i], i=0, 1,…,9, though in real case the representations in layer III for cell assemblies in layer IV can be quite different and involve different numbers of neurons. Thus a cell assembly 1111100000 is also represented 1111100000 in layer III in our network. Array Thresh[10] denotes the thresholds of the layer IV neurons, whose value is 1 initially and 21 after exciting, but returns to 1 after Δt. Intra[10][10] is the learning matrix for association among layer IV neurons, whose value is in [-300, 30]. (Negative means inhibitory. As one pyramidal cell can inhibit another through numerous stellate cells, the inhibitory connection is thought to be much stronger.)

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Inter[10][10] is the learning matrix for association from layer III to layer IV, whose value is in [0,2] (only excitatory, thus the effect of layer III pyramidal cells on layer IV pyramidal cells are not as strong as it of layer IV pyramidal cells on themselves ). The sequence learning process takes discrete steps, and set Δt = 1 step (the delay in a minicolumn cannot be very long). An input sequence is noted as A[10](a), B[10](b), C[10](c),…where A[10], B[10], C[10] are 10 element 0-1 vectors, and a, b, c are integers for the number of steps which the state lasts. At one step when the input is I[10](n) ( n>0 is the remaining time this state lasts), learning starts with setting F[i] = I[i]. The learning rule for updating intra[i][j] is intra[i][j]=(intra[i][j]>=0) × (F[i]F[j] × 0.5 × (30-intra[i][j])-F[i]F[j] × 3)+ (intra[i][j]<=0) × (F[i]F[j] × 30-F[i]F[j] × 0.5 × (300 + intra[i][j]))+intra[i][j]

The learning rule for updating inter[i][j] is inter[i][j]=T[i]F[j] × 0.5 × (2-inter[i][j])+inter[i][j]

At each step, after updating both learning matrixes, the signal transmission from layer IV to layer III is denoted by T[i] = F[i]. The threshold refreshing rule is Thresh[i] = 1+20F[i] . Finally set I[10](n) = I[10](n-1), and continue (when n = 1, the next state is loaded in). During retrieval, still suppose the stimulation is I[10](n). Retrieving starts with setting F[i]=I[i]. The evoked neurons is determined by

F[i]=1 × (∑ F[j]intra[j][i] + ∑ T[j]inter[j][i] > thresh[i]) j≠i

j

Notice in each step we need to repeat the above calculation until F[i] no longer changes (as newly evoked neurons can in turn evoke others). Then the result is the final evoked cell assembly. And let T[i] = F[i] simulating the information transmission. Refresh threshold by Thresh[i] = 1+20F[i]. Finally set I[10](n) = I[10](n-1) and continue (when n= 1, set I[i] = 0 and n = 1). Now look at an example. The temporal sequence 1111000000(16), 0000000001(24), 0000111000(13), 1111000000(7), 0000000110(19) is input for 10 or more times (enough repeating times are necessary as the inter-state association can only happen once when one state changes to another). After learning, intra[10][10] approximates ⎛ 30 ⎜ 30 ⎜ ⎜ 30 ⎜ ⎜ 30 ⎜ −300 ⎜ ⎜ −300 ⎜ −300 ⎜ ⎜ −300 ⎜ −300 ⎜⎜ ⎝ −300

30

30

30

30 30 30 −300 −300 −300 −300 −300 −300

30 30 30 −300 −300 −300 −300 −300 −300

30 30 30 −300 −300 −300 −300 −300 −300

Inter[10][10] approximates

−300 −300 −300 −300 −300 −300 ⎞ −300 −300 −300 −300 −300 −300 ⎟⎟ −300 −300 −300 −300 −300 −300 ⎟ ⎟ −300 −300 −300 −300 −300 −300 ⎟ 30 30 30 −300 −300 −300 ⎟ ⎟ 30 30 30 −300 −300 −300 ⎟ 30 30 30 −300 −300 −300 ⎟ ⎟ 30 −300 ⎟ −300 −300 −300 30 30 −300 ⎟ −300 −300 −300 30 ⎟ −300 −300 −300 −300 −300 30 ⎟⎠

A Hypothesis on How the Neocortex Extracts Information

⎛2 ⎜2 ⎜ ⎜2 ⎜ ⎜2 ⎜2 ⎜ ⎜2 ⎜2 ⎜ ⎜0 ⎜0 ⎜⎜ ⎝0

27

2 2 2 0 0 0 2 2 2⎞ 2 2 2 0 0 0 2 2 2 ⎟⎟ 2 2 2 0 0 0 2 2 2⎟ ⎟ 2 2 2 0 0 0 2 2 2⎟ 2 2 2 2 2 2 0 0 0⎟ ⎟ 2 2 2 2 2 2 0 0 0⎟ 2 2 2 2 2 2 0 0 0⎟ ⎟ 0 0 0 0 0 0 2 2 0⎟ 0 0 0 0 0 0 2 2 0⎟ ⎟ 0 0 0 2 2 2 0 0 2 ⎟⎠

Four cell assemblies 1111000000, 0000111000, 0000000110, 0000000001 are formed. The extracted information is (0000000001, 0000111000), and (0000111000, 1111000000), thus input 0000000001 will return sequence 0000111000, 1111000000. 0000000110 retrieves nothing as it is associated to nothing. 1111000000 retrieves nothing either, but it’s because it is associated to both 0000000110 and 0000000001. In this neural network, it is required that cell assemblies do not overlap. IF two cell assemblies in layer IV overlap, their representations in layer III also share a common part. This common part will try to evoke both cell assemblies no matter which of them causes this, leading to undesired results. This can be solved if another feed forward learning is added for constructing the representations for cell assemblies in layer III, ensuring no overlapping (for example, the WTA mechanism used in Wang and Arbib’s model [2]). Rarely oscillation may happen during retrieval. This requires the input sequence itself ends with repeating circles, like the sequence A, B, C, D, C, D, C. Thus after learning this sequence, input C or D will lead to the oscillation with C and D alternatively. But this situation is really rare as if the above sequence doesn’t end with C or D, for example A, B, C, D, C, D, C, A. Then C will not retrieve D (as it is associated to both D and A), and no oscillation can happen.

4 Conclusion and Significance The model proposed in this article deals with the sequence learning problem from a different viewpoint: extracting information. Adopting this idea, what’s important is what information to extract rather than how to remember all information. The most significant advantage of this idea is that the memory capacity required is not proportional to the sequence length and degree, but to the useful information (knowledge) contained in the sequence. Multiple different sequences may contain common knowledge. The common knowledge appears as the same sub-sequences in the certain minicolumns of certain hierarchical levels. For example, a stone, a tire, or a basket ball rolling down a hill appear to be quite different scenes if considering every detail, but all of them are abstracted as the process of an round object rolling down a slope in physics. This is because the essence of abstraction is the process of extracting important common

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features while omitting the other unimportant details. This process is fulfilled in our model by the complex connections among minicolumns of different hierarchical levels, which lead to complicated neural computation. Naturally, along with the increase of abstraction level, the knowledge is more and more general and the amount of information is reduced, represented by decrease of the variation of sequences. It is arguable that in the high enough hierarchical levels, only few sequences repeat frequently. The learning is by forming associative memory in minicolumns. Each minicolumn associates a sub-component to itself and its immediate follower. But through competition and threshold dynamic, A evokes B if and only if B is the only possible follower of A. This means a minicolumn doesn’t consider a temporal sequence’s degree. Every temporal sequence is treated as a simple sequence. Thus a minicolumn can remember only a small portion of the sequence by itself, seemingly useless compared with Wang and Yuwono’s model [3][4] and Stazyk and He’s model [5]. But the advantage is that the neural network for a minicolumn is extremely simple, as described in part 3, with much less cost than Wang’s or Stazyk’s models. Thus it is very proper for being a functional unit. The complex tasks are hoped to be accomplished by the whole network composed of millions of such functional units. Typically a sub-component in a minicolumn can only retrieve one or two following sub-components, and then this minicolumn ceases. But the retrieved sub-components are submitted to higher level minicolumns, and may trigger the retrieving in them. Repeating this activity, and by possible crosses or loops (recurrent activity), a subcomponent might trigger unlimited retrieving. This process must be consciously controlled by concentration (a mysterious cognitive function not discussed in this article). Finally, the model has the following important features: 1. higher hierarchical level minicolumns tend to learn more than lower hierarchical level minicolumns, as in high abstraction level sequence variation is reduced. 2. two seemingly completely different objects may retrieve the same thing, if only they share some common feature and the concentration is on this common feature. For example, an elephant and the glacier both may retrieve the concept of “huge”. Acknowledgments. Thank Bertram E. SHI of dept of electronic & computer engineering HKUST for offering illuminating advice and inspiring discussion.

References 1. Sun, R., Giles, L.C.: Sequence Learning: From Recognition and Prediction to Sequential Decision Making. IEEE Intell. Syst. 16, 67–70 (2001) 2. Wang, D., Arbib, A.M.: Complex Temporal Sequence Learning Based on Short-term Memory. Proc. IEEE 78, 1536–1543 (1990) 3. Wang, D., Yuwono, B.: Anticipation-Based Temporal Pattern Generation. IEEE Trans. Syst. Man Cybern. 25, 615–628 (1995) 4. Wang, D., Yuwono, B.: Incremental Learning of Complex Temporal Patterns. IEEE Trans. Neural Networks 7, 1465–1481 (1996) 5. Starzyk, A.J., He, H.: Anticipation-Based Temporal Sequences Learning in Hierarchical Structure. IEEE Trans. Neural Networks 18, 344–358 (2007)

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6. Squire, R.L., Zola, M.S.: The Medial Temporal Lobe Memory System. Science 253, 1380– 1386 (1991) 7. Thompson, F.R., Kim, J.J.: Memory systems in the brain and localization of a memory. PNAS 93, 13438–13444 (1996) 8. Mayes, A., Montaldi, D., Migo, E.: Associative Memory and the Medial Temporal Lobes. Trends Cogn. Sci. 11, 126–135 (2007) 9. Creutzfeldt, D.O.: Cortex Cerebri: Performance, Structural and Functional Organization of the Cortex. Oxford University Press, USA (1995) 10. Mountcastle, B.V.: The Columnar Organization of the Neocortex. Brain 120, 701–722 (1997) 11. Rumelhart, D.E., McClelland, J.L.: The PDP Research Group: Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Foundations, vol. 1. MIT Press, Cambridge (1986) 12. McClelland, J.L., Rumelhart, D.E.: The PDP Research Group: Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Psychological and Biological Models, vol. 2. MIT Press, Cambridge (1986) 13. Hebb, D.O.: The Organization of Behavior. Wiley, New York (1949) 14. Korner, E., Gewaltig, O.M., Korner, U., Richter, A., Rodemann, T.: A model of computation in neocortical architecture. Neural Networks 12, 989–1005 (1999) 15. Bliss, P.V.T., Collingridge, L.G.: A synaptic model of memory: long-term potentiation in the hippocampus. Nature 361, 31–39 (1993) 16. Bear, F.M.: A synaptic basis for memory storage in the cerebral cortex. PNAS 93, 13453– 13459 (1996) 17. Chen, R.W., Lee, S., Kato, K., Spencer, D.D., Shepherd, M.G., Williamson, A.: Long-term modifications of synaptic efficacy in the human inferior and middle temporal cortex. PNAS 93, 8011–8015 (1996)

MENN Method Applications for Stock Market Forecasting Guangfeng Jia, Yuehui Chen, and Peng Wu School of Information Science and Engineering University of Jinan, 250022 Jinan, China [email protected]

Abstract. A new approach for forecasting stock index based on Multi Expression Neural Network (MENN) is proposed in this paper. The approach employs the multi expression programming (MEP) to evolve the architecture of the MENN and the particle swarm optimization (PSO) to optimize the parameters encoded in the MENN. This framework allows input variables selection, over-layer connections for the various nodes involved. The performance and effectiveness of the proposed method are evaluated using stock market forecasting problems and compared with the related methods. Keywords: Multi Expression Programming, Artificial Neural Network, Stock Market Forecasting.

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Introduction

Stock index forecasting is an integral part of everyday life. Current methods of forecasting require some elements of human judgment and are subject to error. Stock indices are a sequence of data points, measured typically at uniform time intervals.There are several motivations for trying to predict stock market prices. The most basic of these is financial gain. Any system that can consistently pick winners and losers in the dynamic market place would make the owner of the system very wealthy. Thus, many individuals including researchers, investment professionals, and average investors are continually looking for this superior system which will yield them high returns [1][2]. Artificial neural networks (ANNs) represent one widely technique for stock market forecasting. Apparently, White [3] first used Neural Networks for market forecasting. In other work, Chiang, Urban, and Baldridge have used ANNs to forecast the end-of-year net asset value of mutual funds. Trafalis used feedforward ANNs to forecast the change in the S&P (500) index. Typically the predicted variable is continuous, so that stock market prediction is usually a specialized form of regression. Any type of neural network can be used for stock index prediction (the network type must, however, be appropriate for regression or classification, depending on the problem type). The network can also have any number of input and output variables [4]. In addition to stock index prediction, neural networks have been trained to perform a variety of financial related tasks. F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 30–39, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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There are experimental and commercial systems used for tracking commodity markets and futures, foreign exchange trading, financial planning, company stability, and bankruptcy prediction. Banks use neural networks to scan credit and loan applications to estimate bankruptcy probabilities, while money managers can use neural networks to plan and construct profitable portfolios in real-time. As the application of neural networks in the financial area is so vast, we will focus on stock market prediction. However, most commonly there is a single variable that is both the input and the output. Despite the wide spread use of ANNs, there are significant problems to be addressed. ANNs are data-driven model, and consequently, the underlying rules in the data are not always apparent. Also, the buried noise and complex dimensionality of the stock market data make it difficult to learn or re-estimate the ANNs parameters. It is also difficult to come up with ANNs architecture that can be used for all domains. In addition, ANNs occasionally suffer from the over-fitting problem. In this paper, an automatic method for constructing MENN network is proposed. Based on the pre-defined instruction/operator sets, a MENN network can be created and evolved. MENN allows input variables selection, over-layer connections for different nodes. The novelty of this paper is in the usage of multi expression neural network model for selecting the important features and for improving the accuracy. The paper is organized as follows: Section 2 gives a short review of the original MEP algorithm and ANN model. The representation of MENN model and a hybrid learning algorithm for designing the artificial neural network are given in Section 3. Section 4 presents some simulation results for the stock market forecasting problems. Finally in section 5 we present some concluding remarks.

2 2.1

MEP, PSO Algorithms and ANN Model A Short Review of the Original MEP Algorithm

Evolutionary algorithms are defined as randomized search procedures inspired by the working mechanism of genetics and natural selection [5]. There are different types of evolutionary algorithms such as genetic algorithms (GA), genetic programming (GP), evolution strategies (ES), and evolutionary programming (EP). MEP [6][7] is a relatively new technique in genetic programming that is first introduced in 2002 by Oltean and Dumitrescu. A traditional GP encodes a single expression (computer program). By contrast, a MEP chromosome encodes several expressions. The best of the encoded solution is chosen to represent the chromosome. A MEP individual includes some genes which are represented by substrings of variable length. The number of genes per chromosome is constant and this number defines the length of a chromosome [8]. Each gene encodes a terminal or a function symbol which is selected from a terminal set T or a function set F. The two sets for a given problem are pre-defined. A gene that encode a function includes some pointers towards the function arguments. The number of the

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Fig. 1. A valid MEP chromosome

pointers depends on how many arguments the function have. In order to ensure that each chromosome is a valid MEP individual, there are some restrictions to initialize the population [6]: 1) First gene of the chromosome must contain a terminal that is randomly selected from the terminal set T. 2) For all other genes which encodes functions, we need generate pointers toward function arguments. All the pointers must have indices of lower index than the current gene. In this way only syntactically correct chromosome are generated. An example of a chromosome using the sets F = {+, -, ∗, sin} and T = {a, b, c, d} is shown in Fig. 1. The MEP chromosomes are read in a top-down fashion starting with the first gene. A gene that encode a terminal specifies a simple expression. And a gene that encode a function specifies a complex expression (formed by linking the operands specified by the argument position with the current function symbol) [5]. For instance, genes 1, 2, 4 and 5 in Fig. 1 encode simple expressions formed by a single terminal symbol. These expressions are: E1 = a; E2 = b; E4 = c; E5 = d. Gene 3 indicates the operation ∗ on the operands located at position 1 and 2 of the chromosome. Therefore gene 3 encodes the expression: E3 = a ∗ b. Gene 6 indicates the operation sin on the operand located at position 4. Therefore gene 6 encodes the expression: E6 = sin c. Gene 7 indicates the operation − on the operands located at positions 3 and 5. Therefore gene 7 encodes the expression: E7 = (a ∗ b) − d. Gene 8 indicates the operation + on the operands located at position 7 and 6. Therefore gene 8 encodes the expression: E8 = (a∗ b)− d+ sin c. The tree representations of these expressions are also shown in Fig. 2. As MEP chromosome encodes more than one genes, it is required to choose one of the expressions to present the chromosome. The chromosome fitness is usually defined as the fitness of the best expression encoded by that chromosome. 2.2

PSO Algorithm

The PSO conducts searches using a population of particles that correspond to individuals in an Evolutionary Algorithm (EA) [9][10]. To get the parameters of SVM, particle swarm optimization (PSO) algorithm is employed. All free parameters in the SVM constitute a particle. Initially, a population of particles is

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Fig. 2. Tree representations of a MEP chromosome

randomly generated. Each particle represents a potential solution and has a position represented by a position vector xi . A swarm of particles moves through the problem space with the moving velocity of each particle represented by a velocity vector vi . At each time step, a function fi - representing a quality measure - is calculated by using xi as input. Each particle keeps track of its own best position, which is associated with the best fitness it has achieved so far in a vector pi . Furthermore, the best position among all the particles obtained so far in the population is kept track of as pg . In addition to this global version, another version of PSO keeps track of the best position among all the topological neighbors of a particle. At each time step t, by using the individual best position, pi (t), and the global best position, pg (t), a new velocity for particle i is updated by Vi (t + 1) = vi (t) + c1 φ1 (pi (t) − xi (t)) + c2 φ2 (pg (t) − xi (t)).

(1)

where c1 and c2 are positive constants and φ1 and φ2 are uniformly distributed random numbers in [0,1]. The term ci is limited to the range of Vm ax (if the velocity violates this limit, it is set to its proper limit). Changing velocity this way enables the particle i to search around both its individual best position, pi , and global best position, pg . Based on the updated velocities, each particle changes its position according to xi (t + 1) = xi (t) + vi (t + 1).

(2)

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A Short Review of the Traditional ANN Model

A typical neural network consists of layers. In a single layered network there is an input layer of source nodes and an output layer of neurons. A multi-layer network has in addition one or more hidden layers of hidden neurons. Some standard three-layer feed-forward networks are used widely [11]. A representative feed-forward neural network consists of a three layer structure: input layer, output layer and hidden layer. Each layer is composed of variable nodes. The type of this network is displayed in Fig. 3. The number of nodes in the hidden layers is selected to make the network more efficient and to interpret the data more accurately. The relationship between the input and output can be non-linear or linear, and its characteristics are determined by the weights assigned to the connections between the nodes in the two adjacent layers. Changing the weight will change the input-to-output behavior of the network.

Output layer

Hidden layer

Input layer Fig. 3. A fully connected feed-forward network with one hidden layer and one output layer

A feed-forward neural network analysis consists of two stages, namely training and testing. During the training stage, an input-to-output mapping is determined iteratively using the available training data. The actual output error, propagated from the current input set, is compared with the target output and the required compensation is transmitted backwards to adjust the node weights so that the error can be reduced at the next iteration. The training stage is stopped once a pre-set error threshold is reached and the node weights are frozen at this point. During the testing stage, data with unknown properties are provided as input and the corresponding output is calculated using the fixed node weights. The feed-forward neural network has been shown to perform well in many areas in previous research.

3 3.1

A Novel MENN Model Representation

In this research, a novel multi expression programming based encoding method with specific instruction set is selected for representing a MENN model. The

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Output layer

Hidden layer

Input layer

D

35

E

 

 Fig. 4. A valid MENN chromosome

reason for choosing the representation is that the tree can be created and evolved using linear chromosome structure of MEP. The used function set F and terminal set T for generating a MENN model are described as F = {+, -, ∗, sin} and T = {a, b, c, d}. A gene that encode a function includes some pointers towards the function arguments. The number of the pointers depends on how many arguments the function have. The value of a MENN gene expression is calculated by a recursion way. The multi expression neural network is shown in Fig. 4, where the From this point of view, the MENN is also viewed as a flexible neural network chromosome. 3.2

Initialization

Initial population is generated according to predefined population size parameter which determines the number of MENN chromosome in the population. Individuals of population are repeatedly generated by employing the following procedure. 1) The function symbol or terminal symbol of each gene is selected from a function set F or a terminal set T. According to the proposed representation scheme, the first symbol of each MENN chromosome must be a terminal symbol. For all genes which encodes function, pointers have to be generated to address function arguments. All the pointers must have indices of lower index than the current gene. 2) The second part of the each gene consists of MENN parameters which includes weight parameters and activation function parameters. These real parameters are randomly generated in [0, 1].

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Procedure of the General Learning Algorithm

The general learning procedure for constructing the MEP-NN model can be described as follows: 1) Initial population is generated randomly. All the learning parameters in MENN model should be assigned in advance. 2) The fitness value is calculated for each individual by using PSO algorithm to optimize the parameters encoded in the chromosome. 3) Implementation of the genetic operators: crossover and mutation operators. 4) If maximum number of generations is reached or no better solution is found for a significantly long time (300 steps), then stop, otherwise goto step 2).

4

Experiment Setup and Result

To test the efficacy of the proposed method ,the MEP-NN model is applied to a stock index prediction problem. We have used stock prices in the IT sector: the daily stock price of International Business Machines Corporation (IBM), Dell Inc [12], collected from www.finance.yahoo.com. The daily stock prices of IBM and Dell Inc., training data is from February 10, 2003 to September 10, 2004 and the test data is from September 13, 2004 to January 21, 2005. The two stock index data sets were represented by ’opening value’, ’low value’, ’high value’ and ’closing value’. Also, the experiments for S&P CNX NIFTY stock index [13] are established for evaluating the performance of the proposed methods. S&P CNX NIFTY is a well-diversified 50 stock index accounting for 25 sectors of the economy. It is used for a variety of purposes such as benchmark fund portfolios, index based derivatives and index funds. The CNX Indices are computed using market capitalization weighted method, wherein the level of the Index reflects the total market value of all the stocks in the index relative to a particular base period.

Fig. 5. Test results of IBM

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Fig. 6. Test results of DELL

Fig. 7. Test results of NIFTY

The performance of the method is measured in terms of RMSE. Parameters used by MENN in these experiments are presented in Table 1. For comparison purpose, the forecast performances of a traditional artificial neural network (ANN) mode and an support vector machines (SVM) model [14] are also shown Table 1. Empirical comparison of RMSE result for three methods Model Name SVM model [14] ANN model [14] MENN model (This paper)

IBM Corp. 0.02849 0.03520 0.02887

Dell Inc. 0.03665 0.05182 0.02786

NIFTY 0.03220 0.01857 0.01587

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in Table 2. The actual stock price and the predicted ones for three stock index are shown in Fig. 5, Fig. 6 and Fig. 7. From Table 2, it is observed that the proposed MENN models are better than the traditional neural network.

5

Conclusion

In this paper, a new approach for designing artificial neural networks using multi expression programming is proposed. In the viewpoint of calculation structure, the MENN model can be viewed as a flexible multi-layer feedforward neural network with over-layer connections and free activation function parameters. The work demonstrates that it is possible to find an appropriate way to evolve the structure and parameters of artificial neural networks simultaneously by using multi expression programming. Simulation results for the stock market forecasting problems shown the feasibility and effectiveness of the proposed method. Acknowledgments. This work is partially supported by the National Science Foundation of China under grant No. 60573065, the Key Subject Research Foundation of Shandong Province and the Natural Science Foundation of Shandong Province (grant Y2007G33).

References 1. Robert, R., Jae, T., Lee, L.: Artificial Intelligence in Finance and Investing, ch. 10. IRWIN (1996) 2. Wu, Q., Chen, Y.H., Wu, P.: Higher Order Neural Networks for Stock Index Modeling. In: Zhang, M. (ed.) Artificial Higher Order Neural Networks for Economics and Business (in press, 2008) 3. White, H.: Economic Prediction Using Neural Networks. The Case of IBM Daily Stock Returns. In: Proc. of IEEE Int’l Conference on Neural Networks (1988) 4. Hecht-Nielsen, R.: Kolmogorov’s mapping neural network existence theorem. In: Proc. 1st IEEE Int’l Joint Conf. Neural Network (1987) 5. Adil, B., Lale, O.: MEPAR-miner: Multi-expression programming for classification rule mining. European Journal of Operational Research 183, 767–784 (2007) 6. Oltean, M., Dumitrescu, D.: Multi Expression Programming. Technical Report, UBB-01-2002, Babes-Bolyai University, Cluj-Napoca, Romania (2002), www.mep.cs.ubbcluj.ro 7. Crina, G., Ajith, A., Sang, Y.H.: MEPIDS: Multi-Expression Programming for ´ Intrusion Detection System. In: Mira, J., Alvarez, J.R. (eds.) IWINAC 2005. LNCS, vol. 3562, pp. 163–172. Springer, Heidelberg (2005) 8. Oltean, M., Grosan, C.: Evolving Digital Circuits Using Multi Expression Programming. In: Zebulum, R., et al. (eds.) NASA/DoD Conference on Evolvable Hardware, June 24-26, pp. 87–90. IEEE Press, NJ (2004) 9. Kennedy, J.: Particle swarm optimization. In: Proc IEEE Int. Conf. on Neural Networks. IV, pp. 1942–1948 (1995) 10. Yoshida, H., Kawata, K., Fukuyama, Y., Takayama, S., Nakanishi, Y.: A Particle Swarm Optimization for Reactive Power and Voltage Control Considering Voltage Security Assessment. IEEE Trans. Power Syst. 15, 1232–1239 (2000)

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11. Zhang, X.Q., Chen, Y.H., Yang, J.Y.: Stock Index Forecasting Using PSO Based Selective Neural Network Ensemble. In: International Conference on Artificial Intelligence (ICAI 2007), vol. 1, pp. 260–264 (2007) 12. Hassan, M.R.U., Nath, B., Kirley, M.: A Fusion Model of HMM, ANN and GA for Stock Market Forecasting. Expert Systems with Applications 33, 171–180 (2007) 13. National Stock Exchange of India Limited, http://www.nse-india.com 14. Wu, Q., Chen, Y.H., Liu, Z.: Ensemble Model of Intelligent Paradigms for Stock Market Forecasting. In: First International Workshop on Knowledge Discovery and Data Mining (WKDD 2008), pp. 205–208 (2008)

New Chaos Produced from Synchronization of Chaotic Neural Networks Zunshui Cheng School of Mathematics and Physics, Qingdao University of Science and Technology Qingdao 266061, China [email protected]

Abstract. In this paper, we investigates synchronization dynamics of neural networks. Generalized linear synchronization (GLS) is proposed to acquire a general kind of proportional relationships between two-neuron networks. Under the point of synchronization, we can find that the node has complex dynamics with some interesting characteristics, and some new chaos phenomenons can been found. Numerical simulations show that this method works very well of two-neuron networks with identical Lorenz systems. Also our method can be applied to other systems. Keywords: Neural Networks, Chaos, Synchronization, Control system, Numerical simulation.

1 Introduction Recently, dynamical properties of neural networks have been extensively investigated, and many applications have been found in different areas. Most previous literature has mainly been devoted to the stability analysis. However, it has been shown that such networks can exhibit some complicated dynamics and even chaotic behaviors if the networks parameters are appropriately chosen. Motivated by the study of chaotic phenomena, an increasing interest has been devoted to the study of chaos synchronization since the pioneering work of Pecora and Carrol [1]. Synchronization of neural networks have many applications in secure communication and so on. Therefore, the study of synchronization of neural networks is an important step for both understanding brain science and designing neural networks for practical use [2]-[7]. There are different types of synchronization in interacting nodes of chaotic neural networks, such as complete synchronization (CS), generalized synchronization (GS), phase synchronization, lag synchronization and anticipating synchronization, etc [8][12]. Projective synchronization and generalized projective synchronization, which are the special case of generalized synchronization, are becoming one of the most noticeable subjects. Its typical feature is the state variables of the two-coupled system may 

This work was jointly supported by the Doctoral Found of QUST, and the Natural Science Foundation of Henan Province, China under Grant 0611055100.

F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 40–46, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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synchronize up to a scaling factor, but the Lyapunov exponents and fractal dimensions remain unchanged [13]-[22]. Recently, generalized projective synchronization (GPS) has attracted increasing interests from the researchers. The early projective synchronization is usually investigated only in a class of partially linear systems[17]-[19], however, the generalized projective synchronization is studied in a general class of neural systems including non-partiallylinear systems [13]-[15], [20]-[22]. In [16], modified projective synchronization (MPS) was proposed to acquire a general kind of proportional relationships between the drive and response systems. But in practical applications, rotation is also common and interesting. To the best of our knowledge, rotary and projective synchronization still remains open, so the generalized linear synchronization (GLS) is the special case of GLS) will be proposed and considered in this paper. Motivated by the above discussions, by using the active control techniques, we will investigate the generalized linear synchronization in this paper. The remaining of this paper is organized as follows: In Section 2, the definition and theoretic analysis of generalized linear synchronization of chaotic neural systems are provided. In Section 3, chaos produced from linear synchronization of two chaotic neural systems are analyzed. Finally, our paper is completed with a conclusion and some discussions.

2 Generalized Linear Synchronization of Two-Neuron Systems In this section, follow the idea of generalized projective synchronization, we will study the generalized linear synchronization (GLS). Consider the following two-neuron chaotic system:  f (xm ), x˙ m = (1) x˙ s = g(xm , xs ), where xm , xs ∈ Rn are n-dimensional state vectors. The low subscripts m and s stand for the master and slave systems, respectively. f : Rn → Rn and g : Rn → Rn are vector fields in n-dimensional space. If there exists a matrix An×n , such that limt→∞ Axm − xs  = 0, then the generalized linear synchronization (GLS) of the system (1) is achieved, and A was called a transform factor. We take the Lorenz systems as the master system ⎧ a(ym − xm ), ⎪ ⎨ x˙ m = y˙ m = cxm − xm zm − ym , (2) ⎪ ⎩ xm ym − bzm , z˙m = where a = 10, b = 8/3, c = 28, one can find the chaotic attractor. In order to realize the GLS, the following slave system is constructed: ⎧ a(ys − xs ) + u1 , ⎪ ⎨ x˙ s = y˙ s = cxs − xs zs − ys + u2 , (3) ⎪ ⎩ z˙s = xs ys − bzs + u3 ,

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where u1 , u2 and u3 are the control inputs. To determine the appropriate control inputs ui , (i = 1, 2, 3), assume transform factor ⎞ ⎛ a11 a12 a13 A = ⎝a21 a22 a23 ⎠ , a31 a32 a33 the error vector is defined as ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ xm e1 a11 a12 a13 xs ⎝e2 ⎠ = ⎝a21 a22 a23 ⎠ ⎝ ym ⎠ − ⎝ ys ⎠ . e3 a31 a32 a33 zm zs

(4)

Then the error dynamical system can be obtained: ⎧ ⎪ ⎪ e˙ 1 = a11 a(ym − xm ) + a12 (cxm − xm zm − ym ) ⎪ ⎪ +a13 (xm ym − bzm ) − a(ys − xs ) − u1 , ⎪ ⎪ ⎨ e˙ 2 = a21 a(ym − xm ) + a22 (cxm − xm zm − ym ) +a23 (xm ym − bzm ) − (cxs − xs zs − ys ) − u2 , ⎪ ⎪ ⎪ ⎪ e ˙ = a ⎪ 3 31 a(ym − xm ) + a32 (cxm − xm zm − ym ) ⎪ ⎩ +a33 (xm ym − bzm ) − (xs ys − bzs ) − u3 ,

(5)

employing the original methods of active control, the control inputs ui (i = 1, 2, 3) are chosen as follows: ⎧ u1 = −ays + ca12 xm + (aa11 + aa12 − a12 )ym ⎪ ⎪ ⎪ ⎪ +(aa13 − ba13 )zm − a12 xm zm + a13 xm ym , ⎪ ⎪ ⎨ u2 = −cxs + (ca22 − aa21 + a21 )xm − a22 xm zm +(a23 − ba23 )zm + aa21 ym + a23 xm ym + xs zs , ⎪ ⎪ ⎪ ⎪ u = (ca32 − aa31 + ba31 )xm + (aa31 + ba32 ⎪ 3 ⎪ ⎩ −a32 )ym − a32 xm zm + a33 xm ym − xs ys .

(6)

Based on the above choose of inputs, error system (5) becomes ⎧ ⎪ ⎨ e˙ 1 = −ae1 , e˙ 2 = −e2 , ⎪ ⎩ e˙ 3 = −be3 .

(7)

One can find that all eigenvalues of the closed loop system have negative real parts, so error system (7) will be convergent. In other words, the choice of the control inputs will result in a stable system and the generalized linear synchronization of two identical Lorenz systems was realized. Remark. In fact, our method can also applied to the generalized linear synchronization of other neural networks with identical chaotic systems of each node, such as the Chen system, L¨ u system, etc.

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3 Chaos Produced from Linear Synchronization of Two Chaotic Systems If the error vector is chosen as ⎛ ⎛ ⎞ ⎞⎛ ⎞ ⎛ ⎞ e1 −1 1 0 xm xs ⎝e2 ⎠ = ⎝−1 1 0⎠ ⎝ ym ⎠ − ⎝ ys ⎠ , e3 zm zs 0 01

(8)

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−10

−15 −15

−10

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Fig. 1. projection of the response systems onto the x − y plane 50

45

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0 −15

−10

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Fig. 2. projection of the response systems onto the x − z plane

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0 −15

−10

−5

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Fig. 3. projection of the response systems onto the y − z plane

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20 15

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Fig. 4. Phase plot of the response system

then we obtain the following error dynamical system: ⎧ e˙ 1 = −a(ym − xm ) + (cxm − xm zm − ym ) ⎪ ⎪ ⎪ ⎪ −a(ys − xs ) − u1 , ⎪ ⎪ ⎨ e˙ 2 = −a(ym − xm ) + (cxm − xm zm − ym ) −(cxs − xs zs − ys ) − u2 , ⎪ ⎪ ⎪ ⎪ e ˙ = ⎪ 3 ⎪ ⎩ +(xm ym − bzm ) − (xs ys − bzs ) − u3 ,

(9)

New Chaos Produced from Synchronization of Chaotic Neural Networks

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5 0 −5 −10 −15 20 50

10 40

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−10 10 y

−20

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Fig. 5. Phase plot of the response system

the control inputs ui (i = 1, 2, 3) are taken as following form: ⎧ u1 = −ays + cxm − ym − xm zm , ⎪ ⎪ ⎨ u2 = −cxs + (c + a − 1)xm − xm zm −aym + xs zs , ⎪ ⎪ ⎩ u3 = xm ym − xs ys .

(10)

by the above discussions, the synchronization of the two identical Lorenz systems can be realized. Under the point of synchronization, the response system has complex dynamics, and some new chaos phenomenons can been found (see Fig. 1.-Fig. 5.).

4 Conclusions In this paper, the definition of generalized linear synchronization (GLS) is proposed to for a general kind of proportional relationships between the drive and response systems. Under the point of generalized linear synchronization (GLS), we can find that the response system has complex dynamics with some interesting characteristics, and some new chaos phenomenons can been found. It should be noted that our method can be applied to other chaotic neural systems such as Chen system, Lv system etc, and more interesting chaotic characteristics may been found. The two-neuron chaotic system can be the same system, such as Lorenz system. At the same time, we can also choose different system on each neuron. These are beyond the scope of the present paper and can be further investigated elsewhere in the near future.

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Acknowledgments The authors would like to thank these referees for their valuable suggestions and comments.

References 1. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) 2. Lu, W.L., Chen, T.P.: Synchronization of coupled connected neural networks with delays. IEEE Trans. Circuits and System 51, 2491–2503 (2004) 3. Lu, J., Cao, J.: Synchronization-based approach for parameters identification in delayed chaotic neural networks. Physica A 382, 672–682 (2007) 4. Yu, W., Cao, J., Lv, J.: Global synchronization of linearly hybrid coupled networks with time-varying delay. SIAM Journal on Applied Dynamical Systems 7, 108–133 (2008) 5. Cao, J., Wang, Z., Sun, Y.: Synchronization in an array of linearly stochastically coupled networks with time delays. Physica A 385, 718–728 (2007) 6. Sun, Y., Cao, J.: Adaptive synchronization between two different noise-perturbed chaotic systems with fully unknown parameters. Physica A 376, 253–265 (2007) 7. Sun, Y., Cao, J.: Adaptive lag synchronization of unknown chaotic delayed neural networks with noise perturbation. Physics Letters A 364, 277–285 (2007) 8. Yu, W., Cao, J.: Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks. Chaos 16, 023119 (2006) 9. Cao, J., Lu, J.: Adaptive synchronization of neural networks with or without time-varying delays. Chaos 16, 013133 (2006) 10. Cao, J., Lu, J.: Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters. Chaos 15, 043901 (2005) 11. Amritkar, R.E.: Spatially synchronous extinction of species under external forcing. Phys. Rev. Lett. 96, 258102 (2006) 12. Shahverdiev, E.M., Sivaprakasam, S., Shore, K.A.: Lag synchronization in time-delayed systems. Physics Letters A 292, 320–324 (2002) 13. Li, C., Yan, J.: Generalized projective synchronization of chaos: The cascade synchronization approach. Chaos, Solitons and Fractals 30, 140–146 (2006) 14. Li, G.: Generalized projective synchronization of two chaotic systems by using active control. Chaos, Solitons and Fractals 30, 77–82 (2006) 15. Kittel, A., Parisi, J., Pyragas, K.: Generalized synchronization of chaos in electronic circuit experiments. Physica D 112, 459–471 (1998) 16. Li, G.: Modified projective synchronization of chaotic system. Chaos, Solitons and Fractals 32, 1786–1790 (2007) 17. Ronnie, M., Jan, R.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82, 3042–3045 (1999) 18. Xu, D., Li, Z.: Controlled projective synchronization in nonpartially-linear chaotic systems. Int. J. Bifurcat Chaos 12, 1395–1402 (2002) 19. Xu, D., Chee, C., Li, C.: A necessary condition of projective synchronization in discrete-time systems of arbitrary dimensions. Chaos, Solitons and Fractals 22, 175–180 (2004) 20. Rulkov, N.F., Sushchik, M.M., Tsimring, L.S., et al.: Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51, 980–994 (1995) 21. Kittel, A., Parisi, J., Pyragas, K.: Generalized synchronization of chaos in electronic circuit experiments. Physica D 112, 459–471 (1998) 22. Kocarev, L., Parlitz, U.: Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett. 76, 1816–1819 (1996)

A Two Stage Energy Model Exhibiting Selectivity to Changing Disparity Xiaojiang Guo1 and Bertram E. Shi2,* 1

Department of Electronics Engineering Tsinghua University, Beijing, China [email protected] 2 Department of Electronic and Computer Engineering Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong [email protected]

Abstract. We show that by cascading the disparity energy model and the motion energy model, we obtain neurons that are selective for changing disparity, which is a cue that biological systems may use in the perception of stereomotion. We demonstrate that the outputs of this model exhibit joint tuning to disparity and stereo-motion. The output achieves a peak response for an input with a preferred disparity that also changes at a preferred rate. The joint tuning curve in the disparity–change of disparity space is approximately separable. We further demonstrate that incorporating a normalization step between the two stages reduces the variability of the model output. Keywords: Motion Energy, Disparity Energy, Stereo-motion, Changing Disparity, Visual Cortex.

1 Introduction Stereo-motion refers to motion towards or away from a binocular ob-server. There are at least two cues that could be exploited by an observer to detect or estimate this motion: changing disparity (CD) or inter-ocular velocity difference (IOVD)[3]. The CD cue is derived by first combining monocular images to obtain a disparity signal at each time, and then examining the change in disparity over time. The IOVD cue is derived by first examining the change in each monocular image over time to obtain velocity signals, which are then combined across the two eyes. Psychophysical evidence suggests that both signals play a role in the perception of stereo-motion [2] . Here, we present a biologically plausible two-stage model for creating neurons selective for changing disparity. The first stage extracts disparity signals using a population of disparity energy neurons tuned to different disparities via phase shifts[4]. The second stage then establishes selectivity to disparity changes over time by a temporal filtering operation similar to that used in motion energy models[1]. *

This work was supported in part by the Hong Kong Research Grants Council under Grant HKUST 619205.

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Both motion energy and disparity energy have been used to model the outputs of complex cells in the primary visual cortex. However, to our knowledge, they have not been integrated previously to model neurons that are selective to changing disparity. Previous models combining motion and disparity energy models were tuned to fronto-parallel motion, since the preferred velocities for the left and right eyes were assumed to be identical[5][6]. However, for stereo-motion stimuli, the velocities are non-identical. In particular, for motion along the midline between the two eyes, the left and right image velocities are of opposite sign. The paper is organized as follows. Section 2 describes the two-stage stereomotion model. Section 3 explores several characteristics of the two stage model.

2 Two Stage Stereomotion Energy Model The model can be decomposed into the cascade of two stages: a disparity selective stage based on the disparity energy model followed by a temporal filtering stage based upon the motion energy model. In this section, we review both the motion energy and disparity energy models, and show how they may be combined to achieve selectivity to changing disparity. 2.1 Disparity Energy Model The disparity energy model is depicted in Fig. 1(a). For simplicity, we assume onedimensional images that lie along corresponding epipolar lines. Left and right images are first convolved with complex valued spatial Gabor filters that model pairs of spatial receptive fields in phase quadrature. The disparity energy is the squared magnitude of the sum. Mathematically, we denote the left and right input signals by I l ( x) and I r ( x) , where x indexes the distance from the receptive field center. We denote the outputs of the spatial Gabor filters by: U l (ψ l ) = ∫

+∞

−∞

U r (ψ r ) = ∫

+∞

−∞

g ( x)e j ( Ω x x +ψ l ) I l ( x)dx g ( x)e j ( Ω x x +ψ r ) I r ( x)dx

(1)

where g(x) is a Gaussian envelope with standard deviation σ2, Ωx is the spatial frequency of the Gabor function, and ψ l and ψ r are phase shifts applied to the left and right Gabor filters. The disparity energy is the squared magnitude of the sum of the outputs of the left and right Gabor filters: Ed (Δψ ) = U l (ψ l ) + U r (ψ l )

2

= U l (0) + U r (0) + 2 Re(U l (0)U r (0)* e j Δψ ) 2

2

(2)

The disparity energy depends on the input images and the relative phase difference between the left and right Gabor filters: Δψ = ψ l −ψ r , but not the absolute phases,

ψ l and ψ r .

A Two Stage Energy Model Exhibiting Selectivity to Changing Disparity

(a)

49

(b)

Fig. 1. In the standard disparity energy model (a), left and right images are first filtered by complex valued Gabor filters with different phase shifts, and then summed and squared. In the stereo-motion energy model (b), the outputs of several disparity energy neurons with different preferred disparities due to relative phase shifts Δψ between the left and right Gabor filters are combined to obtain a complex valued output whose phase varies with disparity. This output is then normalized and passed through a temporal Gabor filter to obtain an output that responds to changing disparity.

The preferred disparity of a disparity energy neuron depends upon the phase difference Δψ . Suppose that every pixel x in the right image corresponds to pixel x + d in the left image, i.e. I r ( x) = I l ( x + d ). For small d, the output of the right Gabor filter can be approximated by U r (ψ r ) = ∫

+∞

−∞

g ( x)e j ( Ω x x +ψ r ) I l ( x + d )dx = ∫

+∞

−∞

≈ e j ( Ω x d −Δψ ) ∫

+∞

−∞

g ( x − d )e j ( Ω x ( x − d ) +ψ r ) I l ( x)dx

g ( x)e j ( Ω x x +ψ l ) I l ( x)dx = U l (ψ l )e j ( Ω x d −Δψ )

(3)

Thus, a position shift at the input results in a phase change at the output. Substituting this expression into (2), we obtain Ed (Δψ ) ≈ U l (0) + U l (0) + 2 U l (0) U l (0) cos(Ω x d − Δψ ) 2

2

(4)

Thus, we can see that the energy output achieves its maximum when the input disparity is approximately equal to d pref = Δψ / Ω . 2.2 Motion Energy Model

The motion energy model has been used to model the responses of direction selective neurons in the primary visual cortex[1] . As in the disparity energy model, the input

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image is first convolved with a complex-valued spatial Gabor function. As demonstrated by (3), a small shift in the image can be approximated by a shift in the phase of the filter output. Repeating this phase shift over time results in an oscillation in the output with frequency ωt = −vΩ x , where v is the image velocity. By passing the output of the spatial Gabor filter with a temporal Gabor filter tuned to temporal frequencies Ωt, we obtain a filter that responds maximally when the input image has significant energy at spatial frequencies near Ωx moving at velocity v = −Ωt / Ω x . 2.3 Stereomotion Model

The stereomotion model exploits the fact that by combining the disparity energies at different phase shifts appropriately, we can obtain a signal that oscillates as the image disparity changes. Thus, as in the motion energy model, this oscillation can be detected by cascading this output with a temporal Gabor filter. It can be shown that the response of the disparity energy model can be written as[5][7] Ed (Δψ ) = S + P cos(Ψ d − Δψ )

(5)

where S = U l (0) + U r (0) , P = 2 U l (0) U r (0) and Ψ d = arg(U l (0)U r (0)* ) . If we 2

2

define Ed (0) = S + P cos(Ψ d ) Ed (π ) = S − P cos(Ψ d )

(6)

Ed (π / 2) = S + P sin(Ψ d ) Ed (−π / 2) = S − P sin(Ψ d ) then we can express the output of the combination unit in Fig., 1(b) as Ed (0) − Ed (π ) E (π / 2) − Ed (−π / 2) +j d 4 4 jΨd * = Pe / 2 = U l (0)U r (0)

output =

(7)

By passing this output through a temporal filter, we can obtain an energy neuron tuned to changing velocity. Substituting the approximation in (3), we obtain

(

U l (0)U r (0)* ≈ U l (ψ l ) U l (ψ l )e jΩ x d Thus, assuming that U l (ψ l )

2

)

*

= U l (ψ l ) e− jΩ x d 2

(8)

changes slowly, the output of the combination unit

oscillates as the disparity changes, much in the same way that the output of a spatial Gabor filter oscillates as the input translates in the motion energy model. When examining the stereomotion, the model should be insensitive to the scaling of the input intensity. Therefore, in the normalization unit, we normalize the output of the combination unit by the sum of the squared magnitude of the two monocular spatial Gabor outputs. In the Section 3, we will show that the normalization unit helps to improve the stability and reliability of the response.

A Two Stage Energy Model Exhibiting Selectivity to Changing Disparity

51

Right retina velocity

-4 -2 0 2 4 -4

-2 0 2 Left retina velocity

4

Fig. 2. The average energy response of the model, which is tuned to the preferred velocity difference of 1 pixel/frame. The spatial frequency and temporal frequency are both 2π / 20 .

3 Characteristics of the Model In this section, we explore the characteristics of the model. For the results in section 3.1, we omit the normalization operation in Figure 1(b), because the normalization process does not essentially affect the basic characteristics. 3.1 Velocity Difference Tuning and Common Velocity Invariance

Motion towards or away from the observer results in a changing disparity, which can also be expressed in terms of the difference, vd, between the image velocities in the left and right images. For fronto-paralllel motion, the velocities in the left and right images will be the same. We denote the common or average velocity between the left and right images by vc. Here we show that the combined stereo-motion energy neurons are tuned to velocity differences, vd, but are invariant to changes in the common velocity, vc , between the two eyes. We simulated the model using 100 inputs consisting of translating random dots. The model is tuned to velocity difference of 1 pixel/frame. Fig. 2 depicts the average response of the energy output, from which we can see two salient properties. First, the bottom-left to upper-right diagonal cross section shows the vd tuning, and has the greatest energy output along the diagonal line vd = 1 . Second, the bottom-right to upper-left diagonal cross section shows the invariance of vc , since the energy response along the common velocity line remains roughly unchanged. 3.2 Joint Disparity and Velocity Difference Tuning

Fig. 3 plots the average energy outputs of models versus different d and vd for translating random dots, with (a) for model without normalization unit (b) for model with normalization unit. The models are both tuned to 2 pixels/frame preferred velocity

X. Guo and B.E. Shi

-40

-40

-20

-20

Disparity

Disparity

52

0 20 40

0 20

-2

0

2

4

Velocity difference (a)

40

-2

0

2

4

Velocity difference (b)

Fig. 3. Energy responses of model without normalization (a) and with normalization (b). Both are simulated over 200 translating random dots. The disparity range is -40 to 40 pixels; velocity difference range is -3 to 5 pixels/frame. The spatial and temporal frequency of the cell is 2π / 40 and 2π / 20 , respectively. Therefore the preferred velocity difference is 2 pixels/frame. Red indicates large values. Blue indicates low values.

difference and 0 preferred disparity. From the figure we can see that the contours of both plots are ovals with major and minor axes approximately parallel to the coordinate axis. The peak response occurs where the preferred disparity and preferred disparity are located. Taking any horizontal (or vertical) cross section of the plot results in a velocity difference tuning (or disparity tuning) curve with the preferred velocity difference (disparity) being the peak location. From Fig. 3, it appears that the horizontal and vertical cross sections of the tuning surface have little dependence on where the cross section is taken. For example, the velocity difference tuning curve seems to have a similar shape (up to a scaling factor) with the peak location at the preferred velocity difference, no matter which disparity cross section is chosen. This suggests that disparity tuning and vd tuning may be separable. Here we show that this is indeed the case. Denote the energy response of the normalized model by f (vd , d ) , the disparity tuning curve at vd 0 by hvd 0 (d ) , and the velocity difference tuning curve at d 0 by

hd0 (vd ) . We approximate f (vd , d ) using the assumption of separability by f (vd , d ) = k hvd (d )hd (vd )

∀vd , d

(9)

where k is a scaling factor, hvd (d ) and hd (vd ) are the average vertical and horizontal cross sections, respectively. From Fig. 4 we can see that the measured tuning surface and its approximation assuming separability are approximately the same. To evaluate this fit quantitatively, we examine MSE value of the estimation. Here, we only take into account the model with normalization unit, because it is much more stable than non-normalized model, as described below. We quantify the fit using the square root of mean squared error normalized by the average response

-40

-40

-20

-20

Disparity

Disparity

A Two Stage Energy Model Exhibiting Selectivity to Changing Disparity

0 20 40

53

0 20

-2

0

2

40

4

-2

Velocity difference (a)

0

2

4

Velocity difference (b)

Fig. 4. (a) Measured energy response f (vd , d ) for translating noise stimuli. (b) Energy response estimated assuming separability, f (vd , d ) . The color scales of the two plots are identical.

I=

Average( f (vd , d ) − f (vd , d )) 2

(10)

Average( f (vd , d ))

For the data shown, the value of I is 2.66%. This supports the conclusion that the responses of the stereo-motion tuned neurons are approximately separable. 3.3 Comparison between Models with and without Normalization

The characteristics of the two models do not differ by much in terms of average energy response of inputs, as we can see from Fig. 3. However, intuitively, the output of normalized model should exhibits less variation than the non-normalized model. -40

-40 1.95 1.9 1.85

0

1.8 1.75

20

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Disparity

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Disparity

0.5 0.4 0.3

0

0.2 20

0.1

1.7 40

-2

0

2

Velocity difference (a)

4

40

-2

0

2

4

Velocity difference (b)

Fig. 5. (a) Standard deviation of the non-normalized model. (b) Standard deviation of the normalized model. The standard deviation is expressed as multiples of the average response. The color scales of the two figures are different.

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Fig. 5(a) and (b) shows the relative standard deviation of the energy output. The standard deviation for non-normalized model is 1.7~2 times of the average response while that of the normalized model is below 0.6. In particular, in the vicinity of the preferred region, i.e. disparity = 0 and vd = 2 pixels/frame, the standard deviation is below 0.15 times of the average response for the units with normalization. Therefore, the normalization unit greatly improves the stability and reliability of the response.

References 1. Adelson, E.H., Bergen, J.R.: Spatiotemporal Energy Models for the Perception of Motion. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 2, 284–299 (1985) 2. Brooks, K.R., Stone, L.S.: Stereomotion Speed Perception: Contributions from both Changing Disparity and Interocular Velocity Difference Over a Range of Relative Disparities. J. Vis. 4, 1061–1079 (2004) 3. Cumming, B.G., Parker, A.J.: Binocular Mechanisms for Detecting Motion-in-Depth. Vis. Res. 34, 483–495 (1994) 4. Ohzawa, I.: Mechanisms of Stereoscopic Vision: The Disparity Energy Model. Curr. Opin. Neurobiol. 8, 509–515 (1998) 5. Qian, N.: Computing Stereo Disparity and Motion with Known Binocular Cell Properties. Neural Comput. 6, 390–404 (1994) 6. Qian, N., Anderson, R.A.: A Physiological Model for Motion-Stereo Integration and a Unified Explanation of Pulfrich-Like Phenomena. Vis. Res. 37, 1683–1698 (1997) 7. Fleet, D.J., Wagner, H., Heeger, D.J.: Neural Encoding of Binocular Disparity: Energy Models, Position Shifts and Phase Shifts. Vis. Res. 36, 1839–1857 (1996)

A Feature Extraction Method Based on Wavelet Transform and NMFs Suwen Zhang1, Wanyin Deng1, and Dandan Miao2 1 School of Automation, Wuhan University of Technology, Wuhan, 430070, China 2 School of Resource and Environmental Science, Wuhan University, Wuhan, 430070, China [email protected]

Abstract. In this paper, a feature extraction method is proposed by combining Wavelet Transformation (WT) and Non-negative Matrix Factorization with Sparseness constraints (NMFs) together for normal face images and partially occluded ones. Firstly, we apply two-level wavelet transformation to the face images. Then, the low frequency sub-bands are decomposed according to NMFs to extract either the holistic representations or the parts-based ones by constraining the sparseness of the basis images. This method can not only overcome the the low speed and recognition rate problems of traditional methods such as PCA and ICA, but also control the sparseness of the decomposed matrices freely and discover stable, intuitionistic local characteristic more easily compared with classical non-negative matrix factorization algorithm (NMF) and local non-negative matrix decomposition algorithm (LNMF). The experiment result shows that this feature extraction method is easy and feasible with lower complexity. It is also insensitive to the expression and the partial occlusion, obtaining higher recognition rate. Moreover, the WT+NMFs algorithm is robust than traditional ones when the occlusion is serious. Keywords: Feature extraction, Wavelet transformation, NMFs, Face recognition.

1 Introduction In recent years, with the development of some applications such as electronic commerce, face recognition has become one of the most potential biometrics authentication methods. Feature extraction is the most important part of face recognition. There are many methods which can mainly be divided into two big categories: methods based on the geometry characteristic, and methods based on the statistical characteristic. Since the geometry feature extraction is sensitive to illumination, expression and posture, people mainly used methods based on the statistical character in recent years, among which the most frequently used ones are principle components analysis (PCA) and independent components analysis (ICA). However, PCA and ICA do not impose non-negative constraint on the operation object when implementing the matrix decomposition, and the mutual counteract of the positive and negative coefficient will weaken the characteristic and make the recognition accuracy drop. Lee and Seung F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 55–62, 2008. © Springer-Verlag Berlin Heidelberg 2008

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proposed the concept of Non-negative matrix factorization (NMF) [1], whose entries are all non-negative and produces a part-based representation of images because it allows only additive, not subtractive, combinations of basis components. But the problem of NMF is that it dose not always get basis vectors for local representation. The reason is that the sparseness levels for basis vector and coefficients matrix are not high enough. The localized characteristic of LNMF method [2] is obvious, but the convergence of LNMF is time consuming, and LNMF can’t explicitly control the sparseness of the representation. Patrik proposed NMF with sparseness control and realized controlling the sparsity of basis vector and coefficient matrix [3]. In this paper, we combine Wavelet Transformation (WT) and Non-negative Matrix Factorization with Sparseness Constraints (NMFs) together to extract features for face recognition. Firstly, we apply two-level wavelet transformation to the face images and decompose the low frequency sub-bands by using Names. The experimental results show that the two-level wavelet transformation overcomes the influence of the change of posture and expression to a great extent. It can catch the substantive characteristic and reduce the computation complexity effectively. In addition to that, NMFs can not only discover more stable and intuitionistic partial characteristic but also obtain either the holistic or parts-based representations by constraining the sparseness of the basis images freely. When serious obstruct occurs, NMFs has better robustness than NMF algorithm.

2 Wavelet Transformation The wavelet transformation is a kind of time-frequency signal analysis method. By using it, the image signal can be decomposed into many sub-band image signals with different Spatial Resolution, frequency characteristic and directional features. The change of human face expression and small-scale obstruct affect no low-frequency part but the high-frequency part of the image only. Additionally, the wavelet transformation has perfect restructuring ability guaranteeing that the information will not be lost during the decomposition process and the redundancy will not occur either. Therefore, we can use the wavelet analysis to filter out the high frequency information before the feature extraction. And only the low frequency sub-graph is used for recognition. Given a two-dimension signal f (x1, x2 ) , whose square is integrabel viz.

f ( x1 , x2 ) ∈ L2 (R) . The continuous wavelet transformation of f (x1, x2) is defined as WTf (a;b1, b2 ) =

x −b x − b 1 f (x1, x2 )ψ( 1 1 , 2 2 )dx1dx2 a ∫∫ a a

(1)

where the Wavelet Base Function is

1 x1 − b1 x2 − b2 , ) a a a (2) The most frequently used wavelet transform in image processing is dyadic wavelet transform, which can discretize formula (1) with a = 2 n , b ∈ Z . A small change of exponential n may result in obvious change of scale, so dyadic wavelet transform has scale-amplifying character in signal analysis. Applying 2-level wavelets decomposition to the original image, we get the result as shown in Fig. 1. We select the low

ψ a ;b ,b ( x1 , x2 ) = ψ ( 1

2

A Feature Extraction Method Based on Wavelet Transform and NMFs

Fig. 1. Two-level wavelet transformation

57

Fig. 2. Face recognition procedure

frequency component LL in the second level as the wavelet characteristic, which can not only retain the overall shape information of human face, but also weaken the partial detail [4].

3 NMF and NMFs 3.1 NMF

Given a non-negative matrix Vn×m, NMF finds the non-negative matrix Wn×r and H r×m such that V ≈ WH . Where each column is a non-negative vector of dimen-

，

sion n corresponding to a face image, m is the number of training images, and r is the dimension of eigenvector. Each column of matrix W represents a basis vector while each column of H means the weights used to approximate the corresponding column in V using the bases from W. NMF decomposition is a NP problem, which can be taken as an optimization problem, using iterative method to get basis vectors W and coefficient matrix H. One form of objective function is the Euclidean distance: D(V || WH ) = X − WH

2

= ∑ ( X ij − (WH )ij ) 2

. So NMF factorization is a solution to the following optimization problem: ij

min D (V || WH ) s.t. W , H ≥ 0, ∑ Wij = 1 W ,H

i

Lee and Seung presented an iterative approach to reach a local maximum of this objective function [5]. 3.2 NMFs

Non-negative Matrix Factorization with Sparseness constraints algorithm is a matrix factorization algorithm based on NMF. When the primitive matrix V is factorized, it can control sparseness of feature matrix W or encoder matrix H to attain the specific application requirements. When it is regarded as feature extraction algorithm of face recognition, the feature matrix W is desired to be sparsed. So, the differences of the elements in matrix W increase, the feature of characteristic face will be more prominent and a face image waiting for recognition related with fewer characteristic faces. It is easier to be recognized. The objective function in NMFs is defined as follows:

min D (V || WH ) s.t. W , H ≥ 0, ∑ Wij = 1 W ,H

i

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sparseness( wi ) = Sw , sparseness(h j ) = Sh Where

wi is the i-th column of W and h j is the j-th row of H . Here, S w and S h are

the desired sparsenesses of W and H respectively. These two parameters are set by the user. The sparseness of each column wi in matrix W is defined as:

sparseness ( wi ) =

n − ( ∑ wij ) / j

∑w

2

ij

j

n −1 Here, n is defined as the dimension of non-negative vector. According to the definition, it is easy to know only when the non-negative vector X contains a non-zero element, this function is equal to 1; only when each element is equal to others, this function is equal to 0. When the base image matrix W is in low sparseness, the difference among the elements’ gradation in each column of the matrix corresponding to the human face eigenvector is not so obvious. Therefore, it can better reflect relationship among the gradations of overall face image. Then, it can respond the overall characteristic of human face well. This though is similar to the PCA algorithm based on the overall characteristic extraction. When the base image matrix W is in high sparseness, the difference among the gradations of the elements in each column in the matrix corresponding to the human face eigenvector is very obvious. So, it only retains the relationship among the gradation of partial face image. Then, it can reflect better partial characteristic of human face. This thought returns to the traditional NMF and LNMF algorithm [6].

4 Face Recognition The human face recognition can be divided into the training process and the recognition process. The concrete plan is shown in Fig. 2. The algorithm is described as follows: (i) The training sample images are pretreated and all the sizes of the images should be the same. (ii) The images are transformed by the 2-level wavelet. Then, the low-frequency sub-bands LL2 of each image can be acquired. (iii) The low-frequency sub-bands of the M images that obtained above can be constituted to an n×m -dimensional matrix which is called V.

v j ∈ R n is piled up

according to the columns of the low-frequency sub-bands of the j-th training imn

ages [7]. It also satisfies the equation below :, v ij ≥ 1, ∑ v ij = 1, j = 1, 2 , " m . i =1

(iv) The V can be taken into the NMFs factorization. Then, the base image matrix W and the weighted matrix H can be obtained. Under general circumstances, the r usually selects the square numbers within 100 ([7]). Thus, the W and H will be smaller than the primitive matrix. In this way, the compressed model of primitive data matrix will be obtained [8].

A Feature Extraction Method Based on Wavelet Transform and NMFs

59

(v) The low-frequency sub-bands of the training sample images and the testing samples project to the “characteristic subspace” W which is formed by base images respectively. If W + = (W TW ) −1W T is defined, the projection vector (is also weighted vector) of human face sample on base images can be obtained from h = W + v .These projection vectors are exactly the eiqenvectors that are used to depict the face [9]. (vi) The nearest neighbor classifier can be used to classify the testing faces.

5 Simulation Experiment and Comparison The factorization results of the three algorithms that include NMF, NMFs, WT+NMFs are tested respectively. ORL gallery is chosen to do the experiment. It has 40 individuals altogether, each person has 10 person face images, each image has 256 gradation levels, the size is 92×112. The posture and angle of each image are various. So the training gallery has 400 images in all. At the same time, the eyes, nose and mouth of the testing sample faces will be covered up stochastically. The covered partial images can be expressed by gradations with 80×30 pixel.

Fig. 3. Examples of normal face images and partially occluded ones

For each person, 5 face images are selected stochastically to be trained, the remaining five images are taken as facial images that are waiting for recognition. So, there are 200 images in training gallery and testing gallery each. In the experimental process, the images are normalized to 128×128 first; next, apply 2-level wavelet transformation to the images and obtain the low-frequency subbands of each image, the size of it is 32×32. Then, all the low-frequency sub-bands of facial images are elongated into column vectors of 1024 dimensions, all the column vectors of facial images are generated into the decomposition matrix V. Because the number of the training images selected is 200, the decomposition matrix V has 1024 rows and 200 columns. Then, the matrix V is taken into the NMFs decomposition. W and H are obtained. The MATLAB7.0 is used to program on PC machine with P42.2GHz and 512 MB memory. The processes and conclusions of each experiments are as follows: Firstly, comparing the direct NMFs method with the three different wavelet base WT+NMFs methods, the 2-level wavelet transformation results on the basis of the three different wavelets are obtained. The experimental date from the direct NMFs and WT+NMFs methods are shown in Table 1.

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The original image

db6

haar

Db2

Fig. 4. The images decomposed through different wavelet Table 1. The comparison of the average recognition rate and the average computation time

Method

Average identification rate (% ) without mask

NMFs WT+ NMFs

Db2 Haar Db6

with mask

Average computation time (s)

91.5

85.5

218.2

95.0 94.5 93.5

93.5 91.5 92

49.8 48.7 49.3

The experimental results show that using the WT+NMFs method is able to shorten average calculation time and improve the efficiency of recognition, especially when having something covered, the recognition rate can be increased by 8%. However, the recognition results are not sensitive to the choice of the several types of wavelet. Then, under the two kinds of situations that have cover and doesn’t have cover, changing the value of r and the sparseness, doing the experiments on comparing recognition, the results are obtained and shown in Fig. 3(a) and (b). From the Figure, it is clear that, the recognition rate increases with r, but when the r increases to certain degree, the recognition rate decreases. When there is no cover on face and the value of r is relatively low, the recognition rate of WT+NMFs methods under the high sparseness is high. When the face is covered and the value of r is relatively high, the recognition rate of WT+NMFs methods under low sparseness is high.

(a) normal face images

(b) partially occluded face images

Fig. 5. Comparison of recognition rate

A Feature Extraction Method Based on Wavelet Transform and NMFs

sw=0.45

sw=0.55

sw=0.65

61

sw=0.75

Fig. 6. Basis images of NMFs with r=81

Fig. 4 shows the basis images after WT+NMFs decomposition under the different sparseness when r=81. It is also easy to know from the Figure that along with the enhancement of the sparseness, NMFs base image has tendency of the transition from the overall situation to the partial situation. It indicates that NMFs has both the overall characteristic expression and partial characteristic expression in the control of sparseness in the control of sparseness. The face recognition method with low sparseness, based on the overall expression is more insensitive to covers than that with high sparseness, based on the local expression, and has certain robustness.

6 Conclusions In this paper, the human face feature extraction method is brought up based on wavelet transformation and NMFs. The method is easy and feasible. It si insensitive to the changing of facial gestures, expressions and the head ornaments. It is also able to discover the stable and direct-viewing partial characteristics well and control the sparseness of the decomposed matrix more freely. When serious obstruct occurs, the WT+NMFs algorithm has the better robustness than NMF algorithm. The former algorithm accelerates the feature extraction speed greatly and overcomes the weakness of traditional NMF fundamentally in that aspect. This method is not only applicable to the human face feature extraction, but also suitable for other image feature extraction problems.

References 1. Lee, D.D., Seung, H.S.: Unsupervised Learning by Convex and Conic Coding. In: Mozer, M.C., Jordan, M.I., Petsche, T. (eds.) Advances in Neural Information Processing Systems, pp. 515–521. The MIT Press, Massachusetts (1997) 2. Feng, T., Stan, Z., Li, H.Y.S., Zhang, H.J.: Local Non-negative Matrix Factorization as a Visual Representation. In: 2nd International Conference on Development and Learning, Cambridge, pp. 7695–1459 (2002) 3. Patrik, O.H.: Non-negative Matrix Factorization with Sparseness Constraints. Journal of Machine Learning Research 5, 1457–1469 (2004) 4. Manthalkar, R., Biswas, P.K., Chatterji, B.N.: Rotation and Scale Invariant Texture Features Using Discrete Wavelet Packet Transform. Pattern Recognition Letter 24, 2452–2462 (2003)

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5. Lee, D., Seung, H.S.: Learning the Parts of Objects by Non-negative Matrix Factorization. Nature 1401, 788–791 (1999) 6. Pu, X.R., Zhang, Y., Zheng, Z.M., Wei, Z., Mao, Y.: Face Recognition Using Fisher Nonnegative Matrix Factorization with Sparseness Constraints. In: Wang, J., Liao, X.-F., Yi, Z. (eds.) ISNN 2005. LNCS, vol. 3497, pp. 112–117. Springer, Heidelberg (2005) 7. Ouyang, Y.B., Pu, X.R., Zhang, Y.: Wavelet-based Non-negative Matrix Factorization with Sparseness Constraints for Face Recognition. Application Research of Computers 10, 159– 162 (2006) 8. Chen, W.G., Qi, F.H.: Learning NMF Representation Using a Hybrid Method Combining Feasible Direction Algorithm and Simulated Annealing. Acta Electronica Sinica 31, 2190– 2193 (2003) 9. Zhang, Z.W., Yang, F., Xia, K.W., Yang, R.X.: Research on Face Recognition Method Based on Wavelet Transform and NMF. Computer Engineering 33, 176–179 (2007)

Similarity Measures between Connection Numbers of Set Pair Analysis Junjie Yang, Jianzhong Zhou, Li Liu, Yinghai Li, and Zhengjia Wu School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, China [email protected]

Abstract. The Set Pair Analysis (SPA) is a new system analysis approach and uncertainty theory. The similarity measure between connection numbers is the key to applications of SPA in multi-attribute decision-making, pattern recognition, artificial intelligent. However, it is difficult to accurately depict similarity degree between connection numbers. The distance between connection numbers, a group of checking criterions and the similarity degree functions of connection numbers in SPA are presented in this paper to measure the similarity between connection numbers, and the rationality of such measurement is also explained by the well-designed criterions. The result shows the effectiveness of the proposed similarity measures. Keywords: Set Pair Analysis, Similarity measures, Similarity degree function.

1

Introduction

In the real world, there are all kind of uncertainties such as fuzzy uncertainty, random uncertainty, indeterminate-known, unknown and unexpected incident uncertainty, and uncertainty which resulted from imperfective information [1]. The most successful approach to understand and manipulate the uncertainty knowledge is the fuzzy set theory proposed by Zadeh. Set Pair Analysis (SPA) theory provides another way to expressing and processing the uncertainties. The theory overlaps with many other uncertainty theory, especially with fuzzy set theory, evidence theory, Boolean reasoning methods, and rough set theory. SPA theory emphasizes the relativity and fuzziness in information processing, can identify relatively certainty information and relatively uncertainty information from the researched system. The connection number theory, that includes abundant contents and has significant meaning in the development history of mathematics, has been set up. SPA considers the connection number as a kind of number that can depict uncertain quantity, and thinks that the connection number is different from constant, variable, and super uncertain quantity essentially [2,3]. The similarity measure between connection numbers is the key to applications of SPA in multi-attribute decision-making, pattern recognition, artificial intelligent. However, because the connection number contains identity, discrepancy F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 63–68, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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and contrary information of system, it is difficult to accurately depict similarity degree between connection numbers. In this paper, the new similarity measures are proposed, which are presented in Section 2 and Section 3. Finally, conclusions are obtained in Section 4.

2

The Distance between Connection Numbers

In this section, we present the similarity measures between connection numbers by adopting an extension of distance in functional analysis. Definition 2.1. Let μ1 and μ2 be two connection numbers, where μ1 = a1 + b1 i + c1 j, μ2 = a2 + b2 i + c2 j. Weighted Minkowski distance between μ1 and μ2 can be defined as follows:  (1) dq (μ1 , μ2 ) = q ωa (a1 − a2 )q + ωb (b1 − b2 )q + ωc (c1 − c2 )q , where, ωa , ωb and ωc are weight. There are three forms of distance are as follows: (1) Hamming distance (q=1) d1 (μ1 , μ2 ) = ωa |a1 − a2 | + ωb |b1 − b2 | + ωc |c1 − c2 | . (2) Hamming distance (q=2)  d2 (μ1 , μ2 ) = ωa (a1 − a2 )2 + ωb (b1 − b2 )2 + ωc (c1 − c2 )2 .

(2)

(3)

(3) Chebyshev distance (q → ∞) d∞ (μ1 , μ2 ) = max(ωa |a1 − a2 |, ωb |b1 − b2 |, ωc |c1 − c2 |) .

3

(4)

Similarity Measures between Connection Numbers

In this section, a group of rationality checking criterion of similarity measures is presented, and then similarity measures between connection numbers are proposed by employing the idea of similarity degree function [4,5]. 3.1

Checking Criterion

Let μ1 , μ2 and μ3 be three connection numbers, where μ1 = a1 + b1 i + c1 j, μ2 = a2 + b2 i + c2 j and μ3 = a3 + b3 i + c3 j, ρ(μ1 , μ2 ), ρ(μ1 , μ3 ) and ρ(μ2 , μ3 ) denote the similarity degree function between μ1 and μ2 , μ1 and μ3 , μ2 and μ3 respectively. Similarity degree function must satisfy the following criterions: Criterion 3.1: 0 ≤ ρ(μ1 , μ2 ) ≤ 1 . Criterion 3.2:(monotonicity criterion) ρ(μ1 , μ3 ) ≤ min(ρ(μ1 , μ2 ), ρ(μ2 , μ3 )), If μ1  μ2  μ3 . Criterion 3.3:(symmetry criterion) ρ(μ1 , μ2 ) = ρ(μ2 , μ1 ) . Criterion 3.4:ρ(μ1 , μ2 ) = 0 if and only if μ1 = 1+0i+0j and μ2 = 0+0i+1j; ρ(μ1 , μ2 ) = 1 if and only if μ1 = μ2 , that is a1 = a2 , c1 = c2 . − − Criterion 3.5:ρ(μ1 , μ2 ) = ρ(μ− 1 , μ2 ), where μ1 = c + bi + aj is called as complement connection number of μ1 = a + bi + cj .

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3.2

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Similarity Measures

Definition 3.1. Let μ be a connection number, where μ = a + bi + cj. (1) C(μ) = a − c is called as the core of μ . (2) C ω (μ) = ωa · a + ωb · b + ωc · c is called as the weighted core of μ, where ωa , ωb and ωc are weights of a, b and c respectively, ωa ≥ ωc ≥ 0 ≥ ωb . (3) S(μ) = a(1 + α · b) is called as identity degree of μ, D(μ) = c(1 + β · b) is contrary degree of μ, where α, β ∈ [0, 1], can reflect various risk attitudes of decision makers, the more value of α is, the more the probability which discrepancy degree converts into identity degree on connection number μ is. The more value of β is, the more the probability which discrepancy degree converts into contrary degree on connection number μ is. Let μ1 and μ2 be two connection numbers, where μ1 = a1 + b1 i + c1 j, μ2 = a2 + b2 i + c2 j. The similarity measures between two connection numbers are defined as follows: Definition 3.2. The similarity degree function ρ(μ1 , μ2 ) is defined as: ρ(μ1 , μ2 ) = 1 −

dq (μ1 , μ2 ) , 21/q

(5)

where dq (μ1 , μ2 ) denotes the Minkowski distance between μ1 and μ2 . It is obvious that similarity measure (5) meets criterion 3.1, 3.3, 3.4 and 3.5. Now, q = 2 is taken as an example to prove that the similarity measure meets the criterion 3.2 as follows. Let μ1 , μ2 and μ3 be three connection numbers, where μ1 = a1 + b1 i + c1 j, μ2 = a2 + b2 i + c2 j, μ3 = a3 + b3 i + c3 j, and μ1  μ2  μ3 , that is a1 ≤ a2 ≤ a3 and c1 ≥ c2 ≥ c3 . We can derive equation (6) as:  (c1 − c3 )2 ≥ (c1 − c2 )2 , (c1 − c3 )2 ≥ (c2 − c3 )2 , (6) (a1 − a3 )2 ≥ (a1 − a2 )2 , (a1 − a3 )2 ≥ (a2 − a3 )2 . Then (6) can be wrotten as:  (c1 − c3 )2 ≥ max{(c1 − c2 )2 , (c2 − c3 )2 } , (a1 − a3 )2 ≥ max{(a1 − a2 )2 , (a2 − a3 )2 } . The following equations can be derived by (7): ((a1 − a3 )2 + (c1 − c3 )2 )1/2 ≥ max{((a1 − a2 )2 + c1 − c2 )2 )1/2 , ((a2 − a3 )2 + (c2 − c3 )2 )1/2 } , d(μ1 , μ3 ) ≥ max{d(μ1 , μ2 ), d(μ2 , μ3 )} , 1 − d(μ1 , μ3 ) ≤ min{1 − d(μ1 , μ2 ), 1 − d(μ2 , μ3 )} .

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So, the following result can be derived: ρ(μ1 , μ3 ) ≤ min{ρ(μ1 , μ2 ), ρ(μ2 , μ3 )} . It indicates that the similarity measure of (5) satisfies the criterion 3.2. Definition 3.3. The similarity degree function between μ1 and μ2 is defined as: ρ(μ1 , μ2 ) = 1 −

dq (μ1 , μ2 ) |C(μ1 ) − C(μ2 )| . − 2 21/q

(8)

Definition 3.4. ρ(μ1 , μ2 ) is defined as: ρ(μ1 , μ2 ) = 1 −

|C(μ1 ) − C(μ2 )| |S(μ1 ) − S(μ2 )| + |D(μ1 ) − D(μ2 )| − . 2 2

(9)

It is obvious that similarity measure (9) meets criterion 3.1, 3.3, 3.4 and 3.5. The similarity measure meets the criterion 3.2 will be proved as follows. Let α = β = 1 , we can derive the equation (10) as follows: ρ(μ1 , μ2 ) = 1 −

Δa12 − Δc12 |[2 − (a1 + a2 )]Δa12 − (a1 c1 − a2 c2 )| −( 2 2 |[2 − (c1 + c2 )]Δc12 − (a1 c1 − a2 c2 )| + ), 2

(10)

where, Δa12 = a1 − a2 , Δc12 = c1 − c2 . Let c1 = c2 = c , (11) can be derived as: ρ(μ1 , μ2 ) = 1 − f1 |Δa12 | .

(11)

Let a1 = a2 = a , (12) can be derived as: ρ(μ1 , μ2 ) = 1 − f2 |Δc12 | ,

(12)

1 +a2 )| 1 +c2 )| f1 = 1+|4−2c−(a ,f2 = 1+|4−2a−(c . 2 2 obvious that |Δa13 | ≥ |Δa12 |, |Δa13 | ≥ |Δa23 |,

where, if μ1  μ2  μ3 , that is It is a1 ≤ a2 ≤ a3 and c1 ≥ c2 ≥ c3 , and (13) can be derived as: |Δa13 | ≥ max{|Δa12 |, |Δa23 |} .

(13)

By substituting (13) into (11) and (12), the following result can be derived: ρ(μ1 , μ3 ) ≤ min{ρ(μ1 , μ2 ), ρ(μ2 , μ3 )} . It indicates that the similarity measure of (9) satisfies the criterion 3.2. A group of examples will be presented to illustrate the effectiveness of the proposed similarity measures between connection numbers. Example 3.1. Let μ1 and μ2 be two connection numbers, where μ1 = a1 + b1 i + c1 j, μ2 = a2 + b2 i + c2 j. The calculated results of the proposed similarity measures are shown in Table 1.

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Table 1. The examples of the proposed similarity measures μ1

μ2



a1

c1

a2

c2

1 2 3 4 5 6 7 8 9 10 11 12 13

0.1 0.0 1.0 0.2 0.4 0.2 0.2 0.2 0.2 0.1 0.1 0.4 0.6

0.5 1.0 0.0 0.5 0.1 0.5 0.5 0.5 0.5 0.5 0.5 0.6 0.4

0.1 1.0 0.0 0.3 0.5 0.1 0.1 0.3 0.3 0.4 0.2 0.2 0.8

0.5 0.0 1.0 0.4 0.2 0.4 0.6 0.4 0.6 0.6 0.8 0.8 0.2

ρ(μ1 , μ2 ) Definition Definition Definition 3.4 Definition 3.4 3.2 3.3 α=β=1 α = 0.8, β = 0.2 1.0000 1.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9000 0.8000 0.7700 0.7850 0.9000 0.9000 0.9200 0.9190 0.9000 0.9000 0.9200 0.9010 0.9000 0.8000 0.7700 0.7850 0.9000 0.8000 0.7700 0.7850 0.9000 0.9000 0.9600 0.9210 0.7764 0.6764 0.7200 0.7360 0.7764 0.6764 0.8200 0.7360 0.8000 0.6000 0.6000 0.6000 0.8000 0.6000 0.6000 0.6000

In Table 1, the calculated results included first to third rows illustrate that the proposed similarity measures satisfy the criterion 3.4. The results included the 12 − th and 13 − th rows satisfy the criterion 3.5. The results included 4−th to 9−th rows illustrate that, when Δa12 is equal to Δc12 of two group connection numbers, the similarity degree functions defined by Definition 3.2 can not depict the difference between the two group connection numbers, and Definition 3.3 and Definition 3.4 overcome the problem. Comparing to the values of similarity degree function calculated by Definition 3.3, the values of Definition 3.4 are well-distributed, it indicates the Definition 3.4 is more effective. Furthermore, the more results can be obtained by changing the value of α and β of Definition 3.4 based on the need of practical problem.

4

Conclusions

The similarity degree depiction between connection numbers is one of the basic connection number theories in SPA. In this paper, the similarity degree between connection numbers is described by adopting the similarity degree function of connection numbers. A group of checking criterion and similarity measures of connection numbers are proposed. The computed results of examples show that, the proposed similarity measures between connection numbers are beneficial attempt to measure the similarity between connection numbers in SPA.

Acknowledgments Project supported by the State Key Development Program for Basic Research of China (No. 2007CB714107); The Special Research Foundation for the Public

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Welfare Industry of the Ministry of Science and Technology and the Ministry of Water Resources (No. 200701008); The Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20050487062).

References 1. Jiang, Y.L., Zhuang, Y.T., Li, Z.X.: Application of Set Pair Analysis in Urban Planning Project Comprehensive Evaluation. In: Proceedings of 2005 International Conference on Machine Learning and Cybernetics, pp. 2267–2271 (2005) 2. Zhao, K.Q., Xuan, A.L.: Set Pair Theory: A New Theory Method of Non-Define and Its Applications. Systems Engineering 14, 18–23 (1996) 3. Jiang, Y.L., Xu, C.F., Yao, Y., Zhao, K.Q.: Systems Information in Set Pair Analysis and Its Applications. In: Proceedings of International Conference on Machine Learning and Cybernetics, pp. 1717–1722 (2004) 4. Cheng, K.Y.: Research in Fuzzy Logic on Set Pair Analysis. Systems Engineering Theory & Practice 32, 210–213 (2004) 5. Zhang, D.F., Huang, S.L., Li, F.: An Approach to Measuring the Similarity between Vague sets. Journal of Huazhong University of Science and Technology 32, 59–60 (2004)

Temporal Properties of Illusory-Surface Perception Probed with Poggendorff Configuration Qin Wang and Marsanori Idesawa Graduate School of Information Systems, The University of Electro-Communications 1-5-1, Chofugaoka, Chofu-Shi, Tokyo, 182-8585, Japan Tel.: +81-424-43-5649 [email protected]

Abstract. Temporal properties of illusory surface perception were investigated by using the probing method of the Poggendorff configuration. We used real lines and an opaque illusory surface to compose the Poggendorff configuration, which was presented in an intermittent display method so that the real lines were displayed continuously and the opaque illusory surface was displayed periodically with various duration and interval times. The results showed that the opaque illusory surface required a minimum duration of approximately 220 msec for sustained perception. An interval of as much as 2200 msec was needed to obliterate the perception of the opaque illusory surface. We decided the intermittent display method was effective to directly examine the time course of illusory surface perception. Furthermore, we concluded that we could achieve better understanding of the surface perception mechanism of the human visual system by utilizing the intermittent display method and probing method of the Poggendorff configuration. Keywords: surface perception, illusory surface, temporal properties, Poggendorff illusion.

1

Introduction

A three-dimensional (3D) illusory surface is perceived by the partial disparity along an object’s contour where no physical visual stimuli make point-by-point correspondence (Fig. 1). In relation to the phenomenon of 3D illusory surface perception, opaque and transparent perception has been discovered, and interaction between them has been reported. [1], [2], [3] Temporal properties are a crucial aspect of illusory surface perception in the human visual system, but the results of conventional studies examining the temporal properties have not been consistent. [4], [5], [6], [7], [8], [9] The purpose of the present study was to investigate the temporal properties of 3D illusory surface perception by using the probing method of the Poggendorff configuration. F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 69–77, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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The Poggendorff illusion is one of geometrical illusions. [10], [11] In this illusion, two collinear line segments obliquely abutting an inducing element are perceived as noncollinear (Fig. 2(a)). In the Poggendorff illusion without physical contact, the gaps are 1.4 degrees between the line segments and the inducing element (Fig. 2(b)). Opaque surface perception can be probed from the occurrence of the illusion in the Poggendorff configuration without physical contact. [12], [13] However, during our daily experiences, the human visual system’s response to any stimulus does not result in a final percept without some time delay. Likewise, the perception of any stimulus decays with some time delay after the stimulus disappears. In the present study, we hypothesized that an illusory surface is the result of responses by a transient system. In other words, some time delay is required for the genesis of illusory surface perception, and illusory surface perception requires some time for decay. On the basis of the hypothesis, we examined the temporal properties of illusory surface perception by using the intermittent display method and the probing method of the Poggendorff configuration.

L

L

R

Fig. 1. An example of illusory surfaces. (L is for left-eye view and R is for right-eye view.) A white square surface is perceived.

L

R (a)

L

L

R

L

(b)

Fig. 2. The Poggendorff configurations. (L is for left-eye view and R is for right-eye view.) (a) The conventional Poggendorff configuration. (b) The Poggendorff configuration without physical contact.

2

Probing Method for Opaque Surface Perception

The Poggendorff configuration without physical contact is devised based on the conventional Poggendorff illusion (Fig. 2(a)). In the configuration without

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Observe Poggendorff configuration without physical contact.

No

Illusion occurs?

Yes The inducing element The inducing element is not perceived is perceived as an opaque surface. as an opaque surface.

L

(a)

R

L

(b)

Fig. 3. Principles of the Poggendorff configuration. (a) Diagram of the probing method of the Poggendorff configuration. When observing the Poggendorff configuration without physical contact, such that the line segments are at a farther depth than the inducing element, if the illusion occurs, the inducing element is perceived as an opaque surface; otherwise, the inducing element is not perceived as an opaque surface. (b) The Poggendorff illusion without physical contact.

gn yia1 pls i D

line displaying testing surface displaying duration

duration interval

0 t0

t1

t2

Time

t3

t4

(a)

L

R

(b)

L

L

R

L

(c)

Fig. 4. Diagram of the intermittent display method. (L is for left-eye view and R is for right-eye view.) (a) The procedure of the display; the real lines are displayed continuously and the testing opaque illusory surfaces are displayed periodically in the duration and interval times. (b) The stimuli presented in the duration time, in which the lines and testing opaque illusory surface are displayed synchronously. (c) The stimuli presented in interval time, in which only lines are displayed.

physical contact, the gaps are 1.4 degrees between the line segments and the inducing element. The nearer perceptual depth of the inducing surface and its opaque property is an indispensable factor for perceiving the illusion without physical contact.

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The probing method for opaque surface perception has been proposed on the basis of the characteristics of the Poggendorff configuration without physical contact. In the probing method of the Poggendorff configuration, opaque surface perception could be probed from the occurrence of the illusion in the Poggendorff configuration without physical contact. Specifically, when we observe the Poggendorff configuration without physical contact, such that the line segments are at a farther depth than the inducing element, if the illusion occurs, the inducing element is perceived as an opaque surface; otherwise, the inducing element is not perceived as an opaque surface (Fig. 3).

3

Intermittent Display Method

In the present study, the lines and the testing surface composing the Poggendorff configuration were used as real lines and test opaque illusory surface. The lines were displayed continuously and the testing surface was displayed periodically with various duration and interval times. In other words, real lines and opaque illusory surfaces were displayed synchronously in the duration time, but only real lines were displayed in intervals (Fig. 4).

4 4.1

Experiments General

We conducted two experiments in the present study. In Experiment 1, real lines and the testing opaque surface were presented in four display sequences. In Experiment 2, we used the intermittent display method, in which real lines were displayed continuously and the opaque illusory surface was displayed periodically with various duration and interval times. Apparatus. The left and right eye views were generated by a Silicon Graphics Octane 2 workstation and presented synchronously on a screen (220 cm x 176 cm) by dual projectors (EPSON ELP-735). By wearing a passive pair of polarized glasses, the left eye view could only be seen by the left eye, and the right eye view could only be seen by the right eye; then, stereopsis could be obtained easily. The subject sat 50 cm from the stimuli. Head movement was restricted by a chinrest, and the passive pair of polarized glasses was fixed on the chinrest. Stimuli. The stimulus used in the experiments was a line (0.06 deg width, 10.2 deg length) and an inducing element (3.4 deg width, 18.8 deg length); the acute angle between them was 33 deg, and the gap was 1.4 deg between the line elements and the inducing element. The line elements were 10 mm farther than the inducing element. The distance from the stimuli to the subject was set at 50 cm. 4.2

Experiment 1

In this experiment, the real lines and test illusory surface were presented in various display sequences and for various display times. Four display sequences

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(a)

73

(b)

Fig. 5. Display sequence of type-I and experimental results. (a) Display sequence of type-I. The Poggendorff configuration with lines and testing surface was presented for the display time in the range of 150-850 msec. (b) The experimental results. The collinear perception was dominant in the range of 450-850 msec.

(a)

(b)

Fig. 6. Display sequence of type-II and experimental results (a) Display sequence of type-II. In the display sequence of type-II, the opaque illusory surface was presented first for the previous surface display times of 150-1650 msec. Subsequently, the Poggendorff configuration was displayed for 450 msec. (b) The experimental results. The collinear perception was dominant when the opaque illusory surface was displayed previously for 150-1650 msec.

were prepared for the display of the lines and testing surface. Several types of display times were used for each display sequence. The subjects’ task was to state their perception of the two line segments: the right line segment was perceived higher than, collinear to, or lower than the left line segment. The subjects responded by pressing the left, center or right button on the mouse. There is a 2000 msec blank screen for subjects’ response after the stimulus displayed. The following trials will start after the 2000 msec blank screen. The experimental room was dark during the experiment, and no feedback was given. Four observers with normal or corrected-to-normal vision participated in the experiment. The rates of upper perception and collinear perception are plotted against various display times in the figures presented for each display sequence and experimental results (Fig. 5, Fig. 6, Fig. 7, Fig. 8). The horizontal axis is the display time for the testing surface.

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In the display sequence of type-I, the Poggendorff configuration with lines and testing surface was presented for display times that varied in the range of 150-850 msec. The results showed that collinear perception was dominant when the configuration with lines and testing surface was displayed for 450-850 msec. That is, the Poggendorff illusion could not be perceived when the display range of the Poggendorff configuration was between 450 msec and 850 msec. This result suggests that the testing surface could not be perceived as an opaque surface in the display range of 450-850 msec (Fig. 5). In details, the illusory surface could not be perceived when the real lines and the illusory surface were synchronously presented for the display range between 450 msec and 850 msec. This results are consistent with that illusory surface needs a bit more time than the real image for perceiving [7], [9]. In the display sequence of type-II, the opaque illusory surface was presented first for the previous surface display times of 150-1650 msec. Subsequently, the Poggendorff configuration was displayed for 450 msec. We noticed the collinear perception was dominant when the opaque illusory surface was displayed first for 150-1650 msec. This result suggests that the Poggendorff illusion could not be perceived in the time range in which the opaque illusory surface was displayed previously (Fig. 6). In the display sequence of type-III, the Poggendorff configuration was first displayed for 450 msec. Next, the opaque illusory surface was presented for the varied display times of 150-1650 msec. The results indicate that the upper perception became dominant when the opaque illusory surface was displayed for 1150-1650 msec. after the configuration was first displayed at 450 msec. That is, the Poggendorff illusion could not be perceived, or the perception of the illusion was ambiguous, when the opaque illusory surface was presented for 150-850 msec after the display of the Poggendorff configuration with the real line and the illusory surface. (Fig. 7) In the display sequence of type-IV, the opaque illusory surface was presented for 100 msec. Then, the Poggendorff configuration was displayed for 450 msec. Subsequently, the opaque illusory surface was presented for the varied surface display times of 150-850 msec. The result shows that the Poggendorff illusion could be perceived when the opaque illusory surface was displayed for 250-850 msec after the display of the Poggendorff configuration. Moreover, the Poggendorff illusion was difficult to perceive when the opaque illusory was presented previously than its posterior display. In this experiment, the results show that when the opaque surface was presented for 100 msec before the Poggendorff configuration, the necessary time for the following surface display decreased for perceiving the Poggendorff illusion (Fig. 8). From these investigations, we thought that the afterimage of the real line remained after the visual stimuli disappeared when the real line and illusory surface disappeared synchronously; however, the illusory surface could not be perceived adequately in this case. We assumed that it was necessary to keep the illusory surface display until the afterimage of the real line disappeared.

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(a)

75

(b)

Fig. 7. Display sequence of type-III and experimental results. (a) Display sequence of type-III. In the display sequence of type-III, the Poggendorff configuration was displayed for 450 msec. Subsequently, the opaque illusory surface was presented for the next surface display time that we varied in the range of 150-1650 msec. (b) The experimental results. The upper perception became dominant when the opaque illusory surface was next displayed for 1150-1650 msec.

(a)

(b)

Fig. 8. Display sequence of type-IV and experimental results. (a) Display sequence of type-IV. In type-IV, the opaque illusory surface was presented for 100 msec, then, the Poggendorff configuration was displayed for 450 msec. Subsequently, the opaque illusory surface was presented for the third surface display time that we varied in the range of 150-850 msec. (b) The experimental results. The upper perception was dominant when the opaque illusory surface was displayed for 250-850 msec after the display of the Poggendorff configuration.

4.3

Experiment 2

In Experiment 2, we examined the duration time for the occurrence of the Poggendorff illusion. The stimuli were randomly displayed with various interval times. The interval time was set in increments of 200 sec from 600 msec to 3200 msec. The subject carried out the adjustment task. The subjects’ task was to adjust the time of duration until they perceived the lines as noncollinear. They adjusted the duration time by pressing the key on the keyboard. There is a 2000 msec blank screen after each time of subjects’ adjustments. The following trials

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Q. Wang and M. Idesawa 218.4 175.4 173.2 141.2 132.6 113.8

UpperPerception CollinearPerception

103.6

2200

2000

1800

1600

1400

1200

800

1000

600

112.2

duration time (msec)

190.6

interval time (msec)

Fig. 9. Experimental results. The duration time for the Poggendorff illusion is plotted against the interval time.

will start after the 2000 msec blank screen. The experimental room was dark during the experiment, and no feedback was given. Five observers with normal or corrected-to-normal vision participated in the experiment. The duration time for the Poggendorff illusion is plotted against the interval time (Fig. 9). The Poggendorff illusion could be perceived when the duration time was more than 220 msec in the case of interval times less than 2200 msec. In the case of an interval time more than 2200 msec, the illusion disappeared even though the duration time was 800 msec. The opaque illusory surface for sustained perception required a minimum of about 220 msec. An interval time as long as 2200 msec was needed to obliterate the perception of the opaque illusory surface.

5

Conclusions

In the present study, we examined the temporal properties of opaque illusory surface perception by using the Poggendorff illusion without physical contact as the probing method for detecting surface perception. We utilized an intermittent method to display the test opaque illusory surface. We observed that the opaque illusory surface for sustained perception required a minimum of about 220 msec of duration time. An interval time as long as 2200 msec was needed to obliterate the perception of the opaque illusory surface. We expect to achieve better understanding of the surface perceiving mechanism of the human visual system by utilizing the intermittent display method and the probing method of opaque surface perception of the Poggendorff illusion without physical contact.

References 1. Idesawa, M.: Perception of 3-D Transparent Illusory Surface in Binocular Fusion. Japanese Journal of Applied Physics 30, 1289–1292 (1991) 2. Idesawa, M.: Two Types of Occlusion Cues for the Perception of 3-D Illusory Objects in Binocular Fusion. Japanese Journal of Applied Physics 32, 75–78 (1993)

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3. Idesawa, M.: A Study on Visual Mechanism with Optical Illusions. Journal of Robotics and Mechatronics 9, 85–91 (1997) 4. Spillman, L., Fuld, K., Gerrits, H.J.M.: Brightness Contrast in Ehrenstein Illusion. Vision Research 16, 713–719 (1976) 5. Gellatly, A.R.H.: Perception of An Illusory Triangle with Masked Inducing Figure. Perception 9, 599–602 (1980) 6. Parks, T.E., Rock, I., Anson, R.: Illusory Contour Lightness: A Neglected Possibility. Perception 12, 43–47 (1983) 7. Susan, P.: The Perception of Illusory Contours. Springer, Heidelberg (1987) 8. Ringach, D., Shapley, R.: Spatial and Temporal Properties of Illusory Contours and Amodal Boundary Completion. Vision Research 36, 3037–3050 (1996) 9. Idesawa, M., Nakane, Y., Zhang, Q., Shi, W.: Spatiotemporal Influence of Preperceived Surfaces on the Perception of Bistably Perceptible Surfaces with Binocular Viewing Perception. In: ECVP, vol. 29 (2000) 10. Ninio, J.: Characterisation of the Misalignment and Misangulation Components in the Poggendorff and Corner-Poggendorff Illusions. Perception 28, 949–964 (1999) 11. Westheimer, G., Wehrhahn, C.: Real and Virtual Borders in the Poggendorff Illusion. Perception 26, 1495–1501 (1997) 12. Wang, Q., Idesawa, M.: Veiled Factors in the Poggendorff Illusion. Japanese Journal of Applied Physics 43, 11–14 (2004) 13. Wang, Q., Idesawa, M.: Surface Perception Detecting Method by Using the Poggendorff Illusion in Binocular Viewing. Perception 34 ECVP, 187 (2005)

Interval Self-Organizing Map for Nonlinear System Identification and Control Luzhou Liu, Jian Xiao, and Long Yu School of Electrical Engineering, Southwest Jiaotong University, Chengdu 610031, China [email protected]

Abstract. The self-organizing map (SOM) is an unsupervised neural network which projects high-dimensional data onto a low-dimensional. A novel model based on interval self-organizing map(ISOM) whose weights are interval numbers presented in this paper differ from conventional SOM approach. Correspondingly, a new competition algorithm based on gradient descent algorithm is proposed according to a different criterion function defined in this paper, and the convergence of the new algorithm is proved. To improve the robustness of inverse control system, the inverse controller is approximated by ISOM which is cascaded with the original to capture composite pseudo-linear system. Simulation results show that the inverse system has superior performance of tracking precision and robustness. Keywords: Interval self-organizing map, Unsupervised learning, Nonlinear system, Inverse control system.

1 Introduction The self-organizing map (SOM)[1] is an unsupervised learning algorithm that clusters and projects potentially high dimensional input data onto a discrete neural grid or map of usually reduced dimensions. SOM is a vector quantization method which can preserve the topological relationships between input vectors when projected to a lower dimensional display space. The SOM was developed to help identify clusters in multidimensional datasets. It is successfully used for a wide range of applications including nonlinear system identification and control[2][3]. However, almost every system integrates a certain amount of uncertainty. To solve this problem, it is frequently assumed that the parameters are represented by intervals. So we firstly introduce interval computing which has become an active research branch of scientific computation[4].To process uncertainty, we propose a novel type of self-organizing map, interval self-organizing map, whose weights are interval numbers. The competitive learning algorithm including initialization, updating neuron weights and the use of prior knowledge and preferential training is different due to adopt a new distance measurement based both on empirical risk and structural risk. The winner is an interval number in every competition. The adaptations of the upper bound and lower bound of weight are F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 78–86, 2008. © Springer-Verlag Berlin Heidelberg 2008

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trained respectively with same scalar α (t ) . The convergence of the new algorithm is proved using Robbins-Monro stochastic approximation principle. Although some expressions of the ISOM are similar with conventional SOM, they are essentially different. It does not only pay attention to the difference between the middle of the interval weight and training data but also to the influence of radius of the interval weight which makes the final interval weight converge to the single training data. The method above is applied to identifying and constructing inverse controller in nonlinear control system described in Narendra[5].The validity of the proposed method is illustrated by simulation examples.

2 Interval Self-Organizing Map(ISOM) 2.1 Interval Preliminaries

R. E. Moore introduced interval computing in the late 1950s. Ever since, it has become an active research branch of scientific computation. An interval is represented by its lower bound and upper bound as X = [ x x ] .The following midpoint, radius and distance are used in this paper for the calculation with lower bound and upper bound of interval weight. mid ( X ) = ( x + x ) 2

,

rad ( X ) = ( x − x ) 2

,

(1) (2)

d ( X , Y ) = mid ( X ) − mid (Y ) + rad ( X ) − rad (Y ) .

(3)

2.2 ISOM and Learning Algorithm

Analogously, the interval self-organizing map learns the topological mapping f : G ⊂ R m → X ⊂ R n by means of self organization driven by samples X in X , where G is an output map containing a set of nodes, each representing an interval element in the m-dimensional Euclidean space. Let x = [ x1 ," , xn ]T ∈ X be the input vector. It is assumed to be connected in parallel to every node in the output map. Every dimension of input vector x = [ x1 , " , xn ]T can be regarded as an interval number whose midpoint is xk k = 1, 2 " n and radius is 0. The k − th dimension of interval weight vector of the node i is denoted wik = [ wik wik ] , wik , wik ∈ R, k = 1, 2" n where wi = [ wi1 , ", win ]T ∈ R n , T n wi = [ wi1 , ", win ] ∈ R . The learning rule is given as follows: wi (t + 1) = wi (t ) + Δwi (t )

,

wi (t + 1) = wi (t ) + Δwi (t )

.

(4) (5)

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Where Δwi (t ) = α1 (t )hci (t )[ x − mid ( wi (t )) − rad ( wi (t ))]

,

(6)

Δwi (t ) = α 2 (t )hci (t )[ x − mid ( wi (t )) + rad ( wi (t ))]

(7) . Where t = 0,1, 2 " is the discrete-time coordinate, with α (t ) being a suitable, monotonically decreasing sequence of scalar-valued gain coefficients, 0 < α (t ) < 1 . and hci is the neighborhood function denoting the coordinates of nodes c and i by the vector rc and ri , respectively, a proper form for hci might be a kernel function as:

hci = h0 exp(− ri − rc

2

σ 2)

. with h0 = h0 (t ) and σ = σ (t ) as suitable decreasing functions of time. The step of the competitive algorithm of ISOM is showed in detail:

(8)

(a) Initialization. Generate N × n midpoints of weights and order equal radii, where N is the size of nodes. (b) Present the j − th input data x j = [ x j1 ," , x jn ]T to every node in parallel. (c) Find the best-matching neuron c according to:

x j − wc = min {ε } i

.

(9)

where 2

2

ε = mid ( x j ) − mid ( wi ) 2 + rad ( x j ) − rad ( wi ) 2 + mid ( x j ) − mid ( wi ) • rad ( x j ) − rad ( wi )

. (10)

Where “ • ”is dot product. The criterion function ε does not only minimize the interval distance (empirical risk) but also the complexity of the structure (structural risk) which makes the model have good ability of generalization. (d) Set sign variable flag ∈ R n , flag k = ( wck − x jk ) × ( wck − x jk ) k = 1, 2" n .If flag k > 0 ,then make wck = x jk + rad ( wck ) , wck = x jk − rad ( wck ) .The adjustment above can make the winner interval weight cover the input point and keep the winner closest to the input data under the definition of the criterion function. (e) The gradient descent optimization of ε to the upper bound and lower bound of interval weight respectively yields the sequence as (4)-(7). (f) Order j = j + 1 and return to step (b) until all the data are trained. 2.3 Convergence Property of ISOM

We analyze convergence properties of the interval self-organizing map (ISOM) with multidimensional input using Robbins-Monro stochastic approximation principle. It is shown that the ISOM algorithm optimizes a well defined energy function and converges almost truly if the input data is from a discrete stochastic distribution.

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For the case of discrete input, we employ an energy function for the ISOM algorithm: 1 J (W ) = ∑ hci (t ) ∑ p j ε (11) 2 c ,i x j ∈Xc . According to Robbins-Monro algorithm, we can give a rigorous proof of the convergence of the ISOM algorithm under the following two hypotheses. • H.2.3.1. The input x j ∈ R n has discrete probability density

p( x) = ∑ j =1 p j δ ( x − x j ) . L

• H.2.3.2. The learning rate α (t ) satisfies the following conditions: (a) limt →∞ α (t ) = 0

(b) ∑ t = 0 α (t ) = ∞ ∞

(c) ∑ t = 0 α 2 (t ) < ∞ ∞

Theorem 3.1. Assume that [H.3.1] and [H.3.2] hold. Then the ISOM algorithm will minimize energy function(11), and converge almost truly. Proof. J is piecewise differentiable. Take derivatives of both sides of (11) to upper bound and lower bound of weight respectively ∂J 1 ∂ε ∂ε = ∑ p j hci (t ) = E[hci (t ) ] (12) ∂wik 2 x j ∈X ∂wik ∂wik , ∂J 1 ∂ε ∂ε = ∑ p j hci (t ) = E[hci (t ) ] ∂wik 2 x j ∈X ∂wik ∂wik

By Robbins-Monro algorithm, order

.

(13)

∂J ∂J = 0 and = 0 . Due to the energy func∂wik ∂wik

tion including absolute value, we consider it in two cases: Case 1: if x jk ≥ mid ( wik ) , then ∂ε 1 = − ( x jk − mid ( wik ) − rad ( wik )) 2 ∂wik

,

∂ε 3 = − ( x jk − mid ( wik ) + rad ( wik )) 2 ∂wik

.

(14)

(15)

Case 2: if x jk < mid ( wik ) , then ∂ε 3 = − ( x jk − mid ( wik ) − rad ( wik )) 2 ∂wik

(16) ,

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∂ε 1 = − ( x jk − mid ( wik ) + rad ( wik )) 2 ∂wik

(17) .

It is obvious that the formulas of upper bound and lower bound of weights are the same except the difference in coefficient in both cases.Without lost of generality,we can denote the learning rule as (6),(7). Notice that Robbins-Monro algorithm ensures that the upper and lower bounds of weights of the ISOM converge to the root of

∂J ∂J = 0 and = 0 almost truly if the ∂wik ∂wik

root exists. In practice, J usually exhibits several local minima. Therefore, it is inevitable that the upper and lower bounds of weights of ISOM obtained would only converge to solutions to local minima. However it has been observed that by introducing the neighborhood function hci which has a very large range in the beginning and gradually decreases during the learning process. The ISOM algorithm is capable in some extent to achieve a good global ordering. In order to avoid some essential error (i.e. wik > wik ) , the learning rate α (t ) and neighborhood function might be chosen equally in (6) and (7). It is essentially different between the conventional SOM and ISOM algorithm. The latter does not only consider dissimilarity between interval weight and training data but also the effect of the interval radius. The interval weight might degenerate to exact weight with t → ∞ . As a result, the ISOM degenerates to conventional SOM. In practical application to identification, the proposed algorithm shows good performance in generalization because the interval weight can control the error effectively which enhances robustness of the network model.

3 Nonlinear System Identification and Inverse Control In nonlinear system identification and control, the approximation precision and ability of generalization should be considered simultaneously. Nevertheless, the ability of generalization should be paid more attention in practical application. It is very significant to found a valid model to constrain control error. ISOM is an effective approach for this. The SOM was previously applied to learn static input–output mappings. VQTAM approach can make SOM and other unsupervised networks able to learn dynamical mappings. We generalize this approach to ISOM to learn dynamic mappings. We are interested in systems which can be modeled by the following nonlinear discrete-time difference equation:

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y(t +1) = f [ y(t ),", y(t − ny +1); u(t ),", u(t − nu + 1)]

(18) . where n y and nu are the (memory) orders of the dynamical model. In many situations, it is also desirable to approximate the inverse mapping of a nonlinear plant: u(t ) = f −1[ y(t +1),", y(t − ny +1); u(t −1),", u(t − nu +1)]

. The weight vector of neuron i , wi (t ) ,has its dimension increased accordingly. These changes are written as:

(19)

⎛ wiin (t ) ⎞ ⎛ xin (t ) ⎞ (20) wi (t ) = ⎜ out and x(t ) = ⎜ out ⎟ ⎟ w t ( ) x t ( ) ⎝ ⎠ ⎝ i ⎠. where wiin (t ) and wiout (t ) are the portions of weight vector which store information about the inputs and the outputs of the mapping being studied. To approximate the forward dynamics in (18) the following definitions apply:

x in (t ) = [ y (t )," , y (t − ny + 1); u (t )," , u (t − nu + 1)] x (t ) = y (t + 1)

,

(21)

out

.

In inverse controller design, one defines:

x in (t ) = [ y (t + 1)," , y (t − n y + 1); u (t − 1)," , u (t − nu + 1)] x (t ) = u (t )

,

out

.

Fig. 1. Identification with ISOM

Fig. 2. Structure of inverse control system with ISOM

(22)

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According to ISOM algorithm, the output of the controller will be an interval number. To transform an interval number to an exact one, we may regard the midpoint of the interval as the exact one. To construct inverse controller, a forward dynamic identification must be built first. The ISOM model will be cascaded with the original to capture pseudo-linear system. Fig.1 and Fig.2 show identification and inverse control respectively.

4 Simulation The nonlinear system is assumed to be the form:

y (t + 1) = y (t ) (1 + y 2 (t )) + u 2 (t )

(23) . The input u (t ) is a random input in the interval [0 1] . Added noises are some white Gaussian noise whose variance equals to 0.05.The results are in Table 1 and Table 2 in the form of mean-squared error(MSE). In addition, the radius of weight can also directly affect the precision of identification and control. Fig. 3 shows the inverse control system tracks some sine wave with radius equal 0.03 with accurate data. The MSE of inverse control influenced with interval radius is shown in Fig. 4. Table 1. Identification precision with ISOM and SOM Type

MSE(accurate data) 0.02081 0.03140

ISOM SOM

MSE(noise data) 0.02990 0.04450

Table 2. Inverse control error with ISOM Type of data Accurate data Noise data

Generalization error 0.02972 0.03135

Tracking error 0.0544 0.0720

2.5

2

1.5

1

reference output

0.5 0

2

4

6

8

10

12

14

16

18

20

Fig. 3. The inverse control system tracking sine wave using ISOM

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0.07

0.06

0.05

MSE

0.04

0.03

0.02

0.01

0 0.0002

0.02

0.04

0.06

0.08 0.1 0.12 Interval radius

0.14

0.16

0.18

0.2

Fig. 4. The MSE of inverse control influenced with interval radius

The results show that the ISOM algorithm can identify nonlinear system effectively. It has better abilities in both approximation and generalization than conventional SOM. The inverse control system constructed using ISOM is preferable in robustness which is a valid approach in nonlinear system control.

5 Conclusion In this paper, the ISOM algorithm is proposed based on SOM algorithm and interval analysis theory. However, the ISOM algorithm is essentially different from SOM algorithm for the ISOM criterion function considering both empirical risk (approximation precision) and structural risk(structure complexity).First, the set of interval weight vectors tends to describe the density function of input vectors via modifying the interval midpoints and radii while the standard SOM can only modify weight vectors. In the self-organizing process, the smaller of the interval weight’s radius suggest the higher distribution density of the input vectors. So the ISOM reveals more accurate in modeling and control(Table 1 and Table 2). Second, local interactions between processing units still tend to preserve continuity of interval weight vectors just like SOM. The interval weight strike a nice balance between describing the density function of the input vectors and preserving continuity of interval weights. The simulation results testify the validity of the ISOM algorithm in system identification and nonlinear control.

Reference 1. Teuvo, K.: Self-Organizing Maps, 3rd edn. Springer, Heidelberg (2001) 2. Guilherme, A.B., Aluizio, F.R.A.: Identification and Control of Dynamical Systems Using the Self-Organizing Map. IEEE Trans. Neural Networks 15, 1244–1259 (2004) 3. Principe, J.C., Erdogmus, D., Motter, M.A.: Modeling and Inverse Controller Design for an Unmanned Aerial Vehicle Based on the Self-Organizing Map. IEEE Trans. Neural Networks 17, 445–460 (2006)

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4. Wang, D.R., Zhang, L.S., Deng, N.Y.: Interval Algorithms for Nonlinear Equations. Shanghai Publishing House of Science and Technology, Shanghai (1987) 5. Narendra, K.S., Parthasarathy, K.: Identification and Control of Dynamical Systems Using Neural Networks. IEEE Trans. Neural Networks 1, 4–27 (1990) 6. Lin, S., Jennie, S.: Weight Convergence and Weight Density of the Multi-dimensional SOFM Algorithm. In: Proceedings of the American Control Conference, pp. 2404–2408. IEEE Press, New York (1997) 7. Gregory, L.P.: Adaptive Inverse Control of Linear and Nonlinear Systems Using Dynamic Neural Networks. IEEE Trans. Neural Networks 14, 360–376 (2003)

A Dual-Mode Learning Mechanism Combining Knowledge-Education and Machine-Learning Yichang Chen1 and Anpin Chen2 1

Department of Information Management, NPIC, No.51, Minsheng E. Rd., Pingtung City, Pingtung County, Taiwan 900, R.O.C 2 Institute of Information Management, NCTU, No. 1001, University Road, Hsinchu, Taiwan 300, R.O.C [email protected], [email protected]

Abstract. From 1956, the definitions of learning according to Artificial Intelligence and Psychology to human mind/behavior are obviously different. Owing to the rapid development of the computing power, we have potential to enhance the learning mechanism of AI. This work tries to discuss the learning process from the traditional AI learning models which are almost based on trial and error style. Furthermore, some relative literatures have pointed out that teaching-base education would increase the learning efficiency better than trial and error style. That is the reason we enhance the learning process to propose a dual-perspective learning mechanism, E&R-R XCS. As for XCS is a better accuracy model of AI, we have applied it as a basement and proposed to develop an intelligence-learning model. Finally, this work will give the inference discussion about the accuracy and accumulative performance of XCS, R-R XCS, and E&R-R XCS respectively, and the obvious summary would be concluded. That is, the proposed dual-learning mechanism has enhanced successfully. Keywords: Artificial intelligence, Psychology, Trial and error, Teaching-base education, Intelligence-Learning.

1 Introduction Traditionally, Artificial Intelligence, according to the definition of Computer Science, works as helpful machines to find solutions to complex problems in a more humanlike fashion [1,2]. This generally involves adopted characteristics from human intelligence, and it applies them as algorithms in a computer friendly way. A more or less flexible or efficient approach can be taken depending on the requirements established, which influences how artificial the intelligent behavior appears. Those researches, for example: Neural Network, Fuzzy Approach, Genetic Algorithm, and so on, all focus on Soft Computing. Of course, XCS (Extend Classifier System) is also a hybrid approach with high performance to the accuracy and the rule evolution on the prediction application. However, up to now, the Artificial Intelligence Techniques based on Soft Computing have all involved the concept, trial and error method or stimulus-response F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 87–96, 2008. © Springer-Verlag Berlin Heidelberg 2008

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method even the series of evolution approaches, such as [1,2] and [3], to construct their learning models. For this aspect, if possible, this example, a Chinese idiomatic phrase-”An Illusory Snake in a Goblet”, is taken into consideration as an input-output pattern to training the learning model. The models are formed for sure. It is actually a wrong model trained by a bad experience. Besides, the parameters of those training models are exactly affected by the input dataset, especially the large difference of the training inputs and testing ones. Usually, in many researches it is chosen the high relation between the input and output datasets or given the strong assumption which is the inputs and outputs are relevant. Thus, a subjective black-box view and the tuning view are easily concluded [4]. The other sub-domain, Expert System, which’s primary goal is to make expertise available to decision makers and technicians who need answers quickly. There is never enough expertise to go around -- certainly it is not always available at the right place in the right time. The same systems in-depth knowledge of specific subjects can assist supervisors and managers with situation assessment and long-range planning. These knowledge-based applications of artificial intelligence have enhanced productivity in business, science, engineering, and even the military. Although, the development of those expert systems is the view of anti-extreme to construct domain knowledge first but, for the reason, they are lack of the flexibility and the adaption. In fact, each new deployment of an expert system yields valuable data for what works in which context, thus fueling the AI research that provides even better applications. Many researches, no matter Soft Computing techniques or Expert Systems try to consider into the human-like thinking way to make the simulation. But, from classic psychology, the human-mind researches are the researches to the human-behavior. Since Plato, Psychology is an unfathomable philosophy and those advanced AI researchers should concern this perfect development of Human Psychology, from simple to complex and from single factor to multiple ones. However, the traditional AI techniques are seldom focused on the high level of human-mind process and just paid attentions to the learning definition from the Empricalism Psychology. According to the development of Modern Psychology, the core of Psychology has been already transferred Empricalism-base into Information Process Theory of Human-Mind, and even Cognitive Psychology-base. As for the knowledge and the model construction, the teaching-base aspect has been involved as well to the learning process. Based on the aspect, this work tries to enhance the learning process of traditional AI techniques whose cognitive scotomas of learning definition, and it develops the novel learning model, involving the concept of Cognitive Psychology, which is utilized the high accuracy-prediction XCS [5] model as the construction basement.

2 Relative Survey 2.1 Information Process Theory Among previous learning artificial intelligence techniques, such as neural network, or its hybrid methods, all the models are formed by trial and error learning way, the traditional definition of learning. However, to enhance the learning style, a cognitive learning, Information Process Theory, would be worth to take into consideration. According

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to the information-processing model of learning (see Fig. 1), there is a series of stages by which new information is learned (Gagne, 1985) [6]. Information is received by receptors (such as the eyes and ears), from which it is passed to the sensory register where all of it is held, but for only a few hundredths of a second. At this point of view, selective perception acts as a filter which causes some aspects of the information to be ignored and others to be attended to. For example, the ears (receptors) receive the sounds comprising “Pi equals 3.14,” along with various other background sounds, and all those sounds are passed on to the sensory register in the brain. Then through the selective perception process, some of the information (hopefully the “Pi equals 3.14”) is attended to the part. That information which is attended to is transformed and passed on to short-term memory, which can only contain a few items of information at a time (depending on their complexity). For instance, if “Pi equals 3.14” is attended to, it is then passed on to short-term memory, where it might be said to “echo” for a few seconds, and the echoing can be prolonged through rehearsal.” Items can persist in short-term memory for up to about 20 seconds without rehearsal, but with constant rehearsal they can be retained indefinitely. Finally, the information may be passed on to long-term memory. This process is called encoding to memorize. For example, if appropriate encoding processes are exercised to link the “Pi equals 3.14” with prior knowledge, then the information is passed on to long-term memory. In the traditional model of human memory (Atkinson and Shiffrin, 1968 [7]; Waugh and D. A. Norman, 1968 [8]), immediate free recall yields items directly retrieved from a temporary short-term memory (STM) and items retrieved by retrieval cues from a more durable storage in long-term memory (LTM).

Fig. 1. Information Process Theory proposed by Gagne [9]

2.2 XCS Most machine learning techniques are developed by information process theory. No matter partial application of IPT concept or applying the entire flow of IPT, they all simulated various operations of memory. For example, those neural network types are applications of neuroanatomy. According to that, it is necessary to define the neural structures of the brain simulated as memory. The others would be evolution computing types, such as GA, GP, and LCSs. Among them, LCSs has flexible outcome on rule generation which represents information about the structure of the world in the form of rules and messages on an internal message list, such as its STM or LTM, John Holland mentioned that. The system can be used as the message list to store information about (a) the current state of the world (response), and (b) about previous states (stimulus). From now on, LCS has the ability to store rule according to the input information.

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Fig. 2. XCS Procedure

However, Wilson’s XCS [10] is a recently developed learning classifier system (LCS) that differs in several ways from more traditional LCSs. In XCS, classifier fitness is based on the accuracy of a classifier's pay-off prediction instead of the prediction itself. As a whole, the genetic algorithm (GA) takes place in the action sets instead of the population. XCS's fitness definition and GA locus together result in a strong tendency for the system to evolve accurate, maximally general classifiers that efficiently cover the state-action space of the problem and allow the system's ‘knowledge” to be readily seen. As a result of these properties, XCS has been considered and focused to the kernel of the proposed model in this work. XCS’s detailed loop is shown in Fig. 2, and the current situation is first sensed and the detector received the input from the environment. Second, the match set [M] is formed from all classifiers [N] that match the situation. Third, the prediction array [PA] is formed based on the classifiers in the match set [M]. [PA] predicts for each possible action ai, the resulting pay-off. Based on [PA], one action is chosen for execution and the action set [A] is formed, which includes all classifiers of [M] that propose the chosen action. Next, the winning action is executed. Then the previous action set [A]-1 (a previous action set) is modified by using the Q-learning-like payoff quantity P which is a combination of the previous reward p-1 and the largest action prediction in the prediction array [PA]. Moreover, the GA may be applied to [A]-1. If a problem ends on the current, time-step (single-step problem or last step of a multistep problem), [A] is modified according to the current reward, p, and the GA may be applied to [A]. The loop is executed as long as the termination criterion is not met. A termination criterion is a certain number of trials/inputs. Finally, XCS’s architecture is much neater development base on IPT than the previous models. However, XCS is not sufficient to represent IPT. The coming discussion would be given to its explanation.

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2.3 Discussion Using a computer as a metaphor for memory, the short-term phase is RAM (highly volatile and easily lost when some others else are entered), but long-term memory is such as a hard drive or diskette (the information is stored there even after the machine is turned off). This metaphor is especially helpful because a computer knows the address of each bit of information because of the manner information is entered. It is essential that information placed into a student's long-term memory be linked in a way that the student can retrieve it later. The teacher who should understand the relationship between memory and retrieval can lay out a lesson plan to assist the student in the process and enhance his learning. As the pre-statement portrayed, while rehearsal is important to short-term memory, it can also be used to transfer information into long-term. Elaborating or making material memorable will also enhance the student's learning process. The effective teacher will elaborate and rehearse material so that the student can remember the information more easily. That is the reason the input material is high relevant to memorize to form valued-information, knowledge. However, it is important to note that most application AI models, even XCS model, have more trouble remembering/learning of what data they should remember/learn. Therefore, the entire learning procedure as the effective teacher help the memory process by introducing the student to various organizational techniques cannot come true.

3 Dual-Mode Learning Mechanism 3.1 Conceptual Framework During the Middle Period (mid 1900s), Knowledge is just thought of as the transformation of sensory inputs into associated thought, and the realization that sensory inputs are transformed prior to storage. In the early twentieth century, Knowledge is still considered as a framework of stimulus and response (S-R). The profound breakthrough of this period is that by studying S-R, one can gain insight into the working of cognitive knowledge. This kind viewpoint of knowledge learning is largely based on narrow term of cognitive psychology, information processing theory. Furthermore, SR of cognitive psychology research is historically analogous to the black box testing. Following these two aspects, this work applied the cognitive learning to modify the learning process of traditional soft techniques to increase the efficiency of forming knowledge storage. That is, combining the information process theory and knowledge learning to initial the concept of the dual learning mode framework is the purpose of this work, shown as Fig. 3. It contains two parts: Knowledge Education learning and Reinforcement-Rehearsal(R-R) learning. 3.2 Proposed Model (E-RR XCS) R-R XCS model, a middle version to develop E-RR XCS, is also an enhanced version from XCS by adding a rehearsal mechanism. Owing to them both adapting GA as an

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Fig. 3. Dual perspective learning process of Education and R-R mechanism

evolution methodology of classifiers and based on XCS, their working accuracy rate should be equivalent by the same training data and testing data. The leverage of R-R XCS to XCS deserves to be mentioned. The rational assumption is that R-R XCS has higher leverage to XCS. This reason originates from R-R XCS considering more value information automatically. But its performance would be decreased and its accuracy ratio might not be better than XCS [11]. Education & R-R XCS, an implantation of proposed learning concept, is to increase the accuracy ratio by concerning the education efficiency of learning. In Fig. 4, there are two starting points which is different from XCS. And E & R-R XCS involves R-R XCS discussed in pre-statement. Besides them, in the additional education learning part, discovered knowledge, verified theory, and defined theorem are all considered as input patterns to the mechanism. Those data should be valued and worthy to “teach” the model or the model should be learned/trained. Thus, we increase the practice route in the education part. (E & R-R XCS is a “model” not a student and the model is not necessary to be practiced for more than twice times.) For this, those input data would be easily memorized/ stored by the receiver and internalized to the knowledge rule base [N1]. Population in knowledge rule base has higher weight or effectiveness than ones in experience rule base. Besides, the detector should consider more about the knowledge rule base [N1] than about the experience rule base [N2]. WM still stores the current situation in advance. Second, the match set [M] is formed from [N1] or [N2], which is either the knowledge rule base or experience rule base. The following steps are the same with R-R XCS ones. The difference is that the initial-picked population is more from knowledge rule base than experience one. In the mechanism, this kind population from knowledge rule base seems to be “principle”. While the entire loop has finished, the new population should be generated from knowledge rule base to the experience one. Some experiences have possibility to produce from the real knowledge, if the knowledge really exists. Furthermore, while a rehearsal population from repeater to detector occurs, detector should verify the repeated population qualification that it may be transferred to receiver. The knowledge population, that is, does come not only from the outside environment but also from internal mechanism. The education knowledge should also be possible increased to the knowledge rule base [N1], while the new knowledge or theory is discovered. As for the other detailed procedure same to XCS, they have already been detailed in pr-section.

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Fig. 4. E&R-R XCS Procedure

4 Inference Discussion Actually, the case of adding new population from detector does not happen easily. E & R-R XCS exactly defines the stern discipline to the knowledge. In fact, the percentage of knowledge from detector to receiver ones is low. That is, the population in knowledge rule base should be maintained spotless correctness. Following the descriptions, three inferences of these models are possibly deduced in this section. Their theoretical accuracy and accumulated performance would respectively be detailed as following. The x-axis, Time, in Fig. 5, 6, and 7 might means time, or times which is the operating times of the model. The y-axis is just the theoretical accuracy or accumulated performance. 4.1 In Fig. 5, γ is defined to the difference of the accuracy ratio of R-R XCS and XCS. λ is defined to the difference of the accuracy ratio of E&R-R XCS and XCS. It is sensible that λ >> |γ| >= 0. The reasonable explanation is R-R XCS with rehearsal learning focused on valuable information. When γ is approximate to zero, the two models are applied to the all original data. When |γ| is large to zero, the two models are applied to identify the result for valuable information. As for λ, due to the education efficiency of learning, λ should be larger which means the accuracy ratio of E&R-R XCS is much better than XCS one. 4.2 In Fig. 6, μ is defined to the difference of the accumulative output of R-R XCS and XCS. It is sensible that |μ| >= 0. The reasonable explanation is just R-R XCS with

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rehearsal learning focused on valuable information, but its accuracy rate is not absolutely better than XCS one. Indeed, the leverage effect of R-R XCS originates from it focused on more valuable information. If the output is correct and positive to the result, the accumulative output should be increased more. Contrary to the wrong one, the accumulative output should be decreased more as well.

Fig. 5. Theoretical Accuracy of XCS, R-R XCS, and E&R-R XCS

Fig. 6. Theoretical-Accumulative Performance of XCS and R-R XCS

Fig. 7. Theoretical-Accumulative Performance of XCS and E&R-R XCS

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4.3 In Fig. 7, μ1 and μ2 are defined to the difference of the accumulative output of E&R-R XCS and XCS. It is sensible that |μ1| >> |μ2| >= 0. The reasonable explanation is that E&R-R XCS has not only the ability with rehearsal learning focused on value information but also involves the education efficiency of learning. Therefore, its accuracy rate is absolutely better than XCS one. Indeed, E&R-R XCS still owns the leverage effect, which originates the same to R-R XCS. Owing to the accuracy ratio increased, the output is usually positive to the result, and the accumulative output should be increased much more. In a word, the learning accuracy of the proposed E&R-R XCS is much better than XCS. R-R XCS comparing with XCS has the leverage effect to the accuracy and the accumulated performance at least.

5 Conclusion Such as the description of this work motivation, much Knowledge discovery, Theory verification and Theorem definition are aggregated and not disregarded by this learning mechanism development. Also, they are all continually historical accumulated. That is still the reason that the civilization is enhanced, the culture is accumulated, and knowledge is transmitted. Finally, this work successfully proposed an efficient dual-mode learning mechanism which combines the passive-learning (knowledge education) and the selflearning (machine learning) [12]. That is, the major contribution of this work is the proposed mechanism. Once, more accuracy ability of AI Techniques invented could be substituted for XCS and the mechanism performance would be more efficient.

References 1. McCarthy, J.: Generality in Artificial Intelligence. Communications of the ACM 30(12), 1030–1035 (1987) 2. Ghirlanda, S., Enquist, M.: Artificial Neural Networks as Models of Stimulus Control. Animal Behavior 56(6), 1383–1389 (1998) 3. Ghirlanda, S., Enquist, M.: The Geometry of Stimulus Control. Animal Behavior 58(4), 695–706 (1999) 4. Chiew, V.: A Software Engineering Cognitive Knowledge Discovery Framework. In: 1st IEEE International Conference on Cognitive Informatics, pp. 163–172. IEEE Press, Calgary (2002) 5. Wilson, S.W.: Classifier Fitness Based on Accuracy. Evolutionary Computation 3(2), 149– 175 (1995) 6. Gagne, R.M.: The Conditions of Learning and Theory of Instruction. Holt, Rinehart & Winston, New York (1985) 7. Atkinson, R.C., Shiffrin, R.M.: Human Memory: A Proposed System and Its Control Processes. The Psychology of Learning and Motivation: Advances in Research and Theory, vol. 2, pp. 89–195. Academic Press, New York (1968)

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8. Waugh, N., Norman, D.A.: Primary Memory. Psychological Review 72, 89–104 (1965) 9. Gagne, R.M., Medsker, K.L.: The Conditions of Learning. Training Applications. Harcourt Brace, New York (1996) 10. Butz, M.V., Wilson, S.W.: An Algorithmic Description of XCS, Soft Computing - A Fusion of Foundations. Methodologies and Applications. In: Deb, K., et al. (eds.) GECCO 2004. LNCS, vol. 3103, pp. 144–153. Springer, Heidelberg (2004) 11. Chen, Y.C.: Applying Cognitive Learning to Enhance XCS to Construct a Dual-Mode Learning Mechanism of Knowledge-Education and Machine-Learning - an Example of Knowledge Learning on Finance Prediction. PhD Thesis. National Chiao Tung University, Taiwan (2005) 12. Piaget, J.: Structuralism. Harper & Row, New York (1970)

The Effect of Task Relevance on Electrophysiological Response to Emotional Stimuli* Baolin Liu1, Shuai Xin1, Zhixing Jin1, Xiaorong Gao2, Shangkai Gao2, Renxin Chu3, Yongfeng Huang4, and Beixing Deng4 1

Department of Computer Science and Technology, Tsinghua University, Beijing 100084, P.R. China 2 Department of Biomedical Engineering, Tsinghua University, Beijing 100084, P.R.China 3 Brigham and Women's Hospital, Harvard Medical School, Boston, MA 02115, USA 4 Department of Electronic Engineering, Tsinghua University, Beijing 100084, P.R. China [email protected]

Abstract. To verify whether or not the emotion processing is modulated by task relevance, in this paper two tasks are performed - Simple Task and Complex Task. In the Simple Task, negative pictures are target stimuli, while in the Complex Task white-framed negative pictures are target stimuli. Subjects are required to respond when a target stimulus is onset. The EEG (electroencephalogram) epochs are averaged and ERP (event-related potential) components are obtained. The P300 amplitude is smaller in the Complex Task than in the Simple Task, which proves that the emotion P300 is significantly modulated by task relevance. As P1 and N1 amplitudes are decreased in the Complex Task comparing with the Simple Task, we can suggest that the P1/N1 components elicited by emotional stimuli are modulated by task relevance, too. Keywords: Emotion, ERP, P300, Attention, Task relevance.

1 Introduction ERPs aroused by emotional stimulus had already been deeply studied before and were illustrated with quite a number of examples, such as the studies on the integration of emotion and working memory [1,2,3,4], the integration of emotion and inhibitory control [5,6,7], and Attentional Blink effect to emotional stimuli [8,9]. Researchers showed a strong interest in P300 components which were influenced by emotional valence. P300 waves aroused by negative pictures were finally proved to be stronger through the previous studies [10,11,12,13,14,15]. However, P300 is not the earliest ERP component related to emotion perception. There was a family of task-relevant ERP components prior to the P300 [16,17,18]. For example, an enlarged P1 component could be observed over posterior scalp sites *

This work is supported by the National Basic Research Program of China (973 program) (No. 2006CB303100 and No.2007CB310806) and by the National High Technology Development Program of China (No.2007AA010306 and No.2006AA01Z444), and by the National Natural Science Foundation of China (No.30630022).

F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 97–106, 2008. © Springer-Verlag Berlin Heidelberg 2008

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contralateral to the attended visual field than unattended [19]; Heinze and Mangun [20] studied the ERP waves elicited by bilateral and unilateral stimulus, and got the result that the early P1 component reflected the facilitation of visual inputs at attended locations on visual processing. Commonly, N1 component was also regarded as an indicator of emotion effect (more positive ERPs for pleasant and unpleasant stimuli than for neutral stimuli). In those studies in which stimuli were presented rapidly and unpredictably to the right and left visual fields, paying attention to the events in one field produced an amplitude enhancement of the early Pl, N1 and/or N2 components elicited by those stimuli over the contralateral occipital scalp [21,22,23,24]. Many ERP researches on the emotional modulation of attention and/or task performance had already been done. For instance, emotional modulation of attention shifting was investigated in Posner’s [25] spatial orienting task by conditioning the attention cued to an aversive white noise; Phelps et al. [26] provided the evidence for emotion potentiating the effects of attention on low-level visual processing in stimulus-driven attention. Furthermore, we could see the stronger ERP components actually reflected the increasing attentional resources devoted to the processing of emotional stimuli [27,28,29,30]. There were also some studies on emotional modulation of task. For example, it was already exemplified that negative emotions had been demonstrated to improve task performance [31]. There were already some researches on emotion perception modulated by task relevance or attention. Gierych et al. [32] designed two experiments to investigate the ERP responses to “smile-provoking” pictures. In the first experiment, both affective stimuli were set as targets in an “oddball” procedure, being presented among the more frequent green disks. Then in the second experiment, they were both non-targets whereas the green disks were task-relevant. Both experiments and all pairs of stimuli produced similar results, which indicated that affective stimuli might produce attentional reallocation of processing resources. However, ERP researches on task-modulated emotion were limited, and most of the current studies were based on fMRI (functional magnetic resonance imaging) or PET (positron emission tomography) method. Harlan et al. [33] did some experiments with task-related fMRI to investigate how attentional focus could modulate the ERPs elicited by scenes that varied in emotional content. They manifested that the response to emotional task-relevant scenes was strengthened. Meanwhile, in another research by fMRI, Lane et al. [34] showed that a higher-arousing effect was aroused when participants attended to their own emotional responses than when participants were attending to the spatial setting of the stimulus (indoor/outdoor/either). These results suggested a higher-arousing effect when subjects attended to the emotional aspect of a stimulus to a greater extent. Based on these experiments, we could tell that if the attention was distracted by added factors, the arousal effect would decrease due to the attention deficit. In the perceptual grouping study by Han et al. [35], the stimulus arrays were either evenly distributed, grouped into rows or columns by proximity or similarity, around by colored dots, or with a fixation cross. As a result, the elicited Pd100 was significantly modulated by the task relevance; in the research of Attentional Blink (AB) effect to emotional stimuli (120 pictures of the IAPS- International Affective Picture System) [9], participants were required to name the black-framed target stimuli aloud. Similarly with the stimuli that they used, we designed a Complex Task, in which the

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subjects were required to identify the white-framed negative stimuli, and a Simple Task, in which the subjects were only required to response when a negative picture was onset. We would like to verify whether or not the emotion processing is modulated by task relevance.

2 Materials and Methods Twenty-seven subjects (13 females and 14 males), mean age 22.3 (±3.4) years, were recruited from undergraduate students of Tsinghua University. All the subjects participated in the two experiments. The participants were screened by phone and written questionnaire for history of neurological and psychiatric illness, drug abuse, and psychotropic medication use. They were all right-handed and had normal or corrected-tonormal vision. Handedness was measured using the Edinburgh Handedness Inventory (EHI) [36]. All participants filled in the Volunteer Screening Form and were paid (RMB20/ hour) for their participation. All experiments were conducted in accordance with the Declaration of Helsinki and all the procedures were carried out with adequate understanding of the subjects, who read and signed the Research Consent Form before participating in this research. All the subjects were required to complete the positive affect- negative affect scales (PANAS) questionnaire [37]. The PANAS was a 30 item questionnaire producing 6 scores of positive affects (PA) and negative affects (NA) altogether and had been correlated with both hemispheric asymmetry in brain processes of emotional perception and sensitivity to affective manipulation [38]. A one-way ANOVA (analysis of variance) analysis showed there was no significant difference between the subjects in their PANAS scores. 84 pictures were selected from the IAPS [39,40] according to the valence dimension (28 pleasant, 28 unpleasant, and 28 neutral). The pictures were divided into 2 groups, each group has 14 negative pictures, 14 positive, and 14 neutral. They were all presented in a pseudo random sequence. 24 pictures (8 pleasant, 8 unpleasant, and 8 neutral) for the Complex Task were selected and white-framed. Other 18 pictures were not framed. White-framed pictures were more than others to avoid oddball paradigm, which might elicit oddball P300. The stimuli were conducted on PC (Intel Pentium D, 3.0 GHz, 1 GB RAM) with a 22-inch color monitor. The screen was at a distance of 100 cm from the subjects, and the resolution was 1440 × 900 pixels. The luminance of the pictures was determined according to the research by Palomba et al. [41]. Judging an EEG epoch to be negative or positive was not only by the emotional valence of picture estimated before the experiment, but also the response of each subject. Therefore, if one stimulus, such as a spider, was usually thought negative to most people, but not so to one subject, then we excluded all the data concerning this stimulus after all the experiments. In the Simple Task, subjects were invited to the laboratory and were required to fill in the Research Consent Form. Then they completed the EHI and PANAS. We used a one-way ANOVA analysis to exclude the left-handed subjects and the affect-deviant subjects.

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Subjects sat in front of the computer screen and received instructions explaining the experimental task. Stimuli were presented on a black background of a 22-inch color monitor at a viewing distance of 100 cm. The room was sound-attenuated and dimly lit. EEG signals were recorded from 19 scalp sites (Fp1/Fp2, F3/F4, C3/C4, P3/P4, O1/O2, F7/F8, T3/T4, T5/T6, Cz, Fz and Pz) according to the International 10/20 System [42], using Neuroscan Synamps2 EEG/ERP system. The reference channel was linked to earlobes. When all the electrodes were attached, we checked each of them and made sure the impedance was < 5 kΩ. When a subject was ready, the vocal introduction of the task was played to inform that 42 pictures would be presented and there would be a break every 14 pictures. He/she was required to pay full attention to each picture, and press "B" when an unpleasant picture was shown. He/she was instructed to wait until the input screen appeared and then responded as accurately as possible. The entire task lasted about 7 minutes; each picture stimulus was presented for 2s and waited for the response, then the response result screen was shown to tell the subject what he/she has pressed; a 6s interval occurred between two trials, during which the screen was black except for a cross at the center of the screen, on which the subject were instructed to fixate. The trials were in a pseudo random order. All subjects used both hands to make responses. The "B" button was selected as the response key to negative stimuli. He/she could select another button on the keyboard for both the positive and the neutral stimuli. We selected "B", which was the first letter of "Bad" and was just above the space button which was disabled to avoid misoperation. The self-selection button could counterbalance the hand differences between the subjects. There were two candidate words in the result screen: "bad" (when "B" was pressed) or "not bad" (when another key was pressed), which was presented at the central position. After that, the white cross was shown for 6s until the next picture appeared. No cross was shown while the pictures were presenting. The Complex Task was modified as follows based on the Simple Task: 42 pictures were used in this task -14 pleasant (numbered 1-14), 14 unpleasant (numbered 15-28) and 14 neutral (numbered 29-42). Subjects could not see the internal numberings of each picture. The following pictures were selected and white-framed: 1-8; 15-22; 2936; a subject was required to press "B" if it is both negative and white-framed. He/she would press the self-selected key in other cases. The EEG from each electrode site was digitalized at 256 Hz with an amplifier band pass of 0.01-40 Hz, including a 50 Hz notch filter and was stored for off-line averaging. Subjects were required to give an integer score between 1 and 9 to each of the pictures as a self-assessment to their emotional reactions after all the experiments, where score 1 indicated very unpleasant picture, 9 indicated very pleasant picture, 5 indicated neutral stimuli. A one-way ANOVA analysis on these scores showed there was no significant difference between the mean arousal ratings of the pictures in two tasks.

3 Results Computerized artifact rejection was performed to discard epochs in which deviation in eye position, blinks, or amplifier blocking occurred [43]. The rejected epochs were

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considered invalid. We selected the datasets of the best 20 subjects (7 females and 13 males), and all datasets of the other 7 subjects were rejected. At last, about 20% of the selected 20 subjects’ trials (the negative epochs) were rejected for violating these artifact criteria. The EEG epochs we selected were 100 ms prior the stimulus onset and 900ms after stimulus onset. Fig. 1 shows the average ERPs to unpleasant, pleasant, and neutral stimuli in the Simple Task. We can see the amplitude differences (P300 amplitude in response to unpleasant pictures is significantly higher than that in response to either pleasant pictures or neutral pictures) clearly from it.

Fig. 1. The average ERPs to unpleasant, pleasant and neutral stimuli in the Simple Task. P300 amplitude in response to unpleasant pictures is significantly higher than that in response to either pleasant pictures or neutral pictures.

Studies using neuroimaging techniques and source modeling analyses had shown that effects of emotional stimuli are strongest in occipital and posterior brain locations [44,45,46], so we paid most of our attention to occipital and posterior electrode sites: parietal electrodes (P3, P4 and Pz), occipital electrodes (O1 and O2), and temporaloccipital electrodes (T5 and T6). Additionally, what we practically cared about was the modulation itself rather than the emotional valence here. Therefore, we averaged the negative EEG epochs, which elicited more robust emotion effects, to obtain the emotional ERPs. We plotted the negative ERPs (ERP waves evoked by negative pictures, at sites T5, T6, P3, P4, O1, O2 and Pz) of both the Simple Task and the Complex Task in Fig. 2. By comparison, we could see the decrease of P300 amplitude in the Complex Task at all sites, as well as the shorter P300 latency of Complex Task at most of the sites clearly.

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Fig. 2. Averaged ERP waveforms evoked by negative pictures: Simple Task vs. Complex Task. We can see the P300 latencies are significantly shorter in the Complex Task than in the Simple Task at parietal electrode sites (P3, P4 and Pz) and temporal-occipital electrode sites (T5 and T6), and the P300 amplitudes are significantly larger in the Simple Task than in the Complex Task at all concerned sites.

The negative P300 amplitude was significantly larger in the Simple Task than in the Complex Task at all concerned sites (P3 [F(1,38)=82.84, p<0.001], P4 [F(1, 38)=104.62, p<0.001], Pz [F(1,38)=316.31, p<0.001], O1 [F(1,38)=14.64, p<0.001], O2 [F(1,38)=53.30, p<0.001], T5 [F(1,38)=28.53, p<0.001] and T6 [F(1,38)=40.07, p<0.001]) (see Fig. 2). The P1 amplitude was significantly decreased in the Complex Task than in the Simple Task at all concerned sites O1 [F(1,38)=46.51, p<0.001], O2 [F(1,38)=9.29, p<0.005], P3 [F(1,38)=15.40, p<0.001], P4 [F(1,38)=62.59, p<0.001], Pz [F(1,38)=54.44, p<0.001], T5 [F(1,38)=7.94, p<0.01] and T6 [F(1,38)=5.13, p<0.05] (see Fig. 2).

Fig. 3. Grand average negative ERPs for sites T5, T6, P3, P4, Pz, O1 and O2 of the Simple Task and the Complex Task

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The N1 amplitude was significantly decreased in the Complex Task than in the Simple Task at all concerned sites O1 [F(1,38)=44.36, p<0.001], O2 [F(1,38)=19.18, p<0.001], P3 [F(1,38)=38.14, p<0.001], P4 [F(1,38)=42.22, p<0.001], Pz [F(1,38)=67.17, p<0.001], T5 [F(1,38)=16.65, p<0.001] and T6 [F(1,38)=33.90, p<0.001] (see Fig. 2). For further comparison, we plotted the grand average negative ERPs (ERP waveforms evoked by negative pictures) of sites T5, T6, P3, P4, Pz, O1 and O2 in the following figure (Fig. 3). Fig. 3 shows the decreased P300 amplitudes and the shorter P300 latencies clearly in the Complex Task. The decrease of P1/N1 amplitudes could be obviously observed in the Complex Task, too.

4 Discussion P300 amplitude in response to unpleasant pictures was significantly higher than that in response to either pleasant pictures or neutral pictures, which was consistent with the known studies [10,11,12,13,14,15]. The negative P300 amplitude was significantly decreased at sites T5, T6, P3, P4, Pz, O1 and O2 in the Complex Task comparing with the Simple Task, which showed that the emotion processing was modulated by task relevance. It was often assumed that P300 amplitude depended on the capacity of processing task relevant stimuli. Meanwhile, Kok [47] suggested that in many tasks, an increase in difficulty influenced the very processes that underlay P300 generation. Thus, we could conclude that in the Complex Task, the task difficulty were enhanced, which resulted in the decrease of P300 amplitude. The arousal level governed the amount of attention that was available for task performance, and P300 amplitude was hypothesized to index attentional resources. The non-task events might engage attentional resources to reduce P300 amplitude, namely, P300 amplitude was relatively large under simple task conditions [48]. Accordingly, in the Complex Task, P300 amplitude was supposed to be smaller because attentional resources were detracted for task performance. In fMRI studies, Lane et al. [34] suggested there was a higher arousing effect when subjects attended to the emotional aspect of a stimulus to a greater extent. Thus, in the Simple Task, subjects might pay more attention to the emotional stimulus, while in the Complex Task, due to the distraction of the additional factor- white frames, subjects might pay less attention to the emotional aspect of the stimuli. Amplitude modulation of the P1 and N1 components reflected the selection of the attended spatial location [49,50]. Stimuli at attended locations elicited larger P1/N1 components than those at unattended locations [51]. Hence, as outlined above, we could see that, because of the additional white frame, the selection of the attended spatial location was more uncertain, and the emotional aspect of the stimuli might become unattended, which resulted in the decrease of P1 (at all concerned sites O1, O2, P3, P4, Pz, T5 and T6) and N1 (at all concerned sites O1, O2, P3, P4, Pz, T5 and T6) in the Complex Task. The N1 component was generally enhanced/reduced along with the P1 to attended stimuli (vs. unattended stimuli) [52,33]. We could see the same result supporting this theory while comparing the Complex Task with the Simple Task.

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To sum up, negative pictures produced a significantly larger P300 than either positive or neutral pictures, which is consistent with the known studies. The P300 amplitude was smaller at all concerned sites T5, T6, P3, P4, Pz, O1 and O2 in the Complex Task than in the Simple Task, which suggested that the emotion P300 was significantly modulated by task relevance. As P1 (at sites T5, T6, P3, P4, Pz, O1 and O2) and N1 (at sites T5, T6, P3, P4, Pz, O1 and O2) amplitudes were decreased in the Complex Task comparing with the Simple Task, we could also suggest that the P1/N1 components elicited by emotional stimuli were also modulated by task relevance. Acknowledgments. The authors would like to thank NIMH Center for the Study of Emotion and Attention (University of Florida, Gainesville) for providing the pictures of IAPS (The International Affective Picture System).

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A Detailed Study on the Modulation of Emotion Processing by Spatial Location* Baolin Liu1, Shuai Xin1, Zhixing Jin1, Xiaorong Gao2, Shangkai Gao2, Renxin Chu3, Beixing Deng4, and Yongfeng Huang4 1

Department of Computer Science and Technology, Tsinghua University, Beijing 100084, P.R. China 2 Department of Biomedical Engineering, Tsinghua University, Beijing 100084, P.R. China 3 Brigham and Women's Hospital, Harvard Medical School, Boston, MA 02115, USA 4 Department of Electronic Engineering, Tsinghua University, Beijing 100084, P.R. China [email protected]

Abstract. In this paper two ERP (event-related potential) experiments – oneposition (1-P) task and three-position (3-P) task are performed to study how emotion processing is modulated by spatial location. In the 1-P task, emotional pictures are presented centrally and subjects are required to respond when a negative stimulus is onset; while in the 3-P task, stimuli are presented randomly either in a central position, or in a left or right position to the central point. P300 with shorter latency as well as P1/N1 with decreased amplitude was detected in the 3-P task as compared to the 1-P task, which illustrates that the emotion P300 is significantly modulated by spatial location. Unexpectedly, the P300 amplitude shows no significant difference between the two tasks, providing evidence that there is little spatial-attention modulation of arousal rating in the brain’s processing of emotional perception. Keywords: Emotion, Attention, Spatial location, ERP.

1 Introduction P300 components aroused by emotional stimulus were widely discussed in many articles, such as the studies on the integration of emotion and working memory [5,33], the integration of emotion and inhibitory control[4,6], and Attentional Blink effect to emotional stimuli[1,55]. One of these important research fields was how P300 components were influenced by emotional valence. P300 waves aroused by negative pictures were proved to be stronger according to the previous studies [9,14,23,40,41]. However, P300 was not the earliest ERP (event-related potential) component related to emotion perception. Commonly, Pl component was regarded as an earliest *

This work is supported by the National Basic Research Program of China (973 program) (No. 2006CB303100 and No.2007CB310806) and by the National High Technology Development Program of China (No.2007AA010306 and No.2006AA01Z444), and by the National Natural Science Foundation of China (No.30630022).

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indicator of emotion effect aroused by emotional stimuli [34,54]. In the visuospatial attention studies, an increased P1 or N1 component reflected the early effect of spatial selective attention on visual processing[17,19]. In studies that stimuli were presented rapidly and unpredictably to the right and left visual fields, focus on the events in one field would enhance the amplitude of the early Pl (75-130 ms) and/or Nl (150-190 ms) components[2,22,31,58]. There had been many researches on the emotional modulation of attention. For instance, emotional modulation of attention shifting was investigated in Posner’s [39] spatial orienting task; Phelps et al.[38] provided the evidence for emotion potentiating the effects of attention on low-level visual processing in stimulus-driven attention; Stormark et al. [52] evidenced a larger N1 component with a parietal distribution of ERPs to the cue and at the frontal regions to targets presented outside the cued location. A stronger ERP component reflected increased attentional resources devoted to emotionally arousing stimuli [37,42,45,46,47,56]. However, the interaction and integration of emotion and attention was still not well understood. Harlan et al. [16] used event-related functional magnetic resonance imaging (fMRI) to investigate how attention could modulate the neural activity elicited by scenes varying in emotional content, and found an emotion-attention interaction effect in the anterior area. Schupp et al. [48] found that P300 target effects were significantly augmented when erotica and mutilation stimuli were focused, and their findings suggested specifying the interaction of attention and emotion in distinct processing stages. To further specify the interactions between emotion and attention, one main purpose of the current work was to study how emotional stimuli processing is modulated by attended spatial location. Fize et al. [12] performed a 3-P animal categorization task in which photographs could appear randomly either in the center or at 3.6° eccentricity (right or left) of the fixation point, and they observed an amplitude enhancement for the N1 component. Accordingly we imagined that the brain’s emotion processing might be modulated by attended spatial location, too. To manifest that, we designed a 3-P visual task to investigate the P300/P1/N1 components modulated/or not modulated by the concerned variable, that was, the attended spatial location. Additionally, the brain area (the position of the electrodes) is also a key variable in emotion perception research. Wright et al. [59] and Glascher and Adolphs [15] suggested that the left amygdala was involved in emotional encoding of more detailed information, while the right amygdala might be more sensitive to fast, short and shallow analysis of emotional information. Therefore, another purpose of our current study was to confirm whether the left or right hemispheric lateralization of the processing of emotion perception is valid.

2 Results Computerized artifact rejection was performed to discard epochs in which deviation in eye position, blinks, or amplifier blocking occurred [44]. The rejected epochs were considered invalid. We selected the datasets of the best 20 subjects (7 females and 13 males), and all datasets of the other 7 subjects were rejected. At last, about 20 percent

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of the selected 20 subjects’ trials (the negative epochs) were rejected for violating these artifact criteria. We used 40Hz low-pass filter to reduce high-frequency noise from the data, and the EEG (electroencephalogram) epochs we selected were 100 ms prior the stimulus onset and 900ms after stimulus onset. The effects of emotional stimuli were strongest in occipital and posterior brain locations [10,25,27] in neuroimaging researches. Hence, we paid most of our attention to occipital and posterior electrodes, that was, parietal electrodes (P3, P4 and Pz), occipital electrodes (O1, O2), central electrodes (C3, C4 and Cz) and temporal-occipital electrodes (T5 and T6). Fig. 1 shows the average ERPs to unpleasant, pleasant, and neutral stimuli in the 1-P task. We can see the amplitude differences (P300 amplitude in response to unpleasant pictures is significantly higher than that in response to either pleasant pictures or neutral pictures) clearly from it.

Fig. 1. The average ERPs to unpleasant, pleasant and neutral stimuli in the 1-P task

We plotted the negative P300 (P300 evoked by negative pictures) at the concerned electrodes in Fig. 2 as follows. The P300 latency was significantly shorter in the 3-P task than in the 1-P task at parietal electrodes (P3 [F(1,38)=96.40, p<0.001], P4 [F(1,38)=107.58, p<0.001] and Pz [F(1,38)=101.88, p<0.001]), central electrodes (C3 [F(1,38)=88.71, p<0.001] and C4 [F(1,38)=56.22, p<0.001]) and temporal-occipital electrodes (T5 [F(1,38)=175.94, p<0.001] and T6 [F(1,38)=170.13, p<0.001]). But there was no significant difference between P300 amplitudes of 1-P task and 3-P task. The P1 (evoked by negative pictures) amplitude was significantly decreased in the 3-P task than in the 1-P task at parietal electrodes P3 [F(1,38)=32.68, p<0.001], P4[ F(1,38)=70.47, p<0.001] and Pz [F(1,38)=63.66, p<0.001]. While the N1 (evoked by negative pictures) amplitude was significantly decreased in the 3-P task than in the 1-P task at central electrodes (C3 [F(1,38) =28.67, p<0.001], C4 [F(1,38)=4.92, p<0.05] and Cz [F(1,38)=26.82, p<0.001]) and occipital electrodes (O1 [F(1,38)=49.74, p<0.001] and O2 [F(1,38)=59.68, p<0.001]).

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Fig. 2. Averaged P300 waveforms evoked by negative pictures (at sites O1, O2, P3, P4, Pz, T5, T6, C3, C4 and Cz) of 1-P task vs. 3-P task in array list

We plotted the ERP scale maps of the two tasks (Fig. 3) at time points 0ms, 200ms and 300ms. We could see the voltage distribution for the emotional ERPs. What was more, left hemisphere lateralization could be observed directly and clearly in the following figures of the two tasks. We tested the ERPs at the electrode site pairs C3 vs. C4, O1 vs. O2, P3 vs. P4 and T5 vs. T6 to analyze the effect.

Fig. 3. ERP scale maps of the 1-P task and the 3-P task

In the 1-P task, the P300 latency was significantly shorter at C3 than at C4 [F(1,38)=156.55, p<0.001], at O1 than at O2 [F(1,38)=44.10, p<0.001] and at T5 than at T6 [F(1,38)=6.06, p<0.05]. The P300 amplitude was larger at P3 than at P4 (F[(1,38)=4.73, P<0.05]), at T5 than at T6 [F(1,38)=46.14, p<0.001]. In the 3-P task, the P300 amplitude was larger at C3 than at C4 [F(1,38)=75.93, p<0.001], at O1 than at O2 [F(1,38)=31.05, p<0.001], at parietal electrode P3 than at P4 [F(1,38)=18.31, p<0.001] and at T5 than at T6 [F(1,38)=55.80, p<0.001]. The P300 latency was significantly shorter at C3 than at C4 [F(1,38)=74.63, p<0.001], at T5 than at T6 [F(1,38)=6.06, p<0.05].

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3 Discussion P300 amplitude in response to unpleasant pictures was significantly higher than that in response to either pleasant pictures or neutral pictures, which was consistent with the known studies [9,14,23,40,41]. In the studies of orienting response P300, the P300 amplitude would be enhanced by orienting processing [43], while Lane et al.[26] suggested there was a higher arousing effect when subjects attended to the emotional aspect of a stimulus to a greater extent. Unexpectedly, the P300 amplitude was neither enhanced nor reduced in the 3-P task here. Firstly, it showed that though spatial attention switch was included in both 3-P task and orienting tasks, the spatial-modulated emotion P300 and the orienting response P300 had different processing mechanisms; Secondly, it seemed that subjects hadn’t reduced their attention to the emotional pictures though they were presented in random locations. The arousal level governed the amount of attention that was available for task performance [24], and the P300 amplitude reflects the arousal rating to emotional stimuli. We could hereby conclude that there was no significant spatial-attention modulation of arousal rating in the 3-P task. In the 3-P task, we found P300 latency obviously shorter than that in the 1-P task. How did the reduction come? A number of studies had suggested that the P300 could be divided into two subcomponents – P3a and P3b [3,7,51]. P3a was regarded as a marker of initial signal evaluation [7], whereas P3b was considered to be related to working memory operations [11]. P3a might reflect an involuntary attention switch towards the distractor [13] and was suggested to be an indicator of the orienting response [8]. The increased P3a indicated more attentional orienting [53] and more anticipatory activation to the stimulus [57]. An imaginable reason for the shorter P300 latency in the 3-P task was that, the increase of the attended spatial location, which resulted in an involuntary attention switch towards the emotional stimuli, aroused a similar effect to orienting response as well as enhanced P3a components. Hence, we could suggest that it was the enhancement of P3a that resulted in the shorter negative P300 latency in the 3-P task. The P1/N1 amplitude enhancement seemed to be a general characteristic of the spatial focusing of attention [19]. No matter whether the stimuli were presented continuously in randomized sequences, or cued on each trial as to the most probable location of the subsequent target stimulus, stimuli at attended locations elicited larger P1/N1 components than those at unattended locations [19,31]. Due to the unpredictability in spatial location, the stimuli might be at unattended locations in the 3-P task and the P1/N1 amplitude was decreased. The N1 component is generally enhanced along with P1 to attended stimuli. Inconsistent changes of N1 and P1 only existed when comparing attended and non-attended locations [18]. Under 3-P task condition, no obvious effect was observed on the P1 amplitude at occipital electrodes, while the N1 amplitude was significantly decreased in this area. We suggested that the generator of the two components differed when elicited by emotional stimuli and this difference needed to be further studied.

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There was significant left hemisphere lateralization in both tasks. Markowitsch [32] and Glascher and Adolphs [15] suggested that the left amygdala was more closely related to emotional encoding of more detailed information, while the right amygdala might be more sensitive to a fast, short and shallow analysis of emotional information than the left amygdala. Similarly, Wright et al. [59] hypothesized that the left amygdala might be specialized for sustained evaluation, whereas the right amygdala was involved in rapid, dynamic emotional stimulus detection. Based on the results of Markowitsch [32], Wright et al. [59] and Glascher and Adolphs [15], and compared the stimulus set of the 1-P task and the 3-P task with those of Laurian et al. [30] and Smith and Bulman-Fleming [49,50], we could find that the emotional stimuli were much more complex in our tasks, and were presented over a much longer duration (2000 ms), so the information was more detailed to subjects. As a result, more activities of left amygdala were aroused, and left hemisphere lateralization was observed. Negative pictures produced a significantly larger P300 than either positive or neutral pictures, which was consistent with the known studies. The P300 latency was shorter in the 3-P task than in the 1-P task at parietal electrodes (P3, P4 and Pz), central electrodes (C3 and C4) and temporal-occipital electrodes (T5 and T6), which showed the emotion P300 was significantly modulated by spatial location. The same P300 amplitude showed that there was no significant spatial-attention modulation of arousal rating in the brain processes of emotional perception. As P1/N1 amplitude was decreased in the 3-P task comparing with the 1-P condition, we could suggest that the P1/N1 components elicited by emotional stimuli were also modulated by attended spatial location. Left hemisphere lateralization proved that the emotional pictures were more complex and detailed stimuli than either emotional words or emotional faces, and aroused more activities in the left amygdala.

4 Experimental Procedure Twenty-seven subjects (13 females and 14 males), mean age 22.3 (±3.4) years, were recruited from undergraduate students of Tsinghua University. All the subjects participated in the two experiments. The subjects had written questionnaire for history of psychiatric and neurological illness, drug abuse, and psychotropic medication use. They were all right-handed and had normal or corrected-to-normal vision. Handedness was measured using the Edinburgh Handedness Inventory (EHI) [35]. All subjects filled in the Volunteer Screening Form and were paid (RMB 20/ hour) for their participation. All experiments were conducted in accordance with the Declaration of Helsinki and all the procedures were carried out with adequate understanding of the subjects, who read and signed the Researches Consent Form before participating in this research. 84 pictures were selected from the IAPS (The International Affective Picture System) [28,29], and the pictures were selected according to the valence dimension (28 pleasant, 28 unpleasant, and 28 neutral). The pictures were divided into 2 groups, each group had 14 negative pictures, 14 positive, and 14 neutral. They were all presented in a pseudo random sequence.

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The stimuli were conducted on PC (Intel Pentium D, 3.0 GHz, 1 GB RAM) with a 22-inch color monitor. The screen was at a distance of 100 cm from the subjects, and the resolution was 1440 × 900 pixels. The luminance of the pictures was determined according to the research by Palomba et al. [36]. Subjects sat in front of the computer screen and received instructions explaining the experimental task. Stimuli were presented on a black background of a 22-inch color monitor at a viewing distance of 100 cm. The room was sound-attenuated and dimly lit. EEG signals were recorded from 19 scalp sites (Fp1/Fp2, F3/F4, C3/C4, P3/P4, O1/O2, F7/F8, T3/T4, T5/T6, Cz, Fz, and Pz) according to the International 10/20 System [21], using Neuroscan Synamps2 EEG/ERP system. The reference channel was linked to earlobes. We checked each of them and made sure the impedance was < 5 kΩ when all the electrodes were attached. When a subject was ready, the vocal introduction of the task was played to inform that 42 pictures would be presented and there would be a break every 14 pictures. He/she was required to pay full attention to each picture, and press "B" button when an unpleasant picture was shown. He/she was instructed to wait until the input screen appeared and then responded as accurately as possible. The entire task lasted about 7 minutes; each picture stimulus was presented for 2000ms and waited for the response, then the response result screen was shown to tell the subject what he/she had pressed; a 6000ms interval occurred between two trials, during which the screen was black except for a cross at the center of the screen, on which the subject were instructed to fixate. The trials were in a pseudo random sequence. All subjects used both hands to make responses. He/she could select another button on the keyboard (self-selected button) for both the positive and the neutral stimuli. This self-selection button could counterbalance the hand differences between the subjects. During the 3-P task, stimuli were presented randomly in a central position or at an eccentricity of 6.8° to the left or right of the central fixation point. Subjects were clearly informed which task (1-P or 3-P task) they would be required to perform. The other requirements were as same as those in the 1-P task. The EEG from each electrode site was digitalized at 256 Hz with an amplifier band pass of 0.01-40 Hz, including a 50 Hz notch filter and was stored for off-line averaging. Acknowledgments. The authors would like to thank NIMH Center for the Study of Emotion and Attention (University of Florida, Gainesville) for providing the pictures of IAPS (The International Affective Picture System).

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MATLAB Simulation and Comparison of Zhang Neural Network and Gradient Neural Network for Time-Varying Lyapunov Equation Solving Yunong Zhang1 , Shuai Yue2 , Ke Chen2 , and Chenfu Yi1 1

Department of Electronics and Communication Engineering 2 School of Software Sun Yat-Sen University, Guangzhou 510275, China [email protected]

Abstract. This paper presents a new kind of recurrent neural network proposed by Zhang et al. for solving online Lyapunov equation with time-varying coefficient matrices. Global exponential convergence could be achieved by such a recurrent neural network when solving the timevarying problems in comparison with gradient neural networks (GNN). MATLAB simulation of both neural networks for the real-time solution of time-varying Lyapunov equation is then investigated through several important techniques. Computer-simulation results substantiate the theoretical analysis and demonstrate the efficacy of such a Zhang neural network (ZNN) on time-varying Lyapunov equation solving, especially when using power-sigmoid activation functions. Keywords: Recurrent neural networks, Time-varying Lyapunov equation, Global exponential convergence, MATLAB simulation.

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The problem of solving Lyapunov (or Lyapunov-like) matrix equations arises in numerous scientific and engineering areas; e.g., control theory [1], optimization [2], signal processing [3], large space flexible structures [4], communications [5]. Especially, when we make use of Lyapunov’s direct method to analyze the stability of nonlinear dynamic systems, linear dynamic systems with uncertainty (or reliability problems) and/or to design optimal control systems [6], we may encounter Lyapunov-like matrix equations. Many numerical algorithms have been proposed for the solution of Lyapunov equation [7][8]. However, considering that the numerical algorithms performed on digital computers may not be efficient enough for large-scale online applications [9][10]. In recent years, researchers have developed a variety of neural-computation methods [11]-[13] for solving algebraic matrix equations, considering the parallel distributed nature of neural networks and their hardware realizability. Note that most approaches presented are developed following the gradient-descent method. The so-called gradient neural network is designed by defining a scalar-valued norm-based error F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 117–127, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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function firstly, of which the minimal point equals the solution of the problem. Then, by evolving along a descent direction resulting from the negative gradient of such an error function, the GNN model is developed, which could be guaranteed to solve effectively the Lyapunov equation with constant coefficient matrices [12]. However, the gradient-based neural network and other aforementioned algorithms may not work well if the coefficients are time-varying. In this nonstationary situation, the negative-gradient direction may no longer guarantee the decrease of the error function due to the side effects of time-varying coefficients. The gradient-based computational approaches thus require a much faster convergence as compared with the change of coefficient matrices. This has to impose very stringent restrictions on hardware realization, or sacrifice the solution precision. In addition, such methods can not be applied to the case when the coefficients vary very quickly or when large-scale complicated systems are involved. Recently, a new kind of recurrent neural networks with implicit dynamics has been proposed by Zhang et al. [14][15] for the online solution of time-varying problems (e.g., matrix inversion) instead of solving time-invariant problems. In this paper, we generalize this neural-design approach to handle the Lyapunov equation with time-varying coefficient matrices. The analysis and computer-simulation results of the new model are provided, which substantiate the efficacy of the ZNN-design approach. Besides, it is worth mentioning that the new points in this paper lie in the interested problem, solution and simulation themselves being quite different from our previous work.

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In the ensuing subsections, Zhang neural networks and gradient neural networks are developed for comparative purposes to solve online the Lyapunov equation with time-varying coefficient matrices. 2.1

Preliminaries

Lyapunov methods could be applied effectively to analyze various kinds of nonlinear and linear control systems, which result in the well-known Lyapunov matrix equation, AT P + P A = −C. For its solution with constant coefficients, many numerical algorithms have been developed [7][8]. Neural approaches such as gradient neural networks could be employed as well [11]-[13]. However, as far as we know, there is little work on the time-varying Lyapunov equation solving: AT (t)P (t) + P (t)A(t) + C(t) = 0, 0  t < +∞,

(1)

where A(t) ∈ Rn×n and C(t) ∈ Rn×n are smoothly time-varying coefficient matrices, while P (t) is the unknown matrix to be obtained. In this paper, the objective is to solve the above time-varying Lyapunov equation (1) in real time t and in an error-free manner.

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To solve time-varying Lyapunov equation (1), its coefficient matrices A(t) ˙ ˙ and C(t), together with their time-derivatives A(t) and C(t), are assumed to be known or could be estimated. Moreover, before solving Lyapunov equation (1), we provide the following solution-uniqueness condition and its related lemma. Solution-Uniqueness Condition: There exists α > 0 ∈ R such that, ∀t  0,       min σ I ⊗ AT (t)A(t) + A(t) ⊗ AT (t) + AT (t) ⊗ A(t) + AT (t)A(t) ⊗ I  α, where symbol σ(·) denotes the eigenvalues set of an input matrix, and I denotes an approximately-dimensioned identity-matrix. Symbol ⊗ denotes the Kronecker product of two matrices [14][16]; e.g., P ⊗ Q is a larger matrix made by replacing   the ijth entry pij of P with matrix pij Q. Lemma 1. Time-varying Lyapunov equation (1) satisfies the solutionuniqueness condition if and only if there are no two eigenvalues of bounded matrix A(t) ∈ Rn×n , λi (A(t)) and λj (A(t)) (with i, j = 1, 2, · · · , n), add up to zero at any time instant t [17]. Furthermore, the solution-uniqueness condition means that √ the minimal sum of any two eigenvalues of A(t) are uniformly no less than α.   2.2

Zhang Neural Network

Conventional gradient-neural-network approaches [11]-[13] have been developed to compute exactly the time-invariant Lyapunov equation AT P + P A = −C, of which the coefficients are constant matrices. Compared to the gradient-based method, by following Zhang et al.’s approach [14][15], we could firstly define a matrix-valued error function associated to the Lyapunov equation (1): L(t) = AT (t)P (t) + P (t)A(t) + C(t) ∈ Rn×n ,

(2)

where, if error function L(t) equals zero, then P (t) becomes the theoretical solution P ∗ (t) of the time-varying Lyapunov equation (1). ˙ Secondly, the error-function derivative L(t) ∈ Rn×n should be made such that n×n every entry Lij (t) ∈ R of L(t) ∈ R converges to zero, i, j = 1, 2, · · · , n. In mathematics, we have to choose L˙ ij (t) such that limt→∞ Lij (t) = 0, ∀i, j ∈ ˙ {1, 2, · · · , n}. A general form of error-function time-derivative L(t) could be d(L(t)) = −Γ F (L(t)) , dt

(3)

where design parameter Γ ∈ Rn×n is a positive-definite matrix used to scale the convergence rate of the solution. For simplicity, we can use γI in place of Γ with γ > 0 ∈ R. In addition, the activation-function-array F (·) : Rn×n → Rn×n is a matrix-valued mapping, in which each scalar-valued processing-unit f (·) is a monotonically-increasing odd function. For instance, the following four basic types of activation functions can be adopted: 1) linear activation function

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f (Lij ) = Lij ; 2) bipolar sigmoid function f (Lij ) = (1 − exp(−ξLij ))/(1 + exp(−ξLij )) with ξ > 2; 3) power activation function f (Lij ) = Lpij with odd integer p  3; and 4) power-sigmoid activation function (with ξ  1 and p  3)  p Lij , if |Lij |  1, f (Lij ) = 1+exp(−ξ) 1−exp(−ξLij ) 1−exp(−ξ) · 1+exp(−ξLij ) , otherwise. Thirdly, expanding the design formula (3) could give the following implicit dynamics of Zhang neural network, which solves Lyapunov equation (1): ˙ − C(t) ˙ AT (t)P˙ (t) + P˙ (t)A(t) = − A˙ T (t)P (t) − P (t)A(t)   T − γF A (t)P (t) + P (t)A(t) + C(t) ,

(4)

where P (t), starting from any initial condition P (0) = P0 ∈ Rn×n , is the activation state matrix corresponding to the theoretical solution P ∗ (t) of (1). In addition, the above ZNN model (4) possesses following convergence properties with proof omitted due to space limitation [14][15]. Proposition 1. Consider time-varying coefficient-matrices A(t) ∈ Rn×n and C(t) ∈ Rn×n of time-varying Lyapunov equation (1), which satisfy the solutionuniqueness condition. If a monotonically-increasing odd activation-functionarray F (·) is used, then the neural-state matrix P (t) of ZNN (4) starting from any initial state P0 ∈ Rn×n could converge to the theoretical solution P ∗ (t) of time-varying Lyapunov equation (1).   Proposition 2. In addition to Proposition 1, if a linear activation-functionarray F (·) is used, then the neural-state matrix P (t) of ZNN (4) starting from any initial state P0 ∈ Rn×n could exponentially converge to the theoretical solution P ∗ (t) of (1) with rate γ. As compared to the situation of using linear activation functions, if the array F (·) of power-sigmoid activation functions (4) is used, then superior global exponential convergence can be achieved for (4).   2.3

Gradient Neural Network

For comparison with the ZNN model presented in the preceding subsection, we would like to develop the following gradient neural network (GNN), which handles intrinsically the situation of constant A ∈ Rn×n and C ∈ Rn×n :   P˙ (t) = −γF A(AT P + P A + C) + (AT P + P A + C)AT , (5) where processing-array F (·) can be defined the same as in Subsection 2.2. Comparing ZNN (4) and GNN (5), we have the following important remarks. 1) ZNN (4) is designed based on the elimination of every entry of the matrixvalued error function L(t) = AT (t)P (t) + P (t)A(t) + C(t). In contrast, GNN (5) is designed through the elimination of scalar-valued norm-based error

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function AT P + P A + C 2F /2, where coefficient-matrices A and C could only be constant. 2) ZNN (4) methodically and systematically exploits the time-derivative information of coefficient matrices A(t) and C(t) during its real-time solving process. This is the reason why ZNN (4) could globally exponentially converge to the exact solution of a time-varying problem. On the other hand, GNN (5) has not exploited such important time-derivative information, and thus may not be effective in solving online such time-varying problems. 3) ZNN (4) is depicted in an implicit dynamics, i.e., AT (t)P˙ (t)+A(t)P˙ (t) = · · · . In comparison, GNN (5) is depicted in an explicit dynamics (without any mass matrix), i.e., P˙ (t) = · · · , which is normally associated with classical Hopfield-type recurrent neural networks.

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MATLAB Simulation Techniques

While Section 2 presents the ZNN and GNN models together with related analysis results, we would like in this section to investigate the MATLAB simulation techniques for such two models. To simulate the implicit-dynamic neural system (4), several important simulation techniques have to be employed, e.g., MATLAB routines “ode45” and “diff”. 3.1

Kronecker Product and Vectorization

For the proposes of MATLAB simulation, we have to transform the matrix-form differential equation (4) to a vector-form differential equation. Theorem 1. The matrix-form ZNN model (4) can be transformed to the following vector-form differential equation:   ˙ (6) − γF M (t)p(t) + vec(C(t)) , M (t)p(t) ˙ = −M˙ (t)p(t) − vec(C(t)) 2

where mass matrix M (t) := AT (t) ⊗ I + I ⊗ AT (t). Operator vec(C(t)) ∈ Rn ×1 generates a column vector obtained by stacking all column vectors of C(t) to2 gether. Integration vector p(t) := vec(P (t)) ∈ Rn ×1 . Activation-function-array 2 F (·) is defined the same as in (4) except for its dimensions changed as Rn ×1 . Given matrix A, to generate the mass matrix M = AT ⊗ I + I ⊗ AT , we could exploit MATLAB routine “kron” in the following code (as an example): function M=MatrixM(t,x) A=[-1-1/2*cos(2*t) 1/2*sin(2*t); 1/2*sin(2*t) -1+1/2*cos(2*t)]; M=kron(A.’,eye(size(A)))+kron(eye(size(A)),A.’);

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Obtaining Matrix Derivatives

To obtain the time derivative M˙ (t) required in vector-form ZNN model (6), without loss of generality, we can use MATLAB routine “diff” for such M˙ (t) and

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˙ C(t) evaluation. However, to do so, a symbolic object “u” has to be constructed firstly, and then the MATLAB command “DM=diff(MatrixM(u))” generates the analytical form of M˙ . Finally, evaluating such an analytical form of M˙ with a numerical value of t will give us the required value of M˙ (t). MATLAB code for ˙ evaluating such a M˙ (t) is shown as follows, and we can generate C(t) similarly. function output=DiffM(t,x) syms u; DM=diff(MatrixM(u)); u=t; output=eval(DM);

3.3

ZNN Right-Hand Side

By using MATLAB routine “reshape”, the vectorization of a matrix could be done readily. For example, the following MATLAB code is used to evaluate the right-hand side of vector-form ZNN model (6). function output=ZnnRightHandSide(t,x,gamma) if nargin==2, gamma=1; end C=MatrixC(t); [m,n]=size(C); dotC=DiffC(t,x); vecC=reshape(C,m*n,1); vecDotC=reshape(dotC,m*n,1); M=MatrixM(t,x); dotM=DiffM(t,x); output=-dotM*x-gamma*AFMpowersigmoid(M*x+vecC)-vecDotC;

In a similar way, we could have the MATLAB code which evaluates the vectorized right-hand side of GNN model (5). 3.4

ODE with Mass Matrix

For the simulation of Zhang neural network (4) [equivalently (6)], the MATLAB routine “ode45” is preferred. This is because “ode45” can solve initial-value ordinary differential equation (ODE) problems with nonsingular mass matrices, i.e., M (t, p)p˙ = g(t, p), p(0) = p0 , where matrix M (t, p) on the left-hand side of such an equation is the so-called mass matrix, and the right-hand side g(t, p) := ˙ in this paper. −M˙ p − γF (M p + vec(C)) − vec(C) To solve the ODE problem with a mass matrix, the MATLAB routine “odeset” is also involved. Its “Mass” property should be assigned to be the function handle “@MatrixM”, which returns the evaluation of mass matrix M (t, p). Note that, if 1) M (t, p) does not depend on state variable p and 2) the function “MatrixM” is to be invoked with only one input argument t, then the “MStateDep” property of “odeset” should be set “none”. The following MATLAB code solves such an initial-value ODE problem with state-independent mass matrix M (t) and from a random initial state p0 ; i.e., ZNN (6) [equivalently, (4)] handling (1). tspan=[0 10]; x0=4*(rand(4,1)-0.5*ones(4,1)); options=odeset(’Mass’,@MatrixM,’MStateDep’,’none’); [t,x]=ode45(@ZnnRightHandSide,tspan,x0,options);

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In this section, for illustrative purposes, a computer-simulation example is presented to show the performance and efficacy of ZNN model (4) solving (1). For comparison, conventional gradient neural network (5) is simulated as well. Let us consider Lyapunov equation (1) with the following time-varying coefficient matrices A(t) and C(t):     −1 − 0.5 cos 2t 0.5 sin 2t sin 2t cos 2t A(t) = , C(t) = . 0.5 sin 2t −1 + 0.5 cos 2t − cos 2t sin 2t Simple manipulations can verify that theoretical solution P ∗ (t) could be   −(sin 2t)(−2 + cos 2t)/3 − (2 cos 2t − 1)(2 + cos 2t)/6 P ∗ (t) = , −(2 cos 2t + 1)(−2 + cos 2t)/6 (2 + cos 2t)(sin 2t)/3

(7)

which is used to check the correctness of the neural-network solutions. 4.1

Neural-State Simulation

To solve Lyapunov equation (1) with the above time-varying coefficients via ZNN (4), the following MATLAB code could be used. Fig. 1(a) is then generated. function ZnnConvergence(gamma) tspan=[0 10]; options=odeset(’Mass’,@MatrixM,’MStateDep’,’none’); for iter=1:8 x0=4*(rand(4,1)-0.5*ones(4,1)); [t,x]=ode45(@ZnnRightHandSide,tspan,x0,options,gamma); subplot(2,2,1);plot(t,x(:,1));hold on subplot(2,2,3);plot(t,x(:,2));hold on subplot(2,2,2);plot(t,x(:,3));hold on subplot(2,2,4);plot(t,x(:,4));hold on end

As illustrated in Fig. 1(a), starting from any random initial state in the range of [-2,2], ZNN could always converge to the theoretical solution (7) rapidly where the design parameter γ is set to be 1. For comparison, GNN could also be employed to solve Lyapunov equation (1) with the same time-varying coefficients and under the same design-parameter value by using MATLAB code below. function GnnConvergence(gamma) tspan=[0 10]; options=odeset(); for iter=1:8 x0=4*(rand(4,1)-0.5*ones(4,1)); [t,x]=ode45(@GnnRightHandSide,tspan,x0,options,gamma); subplot(2,2,1);plot(t,x(:,1));hold on subplot(2,2,3);plot(t,x(:,2));hold on subplot(2,2,2);plot(t,x(:,3));hold on subplot(2,2,4);plot(t,x(:,4));hold on end

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The GNN-convergence is, then, shown in Fig. 1(b). It follows from Fig. 1(b) that the GNN solution (denoted by solid blue curves) does not fit well with the theoretical time-varying solution P ∗ (t) (denoted by dash-dotted red curves). Simply put, there always exists a residual error between the GNN solution and the theoretical solution P ∗ (t) to time-varying Lyapunov equation (1). This is evidently because the time-derivative information of A(t) and C(t) has not been utilized in these gradient-descent computational schemes. 4.2

Computation-Error Simulation

The computational error P (t) − P ∗ (t) F could also be used to monitor the neural-network convergence. The following MATLAB code shows the ZNN convergence of error P (t)−P ∗ (t) F . User-defined MATLAB function “ZnnNormError” depends on another user-defined function “NormError”, which could finally generate Fig. 2(a) and (b). The MATLAB code of applying GNN to generating Fig. 2(c) is similar but omitted here due to space limitation. function NormError(x0,gamma) tspan=[0 10]; options=odeset(’Mass’,@MatrixM,... ’MStateDep’,’none’,’RelTol’,1e-6,’AbsTol’,1e-10); [t,x]=ode45(@ZnnRightHandSide,tspan,x0,options,gamma); xs=[-1/3*sin(2*t).*(-2+cos(2*t)) -1/6*(2*cos(2*t)-1).*(2+cos(2*t))... -1/6*(2*cos(2*t)+1).*(-2+cos(2*t)) 1/3*(2+cos(2*t)).*sin(2*t)]’; err=x’-xs; total=length(t); for i=1:total, nerr(i)=norm(err(:,i)); end plot(t,nerr); hold on

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Fig. 2. Convergence characteristics of ZNN and GNN solution errors P (t) − P ∗ (t)F function ZnnNormError(gamma) if nargin<1, gamma = 10; end for i=1:8 x0=4*(rand(4,1)-0.5*ones(4,1)); NormError(x0,gamma); end

Fig. 2 illustrates the computational errors of ZNN (4) and GNN (5) starting from random initial states. It can be seen from the left two graphs of the figure and other simulation data that, by using ZNN (4) to solve online the time-varying Lyapunov equation (1), the steady-state solution errors of P (t) − P ∗ (t) F are less than 3 × 10−13 (computed as of t = 100s), 2 × 10−14 and 1.4 × 10−14 , which correspond respectively to the usage of design parameter γ = 1, 10 and 100. In contrast, it is seen from the rightmost graph of Fig. 2 and simulation data that, by using GNN (5) to solve online the time-varying Lyapunov equation (1) under the same conditions, the computational error P (t) − P ∗ (t) F is considerably large, with steady-state solution errors being around 0.7947, 0.0971 and 0.00965 respectively for γ = 1, 10 and 100. Furthermore, as seen from the figure, the neural-network convergence can be expedited effectively by increasing the value of design parameter γ, which substantiates the aforementioned exponential-convergence property. For example, the convergence time with no appreciable solution-error is less than 4μs when design parameter γ = 106 . In summary, Figs. 1 and 2 show that ZNN could globally exponentially converge to the theoretical solution P ∗ (t) of time-varying Lyapunov matrix equation (7), whereas the GNN solution-error P (t) − P ∗ (t) F is much larger.

5

Conclusions

By following Zhang et al.’s neural-design approach, an implicit-dynamic neural-network model has been developed and simulated for solving online the time-varying Lyapunov matrix equation. As compared with conventional gradient-neural-networks, the new design approach and its resultant ZNN models could have much superior efficacy on time-varying problems solving. Simulation techniques and results have further substantiated the feasibility of the

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ZNN model having global exponential convergence to time-varying theoretical solution. Acknowledgements. Before joining Sun Yat-Sen University in 2006, the corresponding author, Yunong Zhang, had been with National University of Ireland, University of Strathclyde, National University of Singapore, Chinese University of Hong Kong, since 1999. He has continued the line of this research, supported by those research fellowships and assistantship. His web-page is now available at http://www.ee.sysu.edu.cn/teacher/detail.asp?sn=129.

References 1. Phat, V.N., Niamsup, P.: Stability of Linear Time-Varying Delay Systems and Applications to Control Problems. Journal of Computational and Applied Mathematics 194, 343–356 (2006) 2. Subbotina, N.N.: The Value Functions of Singularly Perturbed Time-Optimal Control Problems in the Framework of Lyapunov Functions Method. Mathematical and Computer Modelling 45, 1284–1293 (2007) 3. Li, Y., Liu, K.J.R.: Static and Dynamic Convergence Behavior of Adaptive Blind Equalizers. IEEE Transactions on Signal Processing 44, 2736–2745 (1996) 4. Delorenzo, M.L.: Sensor and Actuator Selection for Large Space Structure Control. Journal of Guidance, Control, and Dynamics 13, 249–257 (1990) 5. Robert, B., Kushner, H.J.: Control of Mobile Communication Systems with TimeVarying Channels Viastability Methods. IEEE Transactions on Automatic Control 49, 1954–1962 (2004) 6. Zhang, Y., Wang, J.: A Dual Neural Network for Constrained Joint Torque Optimization of Kinematically Redundant Manipulators. IEEE Transactions on Systems, Man, and Cybernetics, Part B 32, 654–662 (2002) 7. Fernando, K.V., Nicholson, H.: Solution of Lyapunov Equation for the State Matrix. Electronics Letters 17, 204–205 (1981) 8. Sreeram, V., Agathoklis, P.: Solution of Lyapunov Equation with System Matrix in Companion Form. IEE Proceedings–D 138, 529–534 (1991) 9. Zhang, Y., Leithead, W.E., Leith, D.J.: Time-series Gaussian Process Regression Based on Toeplitz Computation of O(N 2 ) Operations and O(N )-Level Storage. In: 44th IEEE Conference on Decision and Control, pp. 3711–3716. IEEE Press, Seville (2005) 10. Leithead, W.E., Zhang, Y.: O(N 2 )-Operation Approximation of Covariance Matrix Inverse in Gaussian Process Regression Based on Quasi-Newton BFGS Methods. Communications in Statistics - Simulation and Computation 36, 367–380 (2007) 11. Zhang, Y., Wang, J.: Global Exponential Stability of Recurrent Neural Networks for Synthesizing Linear Feedback Control Systems via Pole Assignment. IEEE Transactions on Neural Networks 13, 633–644 (2002) 12. Zhang, Y.: Revisit the Analog Computer and Gradient-Based Neural System for Matrix Inversion. In: 20th IEEE International Symposium on Intelligent Control, pp. 1411–1416. IEEE Press, Cyprus (2005) 13. Zhang, Y.: A Set of Nonlinear Equations and Inequalities Arising in Robotics and its Online Solution via a Primal Neural Network. Neurocomputing 70, 513–524 (2006)

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14. Zhang, Y., Jiang, D., Wang, J.: A Recurrent Neural Network for Solving Sylvester Equation with Time-Varying Coefficients. IEEE Transactions on Neural Networks 13, 1053–1063 (2002) 15. Zhang, Y., Ge, S.S.: Design and Analysis of a General Recurrent Neural Network Model for Time-Varying Matrix Inversion. IEEE Transactions on Neural Networks 16, 1477–1490 (2005) 16. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis, pp. 239–297. Cambridge University Press, Cambridge (1991) 17. Ding, F., Chen, T.: Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations. IEEE Transactions on Automatic Control 50, 1216–1221 (2005)

Improved Global Exponential Stability Criterion for BAM Neural Networks with Time-Varying Delays Yonggang Chen1 and Tiheng Qin2 1 2

Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China Department of Basic Courses, Henan Mechanical and Electrical Engineering College, Xinxiang 453002, China [email protected]

Abstract. In this paper, the global exponential stability analysis is investigated for a class of bidirectional associative memory (BAM) neural networks with time-varying delays. By using Lyapunov functional method, and by reserving the useful terms when estimating the upper bound of the derivative of Lyapunov functional, the less conservative exponential stability criterion is derived in terms of linear matrix inequality (LMI). Numerical example is presented to show the effectiveness and the less conservativeness of the proposed method. Keywords: Exponential stability, BAM neural networks, Time-varying delay, Linear matrix inequality.

1

Introduction

Bidirectional associative memory (BAM) neural networks were introduced by Kosko[1-2]. This class of networks has been successfully applied to pattern recognition and artificial intelligence due to its generalization of the single-layer autoassociative Hebbian correlator to a two-layer pattern-matched heteroassociative circuit. On the other hand, time delays often occur in many practical neural networks, and its existence will lead to instability, oscillation, and poor performances of neural networks. Therefore, the stability analysis for BAM neural networks with time delays has received great attention during the past years, see for example [3-6]. When it comes to design a neural network, one concerns not only on the stability of the system but also on the convergence rate, that is to say, one usually desires a fast response in the network, so it is important to determine the exponential stability and to estimate the exponential convergence rate, see for example [7-9]. Thus, in recent years, the exponential stability analysis for BAM neural networks with delays has been widely investigated [10-13]. By choosing the new Lyapunov functional, the novel delay-dependent exponential stability analysis is considered in [13] for BAM neural networks with time-varying delays. F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 128–137, 2008. c Springer-Verlag Berlin Heidelberg 2008 

Improved Global Exponential Stability Criterion for BAM Neural Networks

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The exponential stability criterion in [13] improves the results in [10,11]. However, when estimating the upper bound of the derivative of Lyapunov functional, the useful terms are ignored in [13]. Therefore, there is much room to improve the result in [13]. In this paper, the global exponential stability analysis is considered for BAM neural networks with time-varying delay. By constructing the new Lyapunov functional, and by reserving the useful terms when estimating the upper bound of the derivative of Lyapunov functionals, we obtain the novel exponential stability criterion which is expressed in linear matrix inequality (LMI). Finally, numerical example shows that our result is less conservative than some existing ones [10,11,13]. Notation. Throughout this paper,  ·  denotes the Euclidean norm. A real symmetric matrix P > 0 denotes P being a positive definite matrix, λM (·) and λm (·) represent for the maximum and minimum eigenvalues of given square matrix, respectively. I is used to denote an identity matrix with proper dimension. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symmetric terms in a symmetric matrix are denoted by ∗.

2

Problem Description

Consider the following BAM neural network with time-varying delays u˙ i (t) = − ai ui (t) +

m 

wij f˜j (vj (t − τ (t))) + Ii , i = 1, 2, · · · , n,

j=1

v˙ j (t) = − bj vj (t) +

n 

vji g˜i (ui (t − h(t))) + Ij , j = 1, 2, · · · , m,

(1)

i=1

where ui (t) and vj (t) are the activations of the ith neurons and jth neurons, respectively; wji and vij are the connection weights at the time t; f˜j (·) g˜i (·) denote the signal functions of the ith neurons and jth neurons, respectively; Ii and Jj denote the external inputs, and ai > 0, bj > 0; τ (t) and h(t) are time-varying delays which are assumed that ˙ ≤ hd < 1, 0 ≤ τ (t) ≤ τM , τ˙ (t) ≤ τd < 1, 0 ≤ h(t) ≤ hM , h(t)

(2)

where τM , hM , τd and hd are positive scalars. Throughout this paper, we make the following assumption on the activation functions f˜j (·), g˜i (·). (A1) The activation functions f˜j (·), g˜i (·) are bounded on R. (A2) For any ζ1 , ζ2 ∈ R, ζ1 = ζ2 , there exist positive scalars k1j , k2i such that f˜j (ζ1 ) − f˜j (ζ2 ) ≤ k1j , j = 1, 2, · · · , m, ζ1 − ζ2 g˜i (ζ1 ) − g˜i (ζ2 ) ≤ k2i , i = 1, 2, · · · , n. 0≤ ζ1 − ζ2

0≤

(3)

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Y. Chen and T. Qin

It is clear that under Assumptions (A1) and (A2), system (1) has at least one equilibrium. In order to simplify our proof, we shift the equilibrium point u∗ = ∗ T [u∗1 , u∗2 , · · · , u∗n ]T , v ∗ = [v1∗ , v2∗ , · · · , vm ] of system (1) to the origin. Let xi (t) = ∗ ∗ ui (t)−ui , yj (t) = vj (t)−vj , fj (yj (t)) = f˜j (vj (t))−f˜j (vj∗ ), gi (xi (t)) = g˜i (ui (t))− g˜i (u∗i ), then system (1) can be transformed to x˙ i (t) = − ai xi (t) + y˙ j (t) = − bj yj (t) +

m  j=1 n 

wij fj (yj (t − τ (t))), i = 1, 2, · · · , n, vji gi (xi (t − h(t))), j = 1, 2, · · · , m.

(4)

i=1

Obviously, the activation functions fj (·), gi (·) satisfy the following properties (H1) The activation functions fj (·), gi (·) are bounded on R. (H2) For any ζ1 , ζ2 ∈ R, ζ1 = ζ2 , there exist positive scalars k1j , k2i such that fj (ζ1 ) − fj (ζ2 ) ≤ k1j , j = 1, 2, · · · , m, ζ1 − ζ2 gi (ζ1 ) − gi (ζ2 ) ≤ k2i , i = 1, 2, · · · , n. 0≤ ζ1 − ζ2

0≤

(5)

(H3) fj (0) = 0, gi (0) = 0, i = 1, 2, · · · , n, j = 1, 2, · · · , m. Also we assume that system (4) with initial value xi (s) = φi (s), yj (s) = ψj (s), ∀s ∈ [−τ ∗ , 0], i = 1, 2, · · · , n, j = 1, 2, · · · , m, where τ ∗ = max{τM , hM }. For convenience, we rewrite system (4) in the form x(t) ˙ = −Ax(t) + W f (y(t − τ (t))), y(t) ˙ = −By(t) + V g(x(t − h(t))), x(s) = φ(s), y(s) = ψ(s), ∀s ∈ [−τ ∗ , 0],

(6)

where x(t) = [x1 (t), x2 (t), · · · , xn (t)]T , y(t) = [y1 (t), y2 (t), · · · , ym (t)]T , f = [f1 , f2 , · · · , fm ]T , g = [g1 , g2 , · · · , gn (t)]T , A = diag{a1 , a2 , · · · , an }, B = diag {b1 , b2 , · · · , bm }, W = (wij )m×n , V = (vij )n×m , φ(s) = [φ1 (s), φ2 (s), · · · , φn (s)]T , ψ(t) = [ψ1 (t), ψ2 (t), · · · , ψm (t)]T . Before giving the main result, we will firstly introduce the following definition and lemmas. Definition 1. The trivial solution of system (6) is said to be globally exponentially stable, if there exist constants k > 0 and μ ≥ 1 such that   x(t)2 + y(t)2 ≤ μe−2kt φ2 + ψ2 , where k is called the exponential convergence rate, and we denote φ2 =

sup

−τ ∗ ≤s≤0

φ(s)2 , ψ2 =

sup

−τ ∗ ≤s≤0

ψ(s)2 .

Improved Global Exponential Stability Criterion for BAM Neural Networks

131

Lemma 1. [7] Suppose that (5) holds, then  u [fj (s) − fj (v)] ≤ (u − v)[fj (u) − fj (v)], j = 1, 2, · · · , m, v  u [gi (s) − gi (v)] ≤ (u − v)[gi (u) − gi (v)], i = 1, 2, · · · , n. v

Lemma 2. [14] For any constant matrix W ∈ Rn×n , W = W T > 0, scalar γ > 0, vector function ω : [0, γ] → Rn such that the integrations concerned are well defined, then   γ  T  γ   γ ω(s) ds W ω(s) ds ≤ γ ω T (s)W ω(s) ds. 0

3

0

0

Main Result

Now, we are in a position to present the delay-dependent global exponential stability criterion of system (6). Theorem 1. For given diagonal matrix K1 = diag{k11 , k12 , · · · , k1m }, K2 = diag{k21 , k22 , · · · , k2n }, and constants hM , τM , hd , τd . The system (6) is globally exponentially stable with the convergence rate k > 0, if there exist matrices Pi > 0, Qi > 0, Ri > 0, Si > 0, Zi > 0, (i = 1, 2), and diagonal matrices D = diag{d1 , d2 , · · · , dn } > 0, L = diag{l1 , l2 , · · · , lm } > 0, Tj ≥ 0, (j = 1, 2, 3, 4), such that the following LMI holds ⎤ ⎡ 0 0 0 0 Ω110 Ω11 Λ1 0 Ω14 0 ⎢ ∗ Ω22 Λ1 0 K2 T2 0 0 0 0 0 ⎥ ⎥ ⎢ ⎥ ⎢ ∗ ∗ Ω33 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ ∗ Ω44 0 0 0 0 0 DW ⎥ ⎢ ⎢ ∗ ∗ ∗ ∗ Ω55 Ω56 0 0 V T LT 0 ⎥ ⎥ < 0, ⎢ Ω=⎢ (7) 0 ⎥ ∗ Ω66 Λ2 0 Ω69 ⎥ ⎢ ∗ ∗ ∗ ∗ ⎢ ∗ ∗ ∗ ∗ ∗ 0 Ω77 Λ2 0 K1 T4 ⎥ ⎥ ⎢ ⎢ ∗ ∗ ∗ ∗ ∗ 0 0 Ω 0 0 ⎥ 88 ⎥ ⎢ ⎣ ∗ ∗ ∗ ∗ 0 ⎦ ∗ ∗ ∗ ∗ Ω99 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Ω1010 where Ω11 = −P1 A − AT P1T + 2kP1 + h2M AT Z1 A + Q1 + S1 − e−2khM Z1 , Ω14 = −AT DT + 2kDT + K2 T1 , Ω1,10 = P1 W − h2M AT Z1 W, Ω22 = −(1 − hd )e−2khM Q1 − 2e−2khM Z1 , Ω33 = −e−2khM S1 − e−2khM Z1 , Ω44 = R1 − 2T1 , 2 Ω55 = −(1 − hd )e−2khM R1 + τM V T Z2 V − 2T2 ,

132

Y. Chen and T. Qin 2 Ω56 = V T P2T − τM V T Z2 B, 2 Ω66 = −P2 B − B T P2T + 2kP2 + τM B T Z2 B + Q2 + S2 − e−2kτM Z2 ,

Ω69 = −B T LT + 2kLT + K1 T3 , Ω77 = −(1 − τd )e−2kτM Q2 − 2e−2kτM Z2 , Ω88 = −e−2kτM S2 − e−2kτM Z2 , Ω99 = R2 − 2T3 , Ω1010 = −(1 − τd )e−2kτM R2 + h2M W T Z1 W − 2T4 , Λ1 = e−2khM Z1 , Λ2 = e−2kτM Z2 . Proof. Choose a Lyapunov-Krasovskii functional candidate as follows V (t) =

6 

Vi (t),

(8)

i=1

where V1 (t) =e2kt xT (t)P1 x(t) + e2kt y T (t)P2 y(t),  xi (t)  n m   V2 (t) =2 di e2kt gi (s) ds + 2 lj e2kt  V3 (t) = V4 (t) = V5 (t) =

0

i=1 t

e t−h(t)  t t−h(t)  t

x (s)Q1 x(s) ds +

t−hM  0

t−τ (t)



2ks T

x (s)S1 x(s) ds +



t

−τM

t

t−τM

e2ks y T (s)Q2 y(s) ds,



t t−τ (t)

e2ks f T (y(s))R2 f (y(s)) ds,

e2ks y T (s)S2 y(s) ds,

e2ks x˙ T (s)Z1 x(s)ds ˙ dθ

−hM t+θ  0  t

+ τM

t

e2ks g T (x(s))R1 g(x(s)) ds + e

V6 (t) =hM



fj (s) ds, 0

j=1

2ks T

yj (t)

e2ks y˙ T (s)Z2 y(s)ds ˙ dθ.

t+θ

Differentiating V (t) with respect to t along the trajectory of (6) gives that V˙ 1 (t) =e2kt {2kxT (t)P1 x(t) + 2xT (t)P1 x(t) ˙ + 2ky T (t)P2 y(t) + 2y T (t)P2 y(t)} ˙ 2kt T T T 2kx (t)P1 x(t) − 2x (t)P1 Ax(t) + 2x (t)P1 W f (y(t − τ (t))) =e

(9) + 2ky T (t)P2 y(t) − 2y T (t)P2 By(t) + 2y T (t)P2 V g(x(t − h(t))) ,  n x (t) i  V˙ 2 (t) =4 kdi e2kt gi (s) ds + 2e2kt g T (x(t))Dx(t) ˙ i=1

+4

m  j=1

0



klj e2kt 0

yj (t)

fj (s) ds + 2e2kt f T (y(t))Ly(t) ˙

Improved Global Exponential Stability Criterion for BAM Neural Networks

133

≤e2kt 4kg T (x(t))Dx(t) − 2g T (x(t))DAx(t) + 2g T (x(t))DW f (y(t − τ (t))) + 4kf T (y(t))Ly(t)

− 2f T (y(t))LBy(t) + 2f T (y(t))LV g(x(t − h(t))) , V˙ 3 (t) ≤e2kt xT (t)Q1 x(t) − (1 − hd )e−2khM xT (t − h(t))Q1 x(t − h(t))

+ y T (t)Q2 y(t) − (1 − τd )e−2kτM y T (t − τ (t))Q2 y(t − τ (t)) , V˙ 4 (t) ≤e2kt g T (x(t))R1 g(x(t)) + f T (y(t))R2 f (y(t)) − (1 − hd )e−2khM g T (x(t − h(t)))R1 g(x(t − h(t)))

− (1 − τd )e−2kτM f T (y(t − τ (t)))R2 f (y(t − τ (t))) , V˙ 5 (t) =e2kt xT (t)S1 x(t) + y T (t)S2 y(t) − e−2khM xT (t − hM )S1 x(t − hM )

− e−2kτM y T (t − τM )S2 y(t − τM ) ,  2kt 2 T ˙ V6 (t) ≤e ˙ + τM y˙ (t)Z2 y(t) ˙ h2M x˙ T (t)Z1 x(t) −2khM



t

T

− hM e x˙ (s)Z1 x(s) ˙ ds − τM e t−hM  2 T =e2kt h2M x˙ T (t)Z1 x(t) ˙ + τM y˙ (t)Z2 y(t) ˙ − hM e−2khM − hM e

−2khM

 

− τM e−2kτM − τM e



t

t−h(t)

t−τ (t)



−2kτM



t

t−τM

T

(11)

(12) (13)



y˙ (s)Z2 y(s) ˙ ds

x˙ T (s)Z1 x(s) ˙ ds

t−h(t)

t−hM t

−2kτM

(10)

x˙ T (s)Z1 x(s) ˙ ds

y˙ T (s)Z2 y(s) ˙ ds

t−τ (t)



T

y˙ (s)Z2 y(s) ˙ ds ,

t−τM

(14)

where Lemma 1 is utilized in (10). By Lemma 2, we have  t x˙ T (s)Z1 x(s) ˙ ds − hM t−h(t)

 ≤−

T

t

x(s) ˙ ds t−h(t) T

 Z1



t

x(s) ˙ ds t−h(t)

    = − x (t) − xT (t − h(t)) Z1 x(t) − x(t − h(t) ,  t−h(t) x˙ T (s)Z1 x(s) ˙ ds − hM t−hM

 x(s) ˙ ds t−hM t−hM     = − xT (t − h(t)) − xT (t − hM ) Z1 x(t − h(t)) − x(t − hM ) , 

≤−

T

t−h(t)

x(s) ˙ ds



(15)

t−h(t)

Z1

(16)

134

Y. Chen and T. Qin

 − τM

t

t−τ (t)

 ≤−

y˙ T (s)Z2 y(s) ˙ ds T

t

y(s) ˙ ds t−τ (t)

 Z2



t

y(s) ˙ ds t−τ (t)

    = − xT (t) − xT (t − τ (t)) Z1 x(t) − x(t − τ (t) ,  t−τ (t) y˙ T (s)Z2 y(s) ˙ ds − τM t−τM

 ≤−

T

t−τ (t)

y(s) ˙ ds t−τ

 Z2

(17)



t−τ (t)

y(s) ˙ ds t−τ

M M    = − y T (t − τ (t)) − y T (t − τM ) Z2 y(t − τ (t)) − y(t − τM ) .

(18)

By inequalities (5), it is well known that there exist diagonal matrices Ti ≥ 0, (i = 1, 2, 3, 4) such that the following inequalities hold   2e2kt xT (t)K2 T1 g(x(t)) − g T (x(t))T1 g(x(t)) ≥ 0, (19)   2kt T T x (t − h(t))K2 T2 g(x(t − h(t))) − g (x(t − h(t)))T2 g(x(t − h(t))) ≥ 0, 2e (20)   2kt T T y (t)K1 T3 f (y(t)) − f (y(t))T3 f (y(t)) ≥ 0, (21) 2e   2kt T T y (t − τ (t))K1 T4 f (y(t − τ (t))) − f (y(t − τ (t)))T4 f (y(t − τ (t))) ≥ 0, 2e (22) By considering (9)-(22), we can eventually obtain V˙ (t) ≤

6 

 V˙ i (t) + 2e2kt xT (t)K2 T1 g(x(t)) − g T (x(t))T1 g(x(t))

i=1

+ xT (t − h(t))K2 T2 g(x(t − h(t))) − g T (x(t − h(t)))T2 g(x(t − h(t))) + y T (t)K1 T3 f (y(t)) − f T (y(t))T3 f (y(t))

 + y T (t − τ (t))K1 T4 f (y(t − τ (t))) − f T (y(t − τ (t)))T4 f (y(t − τ (t)))

≤e2kt ξ T (t)Ωξ(t),

(23)

where Ω is defined in Theorem 1, and  ξ(t) = xT (t) xT (t − h(t)) xT (t − hM ) g T (x(t)) g T (x(t − h(t)) y T (t) T y T (t − τ (t)) y T (t − τM ) f T (y(t)) f T (y(t − τ (t)) . Thus, if matrix inequality Ω < 0, then we have V˙ (t) < 0, which implies that V (t) < V (0). By some simple computation, we can obtain V1 (0) =xT (0)P1 x(0) + y T (0)P2 y(0) ≤ λM (P1 )φ2 + λM (P2 )ψ2 , n m   V2 (0) ≤2 di xi (0)gi (xi (0)) + 2 lj yj (0)fj (yj (0)) i=1

(24)

j=1

≤2λM (DK2 )φ + 2λM (LK1 )ψ2 , 2

(25)

Improved Global Exponential Stability Criterion for BAM Neural Networks

 V3 (0) ≤λM (Q1 )

0

−hM

xT (s)x(s) ds + λM (Q2 )



0

135

y T (s)y(s) ds,

−τM

≤hM λM (Q1 )φ2 + τM λM (Q2 )ψ2 ,  0  T V4 (0) ≤λM (R1 ) g (x(s))g(x(s)) ds + λM (R2 )

(26) 0

f T (y(s))f (y(s)) ds

−hM

V5 (0)

−τM ≤hM λM (R1 )λM (K22 )φ2 + τM λM (R2 )λM (K12 )ψ2 , ≤hM λM (S1 )φ2 + τM λM (S2 )ψ2 ,



V6 (0) ≤hM λM (Z1 )

0



0

−τM

(28)

x˙ T (s)x(s)ds ˙ dθ

−hM θ  0  0

+ τM λM (Z2 )

(27)

y˙ T (s)y(s)ds ˙ dθ,

(29)

θ

It is well known that x˙ T (s)x(s) ˙ ≤2xT (s)AT Ax(s) + 2f T (y(s − τ (s)))W T W f (y(s − τ (s))) ≤2λM (AT A)φ2 + 2λM (W T W )λM (K12 )ψ2 , T

T

T

T

(30)

T

˙ ≤2y (s)B By(s) + 2g (x(s − h(s)))V V g(x(s − τ (s))) y˙ (s)y(s) ≤2λM (B T B)ψ2 + 2λM (V T V )λM (K22 )φ2 .

(31)

Substituting (30), (31) into (29), and using (24)-(29), then we have V (0) =

6 

Vi (0) ≤ρ1 φ2 + ρ2 ψ2 ,

(32)

i=1

where ρ1 =λM (P1 ) + 2λM (DK2 ) + hM λM (Q1 ) + hM λM (R1 )λM (K22 ) + hM λM (S1 ) 3 λM (Z2 )λM (V T V )λM (K22 ), + 2h3M λM (Z1 )λM (AT A) + 2τM

ρ2 =λM (P2 ) + 2λM (LK1 ) + τM λM (Q2 ) + τM λM (R2 )λM (K12 ) + τM λM (S2 ) 3 λM (Z2 )λM (B T B) + 2h3M λM (Z1 )λM (W T W )λM (K12 ). + 2τM

On the other hand

V (t) ≥ e2kt λm (P1 )x(t)2 + λm (P2 )y(t)2 .

(33)

Thus, we can obtain x(t)2 + y(t)2 ≤

α (φ2 + ψ2 )e−2kt . β

(34)

where α = max{ρ1 , ρ2 }, β = min{λm (P1 ), λm (P2 )}. It is clear that α β ≥ 1, by Definition 1, the system (6) or (1) is globally exponentially stable with the convergence rate k > 0. This completes the proof.

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Y. Chen and T. Qin

Remark 1. The global exponential stability was considered in [13], but the Theorem 1 in [13] is incorrect. The Theorem 1 can be corrected by changing the location of P W from (1, 7) to (1, 8) in LMI (7), where (i, j) denotes the ith row and jth column of the corresponding matrix. t t ˙ ds, − t−τM y˙ T (s)Z2 y(s) ˙ ds Remark 2. In [13], the terms − t−hM x˙ T (s)Z1 x(s) t t T T ˙ ds and − t−τ (t) y˙ (s)Z2 y(s) ˙ ds, were simply enlarged as − t−h(t) x˙ (s)Z1 x(s)  t−h(t) T respectively. However, in this paper, the useful terms − t−hM x˙ (s)Z1 x(s) ˙ ds  t−τ (t) T and − t−τM y˙ (s)Z2 y(s) ˙ ds are reserved and the information on them are used.

4

Numerical Example

Example 1. Consider the delayed BAM neural network (6) with gi (x) = 12 (|xi + 1| − |xi − 1), fj (y) = 12 (|yj + 1| − |yj − 1), h(t) = 0.5, τ (t) = 1, and the following parameters ⎡ ⎤ ⎡ ⎤ 0.05 0.25 0.05 0.75 0.75 0.95 A = I, B = 2I, W = ⎣ 0.1 0.05 0.15⎦ , V = ⎣ 0 0.5 0.15⎦ . 0.15 0.15 0.05 0.15 0.15 0.05 Obviously, the activation functions fj , gi satisfy (5) with K1 = K2 = I. Assume that k = 0.4, it is fund that the criteria in [10,11] can not determined that this system is exponentially stable. However, using the Theorem 1 in this paper and the corrected criterion in [13], the obtained maximum allowable exponential convergence rate is k = 0.48. When delays are time-varying with hM = 0.5, τM = 1, the maximum allowable convergence rates obtained in this paper and comparison with those in [13] are listed in Table 1. For this example, it is obvious that the Theorem 1 in this paper improves some existing results[10,11,13]. Table 1. Maximum allowable convergence rate hd = τd [13] (corrected) Theorem 1

5

0.3

0.5

0.7

0.9

0.39

0.30

0.15

infeable

0.445

0.424

0.408

0.407

Conclusion

This paper considers the global exponential stability for bidirectional associative memory (BAM) neural networks time-varying delays. Based on Lyapunov functional method, and by reserving the useful terms when estimating the upper bound of the derivative of Lyapunov functional, the novel exponential stability

Improved Global Exponential Stability Criterion for BAM Neural Networks

137

criterion is derived in terms of linear matrix inequality(LMI). Finally, numerical example is given to show the less conservativeness of the obtained result. Acknowledgments. This work was supported by natural science basic research plan of Henan Institute of Science and Technology (No. 06053).

References 1. Kosko, B.: Adaptive Bi-directional Sssociative Memories. Appl. Opt. 26, 4947–4960 (1987) 2. Kosko, B.: Bi-directional Associative Memories. IEEE trans. Syst. Man Cybernet 18, 49–60 (1988) 3. Cao, J., Wang, J.: Exponential Stability and Periodic Oscillatory Solution in BAM Networks with Delays. IEEE Trans. Neural Netw. 13, 457–463 (2002) 4. Jiang, M., Shen, Y., Liao, X.: Global Stability of Periodic Solution for Bidirectional Associative Memory Neural Networks with Varying-time Delay. Appl. Math. Comput. 182, 509–520 (2006) 5. Lou, X., Cui, B.: On the Global Robust Asymptotic Stability of BAM Neural Networks with Time-varying Delays. Neurocomputing 70, 273–279 (2006) 6. Park, J.H.: A Novel Criterion for Global Asymptotic Stability of BAM Neural Networks with Time Delays. Chaos, Solitons & Fractals 29, 446–453 (2006) 7. Zhang, Q., Wei, X.P., Xu, J.: Delay-dependent Exponential Stability of Cellular Neural Networks with Time-vaying Delays. Chaos, Solitons & Fractals 23, 1363– 1369 (2005) 8. He, Y., Wu, M., She, J.H.: Delay-dependent Exponential Stability of Delayed Neural Networks with Time-varying Delay. IEEE Trans. Circuits Syst. -II, Exp. Briefs 53, 553–557 (2006) 9. Gau, R.S., Lien, C.H., Hsieh, J.G.: Global Exponential Stability for Uncertain Cellular Neural Networks with Multiple Time-vaying Delays via LMI Approach. Chaos, Solitons & Fractals 32, 1258–1267 (2007) 10. Cao, J., Dong, M.: Exponential Stability of Delayed Bi-directional Associative Memory Networks. Appl. Math. Comput. 135, 105–112 (2003) 11. Huang, X., Cao, J., Huang, D.S.: LMI-based Approach for Delay-dependent Exponential Stability Analysis of BAM Neural Networks. Chaos, Solitons & Fractals 24, 885–898 (2005) 12. Ho, D.C., Liang, J., Lam, J.: Global Exponential Stability of Impulsive High-order BAM Neural Networks with Time-varying Delays. Neural Networks 19, 1581–1590 (2006) 13. Park, J.H., Lee, S.M., Kwon, O.M.: On Exponential Stability of Bidirectional Associative Memory Neural Networks with Time-varying Delays. Chaos, Solitons & Fractals (2007), doi:10.1016/j.chaos.2007.05.003 14. Gu, K., Kharitonov, V.L., Chen, J.: Stability of Time-delay Systems. Birkh¨auser, Boston (2003)

Global Exponential Stability and Periodicity of CNNs with Time-Varying Discrete and Distributed Delays Shengle Fang, Minghui Jiang, and Wenfang Fu Institute of Nonlinear Complex Systems, China Three Gorges University, Yichang, Hubei, 443000, China [email protected]

Abstract. In this paper, based on the Lyapunov stability analysis for functional differential equations, the problems of global exponential stability and periodic solutions for a class of cellular neural networks with time-varying delays and distributed delays are investigated. By constructing a new suitable Lyapunov functional, some simple sufficient conditions ensuring existence, uniqueness, global exponential stability of the equilibrium point and periodic oscillation of the networks are obtained, which are easy to check and apply in practice. In addition, an example is also given to illustrate the effectiveness of the theory. Keywords: Cellular neural network, Distributed delay, Exponential stability, Periodic solution.

1

Introduction

The dynamics of delayed cellular neural networks(CNNs) have been deeply investigated in recent years due to its wide application in signal processing, image processing, pattern classification, associative memories, fixed-point computations, and so on. The models of DCNNs with time-varying delays are described by the following system of functional differential equations: xi (t) = −di xi (t) +

n  j=1

aij gj (xj (t)) +

n 

bij gj (xj (t − τij (t))) + Ii (t) ,

(1)

j=1

for i = 1, 2, · · · , n, where di > 0, aij , bij are constants, Ii (t) is external input, τij (t) is nonnegative delays and gj (·) is activation function. For this system, there exist abundant results on the dynamical behavior analysis such as global asymptotical stability[10], global exponential stability[6,8,9,11,12] and periodic solutions[1,3,4], etc. However, up to now, to the best of our knowledge, few authors have considered the global exponential stability, especially periodic solutions of CNNs with both time-varying discrete and distributed delays[5], which is described in the next section of (2). In [2,7], the authors only considered the asymptotically and F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 138–147, 2008. c Springer-Verlag Berlin Heidelberg 2008 

Global Exponential Stability and Periodicity of CNNs

139

exponentially stability of neural networks with discrete and distributed delays. Motivated by the above discussion, this paper is to investigate the dynamical behaviors of neural networks with mixed delays. Comparing with the previous results in the literatures, our results are new and extend those results in some earlier literature.

2

Preliminaries

In this paper, we consider the model of CNNs with time-varying discrete and distributed delays described by the following functional differential equations: n   aij gj (xj (t)) + bij gj (xj (t − hij (t))) xi (t) = −di xi (t) +  +cij

j=1 t

t−τij (t)

 gj (xj (s))ds + Ii (t) ,

(2)

where x(t) = (x1 (t), x2 (t), · · · , xn (t)) ∈ Rn denotes the state of the networks at time t; g(x) = (g1 (x), g2 (x), · · · , gn (x)) : Rn → Rn is a nonlinear vector-valued activation function; Ii (t) ∈ R denotes the external bias on the i-th unit at time t and Ii (t) is bounded defined as |Ii (t)| ≤ Ii ; D = diag{d1 , d2 , · · · , dn }, di > 0, i = 1, 2, · · · , n and aij , bij , cij are constants, i, j = 1, 2, · · · , n; the nonnegative bounded functions hij (t) and τij (t), respectively, represent time-varying discrete and distributed delays of systems with 0 ≤ hij (t) ≤ hij , 0 ≤ hij (t) ≤ h < 1 and 0 ≤ τij (t) ≤ τij .   Let τ = max

max {hij }, max {τij } . The initial conditions for (2) is

1≤i,j≤n

1≤i,j≤n

defined by xi (t) = φi (t),

t ∈ [−τ, 0], (i = 1, 2, · · · , n) ,

where φi (t)(i = 1, 2, · · · , n) are continuous functions. The activation functions g(·) are assumed to satisfy the following assumptions: (H1) gi (·)(i = 1, 2, · · · , n) is bounded on R, i.e., there exists a number ki > 0 such that |gi (·)| ≤ ki , i = 1, 2, · · · , n; (H2) There exists a number μi > 0 such that |gi (ξ) − gi (η)| ≤ μi |ξ − η| for any ξ, η ∈ R, i = 1, 2, · · · , n; that is, the functions gi (·), i = 1, 2, · · · , n, are global Lpschitz continuous. Note that gi (·) are not necessarily differential, monotonic and nondecreasing. As a special case of (2), the DCNNs with constant input I = (I1 , I2 , · · · , In ) ∈ Rn have been studied by a few scholars and mainly they focus on the asymptotic and exponential stability analysis of the CNNs using Lyapunove stability theory and linear matrix inequality techniques. Let x∗ = (x∗1 , x∗2 , · · · , x∗2 ) be the equilibrium point of the system (2). Denote φ − x∗  =

n   i=1

 sup |φi (s) − x∗i | .

−τ ≤s≤0

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Definition 1. The (2) is said to be globally exponentially stable, if it has only one equilibrium point x∗ , and there exist constants ε > 0 and M ≥ 1 such that n 

|xi (t) − x∗i | ≤ M φ − x∗  exp(−εt),

(i = 1, · · · , n)

i=1

for all t ≥ 0. Definition 2. The system (2) is said to be globally exponentially periodic if there exists one ω−periodic solution of the system and all other solutions of the system converge exponentially to it as t → ∞.

3 3.1

Main Results Global Exponential Stability

Consider the special case of the networks (2) as constant input Ii (t) = Ii , i.e., xi (t) = −di xi (t) +  +cij

n  

aij gj (xj (t)) + bij gj (xj (t − hij (t)))

j=1 t

t−τij (t)

 gj (xj (s))ds + Ii ,

i = 1, 2, · · · , n,

(3)

Theorem 1. If the assumptions (H1)–(H2) are satisfied and there exist constants βj > 0, j = 1, 2, · · · , n such that n n n  ε − dj 1  exp(εhij )  +ajj + βi |aij |+ βi |bij | + β |c |τ exp(ετ ) <0, i ij ij ij μj βj 1 − h i=1 i=1 i=1,i=j

(4) in which μj (j = 1, 2, · · · , n) are Lipschitz constants of the hypotheses (H2), and ε is a small positive constant. Then the equilibrium point x∗ is global exponentially stable. Proof. It is not difficult to show that system (3) with a constant input always has an equilibrium. Let x∗ be any of the equilibrium points of (3). Rewrite (3) as n  d[xi (t) − x∗i ] = −di [xi (t) − x∗i ] + aij [gj (xj (t)) − x∗j ] dt j=1

+

n 



bij gj (xj (t − hij (t))) − gj (x∗j )

j=1

+

n  j=1

 cij

t

t−τij (t)



 gj (xj (s)) − gj (x∗j ) ds

(5)

Global Exponential Stability and Periodicity of CNNs

141

for t ≥ 0(i = 1, 2, · · · , n). Then we have d[xi (t) − x∗i ] dt ≤ −di |xi (t) − x∗i | + aii |gi (xi (t)) − gi (x∗i )| n  + |aij | · |gj (xj (t)) − x∗j |

D+ |xi (t) − x∗i | = sgn(xi (t) − x∗i )

j=1,j=i

+

+

n  j=1 n 

|bij | gj (xj (t − hij (t))) − gj (x∗j )  |cij |

j=1

t

t−τij (t)

gj (xj (s)) − gj (x∗ ) ds . j

(6)

βi > 0,

(7)

Constructing the following Lyapunov functional V (t) =

n 

 βi Vi1 (t) + Vi2 (t) + Vi3 (t) ,

i=1

where Vi1 (t) = |xi (t) − x∗i | exp(εt) ,  t n 1  gj (xj (s)) − gj (x∗ ) exp(ε(s + hij ))ds , Vi2 (t) = |b | ij j 1 − h j=1 t−hij (t)  0  t n  |cij | exp(ετij ) exp(εs) gj (xj (s)) − gj (x∗j ) dsdθ , Vi3 (t) = −τij

j=1

t+θ

and ε is a small positive constant. Calculating the Dini derivatives D+ V (t) of V (t), we obtain D+ Vi1 (t) = exp(εt)D+ |xi (t) − x∗i | + ε exp(εt)|xi (t) − x∗i |  ≤ exp(εt) (ε − di )|xi (t) − x∗i | + aii |gi (xi (t)) − gi (x∗i )| +

+

+

n  j=1,j=i n 

|bij | · gj (xj (t − hij (t))) − gj (x∗j )

j=1 n  j=1

D+ Vi2 (t) =

|aij | · gj (xj (t)) − gj (x∗j )

 |cij | ·

t

t−τij (t)

 gj (xj (s)) − gj (x∗j ) ds ,

n  1  |b | exp(ε(t + hij ))|gj (xj (t)) − gj (x∗j )| ij 1 − h j=1

(8)

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 − exp(ε(t − hij (t) + hij ))(1 − hij (t)) gj (xj (t − hij (t))) − gj (x∗j )  exp(εh ) ij gj (xj (t)) − gj (x∗j ) |bij |  1−h j=1  − gj (xj (t − hij (t))) − gj (x∗j ) ,

≤ exp(εt)

n 

(9)

and  |cij | exp(ετij ) τij exp(εt) gj (xj (t)) − gj (x∗j )

n 

D+ Vi3 (t) =

j=1



−

t−τij

 exp(εs) gj (xj (s)) − gj (x∗j ) ds

 |cij | exp(ετij ) τij exp(εt) gj (xj (t)) − gj (x∗j )

n 

≤

t

j=1



− exp(ε(t − τij )) ≤ exp(εt)  −

n 

t

t−τij

 gj (xj (s)) − gj (x∗j ) ds

 |cij | τij exp(ετij ) gj (xj (t)) − gj (x∗j )

j=1

 gj (xj (s)) − gj (x∗ ) ds .

t

j

t−τij (t)

(10)

Applying (8–10) and noting the condition (4) in Theorem 1, we obtain that D+ V (t) =

n 



βi D+ Vi1 (t) + D+ Vi2 (t) + D+ Vi3 (t)

i=1

≤ exp(εt) +

+

n 

n  i=1

|aij | gj (xj (t)) − gj (x∗j )

j=1,j=i n 



 exp(εhij ) ∗ |bij | + |c |τ exp(ετ ) (x (t)) − g (x ) g ij ij ij j j j j 1 − h j=1

≤ exp(εt)

n  i=1

+

 βi (ε − di )|xi (t) − x∗i | + aii |gi (xi (t)) − gi (x∗i )|

n



j=1,j=i

βi

 ε − d  i + aii gi (xi (t)) − gi (x∗i ) μi

|aij | +

n  j=1

|bij |

 · gj (xj (t)) − gj (x∗j )

n  exp(εhij )  + |cij |τij exp(ετij )  1−h j=1

Global Exponential Stability and Periodicity of CNNs

≤ exp(εt)

n 

βj

j=1

+

n 

βi |bij |

i=1

ε − d

j

μj

+ ajj +

143

n 1  βi |aij | βj i=1,i=j

 exp(εhij ) gj (xj (t)) − gj (x∗j ) + β |c |τ exp(ετ ) i ij ij ij  1−h i=1 n 

≤0

(11)

for t ≥ 0. Then, V (t) ≤ V (0) for t ≥ 0. Since V (t) ≥ exp(εt)

n 

βi |xi (t) − x∗i | ≥ exp(εt) min {βi } 1≤i≤n

i=1

= exp(εt) min {βi }x(t) − x∗ 

n 

|xi (t) − x∗i |

i=1

(12)

1≤i≤n

fort ≥ 0, and V (0) =

n 

 βi |xi (0) − x∗i |

i=1

+

+ 

n  j=1 n 

 |bij |

0

−hij (0)

gj (xj (s)) − gj (x∗ ) exp(ε(s + hij ))ds

|cij | exp(ετij )

j=1

j



0

−τij

 θ

0

 exp(εs) gj (xj (s)) − gj (x∗j ) dsdθ

n  

 1 ≤ max {βi } + 1 + μτ exp(ετ ) βi max {|bij |} + max {|cij |} 1≤i≤n 1≤j≤n 1≤j≤n ε i=1

·φ − x∗  ,

(13)

where μ = max {μj }. Hence by (12) and (13), we obtain 1≤j≤n

n 

|xi (t) − x∗i | ≤ M φ − x∗  exp(−εt)

(14)

i=1

for t ≥ 0, where M=

n  

1 max {βi } + 1 + μτ exp(ετ ) βi max {|bij |} 1≤j≤n min1≤i≤n {βi } 1≤i≤n ε i=1  + max {|bij |} .

1



1≤j≤n

Here, by choosing proper parameters, M ≥ 1 can be easily guaranteed. The system (3) cannot has another equilibrium which differs from x∗ . In fact, if there exists another equilibrium x† , it holds that n  i=1

|x†i − x∗i | ≤ M x† − x∗  exp(−εt)

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S. Fang, M. Jiang, and W. Fu

for t ≥ 0. Clearly, it must hold that x† = x∗ . Thus, the network (3) is globally exponentially stable. The proof is completed.  Since ε is a small positive constant, by applying Theorem 1, we can easily obtain the following result. Corollary 1. If assumptions (H1)–(H2) are satisfied and there exist constants βj > 0, j = 1, 2, · · · , n such that n n n  −dj 1  1  < 0 , (15) + ajj + βi |aij | + β |b | + β |c |τ i ij i ij ij μj βj 1 − h i=1 j=1 i=1,i=j

then the equilibrium point x∗ of network (3) is global exponentially stable. Remark 1. The criteria given are dependent on the time delays, which are less conservative than delay-independent criteria when the delay is small. In addition, the result in Theorem 1 is related to the exponential convergence rate ε, which may be useful in the design of neural networks. 1 Remark 2. Note that 0 ≤ h < 1, then 0 ≤ 1−h  ≤ 1. Hence, by the inequation (15), a simple inequation that is less conservative is obtained as follows: n n n   1  −dj + ajj + βi |aij | + βi |bij | + βi |cij |τij < 0 . μj βj i=1 j=1 i=1,i=j

3.2

Periodic Oscillation of the Network

In this subsection, we will consider the periodicity of the network (2). I(t) = (I1 (t), I2 (t), · · · , In (t)) is an input periodic vector function with period ω; that is, there exists a constant ω > 0 such that Ii (t + ω) = Ii (t), i = 1, 2, · · · , n. Let C = C([−τ, 0], Rn ) be the Banach space of all continuous functions that map of uniform convergence. For any φ ∈ C, we [−τ, 0] into Rn with the topology n define φ = sup−τ ≤s≤0 |φ (s)| . i i=1 Theorem 2. If assumption (H1)–(H2) are satisfied and there exist constants βj > 0, j = 1, 2, · · · , n, such that n n n  1  exp(εhij )  ε − dj +ajj + βi |aij |+ βi |bij | + βi |cij |τij exp(ετij ) < 0 ,  μj βj 1−h i=1 i=1 i=1,i=j

(16) in which μj (j = 1, 2, · · · , n) are Lipschitz constants of the hypotheses (H2), and ε is a small positive constant. Then the the network (2) is globally exponentially periodic. Proof. Given any ϕ, ψ ∈ C, let x(t, ϕ) = (x1 (t, ϕ), x2 (t, ϕ), · · · , xn (t, ϕ)) and x(t, ψ) = (x1 (t, ψ), x2 (t, ψ), · · · , xn (t, ψ)) be the solutions of network (2) starting from (0, ϕ) and(0, ψ), respectively.

Global Exponential Stability and Periodicity of CNNs

145

Define xt (φ) = x(t + θ, φ), θ ∈ [−τ, 0], then xt (φ) ∈ C for all t ≥ 0. Therefore, it follows from system (2) that xi (t, ϕ) − xi (t, ψ) = −di (xi (t, ϕ) − xi (t, ψ)) n  

 aij gj (xj (t, ϕ)) − gj (xj (t, ψ)) + j=1

 +bij gj (xj (t − hij (t, ϕ))) − gj (xj (t − hij (t, ψ)))  t

  gj (xj (s, ϕ) − gj (xj (s, ψ)) ds +cij (17) t−τij (t)

for t ≥ 0, i = 1, 2, · · · , n. Motivated by Theorem 1, we consider another Lyapunov functional V (t) =

n 



βi Vi1 (t) + Vi2 (t) + Vi3 (t) ,

(18)

i=1

where Vi1 (t) = |xi (t, ϕ) − xi (t, ψ)| exp(εt) ,  t n 1  gj (xj (s, ϕ)) − gj (xj (t, ψ)) exp(ε(s + hij ))ds , Vi2 (t) = |b | ij  1 − h j=1 t−hij (t)  0  t n  |cij | exp(ετij ) exp(εs) gj (xj (s, ϕ)) − gj (xj (t, ψ)) dsdθ , Vi3 (t) = −τij

j=1

t+θ

and ε is a small positive constant. By a minor modification of the proof of Theorem 1, one can easily get n 

|xi (t, ϕ) − xi (t, ψ)| ≤ M ϕ − ψ exp(−εt)

(19)

i=1

for any t ≥ 0, where M ≥ 1 is a constant. It follows from the formula above that xt (ϕ) − xt (ψ) =

n 

|xi (t + θ, ϕ) − xi (t + θ, ψ)|

i=1

≤ M exp(−ε(t − τ ))ϕ − ψ .

(20)

We can choose a positive integer m such that M exp(−ε(t − τ )) < 13 . Define a Poincar´e mapping P : C → C by Pφ = xω (φ). Then it is deuced from (20) that P m ϕ − P m ψ ≤

1 ϕ − ψ . 3

(21)

This implies that P m is a contraction mapping. Therefore, there exists a unique fixed point φ∗ ∈ C such that P m φ∗ = φ∗ . Hence, P m (Pφ∗ ) = P(P m φ∗ ) = Pφ∗ .

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This shows that Pφ∗ ∈ C is also a fixed point of P m ; that is, Pφ∗ = φ∗ , i.e., xω (φ∗ ) = φ∗ . Let x(t, φ∗ ) be the solution of (2) through (0, φ∗ ). Obviously, x(t + ω, φ∗ ) is also a solution of (2) since x(t + ω, φ∗ ) = x(t, φ∗ ) for t ≥ 0. Note that xt+ω (φ∗ ) = xt (xω (φ∗ )) = xt (φ∗ ) for t ≥ 0, then x(t + ω, φ∗ ) = x(t, φ∗ ) for t ≥ 0. This shows that x(t, φ∗ ) is a periodic solution of (2) with period ω, and it is easy to see that all other solutions of (2) converge exponentially to it as t → ∞.  Corollary 2. If assumption (H1)–(H2) are satisfied and there exist constants βj > 0, j = 1, 2, · · · , n such that n n n  −dj 1  1  < 0 , (22) + ajj + βi |aij | + β |b | + β |c |τ i ij i ij ij μj βj 1 − h i=1 i=1 i=1,i=j

then the network (2) is globally exponentially periodic. Remark 3. It is well known that an equilibrium point can be viewed as a special periodic solution of neural networks with arbitrary period since for any constant T ≥ 0, I(t) = I(t + T ). Based on this idea, the result in Theorem 1 can be viewed as a special case of that in Theorem 2.

4

Illustrative Example

For numerical experiment, one simple example is presented here to illustrate the usefulness of our main results. For the sake of simplicity, we denote D = diag{d1 , d2 , · · · , dn }, A = (aij )n×n , B = (bij )n×n , C = (cij )n×n in system (2). Example 1. Consider the following neural networks of system (2) with ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1.2 −0.8 0.6 −1.4 0.9 0.5 1.8 0.7 −0.8 A = ⎣ 0.5 −1.5 0.7 ⎦ , B = ⎣ −0.6 1.2 0.8 ⎦ , C = ⎣ 0.6 1.4 1 ⎦ , −0.8 −1.2 −1.4 0.5 −0.7 1.1 −0.4 −0.6 1.2 D = diag{6, 5, 7}, and gi (x) = 0.25(|x + 1| − |x − 1|), (i = 1, 2, 3). Then μi = 0.5 (i = 1, 2, 3). Suppose the time-varying delays hij (t) = 0.5 sin2 t, τij (t) = 0.8 cos2 t for any i, j = 1, 2, 3, then hij = 0.5, τij = 0.8 and τ = max1≤i,j≤3 {hij , τij } = 0.8. Taking βi = 1, i = 1, 2, 3. It is easy to verify the conditions in all Theorems and Corollaries obtained are satisfied. So the equilibrium of system (3) and the periodic solution of system (2) are all exponentially stable.

5

Conclusion

Since delays may change dynamics behavior, it is important to analyze the convergence of networks with delay. In this paper, we have studied a model of

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DCNNs which contains variable, distributed delays and is thus more general. It may have applications in processing of motion related phenomena such as moving images and associative memories. We have obtained simple, checkable conditions for global exponential periodicity and stability by using Lyapunov functionals and mathematics technique. The results can be extended to the applications of more complex system design. Acknowledgments. Supported by the National Natural Science Foundation of China (60574025, 60740430664), the Scientific Innovation Team Project of Hubei Provincial Department of Education (T200809) and partially supported by the Scientific Research Projects of Hubei Provincial Department of Education (Q200713001).

References 1. Cao, J.D.: Periodic Solutions and Exponential Stability in Delayed Cellular Neural Networks. Phys. Rev. E. 60, 3244–3248 (1999) 2. Park, J.H.: A Delay-dependent Asymptotic Stability Criterion of Cellular Neural Networks with Time-varying Discrete and Distributed Delays. Chaos, Solitons & Fractals 33, 436–442 (2007) 3. Cao, J.D.: Periodic Oscillation and Exponential Stability of Delayed CNNs. Phys. Lett. A. 270, 157–163 (2000) 4. Cao, J.D.: On Exponential Stability and Periodic Solutions of CNNs with Delays. Phys. Lett. A. 267, 312–318 (2000) 5. Zhou, L.Q., Hu, G.D.: Global Exponential Periodicity and Stability of Cellular Neural Networks with Variable and Distributed Delays. Appl. Math. Comput. 195, 402–411 (2008) 6. Zhang, Y.: Global Exponential Convergence of Recurrent Neural Networks with Variable Delays. Theor. Comput. Sci. 312, 281–293 (2004) 7. Liu, Y.R., Wang, Z.D., Liu, X.H.: Global Exponential Stability of Generalized Recurrent Neural Networks with Discrete and Distributed Delays. Neural Netw. 19, 667–675 (2006) 8. He, Y., Wu, M.: Delay-dependent Exponential Stability of Delayed Neural Networks with Time-varying Delay. IEEE Trans Circuits Syst. 53, 553–557 (2006) 9. Chen, W., Lu, X., Guan, Z., Zheng, W.: Delay-dependent Exponential Stability of Neural Networks with Varible Delay: an LMI approach. IEEE Trans Circuits Syst. 53, 837–842 (2006) 10. Zhang, Q., Wei, X.P., Xu, J.: Stability of Delayed Cellular Neural Networks. Chaos, Solitons & Fractals 31, 514–520 (2007) 11. Zeng, Z.G., Wang, J.: Improved Conditions for Global Exponential Stability of Recurrent Neural Networks with Time-varying Delays. IEEE Trans. Neural Networks 17, 623–634 (2006) 12. Zheng, Y.X., Chen, T.P.: Global Exponential Stability of Delayed Periodic Dynamical Systems. Phys. Lett. A. 322, 344–355 (2004)

Estimation of Value-at-Risk for Exchange Risk Via Kernel Based Nonlinear Ensembled Multi Scale Model Kaijian He1,2 , Chi Xie1 , and Kinkeung Lai1,2 1

College of Business Administration, Hunan University, Changsha, Hunan, 410082, China [email protected], [email protected] 2 Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong [email protected]

Abstract. Risk level in the exchange rate market is dynamically evolving with complicated structures. To further refine the analysis process and achieve more accurate measurement, this paper proposes a novel kernel based nonlinear ensembled multi scale Value at Risk methodology for evaluating the risk level in the exchange rate market. In the proposed algorithm, wavelet analysis is introduced to analyze the multi scale heterogeneous risk structures across different time scales. The Principle Component Analysis is used to extract principle components from the redundant forecast matrixes. Then the support vector regression technique is integrated into the modeling process to nonlinearly ensemble forecast matrixes and produce more stable and accurate results. Taking Euro market as a typical test case, empirical studies employing the proposed algorithm shows the superior performance than benchmark ARMA-GARCH and realized volatility based approaches. Keywords: Wavelet analysis, Value at risk, Support vector regression, Nonlinear ensemble, Principle component analysis.

1

Introduction

The breakdown of the Bretton Woods System in 1973 opened the door for the exchange rate market characterized by high level of fluctuations and risk exposures. Empirical studies have confirmed that the exchange rate risk has significant impacts on the national economy and the individual firms’ wealth. E.g an unexpected sudden drop of exchange rates could wipe out the investment profit for multinational companies. Therefore, the appropriate measurement of exchange risk is crucial from both academic and industrial perspective [1]. Compared to the large volume of work done in the forecasting of exchange rate level and volatility, research results dealing with the accurate and reliable measurement of exchange risk is still scattered in the literature. Among different risk measures developed over years, Value at Risk receives most attentions due to F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 148–157, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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its simplicity and widespread support. Value at Risk is a summarizing tail statics measuring the maximum losses at the given confidence level and investment horizon [2]. Its estimation, as the central issue, has been researched mainly from parametric and non parametric approaches over years. Each has their unique strengths and weaknesses. The parametric approach fits analytical models to the data. It is computationally and intuitively appealing. But its performance is restricted by the assumptions made. The non parametric approaches are data driven and adaptive. But its black box nature and lacks of explanatory power in practice restrict its development in the field. Thus the semi parametric approach emerges in recent year, aiming to be data driven while computationally tractable. This paper contributes to the relevant literature by proposing a novel semiparametric nonlinear ensembled multi scale methodology for estimating value at risk in measuring the exchange risk. The proposed approach is unique in three aspects. Firstly it models the heterogeneous risk structures across scales based on wavelet analysis. Wavelet method, as a promising semi parametric analytic tool, receive increasing attentions in the finance field in recent years. It has been applied successfully to recover micro structure behaviors in the typical financial markets including the exchange rate market, the stock market, etc [3,4]. The application of wavelet analysis in the financial forecasting applications also shows promising results [4,5]. However, its application in the risk management field is premature and largely scattered. Several exploratory works have been done in applying the concept of wavelet analysis to calculate the historical CAPM and VaR [6,7]. But little work has been done in the literature when it comes to the potential performance improvement brought forth by this tool. Secondly, it utilizes Principle Component Analysis (PCA) to determine uncorrelated principle components of the forecast set, which efficiently transforms the original homogeneous ensemble problem into the heterogeneous ensemble problem, and improves the generalizability of ensemble results. Thirdly the Support Vector Regression (SVR) based ensemble algorithm is introduced to aggregate ensemble members, reduce the estimation biases and improve generalizability. Empirical studies have largely supported SVR as a better machine learning tool with higher generalizability of its forecasts than neural network in the exchange rate forecasting field [8,9,10]. But SVR also suffers from the lack of explanatory power due to its ’black box’ nature. Therefore this paper proposes SVR as the optimization tool for the ensemble forecasting methodology, thus improves the forecasting accuracy with sound economic rationale. Compared to previous works using neural network as the foundation for the ensemble learning environment, the solution of the proposed algorithm is more stable and global optimum, with better explanatory power. The rest of the paper is set out as follows. In section 2, we briefly review the relevant theories including wavelet analysis, principle component analysis and support vector regression based ensemble algorithm. In section 3, we propose the integrated multi scale ensemble Value at Risk methodology. In section 4, we

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carry out empirical studies using the proposed algorithm and conduct detailed analysis of results. The final section summarizes and concludes.

2 2.1

Relevant Theories Wavelet Analysis

Wavelets are continuous functions that satisfy the admissibility and unit energy conditions as in (1).  ∞  ∞ |ϕ(f )| df < ∞, Cψ = |ψ(t)|2 dt = 1 . (1) f 0 −∞ Where ϕ is the Fourier transform of the frequency f . ψ is the wavelet transform. There are different wavelet families with different characteristics such as vanishing moments, compact support, regularity, symmetry and time-frequency window [11]. Typical wavelets include Haar, Daubechies, Coiflets, etc. Utilizing wavelets, the continuous wavelet transform is defined as in (2).  ∞ t−u 1 )dt . (2) W (u, s) = x(t) √ ψ( s s −∞ Where u is the index of the dilation, s is the index of the translation. Together these two parameters shift the original wavelet function along the time and frequency scale. The wavelet synthesis, the inverse operation, is defined as in (3)  ∞ ∞ ds 1 W (u, s)ψu,s (t)du 2 . (3) x(t) = Cψ 0 s −∞ In practice, wavelet bases and coefficients are discretized to reduce computational complexity. 2.2

Kernel Based Nonlinear Ensemble Model Incorporating PCA

Given a set of forecasts, ensemble algorithms aim at finding the functional form φ that effectively transform inputs fi (t) into outputs f (t). However, this task is made more complicated by two issues, i.e. the selection of effective and efficient algorithm to search for the appropriate functional form, and the effective reduction of high dimensionality of data and the removal of additive noises. Firstly ε-insensitive SVR, a promising kernel based machine learning technique, is introduced in this paper to tackle the first issue. ε-insensitive SVR aims to search for the function φ that has the maximum level of flatness and the minimum level of deviations from the data. Compared to the standard parametric regression approach, ε-insensitive SVR is data driven and adaptive, that is not restricted by the data distribution assumptions. To achieve this goal, ε-insensitive SVR firstly transforms the nonlinear data into linear data by mapping them into higher dimension feature spaces using kernel functions satisfying Mercer condition as in (4).

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f (x) = ωφ(x) + b, φ : Rn → F, ω ∈ F .

(4) ∗

Then we need to solve the general form of SVR with slack variables ξi , ξ measuring deviations from data points as in (5)  1 ||ω||2 + C (ξi + ξ ∗ ) 2 i=1 ⎫ ⎧ y − ωφ(x ) − b ≤ ε+ξ⎪ ⎪ i i ⎪ ⎪ ⎬ ⎨ ωφ(xi ) + b − yi ≤ ε + ξ . s.t. ∗ ξi , ξi ≥ 0 ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ C ≥0 n

M inimizeRSV R (ω, ξ ∗ ) =

(5)

Where ε is the accuracy constraint. C is the cost coefficient that pre-specifies the tradeoff between the smoothness of the function and the abrupt deviation larger than the accuracy constraint. In practice the dual problem of the primal problem is often constructed to reduce the size of the problem and variable dimensions [9]. Secondly Principle Component Analysis (PCA) is introduced to tackle the second issue. Ensemble models are often utilized to deal with high dimensional data with additive noises, which may introduce the overfitting issue during training and adversely affect its generalizability. One approach to tackle this issue is to extract main features from the ensemble members before they are ensembled. And PCA is one such approach. PCA aims to produce the linear approximation to the original data with transformed principle components characterized by their variances [10]. Given centered data xt ∈ N , t = 1, ..., M , PCA transforms the original data xt into the principle component st by orthogonal transformation as in st = U T xt , Where U is the m×m orthogonal matrix. U is found by solving the eigenvalue problem as in (6). l 1

λi ui = Cui , i = 1, ..., m .

(6)

Where C = l t=1 xt xTt , C is the sample covariance matrix. λi is the ith eigenvalues of C. ui is the ith eigenvector of C.

3

A Novel Wavelet-PCA-SVR Methodology

As empirical studies suggest, the micro structure of the exchange rate market is very complex, consisting of investors with different investment strategies targeting at different investment time horizons [1]. Participants in the typical exchange rate market can be categorized uniquely into the fundamental investors with relatively long term and stable investment strategies and the investors sensitive to the time horizon chosen. The heterogeneous nature of the market participants has fundamental impact on not only the return distribution, but the volatility

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evolution as well. But previous analysis mostly concentrates on the global macro level, due to the lack of effective analysis tools in the time scale domain. Therefore, the integrated Wavelet-PCA-SVR methodology is proposed in this paper to explore the micro market structure and further improve the performance. The proposed Wavelet-PCA-SVR algorithm involves several steps. Wavelet analysis is used to project the original data into different time scales up to the chosen level using the chosen wavelet families as in (7). f (t) = fAJ (t) +

J 

fDj (t) .

(7)

j=1

Where f (t) refers to the original time series. fAJ (t) refers to the decomposed time series using scaling function at scale J. fDj (t) refers to the decomposed time series using wavelet function at scales j, up to scale J. Taking into account the heterogeneous structure of the return distribution, the conditional means at different time scales are modeled by ARMA processes with different parameters as in (8). r t =

S 

(δs +

s=1

m 

φi,s rt−i,s +

i=1

n 

θj,s εt−j,s + εt,s ) .

(8)

j=1

Where r t is the conditional mean at time t. s refers to the wavelet scales up to the scale S. rt−i is the lag m returns with parameter φi , and εt−j is the lag n residuals in the previous period with parameter θj . δs is the constant. Taking into account the heterogeneous structure of the volatility distribution, the conditional volatilities at different time scales are modeled by GARCH processes with different parameters as in (9) σ t 2 =

S  s=1

(ωs +

p  i=1

2 αi,s σt−i,s +

q 

βj,s ε2t−j,s ) .

(9)

j=1

Where σ t is the conditional volatility at time t, s refers to the wavelet scales up 2 to the scale S. σt−i is the lag p variances with parameter αi , and εt−j is the lag q residuals in the previous period with parameter βj . ωs is the constant. To obtain uncorrelated forecast matrix and reduce data dimension, the original forecasts matrix containing forecasts for both conditional mean and conditional volatility are transformed into the lower dimension forecast matrix using the PCA as in (10). T F t , σ tG = T T σ tF . μ G t =T μ

(10)

Where G refers to the projected principle components using PCA. F refers to the set consisting of wavelet families tried. Then two different SVR networks are trained to nonlinearly integrate conditional mean and volatility estimates based on different wavelet families into the aggregate ones as in (11).

Estimation of Value-at-Risk for Exchange Risk Via Kernel

μ Ensemble = φi∈G ( μi ), σ tEnsemble = φi∈G ( σi ) . t

153

(11)

Where φ refers to the SVR optimized nonlinear mapping function. σ tEnsemble refers to the estimate of conditional standard deviation at time t. μ Ensemble t refers to the estimate of the sample mean at time t. Then VaR is estimated parametrically as in (12). V aR = −F (a) σtEnsemble − μ Ensemble . t

(12)

Where F (a) refers to the corresponding quantile (i.e. 95%, 97.5% or 99%) of the empirical distribution, which refers to the normal distribution in this paper. The normal distribution is adopted as it provides the reasonable approximation in this case. It is well known that when data are significantly large, the empirical distribution would converge to the normal distribution.

4 4.1

Empirical Studies Data and Experiment Design

Empirical data analysis is conducted for the exchange rate Euro against US Dollar (EUR/USD), selected for the purpose of illustration. The data are prepared and obtained from Global Financial Data. The data set covers the period from 1st July, 2002 to 18th January, 2008, which consists of 1448 daily observations. The daily observations are further I transformed into weekly observations by arithmetic averaging as in pw = i=1 pd,i , where I is set to 5 in this paper, corresponding to the 5 trading day in each week, pd,i refers to the daily observations at time index i(i = 1, 2...5). The transformation produces 290 weekly observations. I Meanwhile, the realized volatility for each week is calculated as 2 = i=1 ri2 , where I is set to 5 in this paper. σRV The entire weekly data set is divided into 3 parts. The first 60% observations, i.e. 173 observations, forms the training set. The next 16% observations forms the training set for the SVR based ensemble algorithm. Then the rest 24% observations, i.e. 70 data points, is reserved as the out-of-sample test set for model performance evaluations and comparisons. To remove trend factors, the original pt . weekly price series are log differenced at the first order as in ln pt−1 Model performance is evaluated using formal statistical procedure such as backtesting. Kupiec backtesting procedure is adopted in this paper among numerous backtesting procedures available since it is the most widely accepted one. Kupiec backtesting procedure conducts statistical tests on the distribution of the frequency of exceedances of results, which converges to the binomial distribution at infinity [2]. The likelihood ratio test statistics is constructed following the standard approach in the literature as in (13). LRuc = −2log[(1 − ρ)T −N ρN ] + 2log([1 − (

T T −N T N )] ( ) ]) . N N

(13)

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K. He, C. Xie, and K. Lai Table 1. Descriptive Statistics and Statistical Tests

Data Mean(×10−6 ) Standard Deviation Skewness Kurtosis JB Test BDS test raw

-689.3

0.0157

8.2277

109.6684

0.001

0.0057

S(1)

-1166.97

0.0013

0.2877

1.5851

0.0044

0

D(1)

-1.2678

0.0065

0.0542

3.1873

0.5000

-0.0008

D(2)

-4.2078

0.0057

0.0331

3.6535

0.1579

0

D(3)

5.4547

0.0046

-0.1009

4.4631

0.0052

0

D(4)

2.9937

0.0029

-0.1997

2.7419

0.3890

0

D(5)

7.0524

0.0032

-0.0470

2.3281

0.1405

0

D(6)

-9.0669

0.0006

-0.6471

2.5531

0.0076

0

Where LRUC denotes the test statistics that has an asymptotic χ2 (1) distribution. N is the number of VaR exceedances. ρ is the probability of a VaR exceedances occurrence. T is the number of trials carried out. 4.2

Empirical Results

Simple descriptive statistics of the original data in table 1 clearly indicate the nonlinear and non normal nature of the Euro markets, as confirmed by numerous empirical studies to date. The high level of skewness and kurtosis in return distributions suggest that the market deviates significantly from the normal distribution and exhibits fat tail behavior. This is also confirmed by the rejection of the null hypothesis of Jarque-Bera(JB) test of normality and the further rejection of the null hypothesis of BDS test of independence. However, the same sets of descriptive statistics and statistical tests on wavelet decomposed data reveal that decomposed data series have much smaller scale of fat tailness. The normal distribution applies to the return distribution at different time horizons. Based on this observation, wavelet analysis is treated as providing an alternative tool for further performance improvement. Firstly experiments using traditional ARMA-GARCH model are conducted to serve as the benchmark for further performance evaluations. The model order determined under AIC and BIC minimization principle is ARMA(1,1) with GARCH(1,1), among repeated trails with model orders tested up to 5. The rolling window method is used in the one step ahead forecast process to include the newly revealed information. The windows length is set to 172 to cover the relevant information set. Experiment results in table 2 show the generally inferior performance of VaR estimated using ARMA-GARCH model. The reliability of VaR estimated is relatively low and very conservative, which results in moderate accuracy level and could lead to the allocation of excessive idle capital. Given the limited room ARMA-GARCH approach leaves for further performance improvement, wavelet based methodology offers an important alternative to improve further the accuracy and reliability levels.

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Table 2. Performance of VaR estimated using ARMA-GARCH model Model

Confidence Exceedances Kupiec Test MSE Levels

ARMA-GARCH VaR

RVVaR

(p−value )

(×10−4 )

99.0%

0

0.1268

8.9927

97.5%

1

0.1920

7.7036

95.0%

4

0.4177

6.7644

99.0%

3

0.1521

10.7512

97.5%

4

0.5358

9.0214

95.0%

5

0.7273

7.7499

Interestingly, experiment results in table 2 also show that VaR estimated using the historical realized volatility, referred to as RVVaR in the table, could lead to significant performance improvement in the reliability of VaR estimated, as reflected in the higher p value. But this is achieved at the cost of lower accuracy of VaR estimated, as indicated by the larger MSE value. When conducting experiments using wavelet based models, two additional parameters are set, i.e. the wavelet families and the decomposition level. The wavelet families tried include Haar, Daubechies 2 and Coiflets 2. The decomposition level is tried from 1 to 6, and scale 1 is chosen as it gives the best performance. The model orders determined under AIC and BIC minimization principle vary for different wavelet families at different scales. Experiment results in table 3 show that wavelet based approaches track risk more closely and result in more aggressive VaR estimates. The accuracy level significantly increased despite the deteriorating reliability level, i.e. the analysis of the subtle micro market structure results using the wavelet transforms results in more accurate and aggressive, but less stable estimates. Table 3. Performance of VaR estimated using WDVaR Model with different wavelet families Wavelet Confidence Exceedances MSE Families

Levels 99.0%

Haar

Db(2)

Coiflet(2)

(×10−4 ) 39

123.9242

97.5%

39

61.8744

95.0%

40

26.5888

99.0%

25

6.2197

97.5%

32

5.9495

95.0%

34

5.7456

99.0%

31

5.0618

97.5%

34

4.9227

95.0%

37

4.8157

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K. He, C. Xie, and K. Lai Table 4. Experiment results using Wavelet-PCA-SVR model Confidence Levels Exceedances Kupiec Test(p−value ) MSE(×10−4 ) 99.0%

2

0.2026

4.0889

97.5%

2

0.8515

2.9913

95.0%

5

0.4383

6.3313

Since both ARMA-GARCH and wavelet based models extract certain information set and have their own strengths and weakness, the proposed WaveletPCA-SVR algorithm is used to integrate the individual information set and produce more accurate forecasts. ε-insensitive SVR is used to conduct the ensemble process with the kernel Radial Basis Function (RBF). The parameters are chosen by grid searching method targeting at the minimal in sample MSE. The parameters determined for ensemble return estimates are as follows: C in the cost function is 1, Gamma in the kernel function is 0.5, and ε in the loss function is 0.5. The parameters determined for ensemble volatility estimates are as follows: C is 0.5, Gamma is 0.25 and ε is 0.01562525. Besides, The principle components selected by PCA is 3, which are found to give the best performance. Experiment results in table 4 suggest that the proposed Wavelet-PCA-SVR methodology improves the reliability and accuracy of VaR estimated significantly utilizing additional information concerning the mutli scale micro market structure, as indicated by the higher p value and lower MSE across all confidence levels. The performance improvement is attributed to the more refined modeling attempts on the risk evolutions at different time scales, and the SVR based integration of the partial information set captured by individual models into the global optimal one during the modeling process. Thus the proposed model is more stable and robust, and has better generalizability for its forecasts than traditional approaches.

5

Conclusions

In this study, a novel nonlinear ensembled multi scale value at risk methodology is proposed for complicated risk measurement problem. Three emerging techniques including wavelet analysis, PCA and SVR are incorporated in the proposed Wavelet-PCA-SVR algorithm to improve further the risk measurement accuracy and reliability. Experiment results in the Euro exchange rate market reveal that the proposed approach significantly improves the risk measuring reliability and accuracy. The performance improvement stems from the integrated modeling attempts on different features of risk evolutions including the multi scale heterogeneity and auto correlation, etc, as well as the significant biases reduction during parameter estimation using PCA and the nonlinear ensemble algorithm. Experiment results suggest that the proposed method offers a promising alternative and brings additional flexibility for estimating value at risk measures in the exchange rate market. Besides, the proposed algorithm can

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also be easily generalized to market risk measurement problems in diverse areas of interests, aside from the foreign exchange rate market.

Acknowledgements The work described in this paper was supported by a grant from the National Social Science Foundation of China (SSFC No.07AJL005) and a Research Grant of City University of Hong Kong (No. 9610058).

References 1. Sarno, L., Valente, G.: Empirical Exchange Rate Models and Currency Risk: Some Evidence from Density Forecasts. J. Int. Money Finan. 24, 363–385 (2005) 2. Dowd, K.: Measuring Market Risk. John Wiley & Son, West Sussex (2002) 3. Mitra, S., Mitra, A.: Modeling Exchange Rates Using Wavelet Decomposed Genetic Neural Networks. Stat. Methodol. 3, 103–124 (2006) 4. Li, S.T., Kuo, S.C.: Knowledge Discovery in Financial Investment for Forecasting and Trading Strategy Through Wavelet-based Som Networks. Expert Syst. Appl. 34, 935–951 (2008) 5. Fernandez, V.: Wavelet and SVM-based Forecasts: An Analysis of the u.s. metal and Materials Manufacturing Industry. Resour. Policy 32, 80–89 (2007) 6. Fernandez, V.: The Capm and Value at Risk at Different Time-scales. Int. Rev. Financ. Anal. 15, 203–219 (2006) 7. Fernandez, V., Lucey, B.M.: Portfolio Management Under Sudden Changes in Volatility and Heterogeneous Investment Horizons. Physica A 375, 612–624 (2007) 8. Gavrishchaka, V.V., Ganguli, S.B.: Volatility Forecasting from Multiscale and High-dimensional Market Data. Neurocomputing 55, 285–305 (2003) 9. Ince, H., Trafalis, T.B.: A Hybrid Model for Exchange Rate Prediction. Decis. Support Syst. 42, 1054–1062 (2006) 10. Yu, L., Wang, S.Y., Lai, K.K.: A Novel Nonlinear Ensemble Forecasting Model Incorporating Glar and ann for Foreign Exchange Rates. Comput. Oper. Res. 32, 2523–2541 (2005) 11. Percival, D.B., Walden, A.T.: Wavelet methods for time series analysis. Cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge (2000)

Delay-Dependent Global Asymptotic Stability in Neutral-Type Delayed Neural Networks with Reaction-Diffusion Terms Jianlong Qiu, Yinlai Jin, and Qingyu Zheng Department of Mathematics, Linyi Normal University, Linyi 276005, China [email protected], [email protected] Abstract. In this paper, the global uniform asymptotic stability is studied for a class of delayed neutral-type neural networks with reactiondiffusion terms. By constructing appropriate Lyapunov-Krasovskii functional and using the linear matrix inequality (LMI) approach, several sufficient conditions are obtained for ensuring the system to be globally uniformly asymptotically stable. A numerical example is given in the end of this paper to demonstrate the effectiveness and applicability of the proposed criteria.

1

Introduction

It is well known that neural networks play a great important role in various areas such as image processing, signal processing, optimization and pattern recognition. Recently, dynamical neural networks without reaction-diffusion terms have attracted increasing interest in both theoretical studies and engineering applications. And many important results on the global asymptotical or exponential stability have been reported in the literature, see, e.g., [2-6, 15, 17, 28-30] and the references therein. However, strictly speaking, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields. So we must consider that the state of the neuron vary in space as well as in time. In [9-13, 16], the authors considered the stability of neural networks with reaction-diffusion terms, which are expressed by partial differential equations. Especially, the authors in [10, 11] gave several new sufficient conditions on the global exponential stability of reaction-diffusion recurrent neural networks with time-varying delays. On the other hand, due to the complicated dynamic properties of the neural cells in the real world, the existing neural network models in many cases cannot characterize the properties of a neural reaction process precisely. It is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamic behavior for such complex neural reactions. These systems are called neutral-type neural networks. In fact, the stability analysis of neutral systems with delay has received considerable attention over the decades, see, e.g., [14, 20-25] and the references therein. Here, we especially point out that the authors of [14] investigated the delay-dependent exponential stability of a class of delayed neutral-type neural networks without reaction-diffusion terms. F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 158–167, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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Motivated by the above discussions, the aim of the present paper is to consider a class of neutral-type neural networks with time delays and reaction-diffusion terms described by a nonlinear delay differential equation of neutral type. By constructing appropriate Lyapunov-Krasovskii functional and using the linear matrix inequality (LMI) optimization approach, we obtained some sufficient conditions ensuring the global uniform asymptotical stability of the equilibrium point of such a kind of delayed neural networks. The obtained conditions are dependent on the size of the time delay and the measure of the space, which are usually less conservative than delay-independent and space-independent ones. It is worth pointing out that these conditions do not require that the activate functions are bounded and differentiable. Also, it should be noted that the proposed LMI condition can be checked numerically very efficiently by resorting to recently developed interiorpoint methods. To the best of our knowledge, few authors have considered global uniform asymptotical stability for this class of neural networks with time delays and reaction-diffusion terms. The work will have significance impact on the design and applications of globally stable neural networks with time delays and diffusion terms, and are of great interest in many applications. The remainder of the paper is organized as follows. In Section 2 the model formulation and some preliminaries are given. The main results are stated in Section 3. An illustrative example is given to demonstrate the effectiveness of the obtained results in Section 4. Finally, concluding remarks are made in Section 5. Euclidean Notation. Throughout this paper, Rn denotes the n-dimensional √ space. Vector X ∈ Rn , its norm is defined as X = X T X. L2 (Ω) denotes the space of real Lebesgue measurable functions on Ω. It is a Banach space  1 for the norm X2 = [ Ω X2dx] 2 . The notation A > B (respectively, A ≥ B) means that the matrix A−B is symmetric positive definite (respectively, positive semi-definite ), where A and B are matrices of the same dimensions. AT and A−1 denote the transpose of and inverse of the matrix A. λmax (A) and λmin (A) represent the maximum and minimum eigenvalues of matrix A respectively.  For  A ∗ an arbitrarily real matrix B and two real symmetric matrices A and C, BC denotes a real symmetric matrix, where ∗ represents the elements above the main diagonal of a symmetric matrix. Here an arbitrarily real matrix H  (hij )n×n , + H +  (h+ ij )n×n , where hij = max(hij , 0), i, j = 1, 2, · · · , n.

2

Preliminaries

In this paper, we consider a class of neutral-type delayed neural networks with reaction-diffusion terms described by a nonlinear delay differential equation of neutral type.   m n  ∂ui (t, x) ∂ui (t, x)  ∂ = bij fj (uj (t, x)) Dik − ai ui (t, x) + ∂t ∂xk ∂xk j=1 k=1 n 

+

j=1

cij  gj (uj (t − τ, x)) +

n  j=1

hij

∂uj (t − τ, x) + Ii , ∂t

(1)

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where i = 1, 2, · · · , n, x = (x1 , x2 , · · · , xm )T ∈ Ω ⊂ Rm , Ω is a bounded set with smooth boundary ∂Ω and mesΩ > 0 in space Rm ; u = (u1 , u2 , · · · , un )T ∈ Rn , ui (t, x) denotes the state of the ith neuron at time t and in space x; fj , gj are activation functions of the jth neuron; the scalar ai > 0 is the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t and in space x; bij , cij , hij , i, j = 1, 2, · · · , n are known constants denoting the strength of the ith neurons on the jth neurons; smooth functions Dik = Dik (t, x, u) ≥ 0 are transmission diffusion operator of the ith neurons; τ is non-negative constant, which correspond to finite speed of axonal signal transmission delay; Ii denote the ith component of an external input source introduced from outside the network to the cell i at time t and in space x. The initial and boundary condition of the system (1) is given by  T ∂ui ∂ui ∂ui ∂ui = ,··· , , = 0, ∀t ≥ 0, x ∈ ∂Ω, i = 1, 2, · · · , n, (2) ∂n ∂x1 ∂xk 2 ∂xm ui (s, x) = φi (s, x),

∂ ∂φi (s, x) ui (s, x) = , −τ ≤ s ≤ 0, i = 1, 2, · · · , n, (3) ∂t ∂t

in which φi (s, x) are bounded and first order continuous differentiable on [−τ, 0]. Throughout the paper we assume that the activation function satisfies the following assumption. (H) The neurons activation functions fj ,  gj are continuous, and there exist constants lj > 0, kj > 0 such that 0≤

fj (ξ1 ) − fj (ξ2 ) ≤ lj ; ξ1 − ξ2

0≤

gj (ξ2 ) gj (ξ1 ) −   ≤ kj , ξ1 − ξ2

for any ξ1 , ξ2 ∈ R, ξ1 = ξ2 , j = 1, 2, · · · , n. Definition. A vector U ∗ = (u∗1 , u∗2 , · · · , u∗n )T ∈ Rn is said to be an equilibrium point of system (1) if it satisfies −ai u∗i +

n  j=1

bij fj (u∗j ) +

n 

cij gj (u∗j ) + Ii = 0, i = 1, 2, · · · , n.

j=1

Before we develop the delay-dependent criteria, we note that the following two facts and two lemmas. Fact 1.[3] Suppose W, U are any matrices,  is a positive number and matrix H = H T > 0, then the following inequality holds W T U + U T W ≤ W T HW + −1 U T H −1 U. Fact 2. (Schur complement) The following linear matrix inequality (LMI)   Q(x) S(x) > 0, S T (x) R(x)

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where Q(x) = QT (x), R(x) = RT (x), and S(x) depend affinely on x, is equivalent to (1) R(x) > 0, Q(x) − S(x)R−1 (x)S T (x) > 0, (2) Q(x) > 0, R(x) − S(x)Q−1 (x)S T (x) > 0. Lemma 1. For any constant matrix Σ ∈ Rn×n , Σ = Σ T > 0, Ω ⊂ Rn , mesΩ > 0, if ω : Ω → Rn is vector function such that the integration is well defined, then T

 ω(s)ds





Σ

Ω

ω(s)ds

≤ |Ω|

Ω

ω T (s)Σω(s)ds.

(4)

Ω

Remark 1. Obviously, this inequality is a generalization of Lemma 1[5] in ndimension space. Lemma 2.[8] The equilibrium of the system is globally uniformly asymptotically stable if there exists a C 1 function V : R+ × Rn → R such that (i) V is a positive definite, decrescent and radially unbounded, and (ii) −V˙ is positive definite. Now let yi (t, x) = ui (t, x) − u∗i , i = 1, 2, · · · , n. It is easy to see that system (1) can be transformed into ∂yi (t, x)  ∂ = ∂t ∂xk m

k=1 n 

+

  n  ∂yi (t, x) bij fj (yj (t, x)) Dik − ai yi (t, x) + ∂xk j=1

cij gj (yj (t − τ, x)) +

j=1

n  j=1

hij

∂yj (t − τ, x) , x ∈ Ω, ∂t

(5)

where i = 1, 2, · · · , n, fj (yj (t, x)) = fj (yj (t, x) + u∗j ) − fj (u∗j ), j = 1, 2, · · · , n, gj (yj (t − τ, x)) =  gj (yj (t − τ, x) + u∗j ) − gj (u∗j ), j = 1, 2, · · · , n. Then it is easy to see that fj (0) = 0, gj (0) = 0 and fj (·), gj (·) satisfy assumption (H).

3

Main Results

Theorem 1. Under the assumption (H), the equilibrium point U ∗ of system (1) is globally uniformly asymptotically stable if there exist positive definite matrix Q1 and diagonal matrices P > 0, Q2 > 0 and Q3 > 0 such that ⎛

Γ1 ⎜ Γ2 ⎜ ⎝ Γ3 0

∗ Σ1 Γ4 0

⎞ ∗ ∗ ∗ ∗ ⎟ ⎟ < 0, Σ2 ∗ ⎠ 0 −τ Q3

(6)

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where  + L,  − 2AQB  + L + |Ω|LB + T QB Γ1 = −2P A + KQ1 K + 2P BL + AQA  + , Γ3 = P H − AQH  +,  + + |Ω|LB + T QC  + + |Ω|LB + T QH Γ2 = P C − AQC T  + T  + T  + , Σ1 = −Q1 + |Ω|C + QC , Σ2 = −Q2 + |Ω|H + QH , Γ4 = |Ω|C + QH  Q = Q2 + τ Q3 = diag( q1 , q2 ,· · ·, qn ), L = diag(l1 ,· · ·, ln ), K = diag(k1 , · · · , kn ),

P = diag(p1 , · · · , pn ), Q2 = diag(q21 , · · · , q2n ), Q3 = diag(q31 , · · · , q3n ), and li > 0, ki > 0, i = 1, 2, · · · , n are given in assumption (H). Proof. Consider a Lyapunov-Krasovskii functional as V (t) = V1 (t) + V2 (t) + V3 (t) + V4 (t), where

(7)

Y T (t, x)P Y (t, x)dx, t GT (Y (α, x))Q1 G(Y (α, x))dαdx, V2 (t) = V1 (t) =

(8)

Ω

Ω



t−τ t

V3 (t) =

( Ω

V4 (t) =

∂ ∂ Y (α, x))T Q2 Y (α, x)dαdx, ∂α ∂α

t−τ 0 t

( Ω

−τ

t+β

∂ ∂ Y (α, x))T Q3 Y (α, x)dαdβdx, ∂α ∂α

(9) (10) (11)

where Y (t, x) = (y1 (t, x), · · · , yn (t, x))T , F (Y (t, x)) = (f1 (y1 (t, x)), · · · , fn (yn (t, x)))T , G(Y (t, x)) = (g1 (y1 (t, x)), · · · , gn (yn (t, x)))T . Computing the derivative of Vi (t), i = 1, 2, 3, 4 along the trajectory of system (5), we have ∂ ∂ V˙1 (t) = [ Y T (t, x)P Y (t, x) + Y T (t, x)P Y (t, x)]dx ∂t ∂t Ω     n m n  ∂yi (t, x) ∂ = 2pi yi (t, x) 2pi ai yi2 (t, x)dx Dik dx − ∂x ∂x k k Ω Ω i=1 i=1 k=1  n  n + 2pi bij yi (t, x)fj (yj (t, x))dx Ω i=1 j=1

+

 n  n

2pi cij yi (t, x)gj (yj (t − τ, x))dx

Ω i=1 j=1

+

 n  n Ω i=1 j=1

2pi hij yi (t, x)

∂yj (t − τ, x) dx, ∂t

(12)

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  ˙ V2 (t)= GT (Y (t, x))Q1 G(Y (t, x))− GT (Y (t− τ, x))Q1 G(Y (t− τ, x)) dx,(13)  Ω ∂ ∂ ∂ ∂ T T ˙ ( Y (t, x)) Q2 Y (t, x)−( Y (t−τ, x)) Q2 Y (t−τ, x) dx,(14) V3 (t)= ∂t ∂t ∂t Ω ∂t  t  ∂ ∂ ∂ ∂ T T ˙ V4 (t)= Y (α, x) dx,(15) τ ( Y (t, x)) Q3 Y (t, x)− ( Y (α, x)) Q3 ∂t ∂t ∂α t−τ ∂α Ω From boundary condition (2) and Green formula, we have  m

yi (t, x)

Ω k=1

∂ ∂xk

   2  m ∂yi (t, x) ∂yi (t, x) Dik dx,(16) Dik dx = − ∂xk ∂xk Ω k=1

∂ , ∂ , · · · , ∂x∂m ) is the gradient operator, and in which i = 1, 2, · · · , n, ∇ = ( ∂x 1 ∂x2

  m  ∂yi (t, x) ∂yi (t, x) ∂yi (t, x) ∂yi (t, x) Dik = Di1 , Di2 , · · · , Dim . ∂xk ∂x1 ∂x2 ∂xm k=1 Using (16) and assumption (H), we have V˙ (t) ≤

Ω



Y T (t, x)[−2P A + KQ1 K + 2P BL]Y (t, x) + 2Y T (t, x)P CG(Y (t − τ, x))

∂ Y (t − τ, x) − GT (Y (t − τ, x))Q1 G(Y (t − τ, x)) ∂t  t ∂ ∂ ∂ ∂ −( Y (t − τ, x))T Q2 Y (t − τ, x) − ( Y (α, x))T Q3 Y (α, x) dx ∂t ∂t ∂α t−τ ∂α ∂ T  ∂ + ( Y (t, x)) Q Y (t, x)dx  J1 + J2 , ∂t Ω ∂t +2Y T (t, x)P H

(17) where

J2 =

( Ω

∂  ∂ Y (t, x)dx. Y (t, x))T Q ∂t ∂t

To compute J2 , we multiply system (5) by yi (t, x), and then integral on x in Ω, 1 d 2 dt

Ω

yi2 (t, x)dx =

 m

yi (t, x)

Ω k=1  n

+

+

Ω j=1  n Ω j=1

∂ ∂xk

 Dik

∂yi (t, x) ∂xk



bij yi (t, x)fj (yj (t, x))dx + hij yi (t, x)

dx − ai  n Ω j=1

Ω

yi2 (t, x)dx

cij yi (t, x)gj (yj (t − τ, x))dx

∂yj (t − τ, x) dx, i = 1, 2, · · · , n. ∂t

(18)

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Taking into account (H), (16) and by Schwarz inequality, it is easy to see that the above equality can be transformed into n n   d yi (t, x)2 ≤ −ai yi (t, x)2 + b+ l y (t, x) + c+ j 2 ij j ij kj yj (t − τ, x)2 dt j=1 j=1

+

n 

h+ ij y˙ j (t − τ, x)2 , i = 1, 2, · · · , n.

(19)

j=1

By Lemma 1 and some classical inequality, we have T  n  ∂ d  ∂ Y (t, x)dx ≤ J2 = Q Y (t, x) qi ( ||yi (t, x)||2 )2 ∂t ∂t dt Ω i=1  T  + T +  − 2AQB  Y (t, x)[AQA ≤ L + |Ω|LB + QB L]Y (t, x) Ω

 + ]G(Y (t − τ, x))  + + 2|Ω|LB + T QC +Y T (t, x)[−2AQC  + ] ∂ Y (t − τ, x)  + + 2|Ω|LB + T QH +Y T (t, x)[−2AQH ∂t T +T  + +|Ω|G (Y (t − τ, x))C QC G(Y (t − τ, x)) T  + ∂ Y (t − τ, x) +2|Ω|GT (Y (t − τ, x))C + QH ∂t   T ∂ +T  + ∂ +|Ω| Y (t − τ, x) Y (t − τ, x) dx, H QH ∂t ∂t

(20)

Substituting (20) into (17), we have  1 t  V˙ (t) ≤ Y T (t, x)[−2P A + KQ1 K + 2P BL + AQA τ t−τ Ω  + L]Y (t, x)  + L + |Ω|LB + QB −2AQB  + ]G(Y (t − τ, x))  + + |Ω|LB + T QC +2Y T (t, x)[P C − AQC T

 + ] ∂ Y (t − τ, x)  + + |Ω|LB + T QH +2Y T (t, x)[P H − AQH ∂t T +T  + −G (Y (t − τ, x))[Q1 − |Ω|C QC ]G(Y (t − τ, x)) T  + ∂ Y (t − τ, x) +2|Ω|GT (Y (t − τ, x))C + QH ∂t ∂ T  + ∂ −( Y (t − τ, x))T [Q2 − |Ω|H + QH ] Y (t − τ, x) ∂t ∂t  ∂ ∂ −τ ( Y (α, x))T Q3 Y (α, x) dαdx ∂α ∂α t 1 = ΛT (t, α, x)Ψ Λ(t, α, x)dαdx, τ t−τ Ω  T ∂ ∂ where Λ = Y T (t, x), GT (Y (t − τ, x)), ( ∂t Y (t − τ, x))T , ( ∂α Y (α, x))T .

(21)

Delay-Dependent Global Asymptotic Stability

165

By the condition (6), V˙ (t) < 0. From Lemma 2, we derive that the equilibrium of system (1) is globally uniformly asymptotically stable.  When the smooth operator Dik = 0 (i = 1, 2, · · · , n; k = 1, 2, · · · , m), system (1) degenerated into the following model u˙ i (t) = −ai ui (t) +

n 

bij fj (uj (t))

j=1

+

n 

cij fj (uj (t − τ )) +

j=1

n 

hij u˙ j (t − τ ) + Ii , i = 1, 2, · · · , n. (22)

j=1

We have the following corollary Corollary 1. Under the assumption (H), the equilibrium point U ∗ of system (22) is globally uniformly asymptotically stable if there exist positive definite matrices P > 0, Q1 > 0, Q2 > 0 and Q3 > 0 such that ⎛ ⎞ Γ1

∗

∗

∗

0

0

0

0

⎜  BT P −2L−1 AT QB ∗ ∗ ⎜ T  ⎜ C T P − C T QA   C QB −Q1 + C T QC ∗ ⎜ ⎝ H T P − H T QA     H T QB H T QC −Q2 + H T QH

∗ ∗ ⎟ ⎟ ∗ ⎟ ⎟ < 0, (23) ∗ ⎠ −τ Q3

 + LB T QBL.  where Γ1 = −2P A + LQ1 L + AT QA Proof: Consider the Lyapunov functional as t T V (t) = Y (t)P Y (t) + F T (Y (α))Q1 F (Y (α))dα

t−τ t

Y˙ T (α)Q2 Y˙ (α)dα +

+ t−τ



0

−τ



t

Y˙ T (α)Q3 Y˙ (α)dαdβ. t+β

Similarly, calculate the derivative of V (t) along the degenerated system (22). By a minor modification of the proof of Theorem 1, we can easily obtain that the equilibrium point U ∗ of system (22) is globally uniformly asymptotically stable. 

4

An Example

1 Example: Let k = n = 2, Dik = 1 (i, k = 1, 2), |Ω| = 16 , and fj =  gj (j = 1, 2) with the Lipschtiz constant l1 = k1 = 0.5, l2 = k2 = 0.7. Consider the following delayed neural network in (1) with parameters as     2.8346 0 0.3346 −1.8921 A= ,B = , 0 9.7352 1.7243 0.5769     −0.8765 3.5647 0.3346 0.1745 C= ,H = . −0.7324 −2.0125 −0.0765 0.4217

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By Theorem 1, we can conclude that this delayed neural network is globally asymptotically stable for all 0 < τ ≤ 4.7297. When τ = 4.7297, we use the Matlab LMI Control Toolbox to solve the LMI in (6), and obtain the solution as follows     1.3905 0.2632 0.4671 0 , P = , Q1 = 0.2632 1.0979 0 0.2947     0.1066 0 0.0147 0 , Q3 = . Q2 = 0 0.0297 0 0.0011

5

Conclusion

In this paper, we have investigated the global uniform asymptotical stability for a class of neutral-type neural networks with time delays and reaction-diffusion terms. The obtained conditions are dependent on the size of the time delay and the measure of the space, which are usually less conservative than delayindependent and space-independent ones. Moreover, the obtained results improve and extend several earlier publications and are useful in applications of manufacturing high quality neural networks.

Acknowledgement This work was supported by the Natural Science Foundation of Shandong Province of China under Grant Y2007A17.

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Discrimination of Reconstructed Milk in Raw Milk by Combining Near Infrared Spectroscopy with Biomimetic Pattern Recognition Ming Sun1 , Qigao Feng2 , Dong An1 , Yaoguang Wei1 , Jibo Si1 , and Longsheng Fu1 1 China Agricultural University, College of Information and Electrical Engineering, Beijing, China 2 Henan Institute of Science and Technology, Xinxiang, China

Abstract. A new method for discriminating reconstructed milk in raw milk is proposed by combining NIRS (Near Infrared Spectroscopy) with the theory model BPR (Biomimetic Pattern Recognition). In order to compare its discrimination performance, we also carry out experiments by using a traditional method combining near infrared spectroscopy with DA (Discrimination Analysis). The results indicate that the accuracy of detection is 98.3% using the new method; while 87.5% using the traditional method. Comparison results show that the proposed new method is superior to the traditional method and the combination of NIRS with BPR has good potential to detect adulteration of raw milk with reconstructed milk. Keywords: NIRS (Near Infrared Spectroscopy), BPR (Biomimetic Pattern Recognition), DA (Discrimination Analysis), raw milk, reconstructed milk.

1

Introduction

Along with the increase of living standard day by day, milk wins a large number of people’s favor, and the consumption of milk increases continuously, thanks to its characteristic of sweet, tasty, nourishing and easy to digest and absorb. Some milk suppliers sell the adulteration of raw milk with reconstructed milk mixed in, for commercial profit, which harms the advantage of consumers. At the present time, identification of the adulterated milk is mostly implemented by smelling scent, observing colors, and checking if deposits occur after the milk has been laid for a while. Although these methods are simple, they are not precise enough. NIRS (Near Infrared Spectroscopy) analysis, which is widely applied to the examinations of farm production and the food quality, is a modern technique. It is a fast, non-destructive way with less pollution nuisance, which has been rapidly developed in recent years. At the present time, the analytic technique of NIRS has been extensively used in the dairy farming industry. For instance, the F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 168–175, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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quick identification for physical and chemical index of the raw material milk[1][2] as well as the somatic cell in it[3][4] has been realized by using NIRS. Research on reconstituted milk identification in raw milk has been implemented using a traditional method combining NIRS with Mahalanobis distance discrimination analysis by Han et al. and the results showed that NIRS is suitable to detect adulteration of raw milk with reconstituted milk, though its precision still needs to be enhanced[5]. A new model of pattern recognition principles, which is based on “matter cognition” instead of “matter classification” in traditional statistical pattern recognition has been proposed[6][7]. This new model is closer to the function of human beings, rather than traditional statistical pattern recognition using “optimal separating” as its main principle. Hence, the new model of pattern recognition is called the BPR (Biomimetic Pattern Recognition). Its mathematical basis is topological analysis of sample set in the high dimensional feature space; therefore, it is also called Topological Pattern Recognition. This paper puts great emphasis upon the investigation of the identification of raw milk adulterated with reconstructed milk by combining NIRS with BPR as compared to combination of NIRS and DA (Discrimination Analysis), so that a feasible and effective recognition method for the adulterated milk can be obtained.

2 2.1

Materials Obtaining Milk Samples

Raw milk: It was drawn from three Hesitan milk cows, which were fed in the China and America Milk Industry Research Center of China Agriculture University. The raw milk was sampled every noontime. After collection, it was refrigerated in a portable icebox, whose temperature was set at 4 centigrade, and carried back. Reconstructed milk with different gradients: According to the different gradients (10%, 20%, 33%, 50%, 67%, 80%, and 100%), after adding the water and reconstructed milk powder (Yili fresh whole milk powder) into the raw milk, the milk was kept at a constant temperature of 40 centigrade in a water bath boiler. And the reconstructed milk with different gradients was obtained after being stirred to uniformity. 2.2

The Collection of NIRS

Integrating sphere reflex spectra of samples were collected by using the Integrating sphere in the American Nicolet Fourier transform near infrared spectrometer. The range of the spectra collection is 12000-4000 cm−1 ; with the interval set at 2 cm−1 . The temperatures of the samples were kept at 40 centigrade before collection.

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Methods Combination of NIRS and DA

Each piece of the sample chart was considered to be a point. And every type of the sample chart formed a cluster, whose centroid point is x, after dividing the samples into some types. As the centroid of one cluster is different to each other, the samples are classified by comparing adjacent degree between the position of spectrum point and the centroid of each cluster. There are various methods to calculate the distance, and the way of Mahalanobis distance is chosen in this paper. When this method is adopted, different powers were given in different directions. This experiment distinguished and analysed the raw milk collected before, as well as the reconstructed milk of different gradients. The preprocess and discrimination analysis of the spectra was implemented using the software of TQ Analyst V6.2. 3.2

Combination of NIRS with BPR

In high-dimensional spaces, inhomogeneous samples were excluded from the covering interval after the homogeneous samples processed by the optimal covering in closed interval, so a certain sample could be separated from other samples, as shown in Fig. 1. This experiment divided the raw milk samples into two sets. One is modeling set, and the other is considered as forecast set, together with the reconstructed milk of different gradients. In high-dimensional spaces, the spectrum of the raw milk samples which was used as modeling was processed by the optimal covering in closed interval, and the value d, which represented the distance from the spectrum value of forecast set samples to the closed interval is calculated. The whole closed interval was the union of single closed intervals. Fig. 2 shows the selected model of the single closed intervals. In other words, it was the union of

Fig. 1. Biomimetic pattern recognition in the two dimensional space

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Fig. 2. Model of a single closed interval

toroid and hyper-sphere. The radius r was determined. If d < r, and the sample was inside the closed interval, it would be considered raw milk. Conversely, if d > r, and the sample was outside the closed interval, it would be taken for reconstructed milk. The preprocess and discrimination analysis of the spectra was implemented by the Matlab program.

4 4.1

Results and Discussion Spectrum Analysis of the Reconstructed Milk of Different Gradients

Fig. 3 is the spectrogram of the raw milk and the reconstructed milk of different gradients. It can be seen from the Fig. 3 that the spectrogram has a decreasing trend, along with the increase of the gradients of the reconstructed milk. The differences in spectrum with the raw milk are also enlarged. 4.2

Discrimination Based on Combination of NIRS and DA

Fig. 4 is the discrimination and analysis chart of the raw milk and samples adulterated with 10% reconstructed milk. There were altogether 30 samples. 15

Fig. 3. Spectrogram of the reconstructed milk of different gradients

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Fig. 4. Discrimination and analysis chart of the raw milk and samples adulterated with 10% reconstructed milk

Fig. 5. Discrimination and analysis chart of the raw milk and samples adulterated with 20% reconstructed milk

samples were adulterated with 10% reconstructed milk, and the others were the raw milk. There were 3 samples which were not correctly recognized, so the rate of identification reached 90%. Fig. 5 is the discrimination and analysis chart of the raw milk and samples adulterated with 20% reconstructed milk. There were altogether 30 samples. 15 samples were adulterated with 20% reconstructed milk, and the others were the raw milk. There was no misjudgment, so the rate of identification was 100%. Fig. 6 is the discrimination and analysis chart of the raw milk and the reconstructed milk of different gradients. 15 samples were reconstructed milk of different gradients, and the others were the raw milk. There were 15 samples which were not correctly recognized in the 120 samples, so the rate of identification was 87.5%.

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Discrimination Based on Combination of NIRS and BPR

The modeling set contained 30 samples of the raw milk. 120 samples, in which there were 15 samples of the raw milk and 105 samples of the reconstructed milk of different gradients, were taken as the forecast set. Fig. 7 is the distance diagram from the 120 samples of forecast set to the closed interval. The samples of the forecast set were ranked according to the ascending order of gradients of the reconstructed milk. It can be seen from the Fig. 7 that the distance to the closed interval was gradually enlarged, along with the increase of density value of the reconstructed milk. It is also shown that the reconstructed milk of higher gradients could more easily be distinguished. Table 1 is the distinguished result for the raw material milk, which is adulterated with the reconstructed milk by using the method of combination of NIRS and BPR. When radius r of the selected closed interval was 0.21, the misjudged sample of the raw milk and the reconstructed milk was separately only one, two samples totally. There were 2 samples which failed to be correctly recognized, so the rate of identification was 98.3%. If a different r value was selected, the misjudgment number of the raw milk and the reconstructed milk would be different. When r increased, the misjudgment of the raw milk would be reduced, while the misjudgment of the reconstructed milk would be increased. Conversely, when the r value decreased, the misjudgment of the raw milk would be increased, while the misjudgment of the reconstructed milk would be decreased. The proper r value should be determined, according to different demands. It can be seen from Fig. 7 and Table 1 that the raw milk and the reconstructed milk have been distinguished well. 4.4

Comparison between the Two Kinds of Discrimination Methods

The raw milk and the reconstructed milk with single density value were distinguished well by using the combination of NIRS and DA. However, the rate of the misjudgment would be increased, when the model was built together with

Fig. 6. Discrimination and analysis chart of the raw milk and the reconstructed milk of different gradients

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Fig. 7. Distance from the forecast set samples to the closed interval

the reconstructed milk of different gradients. Since the raw milk and the reconstructed milk of different gradients were regarded as two kinds, the result was obtained primarily by seeking the differences between the two kinds of samples, in order to achieve the best partition of the two kinds. When the reconstructed milk of different gradients was considered as one kind of samples for modeling, the range of this kind would be wide, and the centroid would be deflective. Some reconstructed milk of low density value could not be correctly recognized, and the rate of the misjudgment would be increased. When the combination of NIRS with BPR was used, the sample of the raw milk in the modeling set was properly processed by the covering in closed interval. The reconstructed milk of different gradients does not take part in modeling, so they did not influence each other during the forecast modeling. Hence the results of identification of the raw milk with the reconstructed milk of single density value and different gradients were the same.

5

Conclusions

The results of the experiment show that the reconstructed milk of the single density value and the raw milk can be discriminated by using a traditional method combining the Near Infrared Spectroscopy with Discrimination Analysis. But when the reconstructed milk of different gradients is discriminated from the raw milk, the rate of identification is only 87.5%. When a method combining the Near

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Infrared Spectroscopy with the Biomimetic Pattern Recognition is proposed and used, the rate of identification reaches 98.3%. In our life, as the reconstructed milk added into the raw milk is often of different gradients, it is better to choose method combining the Near Infrared Spectroscopy with the Biomimetic Pattern Recognition than the Near Infrared Spectroscopy with Mahalanobis Distance Discrimination Analysis. Since the adulterated milk does not take part in modelling, it has no influences on the results. Therefore, the Biomimetic Pattern Recognition possesses more comprehensive applicability for the identification of the adulterated milk.

References 1. Wang, Y., Xu, K., Chang, M.: Study on Multi-band NIR Spectroscopy for the Determination of Fat and Protein Contents in Milk. Optical Instruments 3, 3–7 (2006) 2. Wang, L., Xu, K., Guo, J.: Compositional Analysis of Fat, Protein and Lactose in Raw Milk by Using Near Infrared Spectroscopy. Journal of Optoelectronics & laser 4, 468–471 (2004) 3. Tsenkova, R., Atanassova, S., Ozaki, Y., et al.: Near Infrared Spectroscopy for Biomonitoring: Influence of Somatic Cell Count on Cow’s Milk Composition Analysis. International Dairy Journal 11, 779–783 (2001) 4. Pravdova, V., Walczak, B., Massart, D.L., et al.: Calibration of Somatic Cell Count in Milk Based on Near-infrared Spectroscopy. Analytica Chimica Acta. 12, 131–141 (2001) 5. Han, D., Lu, C., Liu, Y., Pi, F.: Identification of Reconstructed Milk in Raw Milk Using near Infrared Spectroscopy. Spectroscopy and Spectral Analysis 3, 465–468 (2007) 6. Wang, S., Wang, B.: Analysis and Theory of High-dimension Space Geometry for Artificial Neural Networks. Chinese Journal of Ecology 30, 1–4 (2002) 7. Wang, S.: Bionic (Topological) Pattern Recognition - A New Model of Pattern Recognition Theory and Its Applications. Chinese Journal of Ecology 30, 1417–1420 (2002)

Data Fusion Based on Neural Networks and Particle Swarm Algorithm and Its Application in Sugar Boiling* Yanmei Meng, Sijie Yan, Zhihong Tang, Yuanling Chen, and Jingneng Liu School of Mechanical Engineering, Guangxi University, Nanning, Guangxi, P.R. China [email protected]

Abstract. This paper utilizes data fusion algorithm to analyze the multi-sensors data, in order to obtain accurate and complete information in the sugar boiling process, and to improve the accuracy and reliability of decision-making. Proposing an improved particle swarm optimization method(BWPSO), compared with the basic particle swarm optimization, each particle adjust its position not only according to its own previous best solution and its group’s previous best, but also its group’s previous worst. This paper integrates BP neural network and particle swarm optimization, utilizes BWPSO algorithm to optimize BP neural network’s connection weights, and exerts BWPSO’s capability in global optimization and BP’s advantage in local search sufficiently. The experimental results show that utilizing BWPSO to optimize BP can get faster convergence, higher precision and better effect. Keywords: particle swarm optimization; neural networks; data fusion; sugar boiling

1 Introduction Sugar boiling process is a very significant section in the production of sugar industry [1]. Having been pressed, evaporated and disposed by other processed, the syrup will transform to solid from liquid in the crystallizer pan and then the crystallized sugar would be obtain under the state of super saturation of syrup in the right stable zone. In order to get better quality in refining sugar, it is most pivotal to control the super saturation of syrup and keep it in the appropriate range. The super saturation is affected by the change of material’s quantity, concentration and other technical condition (such as Brix, temperature or pressure etc). The super saturation can be measured on line by electrical-conductivity sensor, which is always influenced by Brix and temperature of syrup and will be easily begrimed, which would affect measurement accuracy seriously. Because there are lost of control parameters and lots of interactions between each other, the traditional instruments have larger limitation to control the whole sugar boiling process. Except that, a single sensor accesses to limited information, which is only about the local and one-sided part of *

The research is supported by Guangxi Province Nanning Key Technology Research Project (200501007A).

F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 176–185, 2008. © Springer-Verlag Berlin Heidelberg 2008

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the surroundings. Moreover, each sensor may be affected by its quality, performance and noise, so the collected information is often incomplete, uncertain and occasionally even wrong. Therefore, in order to obtain more accurate measurement, it is necessary to integrate large amount and different sources of information, and better results would come out after the integration of data which collected by multi sensors [2]. Artificial neural networks has strong ability to adjust and learn, and can comprehend to various information, so it has great advantages in data fusion. Though BP neural network is the most mature and widely used network, its convergence speed is too slow and it can be easy stuck in local minima. Therefore, the study of alternative learning algorithm to BP neural network has been researched by many researchers. In recent years, with the emergence of evolutionary computation research, researcher begins to consider the possibility of using evolutionary computation algorithm to train BP neural network. Because evolutionary algorithm has strong convergence capability and does not need the gradient information, it can not only enhance the generalization ability of BP neural network, but also improve BP neural network’s convergence ability and learning ability in training BP neural network [3]. Particle swarm optimization (PSO)[4][5], one of the intelligent optimizations, receives widespread attention in recent years. Due to the simple concept and easy-realized ability, the standard PSO and improved PSO have been widely used in neural networks training [6]. This paper researches on the data fusion algorithm of the main parameters in sugar boiling process, which integrates BP neural network and PSO algorithm, and makes an improvement on standard PSO. Utilizing PSO to optimize BP neural network’s connection weights is powerful to exert PSO’s capability in global optimization and BP’s advantage in local search sufficiently.

2 Improvement on Standard PSO 2.1 Standard PSO Algorithm and Its Shortage Supposed that the search space is D-dimensional and there are m particles. The position of the i-th particle can be represented as X i = ( xi1 , xi 2 ," , xiD ) and the velocity of this particle is

Vi = (vi1 , vi 2 ,", viD ) . The fitness of each particle can be evaluated

through putting its position into the objective function. The best previously visited position of the i-th particle is represented by Pi = ( pi1 , pi 2 ,", piD ) and the global best position of the swarm found so far is denoted by

Pg = ( p g1 , pg 2 ,", p gD ) [7].

Each particle moves in the search space with a velocity according to its own previous best solution and its group’s previous best solution. The particle’s velocity and its new position are updated as follows:

vid = wvid + c1 r1 ( p id − xid ) + c 2 r2 ( p gd − xid ) .

(1)

xid = xid + vid

(2)

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d ∈ {1,2,", D}, i ∈ {1,2,", m} , w is the inertia weight, acceleration coefficient c1 and c 2 are the positive constants, r1 and r2 are random numbers between where

0 and 1. The PSO algorithm is simple, has little parameter which relying on experience, and its convergence speed is fast and efficient, but it also has some disadvantages, such as low search accuracy and easily get stuck in local optimal. In PSO’s optimization process, all particles are flying towards optimal solution, the swarm trend to be the same and loss the diversity between particles, thus it is easy to stack in the local optimum [8, 9]. 2.2 An Improvement on PSO(BWPSO) 2.2.1 The Inspired Thinking of BWPSO From reference [10] we know that, Chunming Yang and Dan Simon proposed a new particle swarm optimization-NPSO, in which each particle adjusts its speed according to its previous worst position and its group’s previous worst position, but not its previous best position and its group’s previous best position, and the result show that NPSO better than PSO. From our personal experience we know that an individual learns from his or her own and other individuals’ previous success and failure. Based on this social behavior, in order to take full advantage of particle swarm’s information, we combine group’s previous best and worst together, each particle adjusts its speed not only according to its previous best position and its group’s previous best position, but also its group’s previous worst position, we call it BWPSO. 2.2.2 Formula for Calculating BWPSO In order to avoid stacking in local optimum, BWPSO not only consider each particle’s previous best position and group’s previous best position, but also group’s previous worst position, so when the particle moves toward the best position, it also moving gets away from the worst position as far as possible. This can avoid particle swarm stacking in local optimum to some extent. The particle’s position updated equation is similar to those of PSO, and its velocity is updated as follows:

vid = wvid + c1r1 ( pid − xid ) + c2 r2 ( pgd − xid ) + c3r3 ( xid − pwd ) . where acceleration coefficient

(3)

c3 is the positive constants, r3 is random numbers

between 0 and 1. 2.2.3 Validation Based on Goldstein-Price Function In order to verify the effectiveness of BWPSO algorithm, Goldstein-Price function was adopted to test the performance of BWPSO, comparing with PSO at the same time. The definition of Goldstein-Price function is f ( x ) = [1 + ( x1 + x 2 + 1) 2 (19 − 14 x1 + 3 x12 − 14 x 2 + 6 x1 x 2 + 3 x 22 )] × [30 + (2 x1 − 3x 2 ) 2 (18 − 32 x1 + 12 x12 + 48 x 2 − 36 x1 x 2 + 27 x 22 )]

where − 2 ≤ x1 , x2 ≤ 2 .

(4)

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To make a fair comparison, equations of both PSO and BWPSO set the same values of c1 and c 2 , the only difference is that whether c3 exists. The comparison is that: take BWPSO and PSO as a subprogram respectively, the max iteration is 500, then make a main program called the subprogram 1000 times respectively, preserve the results of operations for each subprogram, at last get the average of 1000 times results. Each test run 1000 times and loops 500 generations. Table1 shows the results of the comparisons. Table 1. Comparison of PSO and BWPSO for Goldstein-Price function Algorithm

Minimum

Maximum

Average

Coefficient

PSO

3.0000

3.2551

3.0148

c1=2.5,c2=1.5

BWPSO

3.0000

3.7862

3.0790

c1=2.5,c2=1.5,c3=0.25

BWPSO

3.0000

3.2765

3.0147

c1=2.5,c2=1.5,c3=0.0032

BWPSO

3.0000

3.3011

3.0145

c1=2.5,c2=1.5,c3=0.00312

PSO

3.0000

3.3428

3.0159

c1=2,c2=2

BWPSO

3.0000

3.4779

3.0214

c1=2,c2=2,c3=0.025

BWPSO

3.0000

3.2274

3.0151

c1=2,c2=2,c3=0.003

BWPSO

3.0000

3.2957

3.0148

c1=2,c2=2,c3=0.0033

PSO

3.0000

4.1693

3.0330

c1=2,c2=2.5

BWPSO

3.0000

3.4145

3.0317

c1=2,c2=2.5,c3=0.0073

BWPSO

3.0000

3.5706

3.0299

c1=2,c2=2.5,c3=0.003

BWPSO

3.0000

3.3820

3.0302

c1=2,c2=2.5,c3=0.00123

PSO

3.0000

3.4367

3.0206

c1=1.5,c2=2.5

BWPSO

3.0000

3.4647

3.0213

c1=1.5,c2=2.5,c3=0.005

BWPSO

3.0000

4.0631

3.0233

c1=1.5,c2=2.5,c3=0.00132

BWPSO

3.0000

3.3997

3.0196

c1=1.5,c2=2.5,c3=0.00435

From Table1 we can see that, as long as the values of can set a suitable value for

c1 and c 2 are set, BWPSO

c3 to find better solutions than PSO does. Thus, it is feasi-

ble for BWPSO to consider group’s previous worst positions, BWPSO it can effectively overcome the problem caused by particles assimilation to some extent, maintain a very good ability to explore and improve the quality of optimal solution.

3 Integration of BP Neural Network and BWPSO Utilizing PSO to optimize the weight and threshold of BP neural network, we must solve two problems [11]: First, how to build the mapping between particle’s dimension

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space and BP neural network’s connection weight? In this paper, we correspond each particle dimension component of BWPSO to a connection weight of BP neural network, that is to say the number of connection weights(including threshold) in BP neural network and the dimension of particle are the same; Second, what kind of Evaluation Function should be chosen to fitness function of PSO? This paper, we take the normalized error of BP neural network as the fitness function of BWPSO. Utilizing BWPSO to optimize BP neural network, first we must determine BP’s architecture and establish a reasonable model of BWPSO, and then establish the mapping between BWPSO and BP; meanwhile we take the normalized error as the fitness function of BWPSO, so that the training of BP neural network can transform to the matter of finding a group of best connection weights to minimize the normalized error. Finally we use BWPSO algorithm to train BP neural network. 3.1 BWPSONN—Training BP Neural Network by BWPSO In order to optimize BP, utilizes the advantage of BWPSO in global optimization and its fast convergence speed to train the connection weights of BP neural network. Supposed that the dimension of connection weights of BP neural network is D, we denote it as W, so the training process of BP neural network is the matter of finding optimized D-dimensional parameter one[12]. N particles generates randomly (the number of W is N), the initialized weights and thresholds are taken as the initialized velocity and location of the particles, and then through the BWPSO algorithm to reach the optimization to find a particle that can make the fitness function as the smallest one(the parameter W). Set each particle’s location parameter with each the connection weights of BP network, and calculate the fitness of each particle with the objective function, and then search the best connection weights of BP network with BWPSO algorithm. Once finished, the location of the best particle is the best connection weights of BP network [13, 14]. With this, it is sure to get the D-dimensional parameter W according to the optimization of BWPSO, enable the BP neural network error relatively smaller. 3.2 BWPSONNBP—The Combination of BWPSO and BP BWPSONN is using BWPSO to train BP neural networks, while the BWPSONNBP combine BWPSO with BP, utilize BWPSONN to do the pre-search, and then take advantage of BP in local optimization to do further search. BWPSONNBP first uses BWPSO algorithm to optimize the connection weights of BP neural network, till the fitness of the optimal particle no longer do the meaningful increasing, getting the parameter that is close to optimal, supposed that ε 1 is the current objective function error, and then use BP to do more deeply and precisely search with the network parameter got before until the termination criteria are met, supposed that the ε 2 is the final objective function error, keeping in mind that the objective function error here is ε 1 ≤ ε 2 . As BWPSO does the preliminary search instead BP neural network, BP network does the search on the basis of near optimal, which improves the network’s optimization accuracy and speed effectively.

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According to [15, 16], BWPSONNBP is mainly divided into two steps: (1) Take advantage of the global optimization and fast convergence of BWPSO to train the connection weights of BP neural network. Corresponding particle’s position vector to the connection weights of BP network, then BWPSO is initialized with a group of random particles and search for optima by updating generations, got an near optimal position vector as the initial connection weights of BP neural network. (2) Utilizing BP’s advantage in infinite approach and local optimization, on the basis of near optimal solution got by BWPSO, to do further search until got the optimal connection weights of BP neural network. Using this two-step to train neural networks, we can improve the accuracy of network training and the speed of convergence.

4 Data Fusion Based on BP and PSO and Its Application in Sugar Boiling 4.1 The Determination of Neural Network Model and Its Relevant Parameters The sugar crystallization process is a complex nonlinear one, it is difficult to describe with mathematical model, and we can only get a group of input and output data. In this paper, we establish a sugar boiling process model based on 60 sets of data, which were from a sugar factory. Based on the data, we take the vacuum, the average pressure of steam, and the average temperature as the inputs and the super saturation as output for the training neural networks [17], and its structure has been shown in Fig.1. After trial and error, a 3-3-1 neural network topology structure is obtained, and 0.1840 is the objective error, and 5000 is the max iteration. In order to evaluate the convergence objectively, this paper take the normalized error

E=

1 mn

m

n

∑∑ (d p =1 j =1

pj

− y pj ) 2 [18] as the evaluation function.

Fig. 1. BP neural network model for sugar boiling process

4.2 The Determination of Relevant Parameters for BWPSO After trial and error, the parameters of BWPSO are set as follow: the objective error is 0.1960; max iteration 20; vmax = 10 , xmax = 5 ; the number of particles is 20; the

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dimension of particles is D = 3 * 3 + 3 + 1* 3 + 1 = 16 ; the learning factors

c1 = 2.4

，c

2

= 1.6 , c3 = 0.00032 .

4.3 Results and Discussions Now compare the matlab toolbox functions traingd [19] and trainlm, BWPSONN and BWPSONNBP according to the 60 sets of data, among which the data are divided into two groups. The training set is 40, and the testing set is 20. The compared method is similar to previous. When utilizing BWPSO to train BP neural network, we set the parameters as follow: the objective error is 0.1960; the number of particles is 20; the max iteration is 20; the dimension of particles is 16. After utilizing BWPSO to find the optimization, we get a smallest fitness particle, and then using BP algorithm to train, the parameters are set as follow: the objective error is 0.1840; the max iteration is 3000. Table 2 shows the comparison of traingd, trainlm, BWPSONN and BWPSONNBP, System configuration is as follows: Windows XP system, Pentium 4 CPU, two 512 memories, MATLAB6.5. From table 2we can see that the BP trained by BWPSO got better result than BP, and BWPSONNBP better than BWPSONN. Fig. 2 shows the error curve of traingd, parameters on the left one are as follow: the training error is 0.1; the testing error is 0.0235; the time required for testing is 62.5780 second. Parameters on the right one are as follow: the training error is 0.1840; the testing error is 0.0231; the time required for testing is 30.3120 second. Fig. 3 shows the error curve of trainlm, parameters on the left one are as follow: the training error is 0.1; the testing error is 0.0184; the time required for testing 3.3750 second. Parameters on the right one are as follow: the training error is 0.1840; the testing error is 0.0197; the time required for testing is 3.8590 second. Table 2. Comparison with other algorithm in terms of the average test error and time Algorithm traingd trainlm

Average Time(s) test error 0.0232 1.8661e+004 0.0228 2.6960e+003

BWPSONN

0.0228

253.5000

BWPSONN

0.0233

241.6410

BWPSONNBP

0.0226

1.0309e+003

BWPSONNBP

0.0224

1.0472e+003

Coefficient

c1=2.6;c2=1.4; c3=0.00032; c1=2.6;c2=1.4; c3=0.00032; c1=2.6;c2=1.4; c3=0.00032; c1=2.6;c2=1.4; c3=0.00032;

xmax

vmax

5

10

1.11

0.8

5

10

1.11

0.8

Fig. 4 shows the training and testing error curve of BWPSONN, parameters on the left one are as follow: the testing error is 0.0236; the time required for testing is 8.59 second; vmax = 0.8 ; xmax = 1.11 . Parameters on the right one are as follow: the testing error is 0.0210; the time required for testing is 5.63 second; vmax

= 10 , xmax = 5 .

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Fig. 2. Error curve of traingd

Fig. 3. Error curve of trainlm

Fig. 4. Training and testing error curve of BWPSONN

Comparing all the preceding charts, we find that with the same algorithm parameters, BWPSONNBP get a better result than BP and BWPSONN do, it is more easily and obviously for BWPSONNBP to find the global optimum and the optimize performance. BWPSONNBP gets the highest convergence accuracy, and traingd needs longest time to train. Only utilizing BP to train neural network has very slow convergence speed, needs much time to convergence; utilizing BWPSO to train neural network has fast convergence speed, but is easy to get stuck in local minimum. Utilizing both BWPSO and BP to train neural network can overcome the drawbacks mentioned above, and get better convergence speed and optimize accuracy, BWPSONNBP will get a satisfactory result.

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The results show that, the combination of BWPSO and BP, overcome BP’s disadvantage of slow convergence speed and easily stuck in local minimum, makes the speed and accuracy of convergence much better than traditional BP neural network training algorithm.

5 Conclusions In this paper, we study the neural networks optimized by BP and BWPSO. The results show that the neural network model built here is reasonable, it is feasible to the forecast of super saturation, and the BWPSO trained neural networks has faster convergence speed and better forecast accuracy. It can more effectively search for the optimal connection weights, enjoying higher accuracy, better performance and good prospects. Particle swarm optimization algorithm will be applied in more domain as broad adaptability and a large potential for optimization algorithm.

References 1. Chen, W.J.: Sugar Cane Principle and Technology (volumes 4) [M]. China Light Industry Press, Beijing (2001) 2. He, Y.: Multi-sensor Data Fusion and Applications [M]. Electronic Industry Press, Beijing (2000) 3. Zhao, H., Liu, L.Y., Zhang, G.X.: Neural Network Training Based on Compound Fitness Particle Swarm Optimization [J]. Control and Decision 20(8), 958–960 (2005) 4. Kennedy, J., Eberhart, R.C.: Particle Swarm Optimization[C]. In: Proceedings of IEEE International Conference on Neural Networks, pp. 1942–1948 (1995) 5. Eberhart, R.C., Kennedy, J.: A New Optimizer Using Particle Swarm Theory [C]. In: Proceedings of the Sixth International Symposium an Micro Machine and Human Science, Nagoya, Japan, pp. 34–43 (1995) 6. Wang, J.N., Shen, Q.T., Chen, X.Z.: Evolutionary Design of Adaptive Neuro-Fuzzy Inference System Based on Hybrid Cooperative Particle Swarm Optimization. Systems [J] Engineering-theory & Practice 8, 48–54 (2006) 7. Zeng, J.C., Jie, J., Cui, Z.H.: Particle Swarm Optimization [M]. Science press, Beijing (2004) 8. Sheng, H.H.: Reactive Power Optimization Based on Adaptive Mutation Multi-group Particle Swarm Optimization Algorithm [D]. North China Electric Power University, Beijing (2006) 9. Li, C.H., Ji, C.M., Li, W.W.: Modified Particle Swarm Algorithm and Its Application in Reservoir Operation Optimization[J]. China Rural Water and Hydropower 2, 54–56 (2006) 10. Yang, C.M., Simon, D.: A New Particle Swarm Optimization Technique [C]. In: Proceedings of the 18th international Conference on Systems Engineering (ISCEng 2005), IEEE Computer Society, Washington (2005) 11. Li, F.F., Zhao, Y.K., Jia, Y.Y.: Application of Neural Network Based on Particle Swarm Algorithm for the Prediction of Oil Quality [J]. Journal of Computer Applications 26(5), 1122–1124 (2006) 12. Sun, J.: Sensor Nonlinear Error Compensation Evolved by Neural Network and Particle Swarm Algorithm [J]. Electronic Components & Materials 24(12), 17–19 (2005)

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13. Chen, G.C., Yu, J.S.: Enhanced Particle Swarm Optimization and Its Application in Softsensor [J]. Control and Decision 20(4), 377–381 (2005) 14. Zhi-hong, Y., Chun-guang, W., Jun-qing, F.: SAPSO Neural Network for Inspection of Non-development Hatching Eggs[C]. In: Jiao, L., Wang, L., Gao, X.-b., Liu, J., Wu, F. (eds.) ICNC 2006. LNCS, vol. 4221, pp. 85–88. Springer, Heidelberg (2006) 15. Gao, Y.X., Li, Y.S.: Image Recognition by BP Neural Networks Model Based on Particle Swarm Optimization [J]. Journal of Wuhan Polytechnic University 25(4), 35–38 (2006) 16. Angeline, P.J.: Using Selection to Improve Particle Swarm Optimization [A]. In: Proceedings of the 1998 International Conference on Evolutionary Computation [C], pp. 84–89. IEEE, New York (1998) 17. Wen, S.C.: Research on Neural Networks and Its Application in Sugar Boiling [D]. South China University of Technology, Guangzhou (2002) 18. Wei, H.K.: Neural Network Architecture Design Theories and Methods [M]. National Defense Industry Press, Beijing (2005) 19. Demuth, H., Beale, M.: Neural Network Toolbox User’s Guide-Neural Network Toolbox for Use with MATLAB. The MathWorks, Inc. (1992)

Asymptotic Law of Likelihood Ratio for Multilayer Perceptron Models Joseph Rynkiewicz CES-SAMOS-MATISSE Universit´e de Paris 1, 90 rue de Tolbiac, 750013 Paris [email protected]

Abstract. We consider regression models involving multilayer perceptrons (MLP) with one hidden layer and a Gaussian noise. The data are assumed to be generated by a true MLP model and the estimation of the parameters of the MLP is done by maximizing the likelihood of the model. When the number of hidden units of the model is over-estimated, the Fischer information matrix of the model is singular and the asymptotic behavior of the LR statistic is unknown or can be divergent if the set of possible parameter is too large. This paper deals with this case, and gives the exact asymptotic law of the LR statistic. Namely, if the parameters of the MLP lie in a suitable compact set, we show that the LR statistic converges to the maximum of the square of a Gaussian process indexed by a class of limit score functions. Keywords: Multilayer Perceptron, Likelihood ratio statistic, Donsker class, Gaussian process.

1

Introduction

Feedforward neural networks are well known and are popular tools to deal with non-linear statistic models. We can describe MLP as a parametric family of probability density functions. If the noise of the regression model is Gaussian then it is well known that the maximum likelihood estimator is equal to the leastsquare estimator. Therefore, Gaussian likelihood is the usual assumption when we consider feedforward neural networks from a statistical viewpoint. White [12] reviews statistical properties of MLP estimation in detail. However he leaves an important question pending: the asymptotic behavior of the estimator when an MLP in use has redundant hidden units and the Fisher information matrix is singular. Amari et al. [1] give several examples of behavior of the LR statistic in such cases. Fukumizu [5] shows that, for unbounded parameters, the LR statistic can have an order lower bounded by O(log(n)) with n the number of observations instead of the classical convergence property to χ2 law. However, a fairly natural assumption is to consider that the parameters are bounded. Indeed, computer calculations always assume that numbers are bounded. Moreover a safe practice is to bound the parameters in order to avoid numerical problems. In such context, different situations can occur. In some F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 186–195, 2008. c Springer-Verlag Berlin Heidelberg 2008 

Asymptotic Law of Likelihood Ratio for Multilayer Perceptron Models

187

cases, such as mixture models, the LR statistic is tight and the calculation of the asymptotic distribution is possible ([9]). In other cases it may occur that even if the parameters are bounded the likelihood ratio diverges this is for example the case in hidden Markov models (Gassiat and Keribin ([6]). So the behavior of likelihood ratio in the case of MLPs with bounded parameters is still an open question. In this paper, we derive the distribution of the likelihood ratio if the parameters are in a suitable compact set (i.e. bounded and closed). To obtain this result we use recent techniques introduced by Dacunha-Castelle and Gassiat [3] and Liu and Shao [9]. These techniques consist in finding a parameterization separating the identifiable part and the unidentifiable part of the parameter vector, then we can obtain an asymptotic development of the likelihood of the model which allows us to show that a set of generalized score functions is a Donsker class and to find the asymptotic distribution of the LR statistic. The paper is organized as follows. In section 2 we state the model and the main assumptions. Section 3 presents our main theorem and explains its meaning with a brief summary and a statement of significance of this work in the conclusion. Finally, we prove the theorem in the appendix.

2

The Model

We consider the model of regression for i ∈ IN∗ : Yi = Fθ0 (Xi ) + εi ,

(1)

where Xi ∈ IRd are observed exogenous variables and Yi ∈ IR is the variable to explain. The data (Yi , Xi ) are assumed to be generated by this true model. The noise (εi )i∈IN∗ is a sequence of independent and identically distributed (i.i.d.) N (0, σ 2 ) variables. 2.1

The Regression Function T

Let x = (x1 , · · · , xd )T ∈ IRd be the vector of inputs and wi := (wi1 , · · · , wid ) ∈ IRd be paramater vector of the hidden unit i. The MLP function with k hidden units can be written : Fθ (x) = β +

k 

  ai φ wiT x + bi ,

(2)

i=1

with θ = (β, a1 , · · · , ak , b1 , · · · , bk , w10 , · · · , w1d , · · · , wk0 , · · · , wkd ) ⊂ IRk×(d+2)+1 the parameters of the model. The transfer function φ will be assumed bounded and three times derivable. We assume also that the first, second and third deriva   tives of the transfer function φ: φ , φ and φ are bounded. In order to simplify the presentation, we assume that the variance of the noise σ 2 is known. Note that it is assumed that the true model (1) is included in the considered set of parameter Θ. Let us  as the smallest integer  define the true number of hidden units k 0 so that θ0 = β 0 , a01 , · · · , a0k0 , b01 , · · · , b0k0 , w10 , · · · , wk00 exists with Fθ0 equal to the true regression function of model (1).

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J. Rynkiewicz

Parameterization of the Model

Let us write . for the Euclidean norm. Let us consider the variable Zi = (Xi , Yi ) where Xi and Yi follow the probability law induced by the model (1). We assume that the law of Xi will be q(x)λd (x) with λd the Lebesgue measure on IRd and q(x) > 0 for all x ∈ IRd . The likelihood of the observation z := (x, y) for a parameter vector θ = (β, a1 , · · · , ak , b1 , · · · , bk , w11 , · · · , w1d , · · · , wkd ) will be written: 2 1 1 fθ (z) = √ e− 2σ2 (y−Fθ (x)) q(x) . (3) 2 2πσ Let η > 0 be a small constant and M a huge constant, the set of possible parameters will be Θk := {θ = (β, a1 , · · · , ak , b1 , · · · , bk , w10 , · · · , w1d , · · · , wk0 , · · · , wkd ) , ∀1 ≤ i ≤ k, wi  ≥ η, ai  ≥ η and θ ≤ M } . Constraints on the parameter set. The constraint wi  ≥ η is introduced in order to avoid the hidden unit from being constant like the bias β, instead of being a function of x. The constraint ai  ≥ η forces the parameters of the hidden units to converge to one of the parameter vector wj0 , j ∈ {1, · · · , k 0 } when they maximize the likelihood. Finally, with the constraint θ ≤ M , the parameters are bounded and the set Θk compact. Note that these constraints are very easy to set in practice. The true density of the observation will be denoted f (z) := fθ0 (z). The main goal of the parametric statistic is to give an estimation of the true parameter θ0 thanks to the observations (z1 , · · · , zn ). This can be done by maximizing the log-likelihood function : n  log fθ (zi ) . (4) ln (θ) := i=1

The parameter vectors θˆn realizing the maximum will be called Maximum Likelihood Estimator (MLE). However,the MLE belongs to a non-null dimension submanifold if the number of hidden units n is overestimated. In the next section we will study the behavior of supθ∈Θk i=1 log fθ (zi ) − log f (zi ), where k > k 0 , which is the key point to guess the true architecture of the MLP model.

3

Asymptotic Distribution of the LR Statistic

 We will use the abbreviation P g = gdP for an integrable  function g and a P g 2 and the map measure P . We will define the L2 (P ) norm as g2 = g 2 2 Ω : L (P ) → L (P ) as Ω(g) = g2 if g = 0. The maximum of the log-likelihood will be denoted : n  log fθ (zi ) − log f (zi ) . (5) λkn = sup θ∈Θk i=1

Asymptotic Law of Likelihood Ratio for Multilayer Perceptron Models

Finally, let us note

⎛

e(z) :=

⎛

k 

⎞⎞

0

1 ⎝ y − ⎝β 0 + σ2

189

T a0i φ(b0i + wi0 x)⎠⎠ .

(6)

i=1

For what follows, we will assume the properties: H-1 : The parameters of Θk realizing the true regression function Fθ0 lie in the interior of Θk . H-2 : Let k be an integer greater or equal to k 0 and δ(wi ,bi ) be the indicator function of the parameters (wi , bi ). The model is identifiable in the weak following sense: k  0

Fθ0 = Fθ ⇔ β = β and 0

a0i δ(wi0 ,b0i ) =

i=1

k 

ai δ(wi ,bi ) .

(7)

i=1

Note that, it is possible that some new constraint on the parameters have to be set to fulfill this assumption. For example, if the transfer function is the hyperbolic tangent (or any odd function), the constraints on the parameters ai will be : ai ≥ η, in order to avoid a symmetry on the sign (because tanh(−t) = − tanh(t)). H-3 : E(X6 ) < ∞. H-4 : the functions of the set 

   T 0T 0 1, xj xl φ (wi0 x + b0i ) , x φ (w x + b ) j i i 1≤i≤k0 1≤j≤d, 1≤i≤k0

1≤l≤j≤d,   T 0T 0 0 0 φ (wi x + bi )1≤i≤k0 , xj φ (wi x + bi ) 0

 

 1≤j≤d,1≤i≤k T T φ (wi0 x + b0i ) , φ(wi0 x + b0i ) 1≤i≤k0

1≤i≤k0

are linearly independent in the Hilbert space L2 (qλd ). We get then the following result: Theorem 1. Under the assumptions H-1, H-2, H-3 an H-4, a centered Gaussian process {WS , S ∈ Fk } with continuous sample path and covariance kernel P (WS1 WS2 ) = P (S1 S2 ) exists so that 2

lim 2λkn = sup (max(WS , 0)) .

n→∞

(8)

S∈Fk

The index set Fk is defined as Fk = ∪t Fkt , the union runs over any possible 0 t = (t0 , · · · , tk0 ) ∈ INk +1 with 0 = t0 < t1 < · · · < tk0 ≤ k and   0 T Fkt = Ω γe(z) + ki=0 i e(z)φ(wi0 x + b0i )  0  T + ki=0 e(z)φ (wi0 x + b0i )(ζiT x + αi )   ti  k0  T tT T t tT t2 + i=1 e(z)sg(a0i )φ (wi0 x + b0i ) δ(i) j=ti−1 +1 νj xx νj + ηj νj x + ηj γ, 1 , · · · , k0 , α1 , · · · , αk0 , η1 , · · ·, ηk0 ∈ IR ; ζ1 , · · · , ζk0 , ν1t , · · · , νttk0 ∈ IRd+1 ,

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where (6), δ(i) = 1 if a vectorq exists so that: ti e(z) is defined by √ t √ ti ti qj ηj = 0, otherwise j=ti−1 +1 qj = 1, j=ti−1 +1 qj νj = 0 and j=ti−1 +1 δ(i) = 0. The function sg is defined by sg(x) = 1 if x > 0 and sg(x) = −1 if x < 0. This theorem is proved in the appendix. Note that this theorem prove that the LR statistic is tight so penalized likelihood like BIC yields a consistent method to identify the minimal architecture of the true model, practical applications of such method can be found, for example, in Mangeas [8].

4

Conclusion

We have computed the asymptotic distribution of the LR statistic for parametric MLP regression. Note that the assumption H-4 can be proved for sigmoidal transfert function by using a method similar to Fukumizu [4]. So this theorem can be applied to the most widely used transfer functions for MLP.Finally, the limiting distribution uses the framework proposed by Dacunha-Castelle and Gassiat [3]. It is known that the convergence to the limit is very slow, but this theorem shows that the LR statistic is tight, so information criteria such as the Bayesian information criteria (BIC) will be consistent in the sense that they will select the model with the true dimension k 0 with probability 1, as the number of observations goes to infinite. This is the main practical application of the results obtained in this paper.

References 1. Amari, S., Park, H., Ozeki, T.: Singularities Affect Dynamics of Learning in Neuromanifolds. Neural computation 18, 1007–1065 (2006) 2. Cottrell, M., Girard, B., Girard, Y., Mangeas, M., Muller, C.: Neural Modeling for Time Series: a Statistical Stepwise Method for Weight Elimination. IEEE Transaction on Neural Networks 6, 1355–1364 (1995) 3. Dacunha-Castelle, D., Gassiat, E.: Testing the Order of a Model Using Locally Conic Parameterization: Population Mixtures and Stationary ARMA process. The Annals of Statistics 27, 1178–1209 (1999) 4. Fukumizu, K.: A Regularity Condition of the Information Matrix of a Multilayer Perceptron Network. Neural networks 9, 871–879 (1996) 5. Fukumizu, K.: Likelihood Ratio of Unidentifiable Models and Multilayer Neural Networks. The Annals of Statistics 31, 833–851 (2003) 6. Gassiat, E., Keribin, C.: The Likelihood Ratio Test for the Number of Components in a Mixture with Markov Regime. ESAIM Probability and statistics 4, 25–52 (2000) 7. Gassiat, E.: Likelihood Ratio Inequalities with Applications to Various Mixtures. Annales de l’Institut Henri Poincar´e 38, 897–906 (2002) 8. Mangeas, M.: Neural Model Selection: How to Determine the Fittest Criterion. In: Gerstner, W., Hasler, M., Germond, A., Nicoud, J.-D. (eds.) ICANN 1997. LNCS, vol. 1327, pp. 987–992. Springer, Heidelberg (1997)

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9. Liu, X., Shao, Y.: Asymptotics for Likelihood Ratio Tests Under Loss of Identifiability. The Annals of Statistics 31, 807–832 (2003) 10. Sussmann, H.J.: Uniqueness of the Weights for Minimal Feed-Forward Nets with a Given Input-Output Map. Neural networks 5, 589–593 (1992) 11. Van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998) 12. White, H.: Artificial Neural Networks: Approximation and Learning Theory. Oxford, Basil Blackwell (1992) 13. Yao, J.: On Least Square Estimation for Stable Nonlinear AR Processes. The Annals of Institut. of Mathematical Statistics 52, 316–331 (2000)

A

Appendix: Proof of Theorem 1

Let sθ (z) be the generalized score function defined by sθ (z) :=

fθ f (z) − 1 fθ  f (z) − 12

.

(9)

Firstly, we will get an asymptotic development of the generalized score when the model is over-parameterized. We will reparameterize the model using the same method as Liu and Shao [9] for the mixture models. A.1

Reparameterization

If − 1 = 0, we have β = β 0 and a vector t = (ti )1≤i≤k0 exists so that 0 = t0 < t1 < · · · < tk0 ≤ k and, up to permutations, we have wti−1 +1 = · · · = wti = wi0 , ti ti aj = a0i . Let us define si = j=t aj −a0i bti−1 +1 = · · · = bti = b0i , j=t i−1 +1 i−1 +1 ti  aj ti and, if ti−1 +1 aj = 0, let us write qj =  ti . If ti−1 +1 aj = 0, qj will fθ f

ti−1 +1

aj

be set to 0. We get then the reparameterization θ = (Φt , ψt ) with  

0 tk0 tk0 tk0 . Φt = β, (wj )j=1 , (bj )j=1 , (si )ki=1 , ψt = (qj )j=1

(10)

With this parameterization, for a fixed t, Φt is an identifiable parameter and all the non-identifiability of the model will be in ψt . Namely, ffθ (z) will be equal to  2 

 0 0  ti T exp − 2σ12 y− β+ k i=1 (si +ai ) j=ti−1 +1 qj φ(wj x) .  2 

 0 0 0T exp − 12 y− β 0 + k i=1 ai φ(wi x) 2σ

So

fθ f (z)

= 1 if and only if

0

Φ0t

= (β 0 , w10 , · · · , w10 , · · · , wk00 , · · · , wk00 , b01 , · · · , b01 , · · · , b0k0 , · · · , b0k0 , 0, · · · , 0) .                t1 tk0 − tk0 −1 t1 tk0 − tk0 −1 k0

Now, the third derivative of the transfer function is bounded and a constant C exists so that we have the following inequalities: ∀θi , θj , θl ∈ {b1 , · · · , bk , w11 , · · · , wkd } , sup  θ∈Θk

∂ 3 Fθ (X)  ≤ C(1 + X3 ) . (11) ∂θi ∂θj ∂θl

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So, thanks to the assumption H-3, the third order derivative of the function ffθ (z) with respect to the components of Φt will be dominated by a square integrable function. Then, by the Taylor formula with an integral remainder around the identifiable parameter Φ0t , we get the following expansion for the likelihood ratio: f

Lemma 1. For a fixed t, let us write D(Φt , ψt ) :=  (Φtf,ψt ) − 12 . In the neighborhood of the identifiable parameter Φ0t , the following approximation is true:   fθ (z) = 1+(Φt −Φ0t )T f(Φ0 ,ψt ) (z)+0.5(Φt −Φ0t )T f(Φ0 ,ψt ) (z)(Φt −Φ0t )+o(D(Φt , ψt )), t t f

with

k0  T (Φt − Φ0t )T f(Φ0 ,ψt ) (z) = e(z) β − β 0 + i=1 si φ(wi0 x) t  T k0 ti  T qj wj − wi0 xa0i φ (wi0 x + b0i ) + i=1 j=t i−1 +1    0  0T k0 ti 0 0 b a q − b φ (w x + b ) + i=1 j=t j j i i i i +1 i−1

and 

(Φt − Φ0t )T f(Φ0 ,ψt ) (z)(Φt − Φ0t ) =

 t  T   1 − e21(z) (Φt − Φ0t )T f(Φ0 ,ψt ) (z)f(Φ0 ,ψt ) (z)(Φt − Φ0t ) t t

 0  k ti 0 T T 0 0  0T 0 q (w − w +e(z) × j j i ) xx (wj − wi )ai φ (wi x + bi ) i=1 j=ti−1 +1 0  i  T + ki=1 tj=t qj (wj − wi0 )T x(bj − b0i )φ (wi0 x + b0i ) i−1 +1 0 k ti  T + i=1 j=t qj (bj − b0i )2 φ (wi0 x + b0i ) i−1 +1 0 k ti  T + i=1 j=t qj (wj − wi0 )T xsi φ (wi0 x + b0i ) i−1 +1   0 i  0 0T 0 q (b − b )s φ (w x + b ) . + ki=1 tj=t j j i i i i +1 i−1 The development, obtained by a straightforward calculation of the derivatives of ffθ (z) with respect to the components of Φt up to the second order. Now, the convergence to a Gaussian process will be derived from the Donsker property of the set of generalized score functions S = {sθ (z), θ ∈ Θk }. Let an ε-bracket [l, u] be a set of function h with l ≤h ≤ u with   P (l − u)2 < ε. The bracketing number N[] ε, S, L2 (f λd+1 ) is the minimum number of ε-brackets needed to cover S. The entropy with bracketing is the logarithm of the bracketing number. It is well known (Van der Vaart [11]) that the class of functions S will be Donsker if its entropy with bracketing grows with 2 a slower order than 1ε . A sufficient condition for Donsker property is then that the bracketing number grows as a polynomial function of 1ε . A.2

Polynomial Bound for the Growth of Bracketing Number f

Let us write D(θ) :=  (θ) f − 12 , for all ε > 0, the set of parameters can be divided in two sets: Sε and S0 with Sε = {θ ∈ Θk so that D(θ) ≥ ε} and S0 = {θ ∈ Θk so that D(θ) < ε} .

Asymptotic Law of Likelihood Ratio for Multilayer Perceptron Models

For θ1 and θ2 belonging to Sε , we get:     fθ  fθ  fθ fθ fθ 2 −1 1 −1 2 −1 2 −1  f1 −1      −  f   =  f  −  f  +  f  −  fθ  fθ  fθ  fθ        fθ1 −1   2 −1   1 −1  1 −1  f   f1 −1  f  f  f     2 2 2 2 2 2       fθ  fθ fθ fθ 2 −1 2 −1 2 −1    f1 −1   f    f  f      ≤   fθ1 −1 −  fθ1 −1  +   fθ1 −1 −  fθ2 −1   f  f  f  f       ≤

2   fθ  f  1 − θ2   f f  2   2  fθ   1 −1  f 

≤2

2 2   fθ f  1 − θ2  f f

ε

   

2

2

2

   

fθ 2 f −1  fθ 2  f −1

193

2

     

2

2

.

2

 f Hence, on Sε , it is sufficient that  fθ1 −

fθ2 f

   < 2

  fθ1 fθ2  f −1 f −1     −   fθ1  fθ2    f − 1  f − 1  2

2

ε2 2

for

    < ε.   2

Now, Sε is a parametric class. Since the derivatives of the transfer functions are bounded and EX < ∞ a function m(z) exists , with E[m(z)] < ∞, so that  f  ∂ θ   f  ∀θi ∈ {β, a1 , · · · , ak , w10 , · · · , w1d , · · · , wkd } ,  (z) ≤ m(z).  ∂θi  According to the example 19.7 of Van der Vaart [11], a constant K exists so that the bracketing number of Sε is lower than √ k×(d+2)+1 k×(2d+4)+2 diamΘk diamΘk =K , (12) K ε2 ε where diamΘk is the diameter of the smallest sphere of IRk including Θk . For θ belonging to S0 , fθf(z) − 1 is the sum of a linear combination of 

 T , V (z) := e(z), e(z)xj xl φ (wi0 x + b0i ) 1≤l≤j≤d, 1≤i≤k0 

 T , e(z)xj φ (wi0 x + b0i )

 1≤j≤d, 1≤i≤k0    T 0T 0 e(z)φ (wi0 x + b0i ) , e(z)x φ (w x + b ) , j i i 1≤i≤k0 1≤j≤d,  1≤i≤k0



  T T e(z)φ (wi0 x + b0i ) , e(z)φ(wi0 x + b0i ) 1≤i≤k0

1≤i≤k0

and of a term whose L2 (f λd+1 ) norm is negligible compared to the L2 (f λd+1 ) norm of this combination when ε goes to 0. By assumption H-4, a strictly positive number m exists so that for any vector of norm 1 with components

C = c, c1 , · · · , ck0 × d(d+1) , d1 , · · · , dk0 ×d , e1 , · · · , ek0 , 2 f1 , · · · , fk0 ×d , g1 , · · · , gk0 , h1 , · · · , hk0 )

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and ε sufficiently small: C T V (z)2 > m + ε . Since any function



fθ f −1 fθ f −12

can be written:

C T V (z) + o(C T V (z)2 ) , C T V (z) + o(C T V (z)2 )2 S0 belongs to the set of functions:     1 1 DT V (z) + o(1), D2 ≤ ⊂ DT V (z) + γ, D2 ≤ , |γ| < 1 , m m  k0 ×( d(d+1) +2d+3)+2 2 whose bracketing number is smaller or equal to O 1ε . This proves that the bracketing number of S is polynomial, hence S is a Donsker class. A.3

Asymptotic Index Set

Since the class of generalized score functions S is a Donsker class the theorem follows from theorem 3.1 of Gassiat [7] or theorem 3.1 of Liu and Shao [9]. Following these authors, the set of limit score functions Fk is defined as the set of functions d so that one can find a sequence gn := fθkn , θkn ∈ Θk satisfying

gn f (z)−1  gfn −12 . Note that, for a particular sequence of maximum likelihood estimators (θn )n∈N , the partition of 0 the indices t = (t0 , · · · , tk0 ) ∈ INk +1 can depend on n, but (θn )n∈N will be the

 gnf−f 2 → 0 and d − sgn 2 → 0, where sgn =

union of converging sub-sequences belonging the set of limit score functions. Let us define the two principal behaviors for the sequences gn which influence the form of functions d : – If the second order term is negligible behind the first one : fθkn  (z) − 1 = (Φnk − Φ0 )T f(Φ0 ,ψn ) (z) + o(D(Φnk , ψkn )) . t k f – If the second order term is not negligible compared to the first one : fθn k

f



(z) − 1 = (Φnk − Φ0 )T f(Φ0 ,ψn ) (z)+ t



k

0.5(Φnk − Φ0t )T f(Φ0 ,ψn ) (z)(Φnk − Φ0t ) + o(D(Φnk , ψkn )) . t

k

In the first case, each sequence gn is the finite union of convergent subsequences 0 gk (n) and for each subsequence a set t = (t0 , · · · , tk0 ) ∈ INk +1 (with 0 = t0 < t1 < · · · < tk0 ≤ k) exists so that the limit functions d of sgk (n) will be:  k 0 T Dt1 = Ω γe(z) + i=0 i e(z)φ(wi0 x + b0i )   k0  T + i=0 e(z)φ (wi0 x + b0i ) ζiT x + αi ,  γ, 1 , · · · , k0 , α1 , · · · , αk0 ∈ IR ; ζ1 , · · · , ζk0 ∈ IRd+1 .

Asymptotic Law of Likelihood Ratio for Multilayer Perceptron Models

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In the second case, each sequence gn is the finite union of convergent subsequences gk (n) and for each subsequence, an index i exists so that : ti 

qj (wj − wi0 ) = 0 and

j=ti−1 +1

ti 

qj (bj − wi0 ) = 0 ,

j=ti−1 +1

otherwise the second order term will be negligible compared to the first one, so ti 

√

qj ×

√ qj (wj − wi0 ) = 0 and

j=ti−1 +1

ti 

√ √ qj × qj (bj − wi0 ) = 0 .

j=ti−1 +1 0

Hence, a set t = (t0 , · · · , tk0 ) ∈ INk +1 exists, with 0 = t0 < t1 < · · · < tk0 ≤ k so that the set of functions d will be:

k 0 k0  T T Ω γe(z) + i=0 i e(z)φ(wi0 x + b0i ) + i=0 e(z)φ (wi0 x + b0i )(ζiT x + αi ) 

  0  T ti tT T t tT t2 + ki=1 e(z)sg(a0i )φ (wi0 x + b0i ) δ(i) j=ti−1+1 νj xx νj +ηj νj x+ηj γ, 1 , · · · , k0 , α1 , · · · , αk0 , η1 , · · ·, ηk0 ∈ IR ; ζ1 , · · · , ζk0 , ν1t , · · · , νttk0 ∈ IRd+1

,

i i √ where δ(i) = 1 if a vector q exists with tj=t qj = 1 and tj=t qj νj = 0 i−1 +1 i−1 +1 ti √ and j=ti−1 +1 qj ηj = 0, otherwise δ(i) = 0. So, the limit functions d will belong to Fk . Conversely, for x ∈ L2 (λd+1 ), let d be an element of Fk :

k0 k 0  T T d = Ω γe(z)+ i=0 i e(z)φ(wi0 x + b0i ) + i=0 e(z)φ (wi0 x + b0i )(ζiT x + αi ) 

0 k ti  T tT T t t tT t2 . + i=1 e(z)sg(a0i )φ (wi0 x+b0i ) δ(i) j=ti−1+1 νj xx νj +ηj νj x+ηj As functions d belong to the Hilbert sphere, one of their components is not equal to 0. Let us assume that this component is γ, but the proof would be similar with any other component. The norm of d is 1, so any component of d is determined by the ratio: γ1 , · · · , γ1 ηkt 0 . Then, we can chose θkn = (β n , an1 , · · · , ank , w1n , · · · , wkn , bn1 , · · · , bnk ) so that: ∀i ∈ {1, · · · , k 0 } : ∀i ∈ {1, · · · , k 0 } : ∀i ∈ {1, · · · , k 0 } : ∀j ∈ {1, · · · , tk0 } : ∀j ∈ {1, · · · , tk0 } :

n→∞ i sn i βn −β 0 −→ γ n,  n  ti qj 0 n→∞ 1 −→ γ ζi , j=ti−1 +1 βn −β 0 wj − wi  n  ti qjn 0 n→∞ 1 −→ γ αi , j=ti−1 +1 βn −β 0 bj − bi √  qjn  n 0 n→∞ 1 wj − wi −→ γ νj , 0 β√ n −β  qjn  n 0 n→∞ 1 −→ γ ηj , βn −β 0 bj − bi

since Θk contains a neighborhood of the parameters realizing the true regression function Fθ0 . 

An On-Line Learning Radial Basis Function Network and Its Application Nini Wang1,2, , Xiaodong Liu1,2 , and Jianchuan Yin3 1

Research Center of information and Control, Dalian University of Technology, 2 Linggong Road, 116024 Dalian, China 2 Department of Mathematics, Dalian Maritime University, 1 Linghai Road, 116026 Dalian, China 3 College of Navigation, Dalian Maritime University, 1 Linghai Road, 116026 Dalian, China [email protected], [email protected], [email protected]

Abstract. To improve the on-line predictive capability of radial basis function (RBF) networks, a novel sequential learning algorithm is developed referred to as sequential orthogonal model selection (SOMS) algorithm. The RBF network is adapted on-line for both network structure and connecting parameters. Based on SOMS algorithm, a multi-step predictive control strategy is introduced and applied to ship control. Simulation results of ship course control experiment demonstrate the applicability and effectiveness of the SOMS algorithm. Keywords: Radial basis function network, sequential learning, predictive capability, ship control.

1

Introduction

Neural networks have gained much popularity in modelling and controlling of nonlinear dynamic systems in recent years. The learning algorithms of neural networks are mainly based on fixed network models. However, it is realized that a considerable part of industry processes, such as ship motion, have time-varying dynamics in their nature. Such processes are also subject to time-varying environment. Thus, robust control of these plants could not be achieved using fixed network models. One solution to cope with processes with time-variations is to develop adaptive neural network models for identification of these processes. Among various type of neural networks, radial basis function (RBF) networks are found to be suitable for on-line adaptation because their linear feature allows linear optimization methods, e.g. least squares method, to be applied [1]. RBF networks possess other merits like best approximation property and simple topological structure. Based on these merits, sequential learning algorithms are developed for RBF networks in recent years, which are suitable for on-line 

This work is supported by the NSF of China under Grant 60575039 60534010 and NSF of Liaoning under Grant 20062169.

F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 196–203, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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applications. They does not need retraining whenever a new observation is received, with advantages of low computation burden and adaptive capability. The most widely used sequential learning algorithms are RAN, RANEKF, MRAN and GGAP-RBF algorithms [2], [3], [4], [5]. We develop in this paper a novel sequential learning algorithm for RBF network referred to as sequential orthogonal model selection (SOMS) algorithm. It adds the trend of new observation as a hidden unit directly, and makes use of the normalized error reduction ratio as criterion for model selection. The algorithm is adaptive to the changes of system dynamics with fast learning speed, while employs only a small number of parameters. Results of ship course control simulation demonstrate the applicability and effectiveness of SOMS algorithm and the proposed predictive control strategy.

2

SOMS Algorithm for RBF Networks

By combining the subset selection scheme of orthogonal least squares (OLS) algorithm [6] with a sequential learning mode, we propose the sequential orthogonal model selection (SOMS) algorithm. For identification of nonlinear systems involving variants of local mean and trend, differences of system input and output are applied to the network. The network structure is variable by selecting units which can represent the trend of system dynamics effectively. The contribution of each hidden unit is measured by its normalized error reduction ratio, which is generalized from the conception of error reduction ratio in OLS algorithm [6], [7]. We introduce a sliding window which is a first-in-first-out sequence. When a new observation is received, the sliding window is updated by incorporating the new observation and discarding the foremost one. The width of sliding window remains fixed. (1) window = [(x1 , y1 ), (x2 , y2 ), · · · , (xN , yN )] where N is the width of the sliding data window. The data in the sliding window are used to represent the dynamics of system with input X ∈ Rn×N and output Y ∈ RN ×m . n and m are dimensions of the input and output, respectively. For nonlinear systems involving variants of local mean and trend, the dynamics of system can be easily tracked by applying a suitable difference on the signal [8]. So we use the differences of the sliding window input X  ∈ Rn×N and output Y  ∈ RN ×m as the input and output to the network, respectively. ⎞ ⎛ x1,1 − x1,0 · · · x1,N − x1,N −1 ⎟ ⎜ .. .. .. X  = [x1 − x0 , . . . , xN − xN −1 ] = ⎝ (2) ⎠ . . . xn,1 − xn,0 · · · xn,N − xn,N −1 is the difference of sliding data window input. ⎛ y1,1 − y0,1 ⎜ ..  T Y = [y1 − y0 , . . . , yN − yN −1 ] = ⎝ . yN,1 − yN −1,1

⎞ y1,m − y0,m ⎟ .. ⎠ . . . . yN,m − yN −1,m ... .. .

is the difference of desired output matrix of the sliding data window.

(3)

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The learning procedure begins with no hidden unit. At each step, the difference of new observation input is added as a new hidden unit directly. The set of candidate units C ∈ Rn×M is then formed together with the existing hidden units. M is the number of candidate hidden units. n is the dimension of the candidate hidden units. Calculate the Gaussian functions of the Euclidean distance between X  and the candidate hidden units, we have response matrix of the hidden units to the input of sliding data window Φ ∈ RN ×M , with φj,k = exp(−

xj − ck 2 ) 2σ 2

1 ≤ j ≤ N, 1 ≤ k ≤ M

(4)

where ck are known as the k-th hidden units, σ a width constant and  ·  the Euclidean norm. We note that the dimension of Φ is N × M , which is different from the square response matrix in OLS algorithm. Transform the set of basis vectors Φ into a set of orthogonal basis vectors using Gram-Schmidt method by decomposing Φ into Φ = W A. The space spanned by the set of wk is the same space spanned by the set of φk . The orthogonal columns of W ∈ RN ×M satisfy Tw } W T W = diag{w1T w1 , . . . , wM M

(5)

Calculate the error reduction ratio of each vector wk : [err]ki =

(wkT yi )2 (wT wk )(y T yi ) k

(6)

i

For a multi-input, multi-output (MIMO) process, m T 2 i=1 (wk yi ) [err]ki = (wkT wk )trace(Y T Y )

(7)

The geometrical interpretation of error reduction ratio is: (wkT yi )2 = cos2 θki wk 2 yi 2

(8)

where θki is the angle between the basis wk and desired output vector yi . vector M According to vector space theory, cos2 θk = 1 for single-output conk=1 M dition. This explains why k=1 [err]k = 1 in OLS algorithm in single output condition. Different from the square response matrix in OLS algorithm, the response matrix in SOMS algorithm is generally not square because the size of sliding data window input Mnumber of hidden  is generally not the same as the [err] > 1 when M > N and units. We can find that M k k=1 k=1 [err]k < 1 when M < N . For evaluating the contribution of hidden units to the trend of system dynamics directly, the normalized error reduction ratio (nerr) is obtained by [err]k [nerr]k = M k=1 [err]k

(9)

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The selection is made for units whose sum of nerr value reaches an accuracy threshold ρ at each step. Assume that [nerr]k1 = max{[nerr]k , 1 ≤ k ≤ M }. If [nerr]k1 < ρ, then select [nerr]k2 = max{[nerr]k , 1 ≤ k ≤ M, k = k1 }. k=k +1 The selection procedure continues until the sum k=kS1 [nerr]k ≥ ρ. Select k1 , . . . , kS and the corresponding hidden units are selected as hidden units. After hidden units being added or pruned at each step, the weights between the hidden layer and output layer are adapted to the difference of the sliding data window output using the linear least mean squares estimation (LLSE) [9]: Θ = Φ+ Y  = (ΦT Φ)−1 ΦT Y 

(10)

  When the difference of output yN +1 is obtained according to input xN +1 , the output can be finally achieved by  yN +1 = yN + yN +1

3

(11)

The Neural Predictive Controller

Predictive control has been widely used in industry in recent years. It can be used to control open loop unstable systems and systems with variable dead time [10], [11]. The structure of the neural predictive controller consists of the controlled plant, the multi-step-ahead predictor, and the optimizer that determining the control action. RBF network derived from SOMS algorithm is implemented on-line to estimate the dynamics of a nonlinear system. The predictive control strategy is featured by implementing the adaptive RBF networks in system identification. The general configuration of the controller is shown in Fig.1. To describe the complexities of systems, concept of nonlinear autoregressive model with exogenous inputs (NARX) system representation is used [12]: yˆ(k + 1) = f (y(k), ..., y(k − ny ), u(k − d), ..., u(k − d − nu ))

(12)

where y(·) and u(·) are discrete sequences of output and input. ny and nu are the maximum lags in the output and input, d is the delay of system, f (·) is the unknown function. Thus the model given by (12) represents a system that may be both dynamic and nonlinear.

Fig. 1. Configuration of RBF network-based predictive control strategy

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Based on the results from the one-step-ahead predictor, multi-step-ahead predictor yields long-range predictions of the output. Since future measurements needed in the computation are not available, (12) cannot be directly employed to provide desired long-range predictions. However, the outputs can be cascaded to itself, so with the previous cycle’s information can predict the future status of the system. Based on (12), the multi-step predictions can be expressed as: yˆ(k + d + 1) = f (y(k + d), ..., y(k + d − ny ), u(k), ..., u(k − nu ))

(13)

yˆ(k + d+ 2) = f (ˆ y(k + d+ 1), ..., y(k + d+ 1 − ny ), u(k + 1), ..., u(k + 1 − nu)) (14) Replace yˆ(k + d + 1) in (14) by (13), it is observed that yˆ(k + d + 2) can be written as a function of y(k+d), ..., y(k+d−ny ), u(k+1), u(k), ..., u(k−nu ). Make successive substitutions up to reach the term y(k + dˆ + Np ) that can be written as a function of y(k + d), ..., y(k + d − ny ), u(k + Np − 1), ..., u(k), ..., u(k − nu ). Np is a natural number defined as prediction horizon. For simplicity, the control horizon Nu is set to 0, that is, the control action remains constant after the k time step, u(k) = u(k + 1) = · · · = u(k + Nu ). Then the value yˆ(k + i), i = 1, ..., Np can be expressed in terms of the currently available sequence y(k), ..., y(k − ny ), u(k), ..., u(k − d − nu ). That is, yˆ(k + i) = g(y(k), ..., y(k − ny ), u(k − d), ..., u(k − d − nu )) i = 1, ..., Np (15) The problem of finding an expression for the function g(·) can be interpreted as a functional approximation problem. Here we consider using RBF network with tapped time delays to approximate the function governed by (15). The RBF network is constructed by the proposed SOMS algorithm. y(k), ..., y(k − ny ), u(k − d), ..., u(k − d − nu ) and y(k + i) are set as the input and output data pairs of network, respectively. The purpose of predictive control is to calculate a control action sequence u(k), ..., u(k + Nu ) such that the difference between the desired output and system output is minimized on a time sequence into the future. The quadratic cost function incorporates a penalty for deviation from nominated set point and a penalty for rapid control action changes. The desired control signal can therefore be selected by minimizing J: J=

Np i=1

[yr (k + d + i) − yˆ(k + d + i)]2 +

Nu

λ[u(k + j) − u(k + j − 1)]2

(16)

j=1

where yr (·) is the desired process output, yˆ(·) is the predicted output, u(·) is the process input, λ is a control weighting factor. Based on the above-mentioned Np -step-ahead neural predictive model, control signal is determined by the cost function minimization (CFM) algorithm. Newton-Raphson (N-R) algorithm is a widely used quadratically converging algorithm for cost function minimization. The low iteration numbers make N-R algorithm faster than other techniques. The computational issues of the NewtonRaphson algorithm are addressed in [13].

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201

Ship Course Control Application

The design of ship control system presents challenges cause ship’s motion is a complex nonlinear system with time-varying dynamics and long time delay. The dynamics of ship vary in case of any changes in sailing conditions such as speed, loading conditions, trim, etc. Similar changes may also be caused by environmental disturbances, such as waves, wind, current, water depth under heel, etc. So we examine the performance of the control strategy by applying it in ship course-tracking control. The ship model in this application is based on the model of ”MARINER” [14]. Simplified nonlinear expressions of 3 degrees-offreedom ship motion are: surge : m(u˙ − vr − xG r2 ) = X

(17)

sway : m(v˙ + ur − xG r) ˙ =Y

(18)

yaw : Iz r˙ + mxG (v˙ + ur) = N

(19)

where m is mass of ship, u and v are surge and sway velocities, r is the yaw rate, Iz is moment of inertia about the z-axis, X and Y are forces in the direction of x and y-axis, respectively, N is the moment of around the z-axis and xG is the center of gravity along x-axis. The objective of our simulation was to steer a ship on setting courses with small deviations as well as avoiding large control actions. The desired courses were set as 10◦ during [0s, 360s], 20◦ during [361s, 720s], and −20◦ during [721s,1200s]. To make the simulation more realistic, influence of wind and random measurement noises were also considered [15]. The wind force was set as Beaufort scale 6. The changes of wind speed and course were illustrated in Fig. 2. Noises were added with standard deviation of 0.1. In the simulation, ship speed was set to 15 knot, rudder angle and rate were constrained to ±20◦ and ±5◦ /s, respectively. The parameters were chosen as

Wind Speed (m/s)

30 25 20 15 10 5 0 0

200

400

600

800

1000

1200

800

1000

1200

Time (s)

Wind Direction (degree)

100

50

0

−50

−100 0

200

400

600

Time (s)

Fig. 2. Wind speed and wind force curve

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Fig. 3. Ship heading course and rudder angle (RBFN predictive control)

Fig. 4. Ship heading course and rudder angle (PID control)

follows: ny = nu = 2, Np = 2, Nu = 0, N = 20, ρ = 0.95. SOMS algorithm was implemented on-line. For comparison, the traditional PID control was also implemented under the same experiment condition. The parameters were tuned as: KP = 8, KI = 0.01, KD = 80. Simulation results are shown in Figs. 3 and 4. We compare Fig. 3 and 4 and find that both neural predictive controller and PID controller can steer ship continues to follow the desired courses with small tracking errors. Although both methods can track the desired course well, the proposed neural controller uses much less rudder actions. It indicates that the neural controller reacts fast to the environmental changes with smooth rudder actions, which shows that the RBF network constructed by SOMS algorithm can react to the change of ship dynamics adaptively, and the predictive control strategy can minimize the effect of long time delay to a considerably low level.

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203

Conclusion

A sequential learning algorithm for RBF network is developed referred to as SOMS algorithm. A multi-step predictive control algorithm is introduced based on RBF network constructed by SOMS algorithm. Simulation results show that the proposed control strategy can control a time-varying nonlinear ship model with quick response and satisfactory anti-interference capability.

References 1. Ghosh, J., Nag, A.: An Overview of Radial Basis Function Networks: New Advances in Design. Physica-Verlag, Heidelberg (2001) 2. Platt, J.: A Resource Allocating Network for Function Interpolation. Neur. Comput. 3, 213–225 (1991) 3. Kadirkamanathan, V., Niranjan, M.: A Function Estimation Approach to Sequential Learning with Neural Network. Neur. Comput. 5, 954–975 (1993) 4. Lu, Y.W., Sundararajan, N., Saratchandran, P.: A Sequential Learning Scheme for Function Approximation by Using Minimal Radial Basis Function Neural Networks. Neur. Comput. 9, 461–478 (1997) 5. Huang, G.B., Saratchandran, P., Sundararajan, N.: A Generalized Growing and Pruning RBF (GGAP-RBF) Neural Network for Function Approximation. IEEE Trans. Neur. Netw. 16, 57–67 (2005) 6. Chen, S., Cowan, C.F.N., Grant, P.M.: Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks. IEEE Trans. Neur. Netw. 2, 302–309 (1991) 7. Chen, S., Grant, P.M., Cowan, C.F.N.: Orthogonal Least Squares Algorithm for Training Multi-Output Radial Basis Function Networks. IEE Proc. 139, 378–384 (1992) 8. Chng, E.S., Chen, S., Mulgrew, B.: Gradient Radial Basis Function Networks for Nonlinear and Nonstationary Time Series Prediction. IEEE Trans. Neur. Netw. 7, 190–194 (1996) 9. Jang, J.S.R., Sun, C.T., Mizutani, E.: Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence. Prentice-Hall, Upper Saddle River (1997) 10. Qina, S.J., Badgwell, T.A.: A Survey of Industrial Model Predictive Control Technology. Contr. Eng. Pract. 11, 733–764 (2003) 11. Pottmann, M., Seborg, S.E.: A Nonlinear Predictive Control Strategy Based on Radial Basis Function Models. Comp. Chem. Eng. 21, 965–980 (1997) 12. Chen, S., Wang, X.X., Harris, C.J.: NARX-Based Nonlinear System Identification Using Orthogonal Least Squares Basis Hunting. IEEE Trans. Contr. Syst. Tech. 16, 78–84 (2008) 13. Soloway, D., Haley, P.: Neural Generalized Predictive Control: A Newton-Raphson Implementation. In: IEEE CCA/ISIC/CACSD, pp. 277–282. IEEE Press, New York (1996) 14. Chislett, M.S., Strom, J.T.: Planar Motion Mechanism Tests and Full-Scale Steering and Manoeuvring Predictions for a Mariner Class Vessel. Technical Report, Hydro and Aerodynamics Laboratory (1965) 15. Zhang, Y., Hearn, G.E., Sen, P.: A Neural Network Approach to Ship TrackKeeping Control. IEEE J. Ocean. Eng. 21, 513–527 (1996)

A Hybrid Model of Partial Least Squares and RBF Neural Networks for System Identification Nini Wang1,2, , Xiaodong Liu1,2 , and Jianchuan Yin3 1

Research Center of information and Control, Dalian University of Technology, 2 Linggong Road, 116024 Dalian, China 2 Department of Mathematics, Dalian Maritime University, 1 Linghai Road, 116026 Dalian, China 3 College of Navigation, Dalian Maritime University, 1 Linghai Road, 116026 Dalian, China [email protected], [email protected], [email protected]

Abstract. A novel learning algorithm is presented to construct radial basis function (RBF) networks by incorporating partial least squares (PLS) regression method. The algorithm selects hidden units one by one with PLS regression method until an adequate network is achieved, and the resulting minimal RBF-PLS (MRBF-PLS) network exhibits satisfying generalization performance and noise toleration capability. The algorithm provides an efficient approach for system identification, and this is illustrated by modelling nonlinear function and chaotic time series. Keywords: Radial basis function network, Partial least squares, System identification, Generalization capability.

1

Introduction

The radial basis function (RBF) networks are efficient tools for interpolation problems and achieved considerable success in application to system identification [1]. RBF networks possess merits like simple topological structure, quick convergence speed and best approximation property [2]. Since there is only one single layer of adjustable parameters in RBF network, they can be trained efficiently by linear estimation techniques. However, in the cases of highly correlated inputs where an ordinary least squares approach usually results in ill-conditioned problems, the RBF networks may subsequently suffer from the problem of large variance in generalization. To eliminate the unfavorable effects of multicollinearity and improve the generalization capability of resulting RBF networks, many research activities have been performed. One effective way is to implement partial least squares (PLS) regression in constructing RBF networks. PLS regression has been successfully implemented in many applications such as identification and control [3]. Taking 

This work is supported by the NSF of China under Grant 60575039 60534010 and NSF of Liaoning under Grant 20062169.

F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 204–211, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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joint advantages of the linear nature of RBF network and the power of PLS in processing data, the PLS method is introduced to construct RBF networks (RBF-PLS) [4], [5], [6], [7]. The above-mentioned RBF-PLS regression methods [4], [5], [6], [7] set the full data set as hidden units. Such methods have several drawbacks such as large network dimension and heavy computational burden. The large dimension of network may also cause over-fitting and deteriorate the generalization capability of network. In this paper, we develop a novel RBF-PLS learning algorithm for constructing RBF networks referred to as minimal RBF-PLS (MRBF-PLS) algorithm. In the algorithm, hidden units are added sequentially, the resulting network possesses parsimonious structure and satisfactory generalization performance. By combining the advantages of PLS method and parsimonious network structure, the network is able to reduce generalization variance to a considerably low level. Identification experiments of nonlinear system and chaotic time series are conducted and results demonstrate the effectiveness of the algorithm.

2

Minimal RBF-PLS Algorithm

2.1

RBF-PLS Algorithm

For a three-layer RBF network, assume that the input vector x and output vector y consist of n and l independent variables respectively, and the data set available for modelling consists of m observations. So the dimension of the input matrix X is m × n, and that of the output matrix Y is m × l. In this paper, Gaussian functions are used as RBFs to carry out nonlinear transformation of X and form activation matrix A ∈ Rm×m , the elements of A are defined as aij = exp(−cj − xi 2 /σj 2 ) i, j = 1, 2, . . . , m

(1)

where xi is a vector consisting of the values of variables taken from the i-th observation, aij is the element of A at the i-th row and j-th column,  ·  is a norm which denotes the Euclidean distance here, cj and σj are the center and width of the j-th RBF hidden unit, respectively. The parameter cj here is set by cj = xj ,

j = 1, 2, . . . , m .

(2)

The parameters σj in this paper are calculated by e  xi − xj  m i=1 m

σj1 = σj2 = · · · = σjn =

(3)

where e is a assigned positive number. Thus the resulting symmetrical matrix A has ones on its diagonal. Then, the PLS procedure will be applied to the matrices A and Y . A and Y are projected on the low-dimensional score matrix T , respectively:

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A = t1 p1T + A1 ,

Y = t1 r1T + Y1

(4)

where t1 is the first component of A, p1 and r1 is the first loading vector of A and Y , A1 and Y1 are the residual matrix of A and Y , respectively. After N -iterations of projection, we get T +A A = t1 p1T + t2 p2T + . . . + tN pN N

(5)

T +Y Y = t1 r1T + t2 r2T + . . . + tN rN N

(6)

where ti is the N -th score vector of A, pi is the N -th loading vector of A, ri is the N -th loading vector of Y , AN and YN are the residual matrices of A and Y after N -th iteration of projection, respectively. Each ti (1 ≤ i ≤ N ) can be represented as the linear combination of A, so the linear PLS model is set up as: Y = T R + FN = AW R + FN ,

(7)

where T ∈ Rm×N is the low-dimensional score matrix of A, R ∈ RN ×l represents the regression coefficient matrix, W ∈ Rm×N is the transformation matrix of A, FN is the residual matrix with the dimension of m× l. Because of the fact that T is a linear combination of Gaussian functions that will maximize the covariance between A and Y , N plays an important role in the RBF-PLS network. When N is determined, the RBF-PLS network can be obtained and used for prediction or other purposes. So, by combining PLS algorithm with RBF network, the nonlinear relation between A and Y is transformed to problem in linear algebra. After training, the RBF-PLS network can be used to make predictions on new observations Yp = Ap W R,

(8)

where Ap is the activation matrix of Xp which is used for prediction, Yp is the resulting dependent matrix. 2.2

MRBF-PLS Algorithm

The RBF-PLS methods [4], [5], [6], [7] construct RBF networks by projecting raw data into latent variables to reduce high dimensional and strong correlated data to a much smaller data dimension which can be easily interpreted. In general, the number of data can be very large. An adequate model with acceptable accuracy, however, may need only a few hidden units. To achieve a parsimonious network, we present in this paper a new learning scheme to implement PLS method referred to as minimal RBF-PLS (MRBF-PLS) method. In the proposed method, first select hidden units from data set one by one, then PLS regression method is implemented to the selected data. Error reduction ratio (err) [8] is an efficient index to evaluate the contribution of each sample to the output, here we choose hidden units from samples by their err values. Samples with large err are chosen as hidden units for each step. The selected samples

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and iterations of extraction are used to determining the centers and weighting matrix of RBF network. The procedure stops when the accuracy threshold is reached. This forward center selection procedure leads to smaller dimension of network and consequently achieves better generalization capability [9]. The procedure of MRBF-PLS algorithm is illustrated as follows: 1. Calculate and permute all the training samples in descending order according to the err value of each training sample. 2. At the k-th step, sort the err value in descending order. Take the sample with largest err from input set X and the corresponding output from output set Y , form the training input-outpu sets U and V . The selected input uj (j = 1, 2, ..., k) in U is selected as a RBF hidden unit. 3. Perform PLS method to data in U and V , we get the generalization model: vp1 = Ap1 W1 R1 ,

(9)

where the dimensions of activation matrix Ap1 and W1 are k × k and k × N , respectively. The dimension of R1 is the same as in (8). In implementing PLS regression, N is determined by applying the accuracy threshold e1 as required. For example, the selected threshold of root mean square error (RMSE):  k 2 j=1 [v(j) − vp1 (j)] RM SE = < e1 . (10) k 4. Validate the acquired model by calculating the value of RM SE by the full data set X under the current N by:  m RM SEN =

i=1 [vp2 (i)

− y(i)]2

m

(11)

in which vp2 = Ap2 W1 R1 ,

(12)

where the dimension of Ap2 becomes m × k, with cj = u j ,

j = 1, 2, . . . , k .

(13)

5. As long as RM SEN decreases consistently, repeat step 1-3. 6. Stop the procedure when the value of RM SEN is larger than the previous one, or when k = m. That is, the RM SEN reaches its minimum point [6]. The vectors in U are set as centers of RBF network hidden units.

3 3.1

Modelling Examples Sine Function

Consider using the RBF networks to approximate the scalar function: f (x) = sin(2πx)

0 ≤ x ≤ 1.

(14)

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In this experiment, 100 training samples were generated from y = f (x) + e, with x was uniformly distributed in (0, 1). Another 100 noise-free data f (x) with equally spaced x were also generated as testing data set for model validation. The sine function provides an example for illustrating the construction of a mapping from R1 to R1 . The training data were set with extremely noisy, with zero mean and standard deviation of 0.4. For comparison, the RBF-PLS method and the widely used OLS method [8] were also performed. All the algorithms were performed under the same conditions with e1 = 0.4 for RM SE as accuracy thresholds. The comparison of the network mapping constructed by the MRBFPLS method, the RBF-PLS method and OLS method is shown in Fig. 1. The noisy training points and the underlying function f (x) are also plotted in Fig. 1. We notice from Figs. 1 that the network constructed by the MRBF-PLS method is able to eliminate the unfavorable effect of over-fitting and unfavorable effect of large noise, possesses better generalization performance than other two algorithms with smaller predictive error. The standard deviation of noise was increased to 0.5 and another 100 training data were generated as training data. e1 was set to 0.5. The same test was conducted and the results are shown in Tab. 1. It can be seen clearly that under the same training conditions, the network constructed by MRBF-PLS algorithm has better generalization results. The reason is that the MRBF-PLS algorithm takes the advantage of the parsimonious network structure and the power of PLS to eliminate the unfavorable effects of collinearity, reduce generalization variance to a considerably low level. 3.2

Lorenz Time Series

In this example we illustrate the mapping from Lorenz time series, which is a standard benchmark for testing model fitting algorithms. The Lorenz time-delay equation is expressed by a set of three nonlinear differential equations [10]: dx = −ax + ay, dt

dy = −xz + rx − y, dt

dz = xy − bz. dt

(15)

2 Training data Noise-free function OLS RBF-PLS MRBF-PLS

1.5

1

f(x)

0.5

0

−0.5

−1

−1.5

−2 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Fig. 1. Performance of three algorithms in sine function modelling with σ = 0.4

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20 15 10 5

x

0 −5 −10 −15 −20 30 20 10 0 −10 −20 −30

y

5

10

15

20

25

30

35

40

45

z

Fig. 2. Phase diagram of Lorenz attactor time series 20

15

Approximation Curve

10

5

0

−5

−10

−15 Lorenz time series Approximation using MRBF−PLS method −20 1000

1200

1400

1600

1800

2000 1000 Observations

1200

1400

1600

1800

Fig. 3. Approximation curve of Lorenz time series using MRBF-PLS algorithm

Above equations generate chaotic time series with short-range time coherence. The constants a, r, and b determine the system’s behavior. These three equations exhibit chaotic behavior and are sensitive to initial conditions. The values are set as: a = 10, r = 28 and b = 8/3. In this work, the behavior of x(t) is forecasted by RBF network constructed by the MRBF-PLS algorithm. For purposes of comparison, the series was predicted with one step ahead using eight past samples: sn−1 , sn−2 , · · · , sn−8 . Hence, the n-th input-output data for the network to learn are xn = [sn−1 , sn−2 , sn−3 , sn−4 , sn−5 , sn−6 , sn−7 , sn−8 ]T

(16)

y n = sn .

(17)

and The one-step-ahead predicted value at time n is given by zn+1 = f (xn+1 )

(18)

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N. Wang, X. Liu, and J. Yin Table 1. Generalization Performance of Three Learning Algorithms Method

Training error Testing error Number of (RM SE) (RM SE) Hidden Units

Sine Function (σ = 0.4) OLS RBF-PLS MRBF-PLS

0.3819 0.3834 0.3921

0.1156 0.1126 0.0663

5 100 23

Sine Function (σ = 0.5) OLS RBF-PLS MRBF-PLS

0.4899 0.4950 0.4989

0.1275 0.1123 0.0735

5 100 32

Lorenz Series (σ = 0.1) OLS RBF-PLS MRBF-PLS

0.0991 0.0975 0.0981

0.0572 0.0497 0.0428

26 500 41

where f (xn+1 ) is the network output at time n. The one-step-ahead prediction error is (19) e = sn+1 − zn+1 . This Lorenz chaotic time series model provides a good example for illustrating the construction of a mapping from R8 to R1 . The chaotic behavior of this example is represented in Fig. 2. We notice that the adjacent data are strongly correlated after transient behavior in the beginning. Noise was added to series with standard deviation of 0.1. The accuracy threshold e1 was set as 0.1. We present in Fig. 3 the original Lorenz time series and generalization result of network constructed by the MRBF-PLS algorithm. It is shown that the output of the model and the target values are well fitted. To measure the performance of the various RBFs, we compared the training RM SE, the testing RM SE and the number of the hidden units. It is noted that under the same training conditions, MRBF-PLS algorithm performs better than the other two methods, is able to eliminate the unfavorable effects of multicollinearity efficiently. We also include in table 1 the detailed experiment results, such as training error, testing error and number of RBF hidden units for the three algorithms, RBF-PLS, OLS and MRBF-PLS, in the above experiments under the same training conditions.

4

Conclusion

When RBF network is applied to process modelling, it may give large generalization variance under the condition of input collinearity and strong noises.

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Whereas the generalization variance can be greatly reduced by integrating PLS method in constructing RBF networks, and it can be further reduced by forming a parsimonious network structure. The proposed MRBF-PLS method takes joint advantages of these effects. The results of examples indicate that the MRBF-PLS algorithm can achieve parsimonious networks that generalize well, is an efficient means for system identification.

References 1. Chen, S., Wang, X.X., Harris, C.J.: NARX-Based Nonlinear System Identification Using Orthogonal Least Squares Basis Hunting. IEEE Trans. Contr. Sys. Tech. 16, 78–84 (2008) 2. Moody, J., Darken, C.: Fast Learning in Networks of Locally-Tuned Processing Units. Neur. Comp. 1, 281–294 (1989) 3. Wang, H.W.: Partial Least Squares Regression Method and Applications. National Defence Industrial Press, Beijing (1999) 4. Walczak, B., Massart, D.L.: The Radial Basis Functions - Partial Least Squares Approach as a Flexible Non-linear Regression Technique. Anal. Chim. Act. 331, 177–185 (1996) 5. Li, C.F., Ye, H., Wang, G.Z., Zhang, J.: A Recursive Nonlinear PLS Algorithm for Adaptive Nonlinear Process Modeling. Chem. Eng. Tech. 28, 141–152 (2005) 6. Yan, X.F., Du, W.L., Qian, F.: Development of a Kinetic Model for Industrial Oxidation of p-Xylene by RBF-PLS and CCA. Proc. Sys. Eng. 50, 1169–1176 (2004) 7. Wang, Y., Rong, G., Wang, S.Q.: PLS Algorithm for Radial Basis Function Networks. In: 37th IEEE Congress on Decision and Control, pp. 4748–4753. IEEE Press, New York (1998) 8. Chen, S., Cowan, C.F.N., Grant, P.M.: Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks. IEEE Trans. Neur. Netw. 2, 302–309 (1991) 9. Reed, R.: Pruning Algorithms – A Survey. IEEE Trans. Neur. Netw. 4, 740–747 (1993) 10. Dudul, S.V.: Prediction of Lorenz Chaotic Attractor Using Two-Layer Perceptron Neural Network. Appl. Soft Comput. 5, 333–355 (2005)

Nonlinear Complex Neural Circuits Analysis and Design by q-Value Weighted Bounded Operator Hong Hu and Zhongzhi Shi Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Science Beijing 100191, China [email protected]

Abstract. Dynamical analysis of complex neural circuits, especially for chaotic nonlinear neural circuits is a difficult task. In this paper, a novel approach to understand the nonlinear dynamic attributes of a neural circuit by using approximate logical model of q-Value Weighted Bounded Operator is discussed, and we proved that if a neural circuit works in a non-chaotic way, a suitable fuzzy logical framework which is an approximate logical model of neural cells can be found and we can analyze or design such kind neural circuit similar to analyze or design a digit computer, but if a neural circuit works in a chaotic way, fuzzy logical frameworks of neural cells are different under different precisions, we should use a multi scale approximate police(see Def.5) for understanding the function of such neural system with arbitrary small precisions. Keywords: Fuzzy logic, chaos, neural dynamic analysis, neural circuit, fuzzy logical framework of a neural circuit.

1 Introduction There is a close relationship between fuzzy logical and neural systems, and many papers investigate this subject (Canuto AMP, &Fairhurst M(2002)[1];Yager R(1993)[2]; S-B Cho, &J H Kim(1995)[6], Kim S S(1998)[9];). But most of these investigations focus on finding new controlling or classifying abilities of neural systems by combining fuzzy logical and neural systems. Few researchers try to find the equality or similarity between fuzzy logical and neural dynamics(Li H X , &C. L. Chen. P(2000)[4]; JL Castro(1995)[3]). For feed forward system, JL Castro 1995 proved that Fuzzy logic controllers using fuzzy rules are universal approximators, later, Li H X .et.al. [4] shows a proof of the equality between a forward neural network and a fuzzy logical inference. In fact, recurrent neural networks can also be simulated by fuzzy logical systems. In this paper, we first prove that fuzzy logic has same representation ability as neural network to describe a dynamical system, and then we discuss the approaches of hybrid design of neural network based on the fuzzy logical theory and neural network learning approach. Neural circuits can work in dynamical way or static way. Neural circuits which work in a dynamical way should be described by partial differential equations, and are denoted as “dynamical neural circuits”. Static points or stable states are very important for

（

）

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dynamical analysis of a neural circuit. Many artificial neural circuits are just abstract of static points or stable states of real neural circuits, e.g. BP neural circuits, such kind artificial neural circuits work in a static way and are denoted as “static neural circuits”. In a neural circuit, if neural cells take logistical function as their output function, we can prove that such kind neural cells can be simulates by fuzzy logical formulas. For dynamical neural circuits, there are two kinds such neural cells. For the 1st kind neural cells, the fuzzy information of such neurons is encoded in the amplitude of the output pulse. For the 2nd kind neural cells, information in such neural cells is encoded in the phase of output pulses or spikes; the amplitude of spikes doesn’t carry any useful information. In this paper we only discuss the 1st kind neurons. For the 1st kind neurons, one example of them is described in the equation (1). Such cell’s membrane potential is transferred to output by a sigmoid-like function. Only the amplitude of output pluses carries meaningful information. The rising or dropping time (Δt) of output pluses conveys no useful information and is always neglected. According to [5], the neural circuits described by (1) are based on biological data (Barlow et al. 1967; Hubel & Wiesel 1970;Rockland & Lund 1983, Gilbert & Wiesel 1983; White 1989; Poggio 1992; Von der Heydt et al. 2000; Bakin et.al. 2000). If the neural circuit described by (1) has a layered structure, the fixed point of a neuron at layer l is (2). (2) is just a back propagation neural circuit, so a BP neural circuit can be viewed as an abstract of static points or stable states of a real neural circuit described by (1). U i = − a i ⋅ U i +

∑ w ′V ij

j

Vi =

1

j

+ β ∑ w ik I k ; k

.

(1)

(ex p ( − U i + Ti ) + 1)

U

l ,i

= β

∑

w l ,i , k V l − 1,i / a i

k

V l ,i =

1

(e x p ( − U

l ,i

.

(2)

+ Ti ) + 1 )

Only the first kind of neurons is dealt with in this paper. For the second kind neurons, before we can analyze the fuzzy logical framework of the second type of neuron we need to transform phase information to fuzzy value; this can be done by coding phase information into the threshold of membrane potential. The analysis of nonlinear neural circuits is an intricate task. Many studies only deal with simple neural circuits with simple functions. In most cases, the connection matrices should be simplified; for example, a symmetric matrix is needed in [10]. However, this sort of analysis is almost impossible to understand the ability of a complex neural circuit for the difficulty of nonlinear analysis, e.g. the primary visual cortex. On other hand, computer science has made a great progress; nowadays, we can build very complex computers by assembling integrated circuits. Why don’t we use similar techniques to analyze or build complex neural circuits? Or can we analyze a complex neural circuit by separating it into components or basic elements? The answer is that if a neural circuit works in a non-chaotic way, a suitable fuzzy logical framework can be found and we can analyze or design such kind neural circuit similar to analyze or

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design a digit computer, but if a neural circuit works in a chaotic way, every fuzzy logical framework of neural cells which must be an approximate model of neural cells at a definite precision is unable to reveal all details of a chaotic neural circuit, the reason is that for a chaotic system any difference in the initial state will cause unpredictability in future. We should use some approximate police to understand the function of chaotic neural system. An approximate police is a set of multi scale approximations based on equivalence relations on a real vector space which can approximate infinite aperiodic behave of a chaotic system to a finite periodic behave at arbitrary precisions. For neural circuits described by (1), the fuzzy logical framework can be viewed as their basic elements. Such systems can be transformed from one fuzzy logical framework to another by changing coefficients in (1). We proved that the neural circuits described by (1) can simulate almost all neural circuits described by first order differential equations with an arbitrary small error (see the theorem 2). Fuzzy logical approach of a nonlinear dynamic neural circuit designing has two applications. a) The design of nonlinear dynamic neural circuits is an intricate task. In the neural science, it is almost impossible to design a neural model for a complex brain system by using partial differential functions; many studies only deal with simple neural circuits with simple functions. In most cases, the connection matrices should be simplified; for example, a symmetric matrix is needed. However, this sort of design is almost impossible to simulate a complex neural circuit, e.g. the primary visual cortex. But under the help of logical design, the things become simple. Computer science has made a great progress; nowadays, we can build very complex computers by assembling integrated circuits. Why don’t we use similar techniques to build complex neural circuits? The answer is yes. We prove that for neural circuits described by (1) have universal meaning, such that almost all neural circuits described by first order differential equations with continuous functions in it can be simulated by them with arbitrary small error(see the Theorem 2), and the fuzzy logical framework can be viewed as their basic elements. Such systems can be transformed from one fuzzy logical framework to another by changing coefficients in (1). b). The logical relations in a recurrent neural circuit can help us to understand the function of this neural circuit especially in the chaotic case.

2 q-Value Weighted Bounded Operator and the Fuzzy Logical Framework of Neural Circuit Suppose the dynamical behavior of a neural circuit can be described as G G G G y ( t ) = F ( x ( t ) ) where X ( t ) is the input at time t and y ( t ) is the output at t. If G G G ( x ( t ) ) is a fuzzy logical function (denoted as FLF) with variables X ( t ) , we G G again define y G ( t ) = G ( x ( t ) ) as the dynamical behavior of a fuzzy logical function.

，

，

Definition 1 (Difference error of a fuzzy logical function). If XG ( t ) is input and the G region of input X ( t ) is D. The difference error between a fuzzy logical function G and a neural circuit F is defined as:

Nonlinear Complex Neural Circuits Analysis and Design

errG =

D yn am ical case: S tatic case: errG =

G

G

215

G

∫ | G ( x ( t ) ) − F ( x ( t )) |d t

G

G

∑ | G ( x ) − F ( x ) | , w h ere F ( x ) is th e fix ed p o in t o f F G x∈ D

(3)

.

Definition 2 (The structure of a fuzzy logical function). If S 1 is a set of fuzzy logical functions(FLF), and a fuzzy logical function f ( x1 , x 2 , ..., x n ) can be represented by

∧

∨

the combination of FLFs in S 1 with fuzzy operators " " and " " , but with no parenthesis, e.g. can be represented by f (x1, x2, x3, x4) =x1 ∧x2 ∨x2 ∧x3 ∨x4

f1 ( x1 , x2 ) = x1 ∧ x2 and f 2 ( x2 , x3 , x4 ) = x2 ∧ x3 ∨ x4 as f 1 ( x1 , x 2 ) ∨ f 2 ( x 2 , x 3 , x 4 ) , then the FLFs in S 1 is denoted as the 1st layer sub fuzzy logical functions (1st FLF ) of f ( x1 , x 2 , ..., x n ) , similarly, every 1st FLF in S 1 which is not just a variable xi has its own 1st layer sub FLFs which are denoted as the 2nd layer sub FLFs of th f ( x1 , x 2 , ..., x n ) …. , and every k layer non variable sub FLF can have its sub fuzzy logical functions which are (k+1)th layer sub FLFs of f ( x1 , x 2 , ..., x n ) . In this way, f ( x1 , x 2 , ..., x n ) has a layered structure of sub fuzzy logical functions, we denote

such

layered

f ( x1 , x 2 , ..., x n )

may

structure have

as

several

the

structure

equivalent

of

formats,

f ( x1 , x 2 , ..., x n )

so

the

.

structure

of f ( x1 , x 2 , ..., x n ) is not unique. If a sub fuzzy logical function is just f ( x1 , x 2 , ..., x n ) itself, then f ( x1 , x 2 , ..., x n ) has recurrent structures, otherwise, f ( x1 , x 2 , ..., x n ) has tree kind structures.

Definition 3 (The fuzzy logical framework). Suppose Ωis a set of fuzzy logical functions, a fuzzy logical framework for a neural circuit F is the best fuzzy logical function G inΩ satisfied the constrain that G has smallest errG i.e. errG = min errG ' . ' G ∈Ω

If there is a one to one onto mapping from neurons in a neural circuit F to the all layers sub fuzzy logical functions in a G's structure, such kind fuzzy logical framework is denoted as structure keeping fuzzy logical framework. After the theory of fuzzy logic was conceived by Zadeh in 1965, many fuzzy logical systems have been presented; for example, the Zadeh system, the probability system, the algebraic system, bounded operator system, etc. We can prove that a neural circuit with suitable coefficients described by (2) can have a structure keeping fuzzy logical framework of the weighted bounded operator with a small error (for the sake of pages, we omitted the proof. The Theorem 1 proves that q-value weighted fuzzy logical functions can precisely simulate neural circuits described by (2) with arbitrary small error, or vice versa, i.e. every neural circuit described by (2) can have a fuzzy logical framework of the weighted bounded operator with arbitrary small error, but unfortunately, usually, such kind fuzzy logical framework is not structure keeping. Definition 4 (Bounded operator F( ⊕ f , ⊗ f )).

Bounded product: x … f y

max(0, x  y  1) , and Bounded sum: x ⊕ f y = min(1, x + y ) , where 0 ≤ x, y ≤ 1 .

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In order to simulate neural cells, it is necessary to extend the Bounded Operator to Weighted Bounded Operator. The fuzzy formulas defined by weighted bounded operators are denoted as q-value weighted fuzzy logical functions. Weighted Bounded product: p1 ⊗ f p2 = F⊗ ( p1, p2 , w1, w2 ) = max(0, w1 p1 + w2 p2 − (w1 + w2 −1)q) , f Weighted Bounded sum: p1 ⊕ f p2 = F⊕ ( p1, q2 , w1, w2 ) = min(q, w1 p1 + w2 p2 ) where f

0 ≤ p1 , p2 ≤ q . For association and distribution rules, we define: ( p1Δ f p2 )Θ f p3 = FΔ f ( FΘ f ( p1 , p2 , w1 , w2 ), p3 ,1, w3 ) and

p1Δ f ( p2Θf p3) = FΔ f ( p1, FΘf ( p2, p3, w2 , w3 ), w1,1) , Here Δ f , Θf =⊗f or ⊕f . We can prove that ⊕ f and ⊗ f follow the associative condition and the Demorgan Law. We try to show that a layered neural circuit can be simulated by a q-value weighted fuzzy logical function. Theorem 1. If f ( x1 , ..., x m ) is a continuous mapping from [0,1]m to [0,1]q , for anyε>0, we can build a layered neural circuit Ψdefined by equation (1), and its fixed point can be viewed as a continuous map F(x1,…,xm)=(F1(x1,…,xm),…, Fq(x1,…,xm)) from [0,1]m to [0,1]q, such that | F ( x1 , ..., x m ) − f ( x1 , ..., x m ) |< ε , here x1,x2,…,xm are k inputs of the neural circuit, and if there are totally m>2 layers in such a layered neural circuit Ψ, its first m-1 layers use the binary logic , i.e. its first m-1 layers simulates Boolean formulas, and the q-value fuzzy logic is only used in the last layer, such neural circuit is denoted as “Boolean layered neural circuit”. For more, for an arbitrary layered neural circuit Ψ which has a fixed point function is F(x1,…,xm), we can find a q-value fuzzy logical function F’(x1,…,xm)=(F’1(x1,…,xm),…, F’q(x1,…,xm)) of weighted Bounded operator , such that | F ( x1 , ..., x m ) − F ' ( x1 , ..., x m ) |< ε . Note: In this paper, the number of layers in a neural circuit is the number of weights’ layers. Proof. We can use the universal approximation theorem(Simon Haykin(1999)7) and the theorem 2 in 8 to prove this theorem. □ We can extend the Theorem 1 to a much more general case, the Theorem 2 tries to prove that all kind recurrent neural circuits described by the 1st order partial differential equation (PDEs)(4) can be simulated by neural circuits described by (1). The 1st order partial differential equation (4) has a strong ability to describe neural phenomena. The neural circuit described by (4) can have feedback. For the sake of the existence of feedback of a recurrent neural circuit, chaos will occur in such a neural circuit. ⎧ x 1 = − a 1 x 1 + ⎪ x = − a 2 x 2 + 2 ⎪ ⎨ ⎪ ⎪ x = − a n x n + ⎩ n

w

1

w

2

f1 ( x f

2

1

, x

, ..., x

2

) + u

n

1

( x

1

, x

2

, ..., x

n

) + u

2

( x

1

, x

2

, ..., x

n

) + u

n

• • • •

w

n

f

n

(4)

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Where every f i ( x1 , x2 ,..., xn ),1 ≤ i ≤ n , is a continuous , and bounded function in the domain of trajectory space TR and we suppose every 0 ≤| xi |≤ C , i = 1, 2,..., n . Theorem 2. In a finite time range, 0 ≤ t ≤ T , every neural circuit described by the equation (4) can be simulated by a neural circuit described by the equation (1) with an arbitrary small error ε > 0 , and such kind neural circuit takes a Boolean layered neural circuit as its feedback part. Proof. omitted for the sake of pages.

□

3 The Basic Neural Circuits and Nonlinear Complex Neural Circuits Analysis and Design Electronic digit circuits can be classified as combinatorial circuits and time serial circuits, neural circuits can also be classified into similar two kind neural circuitscombinatorial neural circuits and time serial neural circuits. Combinatorial neural circuits: A circuit is defined as combinatorial neural circuit if its output at time t is determined by the inputs at time t. Time serial neural circuits: A circuit is defined as a time serial neural circuit if its output at time t and the next state S(t+1) are determined not only by the inputs at time t, but also by the neural circuit’s state at time t, S(t), and before time t. A time serial neural circuit has some kind feedback. The neural oscillator and neural registers are two kind simplest time serial neural circuits. For the sake of the existence of feedback, chaos will occur in a time serial neural circuit. We can prove that chaos may occur in a neural system described by (1). Theorem 3 Chaos may occur in a neural circuit described by (1). Proof: omitted for the sake of pages.

□

A chaotic behavior is different with ordinary non chaotic behavior, so when we design a time serial neural circuit, two kind approaches should be considered. The first approach tries to design a time serial neural circuit with no chaos under a definite precision, and the other tries to design a chaotic time serial neural circuit under a set of precisions which can continuously control the calculation error to arbitrary small levels. In this section we discuss this problem. All binary time serial circuits in digit computers (with finite bits storage) work in a periodic way, but chaos will cause a time serial neural circuit works in an aperiodic way. In order to understanding the function of a time serial neural circuit which works in a chaotic way, it is necessary to approximate chaotic neural circuits at arbitrary precision and make chaotic neural circuits work in a periodic way(Turing computable). Roughly speaking, the reason is that only finite bits are needed to represent a rational number, but for an irrational number, infinite bits are needed. In order to simulate the neural circuit, neural cells which are simulated by Turing machines should run infinite steps to compute irrational numbers. From engineering point of view, it is impossible to wait a Turing machine to run infinite steps, so approximation

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of irrational numbers is necessary in order to understand the function of a chaotic neural circuit. Definition 5 (Approximate police). We can use an equivalence relation E to approximate real number vectors to rational number vectors. All vectors in same equivalent class of the quotient space [Rk|E] of an equivalence relation in the k dimensional real space Rk are approximated by same rational number vector denoted as a fuzzy granular vector. If Gk is the set of all equivalence relations on the k dimensional real space Rk, then Gk is a semi-order space or lattice. The order in Gk is defined as : if E1 and E2 are two equivalence relations and every equivalent class e1 in the quotient space [Rk|E1] is a subclass of an equivalent class e2 in the quotient space [Rk|E2], then E1 ≤E2 ,where an equivalence relation Ei divides Rk into a set of equivalent classes [Rk|Ei]. If Ck is a subset of Gk and every equivalent class e in [Rk|E] for all E in Ck is a connected convex region in Rk, then Ck is defined as a k dimensional approximate police, and equivalent classes of such Ck are intuitively denoted as granules. we denote d ( D ) = s u p ( | x − y |) as the diameter of a granule D , r ( Ei ) = max (d ( D )) D∈[ R k | Ei ]

x , y∈ D

as the rough-rate and p ( E i ) = min ( d ( D )) as the precision of an equivalence k D∈[ R | Ei ]

relation Ei in the k dimensional approximate police Ck. . In the following pages, we suppose that the number of neural cells is finite. For the sake of simplicity, the initial state of a time serial neural circuit is included in its G initial input I (0) . If there are k neural cells, at least a k dimensional multi scale approximate police should be used for inner computing. Approximate polices can be used in the system time t , the input, inner computing output and feedback of the output of a neural circuit. In the following discussion, we suppose an 1-dimensional multi scale approximate police is used in the system time t, so the system time can be represented as steps t=0,1,2,3,…. under an equivalence relation with precision>0. In this case, we can use an automorphism from Rn to Rn to compute the discrete trajectory of a time serial neural circuit. Such kind automorphism can be achieved by changing the differential equation (4) to the discrete difference approximate equation (5).

，

⎧ x1 ( t + Δ t ) = (1 − a1 ) x1 ( t ) + w1 f 1 ( x1 ( t ), x 2 ( t ), ..., x n ( t )) + u1 ⎪ x ( t + Δ t ) = (1 − a ) x ( t ) + w f ( x ( t ), x ( t ), ..., x ( t )) + u 2 2 2 2 1 2 n 2 ⎪ 2 • ⎨ • • ⎪ • ⎪ x ( t + Δ t ) = (1 − a ) x ( t ) + w f ( x ( t ), x ( t ), ..., x ( t )) + u n n n n 1 2 n n ⎩ n

(5)

Now we try to discuss the situations of trajectories under different equivalence relations. The theorem 5 tells us that any small difference between two equivalence relations of approximate polices may cause unpredictable result in some chaotic neural circuits. The theorem 6 tells us that all kind details of a time serial neural circuit can be revealed under an approximation with enough high precision. Definition 6. (The difference of two equivalence relations of approximate polices (AP)) If E1 and E2 are two equivalence relations and every vector (x1,x2,…xk) in Rk (or in a domain D in Rk) is approximated as e1(x1,x2,…xk) and e2(x1,x2,…xk) by E1 and E2 respectively, then the difference of E1 and E2 in a domain D can be defined as

Nonlinear Complex Neural Circuits Analysis and Design

d if ( E1 , E

2

) =

∫ || e

D

θ

1

( x 1 x 2 ... x k ) − e 2 ( x 1 x 2 ... x k ) ||d x 1 d x 2 ... d x k

θ θ

.

219

(6)

Suppose E1( ) and E2( ) are the equivalence relations in two k dimensional multi scale approximate polices A1 and A2 respectively, where is a parameter of equivalence relations, e.g. can be the rough-rate r ( E i ) = m a x ( d ( D )) of

θ

D ∈[ R k | E i ]

an equivalence relation. For a chaotic dynamical system, similar approximations of irrational numbers maybe cause a very different trajectory, i.e. no matter how smaller the d if ( E 1 (θ ) , E 2 (θ ) ) is , if d if ( E 1 (θ ) , E 2 (θ )) > ε > 0 , here

G

ε

is a constant , under the input I ( 0 ) , the difference of two trajectories G G G G G G G G T ( I (0) | E1 (θ )) = {OE (0), OE (1),..., OE (t ),...} and T (I (0) | E2 (θ )) = {OE (0), OE (1),..., OE (t ),...} which 1

1

2

1

2

2

are calculated under E1 and E2 respectively may be unpredictable when t trends to G G G infinite, i.e. | O E (t ) − O E ( t ) |> g > 0 , here O E ( t ) = ( o E 1 ( t ), o E 2 ( t ), ...o E k ( t )) is a 1 2 fuzzy granular vector of output and g is an arbitrary large constant. Based on this fact we have the Theorem 4. But at other hand, any small difference between two trajectories of a time serial neural circuit can be detectedG under a suitable approximate police G G G which has enoughG small Grough-rate, i.e. if T ( I a ( 0 )) = { O a ( 0 ) , O a (1), ..., O a ( t ), ...} G G

O b ( t ), ...} are two different trajectories of a time and T ( I b ( 0 )) = { O b ( 0 ), O b (1), ..., G G

serial neural circuit under inputs I a ( 0 ) and I b ( 0 ) with no approximate police and they are G indiscriminating G under an equivalence relation E(θ), i.e. G | T ( I a ( 0 ) | E (θ ) ) − T ( I b ( 0 ) | E (θ ) ) | = 0 , where T ( I a ( 0 ) | E (θ ) ) and G θ = r(E ) = T ( I b ( 0 ) | E (θ ) ) ( where G G I a (0 ) I b (0 )

m a x ( d ( D ))

D ∈[ R k | E ]

) are two trajectories at input

and under the equivalence relation E(θ) respectively, then by reducing θ = r ( E ) , the difference of two orbits will appear, i.e. the rough-rate G G l i m | T ( I a ( 0 ) | E (θ ) ) − T ( I b ( 0 ) | E (θ ) ) | > ε > 0

. In this way, we can prove Theorem 5. Strict proofs of these theorems are omitted for the sake of pages. θ → 0

θ

θ

Theorem 4. Suppose E1( ) and E2( ) are the equivalence relations in two k dimensional multi scale approximate polices A1 and A2 respectively. For a time serial neural circuit, if dif ( E1 (θ ), E 2 (θ )) > ε > 0 , here ε is a arbitrary small constant , under the input G G G G G I ( 0 ) , the difference of two trajectories T ( I (0) | E1 (θ )) = {O E (0), O E (1),..., O E ( t ), ...} and 1

G G G G T ( I (0) | E 2 (θ )) = {O E 2(0), O E2 (1),..., O E 2 (t ),...}

1

1

which are calculated under E1 and E2 respectively G G may be unpredictable when t trends to infinite, i.e. | O E ( t ) − O E ( t ) |> g > 0 , here g is 1

an arbitrary large constant.

2

ε

Theorem 5. Suppose Gk is an approximate police and for any small >0, there are and equivalence relations in Gk which have rough-rate less than G G G G G G G G T ( I a ( 0 ) ) = { O a ( 0 ) , O a (1) , ..., O a ( t ) , ...} and T ( I b (0 )) = { O b (0 ), O b (1), ..., O b ( t ), ...}

ε, G

are two different trajectories of a time serial neural circuit under inputs I a ( 0 ) and

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G Ib (0) with no approximate police, we apply Gk in the input, output and inner computing

ε

G

G

of this neural circuit, then there is a positive >0, T ( I a ( 0 ) ) and T ( I b ( 0 ) ) are discriminable under all equivalence relations E in Gk with rough-rates smaller than .

ε/2

Summary. As we know the important characteristics of chaotic dynamics, i.e., aperiodic dynamics in deterministic systems are the apparent irregularity of time traces and the divergence of the trajectories over time(starting from two nearby initial conditions). Any small error in the calculation of a chaotic deterministic system will cause unpredictable divergence of the trajectories over time, i.e. such kind neural circuits may behave very differently under different precise calculations. According to the theorem 4, any small difference between two approximations of a trajectory of a chaotic time serial neural circuit may create two totally different approximate results of this trajectory ,and the theorem 5 tells us that all details of a time serial chaotic neural circuit S can be revealed by an equivalence relation E in a finite precision multi scale approximate police Gk with enough small rough-rates, so it is reasonable to apply a multi scale precision approximate police in the analyzing and design of a neural circuit with chaos and such multi scale precision approximate police should be able to continuously control the calculation error in order to reveal enough detail of functions of such kind neural circuits. According to the above analysis, when we design or analyze a time serial neural circuit, two kind approaches should be considered. (1). For a time serial neural circuit with no chaos, we can use back propagation kind learning to find a suitable fuzzy logical framework for it . (2). For a chaotic time serial neural circuit, we should find a set of fuzzy logical frameworks under different equivalence relations in an approximate police which can continuously control the calculation error, and every fuzzy logical framework can be computed at a definite precision same as (1). But at what kind condition, the fuzzy logical frameworks of a chaotic time serial neural circuit under different equivalence relations in an approximate police have similar or continuous structure changing is still an open problem.

Acknowledgements This work is supported by the National Science Foundation of China (No. 60435010, 90604017, 60675010), 863 National High-Tech Program (No.2006AA01Z128), National Basic Research Priorities Programme(No. 2003CB317004) and the Nature Science Foundation of Beijing (No. 4052025).

References 1. Canuto, A.M.P., Fairhurst, M.: An investigation of fuzzy combiners applied to a hybrid multi-Neural system. In: IEEE Proceedings of the VII Brazilian Symposium on Neural Networks (SBRN 2002), pp. 156–161 (2002) 2. Yager, R.: Families of Owa Operators. Fuzzy sets and systems 59(22), 125–148 (1993)

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3. Castro, J.L.: Fuzzy logic controllers are universal approximators. IEEE Transactions on Systems, Man and Cybernetics 25(4), 629–635 (1995) 4. Li, H.X., Chen, C.L.P.: The equivalence between fuzzy logic systems and feedforward neural networks. IEEE Trans. on Neural Networks 11(2), 356–365 (2000) 5. Li, Z.P.: Pre-attentive segmentation and correspondence in stereo. Philos. Trans. R Soc. Lond. B Biol. Sci. 357(1428), 1877–1883 (2002) 6. Cho, S.B., Kim, J.H.: Multiple network fusion using fuzzy logic. IEEE Trans. on Neural Networks 6(2), 497–501 (1995) 7. Haykin, S.: NEURAL NETWORKS -a Comprehensive Foundation. Prentice-Hall, Inc., Englewood Cliffs (1999) 8. Sartori, M.A., Antsaklis, P.J.: A simple method to derive bounds on the size and to train multilayer neural networks. Neural Networks, IEEE Transactions on Publication 2(4), 467–471 (1991) 9. Kim, S.S.: A neuro-fuzzy approach to integration and control of industrial processes: Part I. J. Fuzzy Logic Intell. Syst. 8(6), 58–69 (1998) 10. Li, Z.P.: A neural model of contour integration in the primary visual cortex, neural computation, vol. 10, pp. 903–940 (1998)

Fuzzy Hyperbolic Neural Network Model and Its Application in H∞ Filter Design Shuxian Lun1,2, Zhaozheng Guo1, and Huaguang Zhang3 1

School of Information Science and Engineering, Bohai University, Jinzhou. 110004 Liaoning, China 2 The Key Laboratory of Complex System and Intelligent Science, Institute of Automation Chinese Academy of Science, Haidian 100080 Beijing, China 3 School of Information Science and Engineering, Northeastern University, Shenyang. 110004 Liaoning, China [email protected]

Abstract. This paper studies H∞ filter based on a new fuzzy neural model for signal estimation of nonlinear continuous-time systems with time delays. First, a new fuzzy neural model, called fuzzy hyperbolic neural network model (FHNNM), is developed. FHNNM is a combination of the special fuzzy model and the modified BP neural network. The main advantages of using the FHNNM over traditional fuzzy neural network are that explicit expression of expert’s experience and global analytical description. In addition, by contrast with fuzzy neural network based T-S fuzzy model, no premise structure identification is need and no completeness design of premise variables space is need. Next, we design a stable H∞ filter based on the FHNNM using linear matrix inequality (LMI) method. Simulation example is provided to illustrate the design procedure of the proposed method. Keywords: Fuzzy hyperbolic neural network, H∞ filter, Linear matrix inequality (LMI), Nonlinear system.

1 Introduction Recently, there have been a lot of interests on the problem of robust H∞ filtering for nonlinear system [1,2,3,4,5,6]. The advantage of using an H∞ filter over a Kalman filter is that no statistical assumption on the noise signals is needed. However, it is in general difficult to design an efficient filter for signal estimation of nonlinear systems. This paper deals with H∞ filtering problem based on the fuzzy hyperbolic neural network model for continuous-time nonlinear systems. There have been some successful examples of fuzzy neural network theory in filtering applications [7,8,9,10] and successes have been achieved in situations where the dynamics of systems are so complex that it is impossible to construct an accurate model. However, the identification of fuzzy neural network based on T-S fuzzy model is difficult. In order to overcome the difficulty, this paper studies H∞ filter design based on fuzzy hyperbolic neural network model for a class of continuous-time nonlinear systems. A new continuous-time fuzzy neural network model based on fuzzy hyperbolic model proposed in [11,12,13], called fuzzy hyperbolic neural network model (FHNNM). A FHNNM is both a kind of valid global F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 222–230, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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description and nonlinear model in nature. Besides the advantage mentioned above, the advantage of using FHNNM over T-S fuzzy neural network model is that no premise structure identification is needed and no completeness design of premise variables space is needed. Thus, H∞ filter using FHNNM can obtain the better estimation performance than using other fuzzy neural network model. The FHNNM can be obtained without knowing much information about the real plant and it can easily be derived from a set of fuzzy rules. The present paper is organized as follows. In Section 2, the principle of FHNNM is described. In Section 3, the H∞ filter design based on fuzzy hyperbolic neural network model is addressed, called fuzzy hyperbolic neural network H∞ filter. The H∞ filter design problem based on FHNNM is converted to the feasibility problem of a linear matrix inequality (LMI), which makes the prescribed attenuation level as small as possible, subject to some LMI constraints. In Section 4, simulation example is employed to demonstrate the design procedure for fuzzy hyperbolic neural network H∞ filters.

2 The Principle of Fuzzy Hyperbolic Neural Network Fuzzy hyperbolic neural network model is composed of two parts corresponding to the premise and the conclusion of hyperbolic type fuzzy rules, respectively. The definition of hyperbolic type fuzzy rules is as follows: Definition 1. [11,12,13] Given a plant with n input variables x = (x1 (t), ..., xn (t))T and n output variables x˙ = (x˙1 (t), ..., x˙n (t))T . For each output variable x˙l , l = 1, 2, · · · , n, the corresponding group of hyperbolic type fuzzy rules has the following form: R j : IF x1 is Fx1 and x2 is Fx2 , ..., and xn is Fxn THEN x˙l = ±cFx1 ± cFx2 ± · · · ± cFxn where Fxi (i=1,..., n) are fuzzy sets of xi , which include Pxi (positive) and Nxi (negative), and ±cFxi (i=1,...,n) are 2n real constants corresponding to Fxi ; (i) The constant terms ±cFxi in the THEN-part correspond to Fxi in the IF-part; that is, if the language value of Fxi term in the IF-part is Pxi , +cFxi must appear in the THENpart; if the language value of Fxi term in the IF-part is Nxi , −cFxi must appear in the THEN-part; if there is no Fxi in the IF-part, ±cFxi does not appear in the THEN-part. (ii) There are 2n fuzzy rules in each rule base; that is, there are a total of 2n input variable combinations of all the possible Pxi and Nxi in the IF-part; We call this group of fuzzy rules hyperbolic type fuzzy rule base (HFRB). To describe a plant with n output variables, we will need n HFRBs. Fig.1 shows that the configuration of fuzzy hyperbolic neural network model. The weights between the layer L1 and L2 , L2 and L3 , L4 and L5 are 1 in the Fig.1, respectively. The first layer L1 is the input layer of fuzzy hyperbolic neural network model. Each neural unit is used to directly transfer each input variable xi (t), i = 1, 2, · · · n, i.e. f1 (xi ) = xi , i = 1, 2, · · · n. Thus, the number of neural unit is the same as the dimension of input variables.

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Fig. 1. The configuration of fuzzy hyperbolic neural network model

The second layer L2 describes the fuzzy sets Fxi (Pxi and Nxi , i = 1, 2, · · · n ) of input variables, which is used to compute the membership function of input variables. The membership function Pxi and Nxi as: 2

μPxi (xi ) = e− 2 (xi − ki ) 1

1

2

μNxi (xi ) = e− 2 (xi + ki )

(1)

⎧ ⎨ μ (x ) = e− 21 (xi − ki )2 , IF - part is P Pxi i xi f2 (·) = 2 1 ⎩ − 2 (xi + ki ) μNxi (xi ) = e , IF - part is Nxi

(2)

where ki are positive constants. Therefore, we have

The third layer L3 shows that product inference. Each neural unit represents the corresponding hyperbolic type fuzzy rule as (1) and computes the fitness of each rule. (k) The output function f3 (·) (k = 1, 2, · · · 2n ) of the kthneural unit of the third layer is (1)

f3 (·) = μPx1 (x1 )μPx2 (x2 ) · · · μPxn (xn ) (2)

f3 (·) = μNx1 (x1 )μPx2 (x2 ) · · · μPxn (xn ) .. . (2n ) f3 (·) = μPN1 (x1 )μNx2 (x2 ) · · · μNxn (xn ) The forth layer L4 executes the normalization. Each weight between L3 and L4 is cFx1 + cFx2 + · · · + cFxn ,. . . , −cFx1 − cFx2 − · · · − cFxn , respectively. The output function (k)

f4 (·) (k = 1, 2, · · · 2n ) of the kthneural unit of the forth layer is

Fuzzy Hyperbolic Neural Network Model and Its Application in H∞ Filter Design

(1)

225

(1)

f4 (·) = (cFx1 + cFx2 + · · · + cFxn ) f3 (·)/G (2)

(2)

f4 (·) = (−cFx1 + cFx2 + · · · + cFxn ) f3 (·)/G .. . (2n ) (2n ) f4 (·) = (−cFx1 − cFx2 − · · · − cFxn ) f3 (·)/G where G = μPx1 (x1 )μPx2 (x2 ) · · · μPxn (xn ) + μNx1 (x1 )μPx2 (x2 ) · · · μPxn (xn ) + · · · μPN1 (x1 )μNx2 (x2 ) · · · μNxn (xn ) The fifth layer L5 is the output layer. The output variable x˙l as: 2n

x˙l =

∑ f4

(k)

(·), l = 1, 2, · · · n.

k=1

we can derive the following model from reference [11,12,13] as follows: n

x˙l = f (x) = ∑

i=1

cFxi eki xi − cFxi e−ki xi eki xi + e−ki xi

n

= ∑ cFxi tanh(ki xi )

(3)

i=1

According to (3), the whole system has the following form: x˙ = A tanh(kx x)

(4)

where P is a constant vector, A is a constant matrix, and tanh(kx x) is defined by tanh(kx x) T  = tanh(k1 x1 ) tanh(k2 x2 ) · · · tanh(kn xn ) . Therefore, we can obtain a analytical description as (4) from the FHNNM, i. e., FHNNM is equivalent to (4). In Fig.1, we need to identify parameters cFxi and ki . In fact, according to (2), we can simplify the configuration of the fuzzy hyperbolic neural network as Figure 2. In fig.2, n

h1 (xi ) = xi , h2 (·) = tanh(·) and h3 (·) = ∑ oi , oi (i = 1, 2, · · · n) the inputs of the neural i=1

unit. Thus, the identification work becomes easier.

3 Fuzzy Hyperbolic Neural Filter Analysis and Design Consider the FHNNM of the nonlinear system is proposed as the following form: x(t) ˙ = Atanh(kx x) + Ad tanh(kx x(t − d)) + Bw(t) y(t) = C tanh(kx x) + Dw(t) s(t) = Lx(t)

(5)

where x(t) = [x1 (t), x2 (t), · · · xn (t)]T ∈ Rn×1 denotes the state vector; y(t) ∈ Rm×1 denotes the measurements vector; s(t) ∈ Rq×1 denotes the signal to be estimated; w(t) ∈ Rn×1 assumed to be bounded disturbance input; B ∈ Rn×n , D ∈ Rm×m and L ∈ Rq×n are con˙ ≤ β < 1 and β stant matrices, d(t) is time-varying delay in the state and satisfies d(t) is a known constant.

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Fig. 2. The simple configuration of the FHNNM

Based on the FHNNM (4), the following fuzzy hyperbolic neural network H∞ filter is addressed ˙ˆ = A tanh(kx x) ˆ + K(y(t) − C tanh(kx x)) ˆ x(t) s(t) ˆ = Lx(t), ˆ x(0) ˆ =0

(6)

Then, the augmented filter error system can be written as the following form:

η˙ (t) = A¯ tanh(kη (t)) + A¯ d tanh(kη (t − d)) + B¯ ω (t) (7) e(t) = s(t) − s(t) ˆ = C¯ η (t)      T T A 0 Ad 0 ¯  T where η (t) = xT (t) xˆT (t) , A¯ = , A¯ d = , B = B (KD)T , KC A − KC 0 0   C¯ = L −L , k = diag(kx , kx ). Theorem 1. For nonlinear system (5) and a prescribed real number γ > 0, if there exist a positive definite diagonal matrices P such that the matrix inequality ⎡ ⎤ PA¯ + A¯ T P + α H + Q PA¯ d PB¯ ⎣ A¯ T P ⎦<0 −H 0 (8) d T 2 ¯ B P 0 −γ Ww then the filtering error system (7) is asymptotically stable and satisfies a guaranteed γ level of noise attenuation. Proof. Choose the following Lyapunov function candidate for the system (7) n

pi 1 ln(cosh ki ηi ) + 1−β i=1 ki

V (t) = ∑

t t−d(t)

tanhT (kη (s))H tanh(kη (s))ds

(9)

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227

where ηi is the ith element of η , and ki is the ith diagonal element of k. Here, ki > 0, 1 1  pi > 0. Because of cosh(ki ηi ) = (eki ηi + e−kiηi ) 2 ≥ (eki ηi ) 2 · (e−ki ηi ) 2 = 1 and ki > 0, pi > 0 , we known V (t) > 0 for all η and V (t) → ∞ as η 2 → ∞. Along the trajectories of system (7) with w(t) = 0, the corresponding time derivative of V (t) is given by n

V˙ =2 ∑ pi tanh(kη )η˙ + α tanhT (kη )H tanh(kη )− i=1

˙ tanhT (kηd )H tanh(kηd ) α (1 − d(t)) =2 tanhT (kη )Pη˙ + α tanhT (kη )H tanh(kη )− ˙ tanhT (kηd )H tanh(kηd ) α (1 − d(t)) ≤2 tanhT (kη )P[A¯ tanh(kη ) + A¯ d tanh(kηd ]+

α tanhT (kη )H tanh(kη ) − tanhT (kηd )H tanh(kηd )  T     tanh(kη ) tanh(kη ) PA¯ + A¯ T P + α H PA¯ d ≤ × A¯ Td P tanh(kηd ) tanh(kηd ) −H  Δ where P = diag(p1 , p2 , · · · pn ) ∈ R2n×2n, ηd = η (t − d(t)), α = 1 1 − β . A sufficient condition for V˙ < 0 is as follows   PA¯ + A¯ T P + α H PA¯ d <0 −H A¯ Td P

(10)

(11)

According to the Schur complements [14], The inequality (8) can be easily verified to ensures that inequality (11) holds. Therefore, it can be concluded that the filtering error system (7) is globally asymptotically stable. if

∞

eT (t)e(t)dt =

0

∞ 0

(C¯ η (t))T (C¯ η (t))dt =

∞ 0

(kη (t))T Q(kη (t))dt

¯ −1 . where Q = (k−1 )T C¯ T Ck Then, we can define H∞ performance index of fuzzy hyperbolic neural network H∞ filter for arbitrary nonzero w(t) ∈ l2 [0, +∞) as follows

∞ 0

tanhT (kη )Q tanh(kη )dt <

∞ 0

γ 2 wT (t)Ww w(t)dt

(12)

Next, assuming zero initial conditions for the filtering error system (7), the performance index

∞

J= 0

∞

= 0

[tanhT (kη )Q tanh(kη ) − γ 2 wT (t)Ww w(t)]dt [tanhT (kη )Q tanh(kη ) − γ 2 wT (t)Ww w(t) + V˙ ]dt+V (0) − V(t) |t→∞

where V (t) is as in (9).

(13)

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Considering that V (t)|t=0 = 0 and V (t)|t→∞ → 0, (13) becomes

∞

J= 0

[tanhT (kη )Q tanh(kη ) − γ 2 wT (t)Ww w(t) + V˙ ]dt ≤ ξ T Θ ξ

(14)

⎤ ⎡ PA¯ + A¯ T P + α H + Q PA¯ d PB¯  T ⎦. −H 0 where ξ = tanhT (kη ) tanhT (kηd ) wT , Θ = ⎣ A¯ Td P 0 −γ 2Ww B¯ T P ifΘ < 0, we can obtain J < 0. Then (13) satisfying H∞ performance index as (12). Hence, the fuzzy hyperbolic neural H∞ filter index is achieved with a prescribed γ 2 . This completes the proof.   P˜1 0 For the convenience of design, we can define P = and M = P˜2 K(K = P˜2−1 M), 0 P˜2 

where P˜1 and P˜2 are positive difinite diagonal matrices. Thus, (8) can becomes    ⎤ P˜1 A+AT P˜1 CT M T P˜1 Ad 0 P˜1 B α H+Q + ⎢ MC 0 0 MD ⎥ P˜2 A+A˜ T P˜2 −MC−CT M T ⎢ ⎥ < 0 (15) ⎣∗ ⎦ −H 0 2 ∗ ∗ −γ Ww ⎡

where ∗ denotes the entries induced by symmetry. In view of Theorem 1, the fuzzy hyperbolic neural network H∞ filtering problem (8) for nonlinear system (5) can be solved in terms of the feasibility of the LMIs of (15). In fact, any feasible solution to (15) yields a suitable robust filter. To obtain a better robust filtering performance against disturbances, the attenuation level γ 2 can be reduced to the minimum possible value such that (15) is satisfied. Then, the fuzzy hyperbolic neural network H∞ filter with the smallest γ attenuation level obtained from Theorem 1 can easily be determined by solving the following convex optimization problem: min δ , subject to(15)with γ 2 = δ

P1 ,P2 ,K,H

(16)

4 Simulation Example Consider the following continuous-time nonlinear system: x˙1 (t) = x2 (t) + x1 (t − d) + w(t) x˙2 (t) = −1.35 sinx1 (t) + 0.0981x2(t) + x1(t)x2 (t) + 0.2w(t) y(t) = x1 (t) + 0.2 sinx2 (t) + 0.1w(t) s(t) = 0.6x1 (t) + 0.2x2(t)

(17)

we assume that the noise is normally distributed with zero mean and variance 0.01 and d = 1.5. It is assumed that the initial condition is: (x1 (0), x2 (0), xˆ1 (0), xˆ2 (0))T = (0, 0, 0, 0)T .

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1.5 1 0.5 0

s(t)

−0.5 −1 −1.5 −2 −2.5 −3 −3.5

0

20

40

60

80

100

120

140

160

180

200

t

Fig. 3. Original signal ( solid line )and restored signal by fuzzy hyperbolic neural filter ( dashdot line with squares)

Use ode45 function of Matlab software to solve state equation (17), then we can obtain original signal s(k), here k is a sequence of discrete sampling value. Next, we utilize state x1 (k) and x2 (k) to obtain x˙1 (k) and x˙2 (k) at some time k. Then FHNNM of (17) can be constructed by using input-output data pairs (x1 (t), x2 (t), x˙1 (t)) and (x1 (t), x2 (t), x˙2 (t)) and  neural network BP  algorithm  [9]. We  obtain FHNNM of (17)  0 37.3909 1.0432 0 1.4388 0 as in (5),where A = , Ad = , kx = , −1.5421 0.3070 0 0 0 0.0247       1 B= , C = 1.4072 0.008 , D = 0.1, L = 0.6 0.2 . Let P = diag(P1 , P2 ), we 0.2 obtain γ 2 = 0.5612, P1 = diag(0.0470, 7.1032), P2 = diag(21.3127,475.9101), K =  T 0.1408 0.0063 . Fig. 3 shows the approximation errors for s(k). Simulation result shows that the proposed fuzzy hyperbolic neural H∞ filter can obtain better estimation.

5 Conclusions In this paper, fuzzy hyperbolic neural network H∞ filter is designed for a class of nonlinear continuous-time systems with delays. The problem of fuzzy hyperbolic neural network H∞ filter design is converted into linear matrix inequalities problem. The proposed design procedure is very simple. Acknowledgments. This work is supported by the Natural Science Foundation of Liaoning Province under grant 20072199, the Postdoctoral Science Foundation of China under grant 20070420071 and Science technology Foundation of the education department of Liaoning province under grant 05L016. In addition, the author gratefully acknowledges the support of K.C.Wong Education Foundation, Hong Kong.

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References 1. Tseng, C.S., Chen, B.S.: H∞ Fuzzy Estimation for a Class of Nonlinear Discrete-time Dynamic System. IEEE Trans. Signal Processing 49, 2605–2619 (2001) 2. Palhares, R.M., de Souza, C.E., Peres, P.L.D.: Robust H∞ Filtering for Uncertain Discretetime State-delayed System. IEEE Trans. Signal Processing 49, 1696–1703 (2001) 3. Zhang, H.G., Lun, S.X., Liu, D.R.: Fuzzy H∞ Filter Design for a Class of Nonlinear Systems with Multiple Time Delays. IEEE Transactions on Fuzzy Systems 49(3), 453–469 (2007) 4. Lun, S.X., Zhang, M.J.: Fuzzy Robust Filtering and Control Theory. Dalian University of Technology Press, Dalian (2007) 5. Lun, S.X., Zhang, H.G.: Fuzzy H∞ Filter Design for a Class of Nonlinear Systems with Time Delays via LMI. Acta Automation Sinica 31(3), 394–401 (2005) 6. Gao, H.J., Wang, C.H.: New Approach to Robust l2 − l∞ Filter Design for Uncertain Continuous-time Systems. Acta Automation Sinica 29(6), 809–814 (2003) 7. Lee, C.H., Lin, Y.C.: An Adaptive Neuro-fuzzy Filter Design via Periodic Fuzzy Neural Network. Signal Processing 85(2), 401–411 (2005) 8. Lee, C.S., Guo, S.M., Hsu, C.Y.: A Neural Fuzzy Filter for Impulse Noise Removal. In: 2004 International Symposium Neural Networks, China (2004) 9. Lun, S.X., Zhang, H.G.: Noise Cancellation Based on Adaptive Fuzzy Neural Networks Algorithm for Multi-sensory Signals. Control and Decision 19(6), 667–670 (2004) 10. Lun, S.X., Zhang, H.G.: Filtering Technique Based on Adaptive Fuzzy-neural Network Applied to Leakage Ddetection of Pipeline. Acta Petrolei Sinica 25(4), 101–104 (2004) 11. Zhang, H.G., Quan, Y.B.: Modeling and Control Based on Fuzzy Hyperbolic Mmodel. Acta Automation Sinica 26(6), 729–735 (2000) 12. Zhang, H.G., Quan, Y.B.: Modeling, Identification and Control of a Class of Nonlinear System. IEEE Trans. Fuzzy Systems 49, 349–354 (2001) 13. Lun, S.X., Zhang, H.G.: Delay-independent Fuzzy Hyperbolic Guaranteed Cost Robust Control Design for a Class of Nonlinear Continuous-time Systems with Uncertainties. Acta Automation Sinica 31(5), 720–726 (2005) 14. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)

On the Domain Attraction of Fuzzy Neural Networks Tingwen Huang1 , Xiaofeng Liao2 , and Hui Huang3 1

Texas A&M University at Qatar, Doha, P.O. Box 5825, Qatar Department of Computer Science and Engineering, Chongqing University Chongqing 400030, P.R. China Fundamental Study Department, Chongqing Logistics and Engineering College [email protected] 2

3

Abstract. In this paper, we study qualitative properties of equilibrium points in a class of interval fuzzy neural networks and obtain an estimate on the domain of robust attraction of locally exponentially stable equilibrium points. Both the conditions and the estimate are formulated in terms of the parameter intervals, so they are verifiable easily. Keywords: Attraction domain, Fuzzy neural networks, Stability.

1

Introduction

Lots of investigations have been conducted since Hopfiled proposed neural networks (NN) nearly twenty years ago because neural networks have been applied to optimization, pattern recognition, artificial intelligence and so on. However, almost most of the published papers on the dynamics of the neural networks have been investigating the global stability of fuzzy neural networks. Qualitative analysis on locally dynamical properties of fuzzy neural networks is another aspect which is important in the study of fuzzy neural networks. Recently, qualitative analysis of the local dynamics of Hopfield neural networks has received certain attention. Cao et al.[1-4] and Yang et al. [5,6] derived various estimates on the domains of attraction of locally exponentially stable equilibrium points for Hopfield neural networks. In this paper, we focus on the estimate the attraction domain of locally stable equilibrium point under easily verified conditions. The rest of the paper is as follows. In Section 2, problem formulation and preliminaries are given. In Section 3, the main result, a criterion on estimate the domain of attraction for fuzzy neural networks is given. Section 4 is a brief conclusion.

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Problem Formulation and Preliminaries

In this paper, we would like to consider fuzzy neural networks described by the following form: F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 231–236, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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T. Huang, X. Liao, and H. Huang n n n    dxi = −di xi (t) + ξij fj (xj (t)) + γij fj (xj (t) + bij μj + Gi dt j=1 j=1 j=1

+

n  j=1

Tij μj +

n 

δij fj (xj (t) +

j=1

n 

Hij μj ,

i = 1, · · · , n

j=1

(1) where γij , δij , Tij and Hij are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed forward MIN template and fuzzy feed forward MAX respectively; bij are elements of feed forward template;  template,  di ≥ 0; and denote the fuzzy AND and fuzzy OR operation, respectively; xi , μi and Gi denote state, input and bias of the ith neuron, respectively; fi is the activation function. In this paper, we assume that the activation functions are bounded and satisfy the following condition: H: f (x) = (f1 (x1 ), · · · , fn (xn )) is a differentiable bounded function defined on R and satisfies 0 < max1≤i≤n sup−∞≤xi ≤∞ |fi (xi )| = Mf < ∞.

(2)

It is noted that bounded activation functions always guarantee the existence of an equilibrium point for model (1). Let x∗ = (x∗1 , · · · , x∗n )T be an equilibrium point of model (1). To simplify the notation later, we shift the equilibrium point x∗ of (1) to the origin by using the transformation y(t) = x(t) − x∗ . Model (1) becomes the following form: n n   dyi = −di yi (t) + ξij (fj (xj (t)) − fj (x∗j )) + γij fj (xj (t)) dt j=1 j=1

−

n  j=1

γij fj (x∗j ) +

n  j=1

δij fj (xj (t)) −

n 

δij fj (x∗j ),

i = 1, · · · , n

j=1

(3) Definition 1. The equilibrium point x∗ of (1) is said to be globally exponentially stable if there exist constants λ > 0 and M > 0 such that ||x(t) − x∗ || ≤ M  x(0) − x∗  e−λt

(4)

for all t ≥ 0. Definition 2. The domain of attraction of equilibrium x∗ is the maximal region Ω such that every solution x(t) to model (1) satisfying (3) with x(0) ∈ Ω approaches to x∗ . In order to prove the main result regarding the attraction domain of model (1), we need the following lemma.

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Lemma 1. ([7]). For any aij ∈ R, xj , yj ∈ R, i, j = 1, · · · , n, we have the following estimations, | and |

n 

aij xj −

j=1

j=1

n 

n 

aij xj −

j=1

3

n 

aij yj | ≤



(|aij | · |xj − yj |)

(5)

(|aij | · |xj − yj |)

(6)

1≤j≤n

aij yj | ≤

j=1

 1≤j≤n

Attraction Domain for Fuzzy Neural Networks

In this section, we will use the Lyapunov method to obtain the attraction domain for the fuzzy neural networks. The main result is presented as the following theorem. Theorem 1. Suppose that x∗ = (x∗1 , · · · , x∗n ) is an equilibrium point of model (1) with coefficients n satisfying (2), and the following holds: di > |fi (x∗i )| j=1 ζji , i = 1, 2, · · · , n, where ζij = |ξij | + |γij | + |δij |. Then, we have the following: (a) x∗ is locally exponentially stable. (b) Let δ =

2 n

min1≤i≤n {





n di −|fi (x∗ i )| j=1 ζji } n Mf ζ j=1 ji ∗

then, the open ball B(x , δ) is contained in the domain of robust attraction of x∗ . n Proof. First, we prove part (a). Since di > |fi (x∗i )| j=1 ζji , i = 1, 2, · · · , n, let be  any positive number such n that  < min1≤i≤n {di − |fi (x∗i )| j=1 ζji }. Considering a Lyapunov function: V (x(t)) = et

n 

|xi (t) − x∗i | = et

i=1

n 

|yi (t)|

(7)

i=1

where yi (t) = xi (t) − x∗i . For i = 1, · · · , n, by (2) and Lemma 1 and Lemma 2, we have the following: n n   dV (x(t)) d+ |yi (t)| = et |yi (t)| + et dt dt i=1 i=1

≤ et

n  i=1

|yi | + et

n n   [−di |yi (t)| + | ξij (fj (xj (t)) − fj (x∗j ))| i=1

j=1

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+| − ≤e

n 

γij fj (xj (t)) −

j=1 n 

n 

γij fj (x∗j )| + |

j=1

n 

δij fj (xj (t))

j=1

δi+|j fj (x∗j )|]

j=1 n  t

( − di )|yi (t)| + e [| t

i=1

n 

[|ξij | + |γij |

j=1

+|δij |]|fj (xj (t)) − fj (x∗j )| n n   = et ( − di )|yi (t)| + et | |ζij ||fj (xj (t)) − fj (x∗j )| i=1

j=1

(8) Since fj (|ξj ) |(xj (t) − x∗j )2 | 2 Mf ≤ |fj (x∗j )(xj (t) − x∗j ) + (xj (t) − x∗j )2 | 2

|fj (xj (t)) − fj (x∗j )| = |fj (x∗j )(xj (t) − x∗j ) +

(9)

By formulas (8) and (9), we have n n n   dV (y(t)) Mf  ≤ et ( − di + |fi (x∗i )| |ζij | + |ζij ||yi (t)|)|yi (t)| dt 2 j=1 i=1 j=1

(10) We evaluate at t0 . If



we have have



n di −−|fi (x∗ i )| j=1 ζji 2 min { }, n 1≤i≤n n Mf j=1 ζji dV (x(t)) < 0, so V ((x(t)) is a decreasing function dt

|xi (t0 ) − x∗i | <

V (x(t)) = et

n i=1

|xi (t) − x∗i | ≤ V (x(0)) =

n i=1

for t > t0 . Thus, we

|xi (0) − x∗i |,

Thus, we have n n ∗ −t ∗ i=1 |xi (t) − xi | ≤ e i=1 |xi (0) − xi | Namely, ||xi (t) − x∗i ||∞ ≤ ne−t ||xi (0) − x∗i ||∞ According to Definition 3, x(0) is in the attraction domain of x∗ . Thus, we have completed the proof of Theorem 1.

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Conclusion

In this paper, we study the local dynamics of fuzzy neural networks. A criterion on the attraction domain for the fuzzy neural networks has been obtained using Lyapunov method. It is believed that that the stability property of neural networks is very important in designing neural networks. Thus, the results present in this paper are useful in the application and design of fuzzy neural networks since the conditions could be easily verified.

References 1. Cao, J.: An Estimation of the Domain and Convergence Rate of Hopfield Associative Memory. J. Electron. 21, 488–491 (1999) 2. Cao, J.: An Estimation of the Domain of Attraction and Convergence Rate for Hopfield Continuous Feedback Neural Networks. Phys. Lett. A. 325, 370–374 (2004) 3. Cao, J., Tao, Q.: An Estimation of the Domain of Attraction and Convergence Rate for Hopfield Continuous Feedback Neural Networks. J. Comput. Syst. Sci. 62, 528– 534 (2001) 4. Cao, J., Tao, Q.: An Estimation of the Domain of Attraction and Convergence Rate for Hopfield Associatiative Memory and an Application. J. Comput. Syst. Sci. 60, 179–186 (2000) 5. Cao, J., Chen, T.: Globally Exponentially Robust Stability and Periodicity of Delayed Neural Networks. Chaos, Solitions and Fractals 22, 957–963 (2004) 6. Cao, J., Wang, J.: Global Exponential Stability and Periodicity of Recurrent Neural Networks with Time Delays. IEEE Transactions on Circuits and Systems-Part I 52, 920–931 (2005) 7. Yang, X., Liao, X.F., Bai, S., Evans, D.: Robust Exponential Stability and Domains of Attraction in a Class of Interval Neural Networks. Chaos, Solitions and Fractals 26, 445–451 (2005) 8. Yang, X., Liao, X.F., Li, C., Evans, D.: New Estimate on the Domains of Attraction of Equilitrium Points in Continuous Hopfield Neural Networks. Phys. Lett. A. 351, 161–166 (2006) 9. Yang, T., Yang, L.B., Wu, C.W., Chua, L.O.: Fuzzy Cellular Neural Networks: Theory. In: Proc. of IEEE International Workshop on Cellular Neural Networks and Applications, pp. 181–186 (1996) 10. Chen, A., Huang, L., Liu, Z., Cao, J.: Periodic Bidirectional Associative Memory Neural Networks with Distributed Delays. Journal of Mathematical Analysis and Applications 317, 80–102 (2006) 11. Chen, A., Huang, L., Cao, J.: Existence and Stability of Almost Periodic Solution for BAM Neural Networks with Delays. Applied Mathematics and Computation 137, 177–193 (2003) 12. Chen, T., Amari, S.: Stability of Asymmetric Hopfield Networks. IEEE Trans. Neural Networks 12, 159–163 (2001) 13. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1999) 14. Huang, T., Cao, J., Li, C.: Necessary and Sufficient Condition for the Absolute Exponential Stability for a Class of Neural Networks with Finite Delay. Phys. Lett. A. 352, 94–98 (2006)

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15. Huang, T.: Exponential Stability of Fuzzy Cellular Neural Networks with Unbounded Distributed Delay. Phys. Lett. A. 351, 48–52 (2006) 16. Li, C., Liao, X., Huang, T.: Global Stability Analysis for Delayed Neural Networks via an Interval Matrix Approach. IET Control Theory Appl. 1, 743–748 (2007) 17. Liao, X., Wang, J., Cao, J.: Global and Robust Stability of Interval Hipfield Neural Networks with Time-varying Delays. Int. J. Neural Sys. 13, 171–182 (2003) 18. Liao, X.F., Yu, J.B.: Robust Stability for Interval Hopfield Neural Networks with Time Delay. IEEE Trans. on Neural Networks 9, 1042–1045 (1998) 19. Song, Q.K., Cao, J.: Global Robust Stability of Interval Neural Networks with Multiple Time-varying Delays. Mathematics and Computers in Simulation 74, 38– 46 (2007) 20. Song, Q.K., Zhao, Z., Li, Y.: Global Exponential Stability of BAM Neural Networks with Distributed Delays and Reactiondiffusion Terms, vol. 335, pp. 213–225 (2005) 21. Wang, L., Lin, Y.: Global Robust Stability for Shunting Inhibitory CNNs with Delays. Int. J. Neural Syst. 14, 229–235 (2004) 22. Yang, T., Yang, L.B., Wu, C.W., Chua, L.O.: Fuzzy Cellular Neural Networks: Applications. In: Proc. of IEEE International Workshop on Cellular Neural Networks and Applications, pp. 225–230 (1996) 23. Yang, T., Yang, L.B.: The Global Stability of Fuzzy Cellular Neural Network. Circuits and Systems I: Fundamental Theory and Applications 43, 880–883 (1996)

CG-M-FOCUSS and Its Application to Distributed Compressed Sensing Zhaoshui He1,2 , Andrzej Cichocki1,3,4 , Rafal Zdunek5 , and Jianting Cao1,6 1

Lab. for Advanced Brain Signal Processing, Brain Science Institute, Wako-shi, Saitama, 351-0198, Japan 2 School of Electronics and Information Engineering, South China University of Technology, Guangzhou, 510641, China 3 System Research Institute, Polish Academy of Sciences (PAN), Warsaw, 00-901, Poland 4 Warsaw University of Technology, Warsaw, 00-661, Poland 5 Institute of Telecommunications, Teleinformatics, and Acoustics, Wroclaw University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, Poland 6 Saitama Institute of Technology, Saitama, 369-0293, Japan {he shui,cia}@brain.riken.jp, [email protected], [email protected]

Abstract. M-FOCUSS is one of the most successful and efficient methods for sparse representation. To reduce the computational cost of M-FOCUSS and to extend its availability for large scale problems, M-FOCUSS is extended to CG-M-FOCUSS by incorporating conjugate gradient (CG) iterations in this paper. Furthermore, the CG-M-FOCUSS is applied to distributed compressed sensing. We illustrate the performance of CG-MFOCUSS by an MRI image reconstruction example, in which CG-MFOCUSS can not only reconstruct the MRI image with high precision, but also considerably reduce the computational time. Keywords: FOCUSS, M-FOCUSS, Compressed representation.

1

sensing, Sparse

Introduction

Consider the sparse representation problem: x(t) = As(t), t = 1, · · · , T, or X = AS,

(1)

where x(t) ∈ Rm is the given vector (observation), s(t) ∈ Rn is an unknown vector representing sparse sources or hidden components, A = [a1 , · · · , an ] ∈ Rm×n is a given full-row rank basis matrix; T is the number of available samples, m is the number of observations, and n is the number of sources. We consider only the overcomplete case: m < n. The main objective is to find the sparse solutions (sparse sources) s(t) satisfying equations (1). Sparse representation has found many applications in compressed sensoring [1], electromagnetic and biomagnetic problems (EEG/MEG), time-frequency F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 237–245, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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representation, image processing, fault diagnosis, etc [2, 3]. Recently, much attention has been paid to this problem due to its importance. Many methods have been developed for this problem: matching pursuit (MP) method [4], orthogonal matching pursuit (OMP) method [5], minimum 1 -norm methods (e.g., linear programming (LP) [6,7,8,9,10], shortest path decomposition (SPD) [11,9]), various FOCUSS methods [12, 13, 14, 2, 15, 16] and M-FOCUSS [17]. Among these methods, M-FOCUSS is one of the most efficient in terms of both speed and precision. In this paper, we extend M-FOCUSS to CG-M-FOCUSS by incorporating the conjugate gradient iterations and apply it to compressive sensing [18, 19, 20].

2

CG-M-FOCUSS

At first, let us consider the sparse representation of multiple measurement vectors: ¯ = AS, ¯ X (2) ¯ = [¯ ¯ = [¯ where usually L  T (e.g., L = 4), X x(1), · · · , x ¯(L)] and S s(1), · · · , ¯ s¯(L)]. Here X is considered to be a block of X in model (1). In this approach, instead of computing all T vectors s(1), · · · , s(T ) in model (1) simultaneously, we split them into blocks (usually strongly overlapped) and attempt to estimate them sequentially block by block. The overlapping blocks are justified by the fact that the sources are usually locally smooth or continuous. In more details, we use M-FOCUSS or CG-M-FOCUSS to process the T samples in model (1) according to the following scheme: firstly, the T samples in model (1) are segmented into T /L blocks with block length L (L  T , e.g., L = 4); secondly, to make the estimated signals be smooth, we set an appropriate percentage of overlapping between two neighboring blocks (typically, 50%-80% overlapping); Then, we apply M-FOCUSS or CG-M-FOCUSS for each block. According to this scheme, 2.1

M-FOCUSS

Using M-FOCUSS, sparse representation for problem (2) can be converted to solve the following optimization problem [17]: ⎧ L  p2 ⎪ n   2 ⎨ ¯ p) = min s¯ (l) min J(S; (3) ¯ ¯ i=1 l=1 i S S ⎪ ⎩ subject to: X ¯ = AS ¯ The iterative formula of M-FOCUSS for solving problem (3) is ¯ = Π −1 (S) ¯ · AT · (A · Π −1 (S) ¯ · AT )−1 · X, ¯ S  

2−p  L 2 2−p L −1 ¯ 2 where Π (S) = diag ¯1 (l) ,··· , ¯n (l) . l=1 s l=1 s

(4)

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239

CG-M-FOCUSS Algorithm

In the iterative formula (4) of M-FOCUSS, it is time-consuming to compute ¯ T because we must the inverse of symmetric positive definite matrix AΠ −1 (S)A compute it separately for each block at each iteration, and also computation of the matrix inverse is usually quite expensive. For this reason, we consider to exploit conjugate gradient (CG) iterations in (4) to speed up M-FOCUSS and extend its availability for large scale problems. The linear CG method is one of the most computationally inexpensive techniques for solving large linear algebraic systems of equations, which is convergent to a minimal norm least square solution in a finite number of iterations for symmetric positive definite coefficient matrix [21, 22]. For the linear equation set Hλ = b,

(5)

where λ is the unknown vector and H is an m × m symmetric positive definite matrix, CG method can find the solution in a very efficient manner. For convenience, we denote the solution obtained by CG method as λ = H −1 b = cg(H, b, λ(0) , ε), where λ(0) is the initialization and ε is the tolerance. It is worth noting that the performance of the linear CG method depends on the distribution of eigenvalues of the coefficient matrix H [22]. In details, if H has r distinct real-valued eigenvalues (r ≤ m), then the CG iterations will terminate at the solution in at most r iterations. In other words, if the matrix H has very few distinct eigenvalues, the CG method will be extremely fast. For example, if r = 1, the CG can find the right solution in only one iteration even for large scale problems. To improve the eigenvalue distribution of H and accelerate the CG method [22], we precondition matrix H in (5) by a linear transform via a nonsingular preconditioning transform matrix C as ˜ = C −T b, (C −T HC −1 )λ

(6)

˜ = Cλ. Then we can equivalently solve (5) by λ = C −1 · cg(C −T HC −1 , where λ −T C b, λ(0) , ε). In this way, the convergence rate of CG method depends on the eigenvalues of the matrix C −T HC −1 . So we can accelerate CG method by choosing an appropriate preconditioning matrix C. Δ Setting H = AΠ −1 AT in (4), we have the iterative formula of CG-MFOCUSS as ¯ · AT · cg(AΠ −1 (S)A ¯ T,x ¯(l), λ(0) (l), ε), l = 1, · · · , L. s¯(l) = Π −1 (S)

(7)

The expression (7) can be implemented by decomposing into the following two expressions:  ¯ T,x ¯(l), λ(0) (l), ε) λ(l) = cg(AΠ −1 (S)A (8) −1 ¯ T s¯(l) = Π (S) · A · λ(l) In this paper, λ(0) (l) in (8) is initialized as λ(0) (l) = λ(l − 1) when l > 1.

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Since the preconditioning plays a crucial role in CG strategies, we discuss how to design an appropriate preconditioning transform matrix C for the CGM-FOCUSS (7). Perform the singular value decomposition (SVD) on A as A = U ΣV T , where ⎞ ⎛ σ1 · · · 0 0 · · · 0 ⎟ ⎜ (9) Σ = [Λ, 0] = ⎝ ... . . . ... ... . . . ... ⎠ , 0 · · · σm 0 · · · 0 where Λ = diag(σ1 , · · · , σm ). Alternatively, we can perform eigenvalue decomposition (EVD) on matrix AAT = U Λ2 U T to obtain matrice U and Λ. Here we choose the preconditioning transform matrix C as C = ΛU T . Premultiplying the transform matrix C −T = Λ−1 U −1 on both sides of the model (2), we have ˜ =A ˜S, ¯ X (10) ¯ and A ˜ = Λ−1 U −1 A. So the problem (2) is precon˜ = Λ−1 U −1 X where X ditioned to problem (10) and can be equivalently solved by (10). Due to the preconditioning transform matrix C, the CG-M-FOCUSS is more efficient for problem (10) than the original problem (2). Based on above discussion, the CGM-FOCUSS for MMV problem (2) can be outlined as follows:

Algorithm 1. CG-M-FOCUSS for MMV problem (2) 1) Perform EVD on matrix AAT (=U Λ2 U T ) or SVD on A (=U ΣV T ) to get U ˜ = Λ−1 U −1 X. ¯ Set the parameter p and ˜ = Λ−1 U −1 A and X and Λ. Compute A −3 ε ≤ 10 . ¯ as S ¯(0) , initialize λ and set k = 0. 2) Initialize S (k) ¯(k) ) · A ˜T . 3) Compute T˜ = Π −1 (S ¯ as follows: 4) Update S for l=1 to L do ˜T˜(k) , x Update the Lagrange multiplier vector λ = cg(A ˜(l), λ, ε); (k) Update s¯(l) by s¯(k+1) (l) = T˜ · λ; end 5) Let k = k + 1 and goto step 3) until the convergence is reached.

For convenience, λ is initialized to be a zero vector λ = 0m×1 in this paper. The CG-M-FOCUSS is more suitable for larger scale problems than the standard M-FOCUSS because the conjugate directions in the CG can be generated in a very economical way. For the standard M-FOCUSS, the conventional method (e.g., Gaussian elimination) is used to calculate the matrix inversion [AΠ −1 AT ]−1 , where its computational complexity is O(m3 ), whereas the computational cost of CG method for (5) is only O(m2 ).

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241

Distributed Compressed Sensing by Overlapping CG-M-FOCUSS

Let Z be an unknown matrix in Rn×T . Suppose that we have m linear measurements X ∈ Rm×T of the unknown signal matrix Z as follows X = Φ · Z,

(11)

where Φ consists of m rows drawn from an n × n orthogonal transform matrix (e.g., a Fourier transform matrix). So m ≤ n. The standard methods require at least n measurements. Suppose Z is compressible or sparse in an appropriate transform domain W, described by the orthogonal sparsifying transform domain W ∈ Rn×n (after extracting the real and imaginary parts if necessary) [1], i.e., S z = W Z is sparse. Then, from (11), we have X = A · Sz ,

(12)

where A = Φ · W −1 . Since S z is sparse, it is possible to reconstruct Z in the transform domain W even if m < n [19, 18, 20]. Given Φ and W , compressed sensing or compressive sampling recovers the true signals Z by exploiting their sparsity or compressibility in the transform domain W. Then the CG-M-FOCUSS proposed in Section 2 can be employed to solve distributed compressed sensing problem (12) [23]. For compressed sensing problem (12), it is worth mentioning some important features that arise in CG-M-FOCUSS [1] due to the orthogonal transform matrix W and the partial orthogonal transform matrix Φ. Firstly, the SVD step for finding the preconditioned transform matrix C can be omitted because AAT = Φ · W −1 · W −T · ΦT = I m×m . Secondly, the computational complexity ¯ T usually can be reduced by fast orthogonal transforms W and of AΠ −1 (S)A Φ (i.e., fast Fourier transform, fast wavelet transform and so on).

4

Experiments

In this section, we give an MRI image reconstruction example to illustrate the performance of the CG-M-FOCUSS and compare it with the standard FOCUSS and the standard M-FOCUSS [24]. All methods are implemented in Matlab 7.2, and run on the Dell PC with Intel Xeon CPU 3 GHz under Windows XP Professional. The algorithm parameters are taken as follows: the sparsity parameter is p = 1; the block length is L = 10 and the overlapping rate is 80%; for CG-MFOCUSS, the tolerance is ε = 0.001. The initializations of all algorithms are set as matrix “1”, in which all entries are 1. The Lagrange multiplier vector is set as zero vector λ = 0m×1 . Consider a 512 × 512 MRI image reconstruction problem by compressed sensing. We extracted 472 from 512 possible parallel lines in the spatial frequency of an image I. The other 40 lines of 512 were removed (see Fig. 1). Thus, a

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Fig. 1. MRI image reconstruction results. Upper left. Removed DFT coefficients (in white). Upper right. Original MRI image. Lower left. Linear reconstruction. Some artifacts are pointed by an arrow. Lower right. Sparse reconstruction by CG-M-FOCUSS.

472 × 512 DFT coefficient matrix I f , the kept DFT coefficient matrix of the original MRI image I after removing, was obtained, whose compressed sensing matrix Φ is a 472 × 512 matrix by randomly removing the corresponding 40 rows of the DFT transform matrix. Considering that usually images have sparse representation in the wavelet domain, we reconstruct the MRI image in the wavelet domain using the Daubechies 4 transform W . Then, we can derive the following complex-valued compressed sensing problem: I f = Φ · I = Φ · W −1 · W · I = A · I W ,

(13)

where A = Φ · W −1 and I W = W · I. The equation (13) can be further represented as a standard real-valued compressed sensing problem with 512 samples (t = 1, · · · , 512; m = 472, n = 512): I R I IR f + I f = (A + A ) · I W ,

(14)

I R I where I R f , I f are respectively the real part and imaginary part of I f and A , A are the real part and imaginary part of A, respectively. Then, we can reconstruct the original MRI image I by Iˆ = W −1 · IˆW , where IˆW is the solution of (14). Similar to the M-FOCUSS, it also takes much time for CG-M-FOCUSS to ¯ T . As mentioned in Seccompute the matrix-matrix multiplications AΠ −1 (S)A tion 3, for this kind of compressed sensing, fast algorithms are usually available [1]. Note that A = Φ · W −1 , where W is an orthogonal wavelet matrix (W −1 = W T ) and Φ is a part of a Fourier transform matrix. So this multi¯ T can be done efficiently by performing the fast inverse plication AΠ −1 (S)A wavelet transform and fast DFT on matrix Π −1 [1]. For any vector v ∈ Rn , the

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Table 1. MRI reconstruction results Method PSNR [dB] Runtime (seconds) Linear reconstruction 25.03 \ Standard FOCUSS 28.88 1913.17 M-FOCUSS 33.91 942.58 CG-M-FOCUSS 33.81 525.98

computational complexity of fast DFT is only O(n log n). In addition, the SVD or EVD step for finding a good preconditioned transform matrix C is omitted in this example because AAT = I is an identity matrix. All algorithms run 30 iterations and converge within 30 iterations. From Table 1, we can see that M-FOCUSS and CG-M-FOCUSS achieved better results than the standard FOCUSS. Moreover, M-FOCUSS and CG-M-FOCUSS almost achieved the similar results (i.e., the PSNRs are a little higher than 33.8dB) and the main difference is the computational time. So we show only the reconstructed MRI by CG-M-FOCUSS in Fig. 1. In addition, Table 1 shows that the sparse MRI method approximately gained 8.8dB compared with linear reconstruction method, which sets the unobserved DFT coefficients to zeros and then directly performs the inverse DFT. We also can compare their results in Fig. 1, where the linear reconstruction suffers from the arc-like streaking artifacts (pointed by the arrow) due to undersampling, whereas the artifacts are much less noticeable in the sparse reconstruction.

5

Conclusions

M-FOCUSS is a very efficient method for the sparse representation and compressed sensing, which can simultaneously process multiple measurement vectors. In this paper, we extended M-FOCUSS to CG-M-FOCUSS by incorporating CG iterations. The CG-M-FOCUSS is computationally less expensive and more suitable for large scale problems in comparison to the standard M-FOCUSS. In addition, the application of CG-M-FOCUSS in compressed sensing is also discussed. An MRI image reconstruction was performed by distributed compressed sensing. We have shown that CG-M-FOCUSS can considerably reduce computation time compared to the standard FOCUSS and M-FOCUSS while achieving almost the same PSNR as M-FOCUSS. In addition, we would like to emphasize that M-FOCUSS and CGM-FOCUSS can achieve better results by applying overlapping. This point is confirmed and supported by the MRI reconstruction example given in this paper.

References 1. Kim, S.J., Koh, K., Lustig, M., Boyd, S., Gorinevsky, D.: An interior-point method for large-scale 1 -regularized least squares. IEEE Journal on Selected Topics in Signal Processing 4(1), 606–617 (2007)

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2. Rao, B.D.: Signal processing with the sparseness constraint. In: Proceedings of the ICASSP, Seattle, WA, vol. III, pp. 1861–1864 (1998) 3. Cichocki, A., Amari, S.: Adaptive Blind Signal and Image Processing: Learning Algorithms and Applications. John Wiley & Sons, New York (2003) 4. Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Processing 41(12), 3397–3415 (1993) 5. Tropp, J.: Greed is good: algorithmic results for sparse approximation. IEEE Trans. Information Theory 50(10), 2231–2242 (2004) 6. Chen, S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM Journal on Scientific Computing 20(1), 33–61 (1998) 7. Donoho, D.L., Elad, M.: Maximal sparsity representation via 1 minimization. In: Proc. National Academy Science, vol. 100, pp. 2197–2202 (2003) 8. Li, Y.Q., Cichocki, A., Amari, S.: Analysis of sparse representation and blind source separation. Neural Computation 16, 1193–1234 (2004) 9. Takigawa, I., Kudo, M., Toyama, J.: Performance analysis of minimum 1 -norm solutions for underdetermined source separation. IEEE Trans. Signal Processing 52(3), 582–591 (2004) 10. Li, Y.Q., Amari, S., Cichocki, A., Ho, D.W.C., Xie, S.L.: Underdetermined blind source separation based on sparse representation. IEEE Trans. Signal Processing 54(2), 423–437 (2006) 11. Bofill, P., Zibulevsky, M.: Underdetermined blind source separation using sparse representations. Signal Processing 81, 2353–2362 (2001) 12. Gorodnitsky, I.F., George, J., Rao, B.D.: Neuromagnetic source imaging with FOCUSS: A recursive weighted minimum norm algorithm. Electroencephalography and Clinical Neurophysiology 95(4), 231–251 (1995) 13. Rao, B.D., Kreutz-Delgado, K.: Deriving algorithms for computing sparse solutions to linear inverse problems. In: Conference Record of the Thirty-First Asilomar Conference on Signals, Systems and Computers, vol. 1, pp. 955–959 (1997) 14. Gorodnitsky, I.F., Rao, B.D.: Sparse signal reconstruction from limited data using FOCUSS: A re-weighted minimum norm algorithm. IEEE Trans. Signal Processing 45(3), 600–616 (1997) 15. Rao, B.D., Kreutz-Delgado, K.: An affine scaling methodology for best basis selection. IEEE Trans. Signal Processing 47(1), 187–200 (1999) 16. Kreutz-Delgado, K., Murry, J.F., Rao, B.D., et al.: Dictionary learning algorithms for sparse representation. Neural Computation 15, 349–396 (2003) 17. Cotter, S.F., Rao, B.D., Engan, K., Kreutz-Delgado, K.: Sparse solutions to linear inverse problems with multiple measurement vectors. IEEE Trans. Signal Processing 53(7), 2477–2488 (2005) 18. Baraniuk, R.: Compressive sensing. IEEE Signal Processing Magazine 24(4), 118– 121 (2007) 19. Donoho, D.: Compressed sensing. IEEE Trans. on Information Theory 52(4), 1289– 1306 (2006) 20. Duarte, M., Davenport, M., Takhar, D., Laska, J., Sun, T., Kelly, K., Baraniuk, R.: Single-pixel imaging via compressive sampling. IEEE Signal Processing Magazine 25(2), 83–91 (2008) 21. Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards 49(6), 409–436 (1952)

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22. Nocedal, J., Wright, S.J.: Numerical optimization, 2nd edn. Springer series in operations research and financial engineering. Springer, New York (2006) 23. Duarte, M., Sarvotham, S., Baron, D., Wakin, M., Baraniuk, R.: Distributed compressed sensing of jointly sparse signals. In: Conference Record of the Thirty-Ninth Asilomar Conference on Signals, Systems and Computers, pp. 1537–1541 (2005) 24. Lustig, M., Donoho, D.L., Santos, J.M., Pauly, J.M.: Compressed sensing MRI. IEEE Signal Processing Magazine 25(2), 72–82 (2008)

Dynamic of Cohen-Grossberg Neural Networks with Variable Coefficients and Time-Varying Delays Xuehui Mei and Haijun Jiang College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China [email protected]

Abstract. In this paper, we study the Cohen-Grossberg neural networks with variable coefficients and time-varying delays. By applying the Young inequality technique, Dini derivative and introducing many real parameters, estimate directly the upper bound of solutions. We will establish new and useful criteria on the boundedness and global exponential stability. The results obtained in this paper extend and generalize the corresponding results existing in previous literature. Keywords: Neural networks; Boundedness; Exponential stability; Variable coefficients; Dealys.

1

Introduction

In recent years, the dynamical characteristic such as stability and periodicity of Hopfield network, cellular neural network and bidirectional associative memory neural network play an important rule in the pattern recognition, associative memory, and combinatorial optimization (see [1-10]). In particular, a general neural network, which is called the Cohen-Grossberg neural network and can function as stable associative memory, was developed and studied. The stability of recurrent neural networks is a prerequisite for almost all neural network application. In this paper, we consider a general form of Cohen-Grossberg neural network model with variable coefficients and time-varying delays: for i = 1, 2, · · · , n; x˙ i (t) = −ai (xi (t))[bi (t, xi (t)) − −

n 

n 

cij (t)fj (xj (t))

j=1

(1)

dij (t)fj (xj (t − τij (t))) + Ii (t)]

j=1

The main purpose of this paper is to study the dynamic behavior of the general Cohen-Grossberg neural networks system (1). In this paper, by applying the Young inequality technique, Dini derivative and introducing many real parameters, estimate directly the upper bound of solutions of system (1). We will F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 246–254, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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establish new and useful criteria on the boundedness and global exponential stability. We will see that the results obtained in this paper will extend and generalize the corresponding results existing in [2, 7, 10].

2

Preliminaries

For system (1), in order to convenient description we introduce the following assumptions. (H1 ) functions ai (u) > 0 are bounded, and satisfy locate Lipschitz condition, there are positive constants αi , α¯i such that 0 < αi < ai (u) < α¯i < +∞, for all u ∈ R, t ∈ (0, +∞), i = 1, 2, · · · , n. (H2 ) functions bi (t, u) are continuous, bi (t, 0) are bounded, and there exist positive bounded continuous function βi (t) such that bi (t, u) − bi (t, v) ≥ βi (t) > 0, u−v for all t ∈ (0, +∞), u, v ∈ R, u = v, i = 1, 2, · · · , n. (H3 ) functions fi (u) satisfy Lipschitz condition, i.e, there exist positive constants ki ,(i = 1, 2, · · · , n) such that |fi (u) − fi (v)| ≤ ki |u − v|, foe all u, v ∈ R, i = 1, 2, · · · , n. (H4 ) there are constants p1 , p2 , · · · , pn , such that r−lij r−gij r−hij r−sij r−1 pj (|cij (t)| r−1 kj r−1 + |dij (t)| r−1 kj r−1 ) r j=1

n

pi βi (t) −

1 l l pj (|cij (t)|hij kjij + |dij (t)|sij kjij ) > σ > 0 r j=1 n

−

for all t ∈ [0, ∞], i = 1, 2, · · · , n. Remark 1. In system (2.1), If cij (t) ≡ cij dij (t) ≡ dij , in which aij , dij are constants, then the assume (H4 ) is transformed into the following form. (H4 ) There are constants p1 , p2 , · · · , pn , such that r−lij r−gij r−hij r−sij r−1  pj (|cij | r−1 kj r−1 + |dij | r−1 kj r−1 ) r j=1

n

pi βi (t) −

1 l l pj (|cij |hij kjij + |dij |sij kjij ) > σ > 0 r j=1 n

−

for all t ∈ [0, ∞], i = 1, 2, · · · , n.

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For system (1), we assume that Ii (t), (i = 1, 2, · · · , n) are continuous bounded functions, τij (t), (i = 1, 2, · · · , n) are nonnegative, continuous, bounded functions. Let τ = sup{τij (t) : t ∈ [0, +∞), i, j = 1, 2, ..., n}. We introduce C([−τ, 0], Rn ) as the initial function space of system (1), which is the Banach space of all continuous functions φ = (φ1 (t), φ2 (t), · · · , φn (t))T ∈ C([−τ, 0], Rn ), 1 with normal φ = sup−τ ≤θ≤0 |φ(θ)|, |φ(θ)| = [max1≤i≤n |φi (θ)|r ] r . Definition 1. System (1) is said to be uniformly ultimately bounded, if there exists a constant B > 0, for each H > 0 there exists a T (H) > 0 such that [t0 ∈ R+ , φ ∈ C[−τ, 0], φ ≤ H, t > t0 + T ] imply |x(t, t0 , φ)| ≤ B. Definition 2. System (1) is side to be globally exponentially stable, if there are constants  > 0 and M ≥ 1 such that for any two solutions x(t) = (x1 (t), x2 (t), · · · , xn (t)) and y(t) = (y1 (t), y2 (t), · · · , yn (t)) of systems (1) with the initial functions φ, ψ ∈ C[−τ, 0], respectively, one has |x(t) − y(t)| ≤ M φ − ψ exp (−t) for all t ∈ R+ . As the preliminaries, we firstly give the following lemma on the Young inequality. Lemma 1. Assume that a ≥ 0, b ≥ 0, p > 1, q > 1 with following inequality hold: ab ≤

3

1 p

+

1 q

= 1, then the

1 p 1 q a + b . p q

Boundedness and Global Exponential Stability

Theorem 1. Suppose that (H1 ) − (H4 ) hold. Then system (1) is uniformly bounded. Proof. For B1 > 0, take ϕ ∈ C([−τ, 0], Rn ) such that ϕ ≤ B1 . Let x(t) = (x1 (t), x2 (t), · · · , xn (t)) be the solution of system (1) and satisfies the initial condition xi (s) = ϕi (s), s ∈ [−τ, 0] (i = 1, 2, · · · , n). Let xi (t) = pi ui (t), i = 1, 2, · · · , n, then system (1) is transformed into the following form n  1 ai (pi ui (t))[bi (t, pi ui (t)) − cij (t)fj (pj uj (t)) pi j=1 n  dij (t)fj (pj uj (t − τij (t))) + Ii (t)] −

u˙ i (t) = −

j=1

We take



ui (t)

Vi (t) = | 0

|s|r−1 ds| ai (pi s)

(2)

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249

|ui (t)|r−1 σ(ui (t))u˙ i (t) V˙ i (t) = ai (pi ui (t)) n  |ui (t)|r−1 [−bi (t, pi ui (t)) + cij (t)fj (pj uj (t)) = σ(ui (t)) pi j=1 n  dij (t)fj (pj uj (t − τij (t))) − Ii (t)] + j=1

|ui (t)|r−1 [−(bi (t, pi ui (t)) − bi (t, 0)) − bi (t, 0) pi n n   + |cij (t)||fj (pj uj (t)) − fj (0)| + |dij (t)||fj (pj uj (t − τij (t)))

≤ σ(ui (t))

j=1

−fj (0)| +

n 

|cij (t)||fj (0)| +

j=1

n 

j=1

|dij (t)||fj (0)| + |Ii (t)|]

j=1

n n   |ui (t)|r−1 [−pi ui (t)βi (t) + |cij (t)|pj |uj (t)|kj + |dij (t)| pi j=1 j=1 n n   pj kj |uj (t − τij (t))| + (|bi (t, 0)| + |cij (t)||fj (0)| + |dij (t)||fj (0)|

≤ σ(ui (t))

j=1

j=1

+|Ii (t)|)] n n   1 ≤ [−pi |ui (t)|r βi (t) + |cij (t)|pj kj |ui (t)|r−1 |uj (t)| + |dij (t)|pj kj pi j=1 j=1 |ui (t)|r−1 |uj (t − τij (t))| + M |ui (t)|r−1 ],

let M = sup {|bi (t, 0)| + 1≤i≤n t≥0

Since

n  j=1

|cij (t)||fj (0)| +

n 

|dij (t)||fj (0)| + |Ii (t)|}.

j=1

kj |cij (t)||ui (t)|r−1 |uj (t)| r−lij r−hij 1 r−1 l |cij (t)| r−1 kj r−1 |ui (t)|r + |cij (t)|hij kjij |uj (t)|r ≤ r r

and kj |dij (t)||ui (t)|r−1 |uj (t − τij (t))| r−gij r−sij r−1 1 g ≤ |dij (t)| r−1 kj r−1 |ui (t)|r + |dij (t)|sij kj ij |uj (t − τij (t))|r r r from above, we have D+ Vi (t) ≤

n r−lij r−gij  r−hij r−1 1 (|cij (t)| r−1 kj r−1 kj r−1 )|ui (t)|r [−pi |ui (t)|r βi (t) + pi r j=1 n  1 1 l g + |cij (t)|hij kjij |uj (t)|r + |dij (t)|sij kj ij |uj (t − τij (t))|r r j=1 r (3) +M |ui (t)|r−1 ].

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r r Take B2 = max{B1 , 4M 3σ }. Next to proof |ui (t)| ≤ B2 for all t ≥ 0, i = 1, 2, · · · , n. If |ui (t)|r ≤ B2r is not true, then there exist i and t1 > 0 such that

|ui (t1 )|r = B2r ,

d|ui (t)|r = r|ui (t)|r−1 σ(ui (t))u˙ i (t) ≥ 0 dt

and urj (t) ≤ B2r , for −τ ≤ t ≤ t1 , j = 1, 2, · · · , n. Thus we have |ui (t)|r V˙ i (t) = σ(ui (t)) u˙ i (t) ≥ 0. ai (pi ui (t)) But from (3) we have D+ Vi (t1 ) ≤

n r−lij  r−hij r−sij r−1 1 pj (|cij (t)| r−1 kj r−1 + |dij (t)| r−1 [−pi B r βi (t) + pi r j=1 n n r−gij 1 1 l g kj r−1 ) + B2r + pj |cij (t)|hij kjij B2r + pj |dij (t)|sij kj ij B2r r j=1 r j=1 3 1 3 σB r + σB2r ] < [−σB2r + σB2r ] = − 2 < 0. 4 pi 4 4pi

This is a contradiction. Thus, we get |ui (t)|r ≤ B2r , for all t ≥ 0. Let B3 = max1≤i≤n {pi (B2 + 1)}, we finally have |xi (t)| < B3

(4)

for all t ≥ 0. Therefore, we obtain that solutions of system (1) are defined on R+ and are uniformly bounded. This completes the proof. In system (1), If cij (t) ≡ cij dij (t) ≡ dij , in which aij dij are constants, then the system (1) is transformed into the following form. x˙ i (t) = −ai (xi (t))[bi (t, xi (t)) − −

n 

n 

cij fj (xj (t))

j=1

(5)

dij fj (xj (t − τij (t))) + Ii ]

j=1

Theorem 2. Suppose that assume (H1 ) − (H3 ) and (H4 ) hold. Then system (5) there is an equilibrium point x∗ = (x∗1 , x∗2 , · · · , x∗n ). The proof of Theorem 2 is easy. Here,we omit it. Let y(t) = x(t) − x∗ , then y˙ i (t) = −ai (yi (t) + x∗i )[bi (t, yi (t) + x∗i ) − bi (t, x∗i ) − −fj (x∗j ) −

n  j=1

n 

cij (fj (yj (t) + x∗j )

j=1

dij (fj (yj (t − τij (t)) + x∗j ) − fj (x∗j ))].

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Theorem 3. Suppose that assume (H1 ) − (H3 ) and (H4 ) hold, then the solution of system (5) is globally exponentially stability.  |yi (t)| r−1 rs  Proof. Under assume (H4 ), let Vi (t) = ds, we easy to obtain ai (s) 0 |yi (t)|r |yi (t)|r ≤ Vi (t) ≤ . α¯i αi Calculating the derivative of V (t), we obtain r|yi (t)|r−1 V˙ i (t) = σ(yi (t)){−ai (yi (t) + x∗i )[bi (t, yi (t) + x∗i ) − bi (t, x∗i ) ai (|yi (t)|) n n   cij (fj (yj (t) + x∗j ) − fj (x∗j ) − dij (fj (yj (t − τij ) + x∗j ) − j=1

j=1

−fj (x∗j ))]} n rα rα¯i  ≤ − i βi (t)|yi (t)|r−1 + [ |cij |kj |yi (t)|r−1 |yj (t)| α ¯i αi j=1 n  |dij |kj |yi (t)|r−1 |yj (t − τij (t))|] + j=1

By Young inequality we further have rαi V˙ i (t) ≤ {−βi (t)αi Vi (t)} α ¯i n r−lij r−gij r−hij r−sij rα¯i r − 1  + [ (|cij | r−1 kj r−1 + |dij | r−1 kj r−1 )α¯i Vi (t) αi r j=1 n n  1 1 l g |cij |hij kjij α¯j Vj (t) + |dij |sij kj ij α¯j V¯j (t)] + r j=1 r j=1 in which y¯j (t) =

sup

t−τ ≤s≤t

|yj (s)|, V¯j (t) =

sup

t−τ ≤s≤t

|Vj (s)|

we take λi = inf{λ : pi λ − pi βi (t)αi + 1 + r

r−1 r

n

pj |cij |

hij

l kjij α¯j

j=1

+e

λτ

1 r

n

pi α¯i (|cij |

r−hij r−1

r−lij

kj r−1 + |dij |

r−sij r−1

r−gij

kj r−1 )

j=1 n g

pj |dij |sij kj ij α¯j = 0}. j=1

Let α = min{λ1 , λ2 , · · · , λn }, then we have r−lij r−gij r−hij r−sij n pi α − pi βi (t)αi + r−1 ¯i (|cij | r−1 kj r−1 + |dij | r−1 kj r−1 ) j=1 pi α r n n 1 1 l g pj |cij |hij kjij α¯j + eλτ pj |dij |sij kj ij α¯j ≤ 0, + r j=1 r j=1

for all t ≥ 0, i = 1, 2, · · · , n.

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We choose constant β > 1 such that βpi eλτ > 1 for all t ∈ [−τ, 0] and i = 1, 2, · · · , n. Let Zi (t) = βpi

n 

V¯j (0)e−ατ .

j=1

Further let constant k > 1 such that Vi (t) < kZi (t) for all t ∈ [−τ, 0]. Next to proof Vi (t) < kZi (t)for allt ∈ [0, ∞]. If above is not true. Then there exist i ∈ {1, 2, · · · , n} and ti > 0 such that Vi (ti ) = kZi (ti ) and Vj (t) < kZj (t) for t ∈ [−τ, ti ], j = 1, 2, · · · , n, D+ Vi (ti ) ≥  kZi (ti ) = −αkZi (ti ). since V¯j (ti ) = sup−τ ≤θ≤0 Vj (ti + θ) and Zj (t) is strictly monotone decreasing function, then there is θ∗ ∈ [−τ, 0] such that V¯j (ti ) = sup−τ ≤θ≤0 Vj (ti + θ) = Vj (ti + θ∗ ) < kZj (ti + θ∗ ) ≤ kZj (ti − τ ). Thus  V˙ i (ti ) − kZi (ti ) = D+ Vi (ti ) + kβPi α

n  i=1

V¯i (0)e−αti

r−lij r−hij rα¯i r−1 < [−βi (ti )αi kZi (ti ) + pi α¯i (|cij | r−1 kj r−1 αi r j=1 n r−gij r−sij 1 l r−1 r−1 +|dij | kj )α¯i kZi (t) + |cij |hij kjij α¯j kZi (ti ) r j=1 n rα¯i 1 g |dij |sij kj ij α¯j kZi (ti − τ ) + [pi βi (ti )αi + r j=1 αi n r−lij r−gij r−hij r−sij r−1  − pi α¯i (|cij | r−1 kj r−1 + |dij | r−1 kj r−1 ) r j=1 n n  1 1 l g − pj |cij |hij kjij α¯j − eατ pj |dij |sij kj ij α¯j ]· r j=1 r j=1 n  V¯i (0) kβe−αti

j=1

n

n  rα¯i = V¯i (0) [−βi (ti )αi kβpi e−αti αi j=1 n r−lij r−gij r−hij r−sij r−1  r−1 (|cij | r−1 kj + |dij | r−1 kj r−1 )· + r j=1 n n  1 l V¯i (0) + |cij |hij kjij α¯j kβpi · α¯i kβpi e−αti r j=1 j=1 n n n    1 g V¯i (0) + V¯i (0)] e−αti |dij |sij kj ij α¯j kβpj e−α(ti −τ ) r j=1 j=1 i=1

Dynamic of Cohen-Grossberg Neural Networks with Variable Coefficients

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r−hij rα¯i r−1 [pi βi (ti )αi − pi α¯i (|cij | r1 αi r j=1

n

+

+|dij |

r−sij r−1

1 l pj |cij |hij Kjij α¯j r j=1 n

r−gij

Kj r−1 ) −

 1 g V¯l (0) pj |dij |sij Kj ij α¯j ]Kβe−αti r j=1 n

−e

ατ

n

l=1

=0 This is a contradiction. So we have Vi (t) ≤ Zi (t) = βpi

n 

V¯l (0)e−αt ,

for all t ≥ 0.

l=1

because of  V¯i (0) = sup

|yi (t)|

0

rsr−1 |yi (t)|r φ − x∗  ds ≤ sup ≤ ai (s) αi min1≤i≤n {αi } −τ ≤t≤0

and Vi (t) ≥

|yi (t)|r α¯i

we further have |xi (t) − x∗i | ≤

nα¯i βpi φ − x∗ e−αt , min1≤i≤n {αi }

for t ≥ 0.

This show that all solutions of system (5) are globally exponentially stable. This completes the proof of Theorem 2.

4

Conclusion

In this paper, we have investigated the boundedness and global exponential stability for Cohen-Grossberg neural networks. Using the Young inequality technique, Dini derivative and introducing many real parameters, estimate directly the upper bound of solutions, we gave a sufficient criterion ensuring the boundedness and global exponential stability of system (1). The obtained result improves and extend several earlier publications and is useful in applications of manufacturing high quality neural networks.

Acknowledgement This work was supported by The National Natural Science Foundation of P.R. China (60764003), The Major Project of The Ministry of Education of P.R. China (207130) and The Scientific Research Programmes of Colleges in Xinjiang (XJEDU2007G01, XJEDU2006I05).

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References 1. Cao, J., Li, X.: Stability in Delayed Cohen-Grossberg Neural Networks: LMI Optimization Approach. Physica D 212, 54–65 (2005) 2. Cao, J., Liang, J.: Boundedness and Stability for Cohen-Grossberg Neural Networks with Time-Varying Delays. J. Math. Anal. Appl. 296, 665–685 (2004) 3. Chen, T., Rong, L.: Delay Independent Stability Analysis of Cohen-Grossberg Neural Networks. Phys. Lett. A 317, 436–449 (2003) 4. Lu, W., Chen, T.: New Conditions on Global Stability of Cohen-Grossberg Neural Networks. Neural Comput. 15, 1173–1189 (2003) 5. Hwang, C., Cheng, C., Li, T.: Globally Exponential Stability of Generalized CohenGrossberg Neural Networks with Delays. Phys. Lett. A 319, 157–166 (2003) 6. Li, Y.: Existence and Stability of Periodic Solutions for Cohen-Grossberg Neural Networks with Multiple Delays. Chaos Solitons & Fractals 20, 459–466 (2004) 7. Wang, L., Zou, X.: Exponential Stability of Cohen-Grossberg Neural Networks. Neural Networks 15, 415–422 (2002) 8. Wang, L., Zou, X.: Harmless Delays in Cohen-Grossberg Neural Networks. Phys. D 170, 163–173 (2002) 9. Yuan, K., Cao, J.: An Analysis of Global Asymptotic Stability of Delayed CohenGrossberg Neural Networks via Nonsmooth Analysis. IEEE Trans. Circuits Syst.I 52, 1854–1861 (2005) 10. Zeng, Z., Wang, J.: Global Exponential Stability of Recurrent Neural Networks with Time-Varying Delays in the Presence of Strong External Stimuli. Neural Networks 19, 1528–1537 (2006)

Six-Element Linguistic Truth-Valued Intuitionistic Reasoning in Decision Making Li Zou1 , Wenjiang Li2 , and Yang Xu3 1

2

School of Computer and Information Technology, Liaoning Normal University, Dalian 116029, P.R. China [email protected] College of Automation, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610054, P.R. China wjl [email protected] 3 Intelligent Control and Development Center, Southwest Jiaotong University, Chengdu, 610031, P.R. China [email protected]

Abstract. A kind of intuitionistic linguistic truth-valued reasoning approach for decision making with both comparable and incomparable truth values is proposed in this paper. By using the lattice implication algebra, an six-element linguistic truth-valued intuitionistic propositional logic system is established which can express both truth degree and falsity degree. The implication operation of linguistic truth-valued intuitionistic propositional logic can be deduced from four times implication of their truth values. Therefore, we can use more information in the process of reasoning and eventually improve the precision of reasoning. As reasoning and operation are directly acted by linguistic truth values in the decision process, the issue on how to obtain the weight for rational decision making results is discussed. An illustration example shows the proposed approach seems more effective for decision making under a linguistic information environment with both truth degree and falsity degree. Keywords: Lattice implication algebra, Linguistic truth-valued intuitionistic propositional logic, Decision making.

1

Introduction

In the real world, people usually do judgement in a natural language with some uncertain words. The truth values of a fuzzy proposition are nature linguistic, e.g., of the form “true”,“very true”,“possible false”, etc. Therefore, truth values of proposition are often not exactly true or false, but accompany with linguistic hedges [1], such as absolute, highly, very, quite, exactly, almost, rather, somewhat, slightly and so on. Different linguistic hedges will lead to different judgments Where as the degree of the assessment will be strengthened or weakened by the linguistic hedges. In recent years, some researchers have paid their attention to linguistic hedges. Ho proposed an algebraic model, Hedge Algebra, for dealing F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 266–274, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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with linguistic information [2][3]. Turksen studied the formalization and inference of descriptive words, substantive words and declarative sentence [4][5]. Huynh [6] proposed a new model for parametric representation of linguistic truth-values [7][8]. Many-valued logic, a great extension and development of classical logic, has always been a crucial direction in the non-classical logic. It is a good tool to deal with linguistic values. There exists incomparable linguistic truth value in the many-valued logic. Lattice-valued logic system is an important case of manyvalued logic. It can be used to describe uncertain information that may be comparable or incomparable. In [9][10], Xu et al. discussed the lattice-valued propositional logic LP(X) and gradational Lattice-valued propositional Logic Lvpl based on lattice implication algebra. Xu et.al. have done some research on characterizing the set of linguistic values by a lattice-valued algebraic structure and investigate the corresponding logic systems with linguistic truth-value based on lattice implication algebra(LIA for short) [11][12]. From the point of lattice-valued logic system view [13][14], linguistic truth-values can be put into the lattice implication algebra(LIA) [15][16]. Zou [17] proposed a framework of linguistic truth-valued propositional logic and developed the reasoning method of six-element linguistic truth-valued logic system. Sometimes, we analysis an event which has both certainty and uncertainty characteristic or has both obverse and inverse demonstration. Therefore, a proposition has two truth values: truth degree and falsity degree. From the view of intuitionistic fuzzy set introduced by K.Atanassov, the true value of a fuzzy proposition p are juxtaposed two two real number (μ(p), ν(p)) on the closed interval [0,1] with the following constraint: μ(p) + ν(p) ≤ 1. In [18] the evaluation function V was defined over a set of propositions S in such a way that V (p) =< μ(p), ν(p) > . Hence the function V : S → [0, 1] × [0, 1] gives the truth and falsity degrees of all propositions in S. which represents its truth degree and its falsity degree [1]. With above work,we will put the linguistic truth-values into intuitionistic fuzzy logic. The truth values of the intuitionistic fuzzy logic are linguistic truthvalues instead of number. Then we discuss the properties of linguistic truthvalued reasoning in intuitionistic fuzzy logic. The rest of this paper is organized as follows: Section 2 outlines Six-element linguistic truth-valued lattice implication algebra, which can express both the comparable and incomparable truth values of linguistic truth-values. Section 3 introduces Linguistic truth-valued intuitionistic logic which truth-value field is Six-element linguistic truth-valued lattice implication algebra and its logic properties are provided. Section 4 illustrates with an example how the proposed

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approach works for the reasoning method of six-element linguistic truth-valued intuitionistic propositional logic. Finally, Section 5 concludes the paper.

2

Six-Element Linguistic Truth-Valued Lattice Implication Algebra

In this section we briefly review the notion of linguistic truth-valued lattice implication algebra and its main properties. Let V be a linguistic truth values set, every linguistic truth value v ∈ V is composed of a linguistic hedge operator h and a basic word c, i.e. V =H × C where the linguistic hedge operator set H is a totally ordered and finite set. According to the characteristic of lattice implication algebra, we can construct a new lattice implication algebra using the product of some lattice implication algebras. When a hedge is added to the sentence P (x), the truth value V (P ) of P will be strengthened or weakened, denoted it by HV (P ). Hence the truth value set is V = {h+ T, 0T, h−T, h+ F, 0F, h− F }, which represents strong true, true, weak true, weak false, false and strong false. Let L=(V , ∨, ∧,  , →), its operation “∨ and “∧” shown in the Hasse diagram of L defined in Fig. 1 and its operations “ −→” and “  ” defined in Table 1 and Table 2 respectively. Then L = (V, ∨, ∧, , →) is a lattice implication algebra.

h+T 0T h -F -

hT 0F h+F

Fig. 1. Hasse diagram of L

We will discuss linguistic truth-valued intuitionistic logic based on six-element lattice implication algebra.(abbreviated to LT V − IP ).

3

Six-Element Linguistic Truth-Valued Intuitionistic Propositional Logic

Since some kinds of truth and falsity are incomparable, we can choose the linguistic truth-values based on six-element LIA as the truth-valued field of intuitionistic logic. We denote the linguistic truth-valued intuitionistic proposition

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Table 1. Implication operator of L = (V, ∨, ∧, , →) →

h+ F 0F

h− F h− T 0T

h+ T

h+ F 0F h− F h− T 0T h+ T

h+ T hT h− T h− F 0F h+ F

h+ T hT h+ T h− F h− F h− F

h+ T h+ T h+ T h+ T h+ T h+ T

h+ T h+ T 0T h− F h− F 0F

h+ T 0T h− T h+ T 0T h− T

h+ T h+ T 0T h+ T h+ T 0T

Table 2. Complementary operator of L = (V, ∨, ∧, , →) v h+ F 0F h− F 0T h+ T v’ h+ T 0T h− T 0F h+ F

by LTV-IP.According to the definition of intuitionistic proposition, the truthvalued field of LTV-IP is as follows: Li = {(h+ T, h− F ), (0T, 0F ), (0T, h− F ), (h− T, h− F ), (h− T, 0F ), (h− T, h+ F )}.

The complementary “  ” hold as follows:

Table 3. Complementary operator of Li v (h+ T, h− F ) (0T, 0F ) (0T, h− F ) (h− T, h− F ) (h− T, 0F ) (h− T, h+ F )

v (h− T, h+ F ) (0T, 0F ) (h− T, 0F ) (h− T, h− F ) (0T, h− F ) (h+ T, h− F )

The conjunction, disjunction and implication are shown as follows: Let G, H ∈ LT V − IP , v(G) = (hi T, hj F ), v(G) = (hm T, hl F ), 1.v(G ∨ H) = (hi T ∨ hm T, hj F ∧ hl F ); 2.v(G ∧ H) = (hi T ∧ hm T, hj F ∨ hl F ). 3.v(G → H) = v(G) → v(H) = (hi T → hm T ) ∧ (hj F → hm T ) ∧ hj F → hl F, (hi T → hl F ); Note that, for 1 and 2 they satisfy the valuation conditions of LTV-IP obviously. For 3, we get v(G → H) = v(G) → v(H) = (hi T → hm T ) ∧ (hj F → hm T ) ∧ (hj F → hl F ), (hi T → hl F ) = (hmin{n,n−i+m} T ∧ hmin{n,j+m} T ∧ hmin{n,n−l+j} T ), hmax{0,i+l−n} F )

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For the truth degree of G → H there are four cases, the subscripts are n, n − i + m, j + m, n − l + j respectively. For the falsity degree of G → H, the subscript is i + l − n. We can prove that the sum is always equal to or less than n. Hence the definitions of conjunction, disjunction and implication of LTV-IP are rational. The symbols in LTV-IP Logic system are (1) The set of propositional variable: X = {p, q, r, ...}; (2) The set of constants: Li ; (3) Logical connectives: →,  ; (4) Auxiliary symbols: ),(. The set F of formulae of LTV-IP is the least set Y satisfying the following conditions: (1)X ⊆ Y ; (2)L ⊆ Y ; (3)If p, q ∈ Y ,then p and p → q ∈ Y. Note that from the viewpoint of universal algebra, LTV-IP is the free algebra on X w.r.t. the type T = Li ∪ { , →}, where α ∈ Li is a 0-ary operation. According to the properties of lattice implication algebra, L and LTV-IP can be looked as algebras with the same type T = Li ∪ { , →} and for any p, q ∈ F , (1)p ∨ q = (p → q) → q, (2) p ∧ q = (p ∨ q  ). Definition 1 A mapping v : LT V − IP → Li , is called a valuation of LTV-IP, if it is a T -homomorphism. Corollary 1 Let v : LT V − IP → Li be a mapping, then v is a valuation of LTV-IP if and only if it satisfies (1)v(hα T, hβ F )=(hα T, hβ F ), for any (hα T, hβ F ) ∈ Li ; (2)v(p ) = (v(p)) for any p ∈ F ; (3)v(p → q) = v(p) → v(q) for any p, q ∈ F . Definition 2 Well-formed formula of LTV-IP or formula for short are defined recursively as follows: (1) LTV-IP atom is a formula; (2) If G, H are LTV-IP formulae, then ∼ G, (G ∨ H), (G ∧ H), ((G → H) and (G → H) are formulae; (3) No expression is a formula unless it is compelled to be one by (1) and (2). Some intuitionistic linguistic truth-valued properties hold as follows: Theorem 1 For any (hi T, hj F ) ∈ Li , (1) (h− T, h+ F ) → (hi T, hj F ) = (h+ T, h− F ), (2) (h+ T, h− F ) → (hi T, hj F ) = (hi T, hj F ), (3) (hi T, hj F ) → (h− T, h+ F ) = (hj T, hi F ), (4) (hi T, hj F ) → (h+ T, h− F ) = (h+ T, h− F ).

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Corollary 2 For any ((hi , T ), (hj , F )) ∈ Li , (1) (h− T, h+ F ) → (h− T, h+ F ) = (h+ T, h− F ), (2) (h+ T, h− F ) → (h+ T, h− F ) = (h+ T, h− F ), (3) (h+ T, h− F ) → (h− T, h+ F ) = (h− T, h+ F ), (4) (h− T, h+ F ) → (h+ T, h− F ) = (h+ T, h− F ). Definition 3 For any (hi T, hj F ), (hm T, hl F ) ∈ Li , (hi T, hj F ) is said to truer than (hm T, hl F ) if and only if hi T ≥ hm T and hj F < hl F or hi T > hm T and hj F ≤ hl F , denoted by (hi T, hj F ) ≥ (hm T, hl F ). Theorem 2 If For any (hi T, hj F ), (hm T, hl F ) ∈ Li , (hi T, hj F ) ≥ (hm T, hl F ), then (hi T, hj F ) → (h− T, h+ F ) ≤ (hm T, hl F ) → (h− T, h+ F ). Proof. From Theorem 1, we get (hi T, hj F ) → (h− T, h+ F ) = (hj T, hi F ); (hm T, hl F ) → (h− T, h+ F ) = (hl T, hm F ). Since(hi T, hj F ) ≥ (hm T, hl F ), and the linguistic hedge set H = hi |i = 1, 2...n is a chain, then we get (hj T, hi F ) ≤ (hl T, hm F ). Note that if the consequence is the most false then the truth degree of the implication will decrease while the truth degree of the premise increases. Conversely, while the truth degree of the premise decreases, the truth degree of the implication will increase. This property is consistent with the classical logic and people’s intuition. Also, the linguistic truth-values based on LIA are special cases of intuitionistic linguistic truth-values. So the linguistic truth-valued intuitionistic logic is an extension of linguistic truth-valued logic. Theorem 3 For any (hi T, hj F ), (hi T, hl F ), (hm T, hj F ) ∈ L, (1)(hi T, hj F ) → (hi T, hl F ) = (hi+j T, h− F ), (2)(hi T, hj F ) → (hm T, hj F ) = (hj+m T, h− F ), Now when we do fuzzy inference in intuitionistic fuzzy logic system based on linguistic truth-value, we must consider the fact that the proposition has the truth degree as well as the falsity degree. So more information is used in the reasoning process, which can improve the precision of reasoning and reduce the loss of information in a sense.

4

A Kind of Decision Making Approach

In this section, we consider a multiple attribute decision making with a pair of linguistic truth value information: Let A = A1 , A2 , ..., An be a finite set of alternatives, and let G = G1 , G2 , ..., Gn be a finite set of attributes and W = (w1 , w2 , ..., wn ) be the Intuitionistic fuzzy linguistic truth-valued weigh vetor of attributes. Let R = (rij )m×n be an Intuitionistic fuzzy linguistic truth-valued decision matrix, where rij = (μij , νij ) ∈

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Li , where μij indicates the degree that the alternative Aj satisfies the attribute Gi , while νij indicates the degree that the alternative Aj does not satisfies the attribute Gi . The conclusion is n 

D=

(wi → Gi (Aj )).

i=1

The optimal alternative is the Aj ∈ A that maximizes D. Example: We consider a simple example to evaluate the set of cars A = {A1 = Chevrolet, A2 = T oyota, A3 = Buick}, the attribute set G = {G1 = comf ort, G2 = price, G3 = repairf requency}. Assume the evaluation set is Li , where ”true” is changed as ”high” and ”f alse” is changed ”low” respectively. Assume the weights of importance for G as w1 = (0T, h− F ), w2 = Table 4. Evaluation table rij A1

A2

A3

G1 (0T, 0F ) (h+ T, h− F ) (h− T, h− F ) G2 (0T, h− F ) (h− T, 0F ) (h+ T, h− F ) G3 (h− T, h+ F ) (0T, h− F ) (0T, 0F )

(h− T, 0F ), w3 = (h+ T, h− F ), then Table 5. Weighted evaluation table rij

A1

A2 −

A3

w1 → G1 (0T, h F ) (h T, h F ) (h− T, h− F ) w2 → G2 (0T, h− F ) (0T, h− F ) (0T, h− F ) w3 → G3 (h− T, h+ F ) (0T, h− F ) (0T, 0F ) +

−

Hence, D = {(h− T, h− F )/Chevrolet, (0T, h− F )/T oyota, (h− T, h− F )/Buick} Finally, we will choose the Toyota. Note that sometimes there exists some incomparable elements in the the result and this is according with people’s intuition.

5

Conclusions

We have found that some properties of lattice-valued logic based on linguistic truth-valued are fit for researching linguistic truth-values. The result is consistent with people’s intuition. The classical logic and linguistic truth-valued logic based on LIA are the special cases of this logic system.

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The problem which has positive evidence and negative evidence at the same time can be dealt with by means of linguistic truth-value intuitionistic fuzzy logic. If a proposition has both credibility and incredibility, then the reasoning method proposed above can be used. It is illustrated by the proposed approach that linguistic truth-valued propositional logic makes intelligent decision making systems more effective. Acknowledgments. This work is partially supported by National Nature Science Foundation of China (Grant No. 60603047)and the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20060613007.

References 1. Herrera, F., Herrera, E., Martinez, L.: A Fusion Approach for Managing Multigranularity Linguistic Term Sets in Decision Making. International Journal of Fuzzy Sets and Systems 114, 43–58 (2000) 2. Ho, N.C., Wechler, W.: Hedge Algebras: an Algebraic Approach to Structure of Sets of Linguistic Turth Values. International Journal of Fuzzy Sets and Systems 35, 281–293 (1990) 3. Ho, N.C., Wechler, W.: Extended Hedge Algebras and Their Application to Fuzzy Logic. International Journal of Fuzzy Sets and Systems 52, 259–281 (1992) 4. Turksen, I.B.: Computing With Descriptive and Verisic Words. In: NAFIP 1999, pp. 13–17 (1999) 5. Turksen, I.B., Kandel, A., Zhang, Y.Q.: Universal Truth Tables and Normal Forms. IEEE Trans. Fuzzy Systems 6, 295–303 (1998) 6. Huynh, V.N., Nam, H.V.: Ordered Structure-based Semantics of Linguistic Terms of Linguistic Variables and Approximate Reasoning. In: Third International Conference on Computing Anticipatory Systems (CASYS 1999), pp. 98–116. AIP Press, New York (1999) 7. Huynh, V.N., Ho, T.B., Nakamori, Y.: A Parametric Representation of Linguistic Hedges in Zadeh’s Fuzzy Logic. International Journal of Approximate Reasoning 30, 203–223 (2002) 8. Nguyen, C.H., Huynh, V.N.: An Algebraic Approach to Linguistic Hedges in Zadeh’s Fuzzy Logic. International Journal of Fuzzy Set and Systems 129, 229– 254 (2002) 9. Xu, Y., Ruan, D., Qin, K.Y., Liu, J.: Lattice-Valued Logic. Springer, Heidelberg (2004) 10. Xu, Y., Ruan, D., Kerre, E.E., Liu, J.: α-Resolution Principle Based on Latticevalued Propositional Logic LP(X). International Journal of Information Sciences 130, 195–223 (2000) 11. Xu, Y., Ruan, D., Kerre, E.E., Liu, J.: α-Resolution Principle Based on First-order Lattice-valued Propositional Logic LF(X). International Journal of Information Sciences 132, 195–223 (2001) 12. Xu, Y., Liu, J., Ruan, D., Lee, T.T.: On the Consistency of Rule Bases Based on Lattice-valued First-Order Logic LF(X). International Journal of Intelligent Systems 21, 399–424 (2006)

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13. Pei, Z., Xu, Y.: Lattice Implication Algebra Model of a Kind of Linguistic Terms and its in Inference. In: 6th Internal FLINS Conference, pp. 93–98 (2004) 14. Xu, Y., Chen, S., Ma, J.: Linguistic Truth-Valued Lattice Implication Algebra and Its Properties. In: 2006 IMACS Multiconference on Computational Engineering in Systems Applications (CESA 2006), Beijing, pp. 1413–1418 (2006) 15. Zou, L., Liu, X., Xu, Y.: Resolution Method of Linguistic Truth-valued Propositional Logic. In: International Conference on Neural Networks and Brain, pp. 1996–2000. IEEE Press, New York (2005) 16. Zou, L., Ma, J., Xu, Y.: A Framework of Linguistic Truth-valued Propositional Logic Based on Lattice Implication Algebra. In: 2006 IEEE International Conference on Granular Computing, pp. 574–577. IEEE Press, New York (2006) 17. Zou, L., Li, J.L., Xu, K.J., Xu, Y.: A Kind of Resolution Method of Linguistic Truth-valued Propositional Logic Based on LIA. In: 4th International Conference on Fuzzy Systems, pp. 32–36. IEEE Press, New York (2007) 18. Atanassov, K.: Elements of Intuitionistic Fuzzy Logic, Part I. International Journal of Fuzzy Set and Systems 95, 39–52 (1998)

A Sequential Learning Algorithm for RBF Networks with Application to Ship Inverse Control Gexin Bi and Fang Dong College of Navigation, Dalian Maritime University, 1 Linghai Road, 116026 Dalian, China [email protected], [email protected]

Abstract. A sequential learning algorithm for constructing radial basis function (RBF) network is introduced referred to as dynamic orthogonal structure adaptation (DOSA) algorithm. The algorithm learns samples sequentially, with both structure and connecting parameters of network are adjusted on-line. The algorithm is further improved by setting initial hidden units and incorporating weight factors. Based on the improved DOSA algorithm, a direct inverse control strategy is introduced and applied to ship control. Simulation results of ship course control simulation demonstrate the applicability and effectiveness of the improved DOSA algorithm and the RBF network-based inverse control strategy. Keywords: Radial basis function network, Sequential learning, Inverse control.

1

Introduction

Neural networks have become a fashionable area of research in recent years. They have been applied in many areas such as modelling and controlling of non-linear dynamic systems [1]. Most of the successful application results, however, are achieved when the network is applied to system with static dynamics. It is realized that a considerable part of industry processes have time-varying dynamics in their nature, and such processes are prone to be influenced by time-varying environment. Thus, adaptive control of such systems could not be obtained merely using neural network models with fixed structure. One solution is to develop adaptive neural network models and control strategies whose structure are variable to process time-variants. As a kind of feed forward neural networks, radial basis function (RBF) networks are found to be suitable for on-line adaptation because of their features of best approximation and quick convergence [2]. Utilizing the merits of RBF networks, sequential learning algorithms are developed in recent years which are suitable for control applications. Sequential learning algorithms overcome drawbacks of RBF networks with fixed structure. They does not need retraining whenever a new observation is received, with advantages of low computation burden and adaptive ability. The most widely used sequential learning algorithms are RAN, RANEKF, MRAN and GGAP-RBF algorithms [3-6]. F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 275–282, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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In this paper, we introduce the dynamic orthogonal structure adaptation (DOSA) algorithm to construct RBF networks online [7]. It adds new observation as a hidden unit directly, and makes use of normalized error reduction ratio to prune units which contribute less to system output. The algorithm is adaptive to the changes of system dynamics with fast learning speed, while employs only a small number of parameters. By combining RBF network with adaptive inverse control mechanism, we present a neural inverse control strategy [8,9]. Ship course control simulation results demonstrate the applicability and effectiveness of the algorithm and control strategy.

2

DOSA Algorithm for RBF Networks

DOSA algorithm combines subset selection scheme with a sequential learning mode, the resulting network structure is variable by adding the new observation as hidden unit directly and pruning units which contribute less to output over a number of observations. The contribution of each hidden unit is measured by its normalized error reduction ratio. The conception of normalized error reduction ratio is generalized from error reduction ratio in OLS algorithm [10]. The sliding window is a first-in-first-out sequence with fixed width. When a new observation is received, the sliding window is updated by adding the new observation and discarding the foremost one. The window is of the form: window = [(x1 , y1 ), (x2 , y2 ), · · · , (xN , yN )]

(1)

N is the width of the sliding data window. The data in the sliding data are composed of input X ∈ Rn×N and output Y ∈ RN ×m . n is the dimension of the input and m is the dimension of the output. The learning procedure begins with no hidden units. At each step, the new observation is added as a hidden unit directly. The candidate units is then formed together with the existing hidden units: ⎞ ⎛ c1,1 · · · c1,M ⎜ .. ⎟ (2) C = [c1 , . . . , cM ] = ⎝ ... ... . ⎠ cn,1 · · · cn,M where M is the number of candidate hidden units. n is the dimension of the candidate hidden units. By calculating the Gaussian functions of Euclidean distance between the sliding window inputs and the candidate hidden units, we have response matrix Φ of hidden units to the input matrix of sliding window: ⎞ ⎛ φ1,1 · · · φ1,M ⎜ .. .. ⎟ Φ = [Φ1 , . . . , ΦM ] = ⎝ ... (3) . . ⎠ φN,1 · · · φN,M

A Sequential Learning Algorithm for RBF Networks

with φj,k = exp(−

xj − ck 2 ) 2σ 2

1 ≤ j ≤ N, 1 ≤ k ≤ M

277

(4)

where ck are known as the k-th hidden units, σ a width constant and  ·  the Euclidean norm. In sequential learning scheme, we pay more attention to the new observations because the information they bring is more capable to represent the dynamics of system. This is more obvious when the data are used to on-line represent systems with time-varying dynamics. In this paper we improve DOSA algorithm by employing forgetting factors. Here we use the linear weighting coefficients: βi =

2i N (N + 1)

1≤i≤N

(5)

The response matrix is transformed into weighted response matrix by multiply βi to the corresponding elements of the matrix. Φ is transformed into a set of orthogonal basis vectors by decomposing Φ into Φ = W A. Here we implement the Gram-Schmidt method. ⎞ ⎛ w1,1 · · · w1,M ⎜ .. .. ⎟ (6) W = [w1 , . . . , wM ] = ⎝ ... . . ⎠ wN,1 · · · wN,M Calculate the error reduction ratio of each vector wk : [err]ki =

(wkT yi )2 T (wk wk )(yiT yi )

(7)

For a multi-input multi-output (MIMO) process, m [err]ki =

T 2 i=1 (wk yi )

(wkT wk )trace(Y T Y )

(8)

The geometrical interpretation of error reduction ratio is: (wkT yi )2 = cos2 θki wk 2 yi 2

(9)

where θki is the angle between the basis  vector 2wk and desired output vector yi . According to vector space theory, M k=1 cos θki = 1 for single-output condiM tions. This explains why k=1 [err]k = 1 in OLS algorithm under single output condition. Unlike OLS algorithm, the response matrix in DOSA algorithm is generally not square because the size of sliding window input is generally not M the same as the number of hidden units. Specifically, k=1 [err]k > 1 when M M M > N and k=1 [err]k < 1 when M < N . k=1 [err]k = 1 hold true only

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when M = N . In order to evaluate the contribution of hidden units directly, the normalized error reduction ratio (nerr) is obtained by [err]k [nerr]k = M k=1 [err]k

(10)

At each step, we select units whose sum of error reduction ratio is less than an accuracy threshold ρ. Select [nerr]k1 = min{[nerr]k , 1 ≤ k ≤ M }. If [nerr]k1 < ρ, then select [nerr]k2 = min{[nerr]k , 1 ≤ k ≤ M, k = k1 }. The selection prok=kS +1 cedure continues until [nerr]k ≥ ρ. Select k1 , . . . , kS and mark the k=k1 corresponding hidden units Sk = {ck1 , . . . , ckS }. Make the selection at each step. If the same hidden units are selected for MS consecutive observations, the particular units will be pruned from the network. That is, remove the units in the intersection of sets selected in the past MS observations. I = {Sk Sk−1 . . . Sk+MS −1 } (11) After the hidden units being added or pruned at each step, the weights between the hidden layer and output layer Θ are adjusted using the linear least mean squares estimation: Θ = Φ+ Y = (ΦT Φ)−1 ΦT Y

(12)

One drawback of sequential learning is that there are no initial hidden units, which will result in more learning time and complexity of algorithm. We further improve the algorithm by setting several initial samples as hidden units directly without pruning. This proved to be helpful to stabilize the learning procedure.

3

The Neural Inverse Controller

Adaptive neural inverse control for unknown discrete-time nonlinear dynamical system has received much attention in recent years [8,9]. Its basic idea is to use a signal that comes from the controller to drive the plant while the model of the controller is the inverse model of the plant, the output of the plant follows the input to the controller and then realizing the anticipate control effects [8]. Therefore, the key of inverse control is how to obtain the inverse model of the plant. In our study, the RBF network constructed by improved DOSA algorithm is introduced to satisfy the requirements of on-line control. The configuration of the proposed direct inverse control strategy is shown in Fig. 1. The controller employ RBF network constructed by the DOSA algorithm. The inputs of the network include the desired output, derivatives and delayed messages from the input and output of system. To describe the inputCoutput dynamics of a nonlinear system, the concept of a nonlinear autoregressive model with exogenous inputs (NARX) system representation is used: y(t + 1) = f (y(t), . . . , y(t + 1 − ny ), u(t), . . . , u(t + 1 − nu ))

(13)

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Fig. 1. Configuration of RBF network-based adaptive inverse control strategy

where y(·) is the system output, u(·) is the system input, ny and nu are the maximum lags in the output and input, f (·) is the nonlinear function to be identified. Thus the model given by (13) represents a process that may be both dynamical and nonlinear. Suppose that the dynamical system represented by (13) is invertible, there exist a function g(·) such that the input can be expressed in terms of a nonlinear expansion with lagged inputs and outputs as follows: u(t) = g(y(t + 1), y(t), . . . , y(t + 1 − ny ), u(t − 1), . . . , u(t + 1 − nu ))

(14)

Assuming that function g(·) is known, the expression given by (14) allows the calculation of the control action at time t such that the value y(t + 1) is reached by the system at time t + 1. Thus, if the objective of the control action is to reach a set point r(t + 1), the control action is obtained by replacing the process output at time t by the desired plant output r(t + 1): u(t) = g(r(t + 1), y(t), . . . , y(t + 1 − ny ), u(t − 1), . . . , u(t + 1 − nu ))

(15)

Here we consider using the RBF network with tapped time delays to approximate the identifier governed by (14). The severity of the system invertibility condition is weakened by constructing a predictive model like (14), which can be approximately inverted to achieve control actions. It can be further weakened by incorporating the gradient messages involving the trend of changes in the time-varying dynamics. Here we incorporate the first-order and second-order differences of system output as input of the RBF network. The configuration of the RBF network is shown in Fig. 2.

4

Ship Course Conrol Application

The design of ship control strategy presents challenges because ship’s motion is a complex nonlinear system with time-varying dynamics. The dynamics of ship also vary in case of any changes in sailing conditions such as speed, loading conditions, trim, etc. Similar changes may also be caused by environmental disturbances, such as waves, wind, current, etc. So we examine the performance

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Fig. 2. Configuration of RBF network as inverse controller

wind speed (m/s)

15

10

5

0

0

200

400

600 time (s)

800

1000

1200

200

400

600 time (s)

800

1000

1200

wind direction (degree)

100

50

0

−50

−100 0

Fig. 3. Wind speed and wind force curve

of the proposed control strategy by applying it in ship control. The ship model in this application is based on the model of ”MARINER” [11]. The simplified nonlinear expression of 3 degrees-of-freedom (DOF) ship motion are: surge : m(u˙ − vr − xG r2 ) = X

(16)

˙ =Y sway : m(v˙ + ur − xG r)

(17)

yaw : Iz r˙ + mxG (v˙ + ur) = N

(18)

where m is mass of ship, u and v are surge and sway velocities, r is the yaw rate, Iz is moment of inertia about the z-axis, X and Y are forces in direction of x and y-axis, respectively, N is the moment around z-axis and xG is the center of gravity along x-axis. The objective of our simulation was to steer a ship on setting courses with small deviations as well as avoiding large control actions. The desired course were set as 10 during [0s, 360s], 20 during [361s, 720s], −20 during [721s,120s]. To make the simulation more realistic, influences of wind and random measurement

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Fig. 4. Ship heading course and rudder angle (RBF Network Inverse control)

Fig. 5. Ship heading course and rudder angle (PID control)

noises were considered [12]. Wind force was set to 4 in Beaufort scale, the changes of wind speed and course were illustrated in Fig. 3. Noises were added with standard deviation of 0.2. Improved DOSA algorithm was implemented on-line. In the simulation, ship speed was set to 15 knot, rudder angle and rate were constrained to ±20◦ and ±5◦ /s, respectively. The parameters were chosen as follows: N = 20, MS = 3, ρ = 0.02. For comparison, traditional PID control was also implemented under the same condition. The parameters are tuned as: KP = 8, KI = 0.01, KD = 80. Simulation results are shown in Figs. 4 and 5. We compare Fig. 4 and 5 and find that although both methods can track the desired course well, the proposed RBF network-based inverse control strategy uses much less rudder action. It also indicates that the controller reacts fast to the environmental changes with smooth rudder actions, which shows that the RBF network constructed by DOSA algorithm can react to the change of ship dynamics adaptively, and the inverse control strategy can minimize the effect of long time delay to a low level.

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Conclusion

A direct inverse control strategy is introduced based on RBF network which is constructed by improved DOSA algorithm. Simulation results show that the proposed control strategy can control a nonlinear ship model with quick response and satisfactory course tracking ability.

References 1. Chen, S., Wang, X.X., Harris, C.J.: NARX-based Nonlinear System Identification Using Orthogoanl Least Squares Basis Hunting. IEEE Tans. Contr. Sys. Tech. 16, 78–84 (2008) 2. Moody, J., Darken, C.: Fast Learning in Networks of Locally-Tuned Processing Units. Neur. Comput. 1, 281–294 (1989) 3. Platt, J.: A Resource Allocating Network for Function Interpolation. Neur. Comput. 3, 213–225 (1991) 4. Kadirkamanathan, V., Niranjan, M.: A Function Estimation Approach to Sequential Learning with Neural Network. Neur. Comput. 5, 954–975 (1993) 5. Lu, Y.W., Sundararajan, N., Saratchandran, P.: A Sequential Learning Scheme for Function Approximation by Using Minimal Radial Basis Function Neural Networks. Neur. Comput. 9, 461–478 (1997) 6. Huang, G.B., Saratchandran, P., Sundararajan, N.: A Generalized Growing and Pruning RBF (GGAP-RBF) Neural Network for Function Approximation. IEEE Trans. Neur. Netw. 16, 57–67 (2005) 7. Yin, J.C., Dong, F., Wang, N.N.: A Novel Sequential Learning Algorithm for RBF Networks and Its Application to Dynamic System Identification. In: 2006 International Joint Conference on Neural Networks, pp. 827–834. IEEE Press, New York (2006) 8. Widrow, B., Wallach, E.: Adaptive Inverse Control. Prentice Hall, Upper Saddle River (1996) 9. Deng, H., Li, H.X.: A Novel Neural Approximate Inverse Control for Unkown Nonlinear Discrete Dynamical Systems. IEEE Trans. Sys. Man Cyber. 35, 115–123 (2005) 10. Chen, S., Cowan, C.F.N., Grant, P.M.: Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks. IEEE Trans. Neur. Netw. 2, 302–309 (1991) 11. Chislett, M.S., Strom, J.T.: Planar Motion Mechanism Tests and Full-scale Steering and Manoeuvring Predictions for a Mariner Class Vessel. Technical Report, Hydro and Aerodynamics Laboratory, Denmark (1965) 12. Zhang, Y., Hearn, G.E., Sen, P.: A Neural Network Approach to Ship TrackKeeping Control. IEEE J. Ocean. Eng. 21, 513–527 (1996)

Implementation of Neural Network Learning with Minimum L1 -Norm Criteria in Fractional Order Non-gaussian Impulsive Noise Environments* Daifeng Zha College of Electronic Engineering, Jiujiang University 332005 Jiujiang, China [email protected]

Abstract. Minimum L1 -norm optimization model has found extensive applications in linear parameter estimations. L1 -norm model is robust in non Gaussian alpha stable distribution error or noise environments, especially for signals that contain sharp transitions (such as biomedical signals with spiky series) or dynamic processes. However, its implementation is more difficult due to discontinuous derivatives, especially compared with the least-squares ( L2 -norm) model. In this paper, a new neural network for solving L1 -norm optimization problems is presented. It has been proved that this neural network is able to converge to the exact solution to a given problem. Implementation of L1 -norm optimization model is presented, where a new neural network is constructed and its performance is evaluated theoretically and experimentally. Keywords: L1 -norm optimization, Neural network, Alpha stable distribution, Non-Gaussian distribution.

1 Introduction In modern signal processing fields such as communication, automatic control, speech, and biomedical engineering, etc., and for most conventional and linear-theory-based methods, linear parametric estimation models, such as linear predictor and auto-regressive moving-average, have been extensively utilized. Generally speaking, it is reasonable to assume that the error or noise is usually assumed to be Gaussian distributed with finite second-order statistics. An L2 -norm (Least squares) solution is easy to find and suitable for the situations where noises or errors are Gaussian. But, in real application environments, the assumption that the distribution of noises or errors is Gaussian is unrealistic. For a non-Gaussian error distribution, the solution to problem using minimum L2 -norm optimization model may be very poor, especially for those signals with sharp outliers. In fact, the L1 -norm optimization model [1][4] is superior to L2 -norm models when the observation vector is contaminated by some large outliers *

This work is supported by National Science Foundation of China under Grant 60772037.

F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 283 – 290, 2008. © Springer-Verlag Berlin Heidelberg 2008

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or impulse noises, such as stable distributions [2] noise, including underwater acoustic, low-frequency atmospheric, and many man-made noises, which is suitable for modeling random variables with tails of the probability density function that are heavier than the tails of the Gaussian density function. Stable distributions is a kind of physical process with suddenly and short endurance high impulse in real world has no its second order or higher order statistics. It has no close form probability density function so that we can only describe it by its characteristic function [3]: α

Φ (t ) = exp{ jμt − γ t [1 + jβ sgn( t )ω (t , α )]}

(1)

απ

2 (if α ≠ 1) or log t (if α = 1) , 0 < α ≤ 2 is the 2 π characteristic exponent, it controls the thickness of the tail in the distribution. The Gaussian process is a special case of stable processes with α = 2 . The dispersion parameter γ > 0 is similar to the variance of Gaussian process and −1 ≤ β ≤ 1 is the symmetry parameter. −∞ < μ < ∞ is the location parameter. The typical lower order alpha stable distribution sequences are shown in Fig.1.

where ω (t , α ) = tan

Fig. 1. Alpha stable distribution sequences

In this paper, we present a new neural network to solve the L1 -norm optimization problem in alpha stable distribution environments. By experimental validations, the proposed network is able to globally converge to the exact solution to a given problem.

2 L1 -Norm Optimization Model Because of its excellent properties, the L1 -norm optimization model has been extensively studied. However, with the increase of the model scale, these numerical algorithms are not adequate for solving real-time problems. One possible and promising approach to real-time optimization is to apply neural networks. Because of the inherent massive parallelism, the neural network-based approach can solve optimization problems within a time constant of the network.

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In fact, many models can be mathematically abstracted as the following over-determined system of linear equations [1][8]: x = As − e

(2)

where A = {a ij } ∈ R M × N ( M > N ) is the model matrix derived from a given data set, s = [ s1 , s 2 ,..., s N ]T ∈ R N is the unknown vector of the parameters to be estimated,

x = [ x1 , x 2 ,..., x M ]T ∈ R M is the vector of observation or measurements containing errors or artifacts, e ∈ R M is the alpha stable distribution error or noise vector. Define the L1 -norm of error vector as follows: || e ||1 =|| As − x ||1

(3)

then, the parameter vector s can be solved via solving the following unconstrained optimization model: s opt = min || As − x ||1

(4)

s

This model is called L1 -norm optimization model, which is generally difficult to be solved because of discontinuous derivatives. Using the following Proposition 1, we turn the problem described in (4) into another form, which is easier to be solved. Proposition 1: The optimization model described in (4) is equivalent to the following optimization model:

(

)

⎫ ⎧ s opt = min ⎨max y T ( As − x) ⎬ s ⎩ y ⎭

(5)

where y = [ y1 , y 2 ,..., y M ]T ∈ R M , | y i |≤ 1 , i = 1,2,..., M . Proof: Let u = ( As − x) ∈ R M , then for any y , we have M

M

M

i =1

i =1

i =1

y T ( As − x) = y T u = ∑ y i u i ≤ ∑ | y i || u i | ≤ ∑ | u i | = || As − x ||1 Thus

(

)

max y T ( As − x) = || As − x ||1 y

(6)

(7)

This completes the proof of Proposition 1.

3 The L1 -Norm Neural Network Now we propose a neural network for solving the problem in (5) whose model is described by the following dynamic system:

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⎧ ds T ⎪ dt = − A P (y + As − x) ⎨ dy ⎪ = −( AA T + I )y + P (y + As − x) ⎩ dt

(8)

where P ( v) = [ P (v1 ), P(v 2 ),..., P(v M )]T ∈ R M , P(vi ) is defined as a projection operator: ⎧ 1 if vi > 1 1 ⎪ (9) P(vi ) = (| vi + 1 | − | vi − 1 |) = ⎨ vi otherwise 2 ⎪− 1 if v < −1 i ⎩

The proposed L1 -norm neural network ( L1 N-NN) described in (8) is shown in Fig.2.

Fig. 2. The L1 -norm neural network

Since the neural network described in (8) is a continuous-time network governed by a set of ordinary differential equations, It can be real-time implemented. In such implementation, the projection operator of P(vi ) is actually a simple limiter with a unit threshold. The matrix or vector multiplications are actually the synaptic-weighting and summing operations and hence can be implemented via a number of adders with a weighting function [5] [6]. And the rest are a number of simple integrators. As a simulation for the implementation, Fig.3 illustrates the implementation of the proposed L1 N-NN under MATLAB SIMULINK. In the following, we will prove that the neural network described in (8) and Fig. 2 globally converges to the exact solution to problem (5), or equivalently to problem (4). Let L(s, y ) = y T ( As − x) , according to K-T theorem [7], If s * ∈ R N is a solution to the problem in (5), we know that (s * , y * ) is a solution if and only if there exists a saddle point of model (5), and L(s * , y ) ≤ L(s* , y * ) ≤ L(s, y * ) . Thus we can easily have

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Fig. 3. The implementation of the proposed L1 N-NN

(y-y * ) T ( As* − x) ≤ 0

(10)

(y * ) T ( As* − x) ≤ (y * ) T ( As − x)

(11)

then there exists y * satisfying

⎧⎪ A T y * = 0 ⎨ * ⎪⎩y = P (y * + As * − x)

(12)

For any y , the following inequality holds:

{P(y + As − x) − y } (x - As ) ≥ 0 * T

*

(13)

The solution set of ⎧ ds T ⎪ dt = − A P(y + As − x) = 0 ⎨ dy ⎪ = −( AA T + I )y + P(y + As − x) = 0 ⎩ dt is just the equilibrium point set of dynamic system (8). Let E1 = A T y , E 2 = y - P(y + As - x) , then

(14)

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⎧ ds T ⎪ dt = −E1 + A E 2 ⎨ dy ⎪ = − AE1 − E 2 ⎩ dt

(15)

Let (s * , y * ) denote the solution set of model (5). By (12), we can get that when dy ds = 0 and = 0 . Thus we can give the relationship between the dt dt solution set of model (5) and the equilibrium point set of dynamic system (8).

(s, y ) = (s * , y * ) ,

4 Numerical Experiments Two numerical experiments are conducted to validate the proposed L1 N-NN and to evaluate its convergence. The experiments are performed under MATLAB 6.5 and its SIMULINK for Windows XP.

(a) s1 , s 2 , s 3

(b) s1

(c) s 2

(d) s3 Fig. 4. The convergence process

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Experiment 1: Let us consider the following problem: s opt = min || As − x ||1

(16)

s

where A = [ 1,0,0;1,1,1;1,2,4;1,3,9;1,4,6 ] , x = [1,2,1,3,3 ] T . The exact solution to problem (16) is that s = [ 1.0000,0.3571, 0.0714 ]T .Under MATLAB SIMULINK, we run the NN as shown in Fig. 2 and start simulation. The solution trajectories corresponding to the state variable in dynamic system (8) are rapidly derived and displayed in Fig.4(a). It is clearly shown that the numerical result is equal to the exact solution. Fig. 4(b-d) demonstrates the convergence process of the neural network with different initial values. It is shown that the proposed network sharply converges to exact solution to the given problem independent of initial values.

(a) k =0.0001

(b) k =0.01

x = [1.0000 6.0000 17.0000 34.0000 27.0000]T x = [0.9828 5.8599 16.8636 33.9385 26.9425] T

(c) k =1

(d) k =10

x = [0.9995 5.9930 16.9959 33.9970 26.9964] T x = [-10.3137 - 9.1728 3.7807 26.5906 13.2231] T

Fig. 5. The solution of s1 , s 2 , s 3 corresponding to different k

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Experiment 2: For x = As − e = As − ke 0 , let us solve the following problem in alpha stable distribution environments: s opt = min || As − x ||1 s (17) = min || e ||1 = min || ke 0 ||1 s

where

s

A = [ 1,0,0;1,1,1;1,2,4;1,3,9;1,4,6 ]

,

, e 0 is a error vector whose e 0 = [3.1026 0.9499 - 2.3361 - 0.1507 1.4950] elements are alpha stable distribution variables ( α = 1.8, β = 0, μ = 0, γ = 1 ). Suppose T

that s = [ 1,2, 3 ]T . For different k , we run the NN and start simulation. The solutions of s1 , s 2 , s 3 corresponding to different k are displayed in Fig.5.

5 Conclusion In this paper, a new neural network for solving L1 -norm optimization problems is presented. It has been proved that this neural network is able to converge to the exact solution to a given problem. This network is a continuous-time network, which is governed by a set of ordinary differential equations and hence can be implemented easily. As a simulation, an implementation of the proposed neural network under MATLAB SIMULINK has been presented. Using this implementation, numerical validation experiments have been presented. It is shown that the proposed network still gives exact solutions with rapid convergence independent of initial values. In addition, the experiments presented in this paper have illustrated that the proposed network has practical applications in alpha stable distribution error problems in non-Gaussian noise environments.

References 1. Cichocki, A., Unbehauen, R.: Neural Networks for Solving Systems of Linear Equation–Part II: Minimax and Least Absolute Value Problems. IEEE Trans. Circuits Syst. II 39, 619–633 (1992) 2. Nikias, C.L., Shao, M.: Signal Processing with Alpha-Stable Distribution and Applications, 1st edn. Wiley, Chichester (1995) 3. Georgiou, P.G., Tsakalides, P., Kyriakakis, C.: Alpha-Stable Modeling of Noise and Robust Time-Delay Estimation in the Presence of Impulsive Noise. IEEE Trans. on Multimedia 1, 291–301 (1999) 4. Bloomfield, P., Steiger, W.L.: Least Absolute Deviations: Theory Applications and Algorithms. Brikhäuser, Boston (1983) 5. Xia, Y.S.: A New Neural Network for Solving Linear Programming Problems and Its Application. IEEE Trans. Neural Networks. 7, 525–529 (1996) 6. Xia, Y.S., Wang, J.: A General Methodology for Designing Globally Convergent Optimization Neural Networks. IEEE Trans. Neural Networks 9, 1331–1343 (1998) 7. Luenberger, D.G.: Introduction to Linear and Nonlinear Programming. Addison- Wesley, New York (1973) 8. Zala, C.A., Barrodale, I., Kennedy, J.S.: High-resolution Signal and Noise Field Estimation Using the L1 (least absolute values) Norm. IEEE J. Oceanic Eng. OE- 12, 253–264 (1987)

Stability of Neural Networks with Parameters Disturbed by White Noises Wuyi Zhang and Wudai Liao Zhongyuan University of Technology, 450007, Zhengzhou, Henan, China [email protected], [email protected]

Abstract. Almost sure exponential stability (ASES) of neural networks with parameters disturbed by noises is studied, the basis of which is that the parameters in the implemented neural networks by very large scale integration (VLSI) approaches is well defined by the white-noise stochastic process, and an appropriate way to impose random factors on deterministic neural networks is proposed. By using the theory of stochastic dynamical system and matrix theory, some stability criteria are obtained which ensure the neural networks ASES, and the convergent rate is estimated. Also, the capacity of enduring random factors of the well-designed neural networks is estimated. The results obtained in this paper need only to compute the eigenvalues or verify the negative-definite of some matrices constructed by the parameters of the neural networks. An illustrative example is given to show the effectiveness of the results in the paper. Keywords: Almost sure, exponential stability, matrix theory.

1

Introduction

Neural networks have been applied to many scientific and engineering fields in the way of VLSI, such as in the neuron-dynamic optimization. As we all know, the white noises occur unavoidably in this type of circuit system, namely, the parameters, such as resistors, capacitors in the circuit system, take random values, the mean of which equal to the designed-value. A new type of neural networks – stochastic neural networks (SNN) – was first proposed and the stability of SNN was discussed in [1]. Up till now, lots of results were presented [2,3,4,5,6,7]. SNN can be treated as nonlinear dynamical systems with stochastically perturbed noise. These systems can be described by Itˆo stochastic differential equations. But little results concerned the problem that in what way the random factors were imposed on the neural networks. In the paper, an appropriate way of such problems was proposed naturally, in other words, each parameter in neural networks was estimated by white noises and it is a stochastic process. On the basis of deriving the mathematical equation of the new type SNN, some ASES conditions were obtained by using stochastic analysis and the matrix analysis, the convergent rate is given. And the capacity of SNN to endure the random intensity is also estimated. An illustrative example will be given to show the effectiveness of the results obtained. F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 291–298, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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This paper is organized as follows: In Section 2, some notations and lemmas are given; some definitions are given, such as the white noise stochastic process, the standard Brownian motion, etc., and the mathematical description of SNN is given by the estimation formula of the parameters; In Section 3, the stability criteria are obtained which need only to compute the eigenvalues or examine the negative-definite of some matrices constructed by the network’s parameters, the Lyapunov-exponent of equilibria are also estimated; In Section 4, an illustrative example is given to show the effectiveness of the results in the paper.

2

Preliminary

In this section, we give some notations, definitions, lemmas, and the mathematical description of SNN. 2.1

Notations, Definition and Lemma

In the paper,IRn denotes the n-dimension real Euclidean space, x ∈ IRn a ndimension column vector. The Euclidean norm of x ∈ IRn is denoted |x|. IRn×n denotes n-dimension real matrix space, for A ∈ IRn×n , λmax (A), λmin (A) denote the maximum and minimum eigenvalue of the matrix A, and |A|1 , |A|∞ the 1-norm and ∞-norm of the matrix A respectively. Definition 1. A scale-valued stochastic process ξt defined on the probability space (Ω, F, P ) is called a standard white noise, if and only if its expectation Eξt = 0 and the co-variation covar(ξt+τ ξt ) = δ(τ ), or equivalently, the spectral density is 1, where δ(τ ) is a Dirac-δ function. Let wt be a 1-dimensional standard Brownian motion, we have formally wt+τ − wt dwt = lim = ξt . τ →0 dt τ

(1)

For x, f (·) ∈ IRn , g(·) ∈ IRn , and wt a 1-dimensional standard Brownian motion, we consider the following Itˆo stochastic differential equations: dx(t) = f (x)dt + g(x)dwt .

(2)

In which, we assume that f (0) = g(0) = 0, this implies that (2) has trivial solution x = 0. For a Lyapunov function V (x) : IRn → IR, we define the differential operator on (2) as the following: 1 LV (x) = Vx (x)T f (x) + g(x)T Vxx (x)g(x) . 2

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Lemma 1. Assume that there exist V (x) ∈ C 2 (IRn ; IR) and constants p > 0, c1 > 0, c2 ∈ IR, c3 ≥ 0 such that for all x = 0, t ≥ 0, 1) V (x) ≥ c1 |x|p , 2) LV (x) ≤ c2 V (x) , 3) |Vx (x)T g(x)|2 ≥ c3 V 2 (x) hold, then the estimation lim sup t→∞

1 c3 − 2c2 log |x(t; x0 )| ≤ − a.s. t 2p

holds for any x0 ∈ IRn . In particular, if c3 > 2c2 , then the trivial equilibrium x = 0 of (2) is ASES with Lyapunov exponent (c3 − 2c2 )/2p. Remark 1. In Lemma 1, Condition 3) holds for c3 = 0 obviously, in this case, if condition 2), 3) hold with c2 < 0, then the conclusions of Lemma 1 is true. 2.2

The Description of SNN

Now, we consider the neural networks with perturbed parameters by white noises as follows: n  dxi (t) = −bi (t)xi + aij (t)fj (xj ) + Ii (t), i = 1, 2, · · · , n . dt j=1

(3)

x = (x1 , x2 , · · · , xn )T ∈ IRn is the state vectors, B(t) = diag(b1 (t), b2 (t), · · · , bn (t)) is feedback gain matrix, A(t) = (aij (t))n×n ∈ IRn×n is the weight matrix between neurons, aij (t) is the connected weight from neuron j to neuron i, I(t) is the bias vector, the vector activation function f (x) = (f1 (x1 ), f2 (x2 ), · · · , fn (xn ))T and fi : IR → IR satisfy local Lipschitz condition, namely, for ∀x0 , ∃ constant li > 0 and a neighborhood B(x0 ), such that for ∀θ, ρ ∈ B(x0 ) |fi (θ) − fi (ρ)| ≤ li |θ − ρ| .

(4)

Condition (4) includes some common activation function used in neural networks, such as Sigmoid (li = 1/4) and linear saturation (li = 1) functions. Assume that the parameters have the following estimations: bi (t) = bi + βi ξt , i = 1, 2, · · · , n , aij (t) = aij + αij ξt , i, j = 1, 2, · · · , n , Ii (t) = Ii + γi ξt , i = 1, 2, · · · , n .

(5)

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Where bi , aij , Ii are well designed parameters in neural networks, ξt is the standard white noise, βi , αij , γi are the noised densities of corresponding parameter. By using (5) and (1), Eq. (3) can be rewritten as the following Itˆ o type stochastic differential equation: dxi (t) = [−bi xi +

n 

aij fj (xj ) + Ii ]dt + [−βi xi +

j=1

n 

αij fj (xj ) + γi ]dwt .

j=1

Or its vector form dx(t) = [−bx + Af (x) + I]dt + [−βx + αf (x) + γ]dwt .

(6)

Where b = diag(b1 , · · · , bn ), A = (aij )n×n , I = (I1 , · · · , In )T , β = diag(β1 , · · · , βn ), α = (αij )n×n , γ = (γ1 , · · · , γn )T . Let x∗ be an equilibrium of (6), take the transformation y = x − x∗ , then (6) has the following form: dy(t) = [−by + Ag(y)]dt + [−βy + αg(y)]dwt .

(7)

In which g(y) = f (y + x∗ ) − f (x∗ ), satisfying |gi (yi )| ≤ li |yi |, i = 1, 2, · · · , n with Lipschitz constants li . So, in order to study the stability of equilibrium x∗ of (6), we only to study the equilibrium y = 0 of (7).

3

Main Results

In this section, we will set up some sufficient algebraic criteria ensuring the equilibrium of (6) to be almost sure exponential stability, and the convergent Lyapunov exponent are estimated. Select the symmetric positive-definite matrix Q and the diagonal matrix P = diag(pi ) with pi > 0, i = 1, 2, · · · , n, construct the following symmetric matrix:   −Qb − bQ + βQβ QA − βQα H= . −P + αT Qα AT Q − αT Qβ Denote −λ the maximum eigenvalue of the symmetric matrix, by using the semipositive definite of the matrix αT Qα, we can easily deduce that −λ + pi ≥ 0, i = 1, 2, · · · , n. Theorem 1. Assume that x∗ is an equilibrium of (6). If there exist a symmetric positive-definite matrix Q, a positive diagonal matrix P = diag(p1 , p2 , · · · , pn ) such that the maximum eigenvalue −λ of the matrix H satisfies λ>

li2 pi , i = 1, 2, · · · , n , 1 + li2

then, the equilibrium x∗ is ASES, the convergent rate is μ/2λmax (Q), μ := min1≤i≤n {λ(1 + li2 ) − pi li2 } > 0. Here, li , i = 1, 2, · · · , n are the Lipschitz constants on activation functions fi at the equilibrium x∗ .

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Proof. In order to examine the stability of equilibrium x∗ of (6), we equivalently verify the trivial equilibrium y = 0 of (7). For (7), construct the Lyapunov function V (y) = y T Qy, then, Vy (y) = 2Qy, Vyy (y) = 2Q, its differential operator on (7) has the following estimation: 1 LV (y) = VyT (y)[−by + Ag(y)] + [−y T β + g T (y)αT ]Vyy (y)[−βy + αg(y)] 2 = 2y T Q[−by + Ag(y)] + [−y T β + g T (y)αT ]Q[−βy + αg(y)] = y T [−Qb − bQ + βQβ]y + 2y T (QA − βQα)g(y) + g T (y)(αT Qα)g(y)     y = y T , g T (y) H + g T (y)P g(y) g(y) n  ≤ −λ(|y|2 + |g(y)|2 ) + pi gi2 (yi ) i=1

=−

n 

λyi2 +

i=1

n 

(−λ + pi )gi2 (yi ) .

i=1

Because of |gi (yi )| ≤ li |yi | and −λ + pi ≥ 0, i = 1, 2, · · · , n, we have LV (y) ≤ −

n n     λ + (λ − pi )li2 yi2 ≤ −μ yi2 ≤ − i=1

i=1

μ V (y) . λmax (Q)

By using Lemma 1 with p = 2, c2 = −μ/λmax (Q), c3 = 0, one can deduce that the trivial equilibrium y = 0 of (7), equivalently, the equilibrium x∗ of (6), is 

ASES, and the Lyapunov exponent is μ/2λmax (Q). How to select an appropriate matrix P in Theorem 1 is the key problem to use this theorem. We give a way  to do this in Corollary 1. Select a positive number r > 0, denote R = r · diag l1−2 , · · · , ln−2 , E ∈ IRn×n is the unit matrix, construct a symmetric matrix H1 :   −Qb − bQ + rE + βQβ QA − βQα H1 = . AT Q − αT Qβ −R + αT Qα Corollary 1. Assume that x∗ is an equilibrium of (6). If there exist a symmetric positive-definite matrix Q, a positive number r > 0 such that the matrix H1 is negative definite, then, the equilibrium x∗ is ASES. Proof. In Theorem 1, we choose pi = (1 + li−2 )r, then, H1 = H + rE2n×2n . From the negative definite of the matrix H1 , we have λmax (H1 ) = λmax (H) + r < 0, that is, −λ + r < 0. So, λ>r=

li2 pi , i = 1, 2, · · · , n . 1 + li2

By using Theorem 1, the equilibrium x∗ is ASES.



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Let Q = E in Corollary 1, and the matrix H1 becomes the following form:   −2b + rE + β 2 A − βα H2 = . AT − αT β −R + αT α we have the following result: Corollary 2. Assume that x∗ is an equilibrium of (6). If there exist a positive number r > 0 such that the matrix H2 is negative definite, then, the equilibrium x∗ is ASES. Remark 2. Obviously, the necessary condition of the matrix H2 being negative definite is that the optional parameter r satisfies li2

n 

α2ji < r < 2bi − βi2 , i = 1, 2, · · · , n .

j=1

In the following, we consider the capability of enduring the perturbation on the well designed neural networks (deterministic neural networks). Denote     −2b + rE A β2 −βα D= , S = . AT −R −αT β αT α Obviously, H2 = D + S, this is a decomposition of matrix H2 according to its deterministic part and random part. Corollary 3. Assume that x∗ is an equilibrium of the well designed deterministic neural network. If the condition λmax (S) < −λmax (D) holds, then, the equilibrium x∗ is also ASES. Proof. By Corollary 2, we need only to verify the negative definite of the matrix H2 . For any z ∈ IR2n , we have z T H2 z = z T Dz + z T Sz ≤ (λmax (D) + λmax (S)) |z|2 This shows that the matrix H2 is negative definite.



From the matrix inequality ρ(A) ≤ |A|1 , which ρ(A) is the spectrum radius of the matrix A and ρ(A) = λmax (A) if A is symmetric, we have the following corollary. Corollary 4. Assume that x∗ is an equilibrium of the well designed deterministic neural network. If the condition |S|1 < −λmax (D) holds, then, the equilibrium x∗ is also ASES.

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Remark 3. |S|1 = maxj { 2n i=1 |sij |} is easier to compute than to compute the maximum eigenvalue of the matrix S, it is convenient in application.

4

An example

Consider the following 2-dimensional SNN dx1 = −b1 (t)x1 + a11 (t)f1 (x1 ) + a12 (t)f2 (x2 ) + I1 (t) dt dx1 = −b2 (t)x2 + a21 (t)f1 (x1 ) + a22 (t)f2 (x2 ) + I2 (t) dt with the Sigmiod activation function f1 (u) = f2 (u) =

1 , u ∈ IR . 1 + e−u

The Lipschitz constants l1 = l2 = fi (0) = 1/4. The parameters are as follows estimated by statistic: b1 (t) = 1 + 0.1ξt , b2 (t) = 1 + 0.1ξt , a11 (t) = 2 + 0.2ξt , a12 (t) = 1 + 0.1ξt , a21 (t) = 1 + 0.1ξt , a22 (t) = 2 + 0.2ξt I1 (t) = I2 (t) = −3 − 0.3ξt , . We have  b=

1 0

Choose r = 1, follows ⎛ −1 ⎜ 0 D=⎝ 2 1

0 1



 ,β=

0.1 0 0 0.1



 ,A=

2 1

1 2



 ,α=

0.2 0.1 0.1 0.2

 .

then R = diag(16, 16). The matrices D, S in Corollary 4 are as ⎞ ⎞ ⎛ 0 2 1 0.01 0 −0.02 −0.01 −1 1 2 ⎟ 0.01 −0.01 −0.02 ⎟ ⎜ 0 ⎠ . ⎠, S = ⎝ 0.04 1 −16 0 −0.02 −0.01 0.05 −0.01 −0.02 0.04 0.05 2 0 −16

λmax (D) = −0.4223, |S|1 = 0.12. The condition |S|1 < −λmax (D) holds in Corollary 4, the equilibrium x1 = x2 = 0 is ASES.

Acknowledgment This work was supported in part by the National Natural Science Foundation of China under grant No. 60774051, 60474001.

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References 1. Liao, X.X., Mao, X.R.: Exponential Stability and Instability of Stochastic Neural Networks. Stochast. Anal. Appl. 14, 165–185 (1996) 2. Liao, W., Wang, D., Wang, Z., Liao, X.X.: Stability of Stochastic Cellular Neural Networks. Journal of Huazhong Univ. of Sci. and Tech. 35, 32–34 (2007) 3. Liao, W., Liao, X.X., Shen, Y.: Robust Stability of Time-delyed Interval CNN in Noisy Environment. Acta Automatica Sinica 30, 300–305 (2004) 4. Blythe, S., Mao, X.R., Liao, X.X.: Stability of Stochastic Delayed Neural Networks. Journal of the Franklin Institute 338, 481–495 (2001) 5. Shen, Y., Liao, X.X.: Robust Stability of Nonlinear Stochastic Delayed Systems. Acta Automatica Sinic. 25, 537–542 (1999) 6. Liao, X.X., Mao, X.R.: Exponential Stability of Stochastic Delay Interval Systems. Systems and Control Letters 40, 171–181 (2000) 7. Liao, X.X., Mao, X.R.: Stability of Stochastic Neural Networks. Neual. Parallel and Scientific Computations 14, 205–224 (1996) 8. Mao, X.R.: Stochastic Differential Equations and Applications. Horwood Pub., Chichester (1997)

Neural Control of Uncertain Nonlinear Systems with Minimum Control Effort Dingguo Chen1 , Jiaben Yang2 , and Ronald R. Mohler3 1

Siemens Power Transmission and Distribution Inc., 10900 Wayzata Blvd., Minnetonka, Minnesota 55305, USA 2 Department of Automation, Tsinghua University, Beijing, 100084, People’s Republic of China 3 Department of Electrical and Computer Engineering, Oregon State University, Corvallis, OR 97330, USA Abstract. A special class of nonlinear systems are studied in this paper in the context of fuel optimal control, which feature parametric uncertainties and confined control inputs. The control objective is to minimize the integrated control cost over the applicable time horizon. The conventional adaptive control schemes are difficult to apply. An innovative design approach is proposed to handle the uncertain parameters, physical limitations of control variables and fuel optimal control performance simultaneously. The proposed control design methodology makes an analysis of the fuel control problem for nominal cases, employs a hierarchical neural network structure, constructs the lower level neural networks to identify the switching manifolds, and utilizes the upper level neural network to coordinate the outputs of the lower level neural networks to achieve the control robustness in an approximately fuel-optimal control manner. Theoretical results are presented to justify the proposed design procedures for synthesizing adaptive, intelligent hierarchical neural controllers for uncertain nonlinear systems. Keywords: Bilinear System, Uncertain nonlinear System, Multiple Input Nonlinear System,Neural Network, Fuel Optimal Control, Neural Control, Switching Manifold, Hierarchical Neural Network.

1

Introduction

Bilinear systems [3] have been widely studied because of their appealing structure that can be utilized for better system controllability. A generalization of the bilinear systems is the affine systems that are linear in control. Numerous research results have been reported in two main categories: (1) in the context of adaptive control where system parametric uncertainties are considered but without considering physical control restrictions and incorporating control performance indexes; and (2) in the context of optimal control where both physical control restrictions and control performance indexes are considered but without considerations in system control robustness. There are many practical systems that are desired to be controlled followed appropriate control designs so that appropriate optimal control performance is achieved with certain control robustness. Rare results have been F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 299–308, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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obtained in the context of adaptive, optimal, and constrained control. It is the motivation of this paper to make a contribution in this direction. It is true that numerous elegant adaptive control techniques are available but they are difficult to apply to this type of problems. A new trend has been witnessed in the recent years that neural networks have been introduced into nonlinear adaptive control design [14,16,15,17,18] to hopefully tackle some of the difficult problems that conventional design approaches can not handle. This is due to the superior function approximation capabilities, the distributed structure, the parallel processing feature, of neural networks, and more important, the appealing structure of the three-layered neural networks exhibiting the linearity in unknown parameters when linearized at unknown optimal parameters [18] or at given sub-optimal parameters [21]. The control designs based on the adaptive control schemes [18,21] however, have the drawback that the control signal is not constrained within a predesignated desired physical range. Further, additional control performance criteria, e.g., optimal control performance, are difficult to incorporate within the framework of the traditional adaptive control schemes. It becomes apparent that the popular adaptive control schemes can not be directly applied to solve the practical problems that require that the control signal be bounded by a given number, and a new theory tailored for the application has yet to be worked out. It is the objective of this paper to make an attempt to address the fueloptimal control problem in the broader context of adaptive, optimal, and constrained control. The intention is to generalize the results from the previous efforts [11,7,10,22] so that the new results can cover a broader class of problems that have the multiple control variables, need to respect constraints of control, need to incorporate broader class of control performance criteria, and contains parametric uncertainties. This paper is organized follows: Section 2 describes the class of uncertain nonlinear systems to be studied in this paper and the control objective and makes several conventional assumptions. The fuel optimal control problem is analyzed and an iterative numerical solution process is presented in section 3. The control problem studied in this paper is decomposed into a series of control problems that do not have parameter uncertainties. This decomposition is utilized in the hierarchical neural control design methodology, which is presented in section 4. The synthesis of hierarchical neural controllers is to achieve (a) near fuel-optimal control of the studied systems with constrained control; (b) adaptive control of the studied control systems with unknown parameters. Theoretical results are developed to justify the fuel-optimal control oriented neural control design procedures and presented in section 5. Finally, some conclusions are drawn.

2

Problem Statement

Although the conventional adaptive control schemes are powerful, they have common drawbacks that include (a) the control usually does not consider the physical control limitations, and (b) a performance index is difficult to incorpo-

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rate. Most practical systems usually need to consider both physical constraints and an optimal performance index and yet require robustness of control with respect to certain system parameter variations. This paper attempts to address these challenges. The systems to be studied in this paper are linear in both control and parameters, and feature parametric uncertainties, confined control inputs, and multiple control inputs. These systems are represented by a finite dimensional differential system linear in control and linear in parameters as shown in [22]. The control objective is to follow a theoretically sound control design methodology to design the controller such that the system is adaptively controlled with respect to parametric uncertainties and yet achieves a desired control performance. To facilitate the theoretical derivations, several conventional assumptions are made same as in [22] except that AS4 is slightly modified to reflect minimum fuel control ans AS9 is added: t  AS4: The control performance criteria is J = t0f [e0 + m k=1 ek |uk |]ds where t0 and tf are the initial time and the final time, respectively, and ek (k = 0, · · · , m) are non-negative constants. The cost functional reflects the requirement of fueloptimal control and the interest of this paper having the total cost as related to the integration of the absolute control effort of each control variable over time. Remark 1: If ek = 0 for (k = 1, · · · , m) and e0 > 0, the control performance t criteria becomes J = t0f e0 ds, which corresponds to the time-optimal control problem. In this sense, the time-optimal control problem can be viewed as a special case of the fuel-optimal control problem studied in this paper. AS9: The total number of switch times for all control components for the studied fuel-optimal control problem is greater than the number of state variables. Remark 2: AS9 is true for practical systems to the best knowledge of the authors. The assumption is made for the convenience of the rigor of the theoretical results developed in this paper.

3

Fuel-Optimal Control Problem and Numerical Solution

Decomposing the control problem (P ) into a series of control problems (P0 ) is an important treatment toward the hierarchical neural control design that is intended for addressing the near fuel-optimal control of uncertain nonlinear systems. The distinction made between the control problem (P ) and the control problem (P0 ) is to facilitate the development of the hierarchical neural control design and for the clarify of the presentation of the paper. The original control problem (P ) is associated with an unknown parameter vector p while the control problem (P0 ) corresponds to a given parameter vector p. The control problem (P) can be viewed as a family of the control problems (P0 ), which together represent an approximately accurate characterization of the dynamic system behaviors exhibited by the nonlinear systems in the control problem (P ).

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The application of the Pontryagin minimum principle gives rise to the socalled two-point boundary-value problem (TPBVP) which must be satisfied by an optimal solution. In general, an analytic solution to the TPBVP is extremely difficult, and usually practically impossible to obtain. It is shown that the iterative solution obtained by the switching-timesvariation method (STVM) [3,4] through successive approximation, converges to the unique solution to the optimal control problem provided that the TPBVP has a unique solution. With some derivations, the optimal control can be written as follows: ∗− u∗k = u∗+ k + uk

(1)

∗− 1 1 ∗ ∗ ∗ where u∗+ k = 2 [sgn(−sk (t) − 1) + 1], uk = 2 [sgn(−sk (t) + 1) − 1] and sk (t) is the kth component of the switching vector. It has been shown in [4] that the number of the optimal switching times must be finite provided that no singular solutions exist. Let the zeros of −sk (t) − 1 + + + (j = 1, · · · , 2Nk+ , k = 1, · · · , m; and τk,j < τk,j for 1 ≤ j1 < j2 ≤ 2Nk+ ) be τk,j 1 2 which represent the switching times corresponding to positive contorl u∗+ k , the − − − zeros of −sk (t) + 1 τk,j (j = 1, · · · , 2Nk− , k = 1, · · · , m; and τk,j < τ k,j2 for 1 − 1 ≤ j1 < j2 ≤ 2Nk ) which represent the switching times corresponding to + − + negative control u∗− k . Altogether τk,j ’s and τk,j ’s (j = 1, · · · , 2Nk , k = 1, · · · , m) ∗ represent the switching times which uniquely determine uk as follows: +

u∗k (t)

Nk 1  + + = { [sgn(t − τk,2j−1 ) − sgn(t − τk,2j )] − 2 j=1 −

Nk 

− − [sgn(t − τk,2j−1 ) − sgn(t − τk,2j )]}.

(2)

j=1

Let the switch vector for the kth component of the control vector be + − + − + + τ · · · τk,2N and τ Nk = τ Nk = [(τ Nk )τ (τ Nk )τ ]τ where τ Nk = [τk,1 +] k

− − + − τ Nk · · · τk,2N is the switching vector [τk,1 − ] . Let Nk = 2Nk + 2Nk . Then τ k

of Nk dimensions. Let the vector of switch functions for the control variable uk be dek k k k k · · · φN φN · · · φN ]τ where φN = fined as φNk = [φN 1 j 2N + 2N + +1 2N + +2N − k

k

k

k

+ − k (−1)j−1 ek (sk (τk,j + 1) (j = 1, · · · , 2Nk+ ), and φN = (−1)j ek (sk (τk,j − 1) j+2N + k

(j = 1, · · · , 2Nk− ). The gradient that can be used to update the switching vector τ Nk can be given by (3) ∇τ Nk J = −φNk .

The optimal switching vector can be obtained iteratively by using a gradientbased method. (4) τ Nk ,i+1 = τ Nk ,i + K k,i φNk

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where K k,i is a properly chosen Nk × Nk -dimensional diagonal matrix with nonnegative entries for the ith iteration of the iterative optimization process; and τ Nk ,i represents the ith iteration of the switching vector τ Nk . When the optimal switching vectors are determined upon convergence, the optimal control trajectories and the optimal state trajectories are computed. This process will be repeated for all selected nominal cases as discussed in Section 4 until all needed off-line optimal control and state trajectories are obtained. These trajectories will be used in training the fuel-optimal control oriented neural networks.

4

Neural Control Design Methodology for Fuel-Optimal Control

In one of the previous endeavors, the hierarchical neural network based control design was applied to the single-machine infinity-bus (SMIB) power system [11]. In a recent attempt [22] to generalize the control design approaches and extend them to a broader class of nonlinear systems, multiple control inputs were considered. However, the control problems studied have been limited to the time-optimal control. This paper presents a design methodology for fuel-optimal control, which in a sense can be considered as a more general control problem that the time-optimal control problem. The proposed design consists of the two major steps: 1. Use neural networks to approximate the switching manifolds for all the control components uk (k = 1, · · · , m) for each selected nominal case; 2. Use neural networks to approximate the coordination function which determines the relative control effort contributions of the lower-level neural controllers. The system dynamic behaviors are affected not only by the initial system state and the control variables, but also by the parameter vector p. The analysis of the effect of p on the system dynamic behaviors is helpful in determining the nominal cases required in the proposed design approach. Based on the qualitative system behavior analysis, the parameter vector space may be tessellated in such a way that an appropriate tessellation granularity level is achieved to meet the desired control performance yet with minimal number of nominal cases. For each individual control problem (P0 ), the bang-off-bang control is resulted. Consequently, the switching manifold can be identified using the optimal control and state trajectories that are obtained using applicable numerical methods and cover the stability region of interest. Mathematically, this is equivalent to say ui = −sgn(Si (x)) for |Si (x)| > 1; and ui = 0 for |Si (x)| = 1 (i = 1, · · · , m). Si (x) is the switching function with |Si (x)| = 1 identifying the switching manifolds. Since the switch functions are functions of state variables and costate variables, the state trajectories and costate trajectories are usually not readily analytically available, direct approximation of Si (x) using neural networks is difficult. Instead, the patterns generated from the off-line calculated optimal

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control and state trajectories are used to determine the relationship between the control variable uk and the state x. Since the fuel-optimal control is a bangoff-bang control, the uk ’s thus obtained need further processing. This includes cascading another neural network to conduct the following computation logic: vk = 12 [sgn(−uk − 1) + 1] + 12 [sgn(−uk + 1) − 1]. This new neural network has the heaviside activation applied, is constructive, hence not requiring any off-line training. This resulted switching manifold identification oriented neural control is shown in Fig. 1.

x

Standard NN

uk

Fig. 1. Switching manifolds identification by neural network

The design of the upper-level neural controllers also utilizes the off-line generated optimal state trajectories. In addition, it makes use of the outputs of the lower-level neural controllers to determine the relative control effort contribution from each lower-level neural controller. Each component of the final control vector is the respective sum of the lowerlevel neural control signals modulated by the corresponding coordinating signals of the upper-level neural networks. The hierarchical neural control diagram is shown in Fig. 2.

5

Theoretical Justification on Construction of Hierarchical Neural Controllers

To validate the proposed fuel-optimal control design methodology for using neural networks to adaptively control the uncertain nonlinear systems studied in this paper, two main theoretical results are presented for the rationale of (a) the use of lower-level neural networks for switching manifolds identification for the control problem (P0 ) and (b) the use of the hierarchical neural networks for adaptive control of the control problem (P ). To address the first issue, we present the following result. Proposition 1. For the control problem (P0 ) with assumptions AS1 through AS4 and AS6 satisfied, let the switching manifolds for the kth component of the control vector u be Sk (k = 1, 2, · · · , m). Define Sk+ = Sk for Sk = 1, and Sk− = Sk for Sk = −1. Let Dk+ , a connected open subset of Ω be constructed in such a way that the switching manifold Sk is a subset of Dk and Dk+ = {x : + + ||x − y|| < + k , y ∈ Sk , x ∈ Ω} where k is a pre-specified positive number − and k = 1, 2, · · · , m. Similarly, Dk is constructed such that Sk− ⊂ Dk− and

Neural Control of Uncertain Nonlinear Systems Upper Level Neural Networks

TDL TDL

x

1

Multiplier Processing Unit

NN TDL NN

2

...

TDL

x

TDL NN TDL

NN

x

NN

x

Lower Level Neural Networks

NN

x2 x1

M

1

u1

2

u2

M

uM

...

x

xM

...

x

305

X X

+

u

X

Fig. 2. Hierarchical fuel-optimal neural control − − Dk− = {x : ||x − y|| < − k , y ∈ Sk , x ∈ Ω} where k is a pre-specified positive + − number and k = 1, 2, · · · , m. If Dk and Dk are constructed such that Dk+ ∩Dk− = ∅, then there exists a state feedback neural controller unn,k = N Nk (x) which only takes -1 or 0 or +1 with x being the state, such that if x ∈ Ω − Dk+ − Dk− , ||uk (x) − unn,k (x)|| = 0 (k = 1, 2, · · · , m).

Proof: First of all, note that Ω − Dk+ − Dk− is a subset of Ω and is also a closed subest, hence compact. The optimal control uk = gk (x) (k = 1, 2, · · · , m) with x ∈ Ω is a discontinuous function only on x ∈ Sk+ ∪ §− k . It can be approximated with a continuous function,say vk = hk (x), with the same support with sufficiently small error γk > 0 such that hk = gk if x ∈ Ω − Dk+ − Dk− , and |hk (.) − gk (.)| < γk for x ∈ Dk+ ∪ Dk− . Then for any ∗ > 0, there exists a neural network N N 1k (x, Θ∗ ) with the optimal parameter vector Θ∗ such that |N N 1∗k (x, Θ∗ ) − hk (x)| < ∗ . It follows from AS6 that the optimal control uk is bang-off-bang type, and therefore hk (.) takes a value of -1 or 0 or +1 for any x ∈ Ω − Dk+ − Dk− . Let γ1,k = ∗ . Then 1 − γ1,k < N N 1k (x, Θ∗ ) < 1 + γ1,k when hk = 1, or −1−γ1,k < N N 1k (x, Θ∗ ) < −1+γ1,k when hk = −1, or −γ1,k < N N 1k (x, Θ∗ ) < γ1,k when hk = 0 for x ∈ Ω − Dk+ − Dk− . As long as ∗ is chosen such that ∗ < 12 , then one of the following three mutually exclusive conditions is satisfied: either 1 < 2 ∗ N N 1k (x, Θ∗ ) < 3, or −3 < 2 ∗ N N 1k (x, Θ∗ ) < −1, or −1 < 2 ∗ N N 1k (x, Θ∗ ) < 1. Consequently, 12 [sgn(−2 ∗ N N 1k (x, Θ∗ ) − 1) + 1] + 12 [sgn(−2 ∗ N N 1k (x, Θ∗ ) + 1) − 1] = hk (x) for x ∈ Ω − Dk+ − Dk− . But 1 1 2 [sgn(−2 ∗ N N 1k (x, Θ∗ ) − 1) + 1] + 2 [sgn(−2 ∗ N N 1k (x, Θ∗ ) + 1) − 1] can be

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constructed as another neural network using the heaviside activation function. Let N Nk (x) = 12 [sgn(−2 ∗ N N 1k (x, Θ∗ ) − 1) + 1] + 12 [sgn(−2 ∗ N N 1k (x, Θ∗ ) + 1) − 1]. Thus, the existence of neural controller unn,k = N Nk (x) is assured. This completes the proof. With the application of the above result along with AS8, it follows that there exists a neural network |N N 1k (x, Θs )−hk (x)| < |N N 1k (x, Θs )−N N 1∗k (x, Θ∗ )|+ |N N 1∗k (x, Θ∗ ) − hk (x)| < s + ∗ . As long as the off-line trained neural network and the neural network with the ideal parameters are sufficiently close, i.e., if s < 12 − ∗ , then as shown in the above, this off-line trained neural network, even though not in optimal configuration, is good enough for approximating the switching manifolds. From the practical implementation point of view, this is particularly meaningful in the sense that it justifies that the lower-level neural controllers can be constructed using the optimal control and state trajectories yet achieving desired, sufficiently accurate switching manifold approximations. To address the second issue highlighted at the beginning of this section, we present the following result. The following result for fuel-optimal control is a generalization of a result in [22] in which time-optimal control is considered. Proposition 2. For the control problem (P ) with the assumptions AS1 through AS9 satisfied, suppose Ω is a compact region where with the bang-off-bang control the optimal trajectories starting in the compact region will still remain in it. Then 

1. for any 1 > 0 and 2 > 0, there exists 3 > 0 such that if ||x0 − x0 || < 3 ,  there exists the terminal time tf such that |tf − t∗f | < 1 ,and ||x(x0 , tf ) − ∗ ∗ ∗ x (x0 , tf )|| < 2 where tf is the optimal terminal time for the initial state  x0 ; x∗ (x0 , t) is the optimal trajectory starting from x0 ; and x0 is a perturbed initial condition of x0 .  2. for any 4 > 0 and 5 > 0, there exists 6 > 0 such that if p − p|| < 6 ,  there exists the terminal time tf such that ||tf −t∗f || < 4 , and ||x(x0 , p , tf )− x∗ (x0 , p, t∗f )|| < 5 where t∗f is the optimal terminal time for the initial state x0 ; x∗ (x0 , p, t) is the optimal trajectory starting from x0 for the control prob lem (P ) with the parameter vector p; and p is a perturbed parameter vector of p. Proof: Due to the page limit, only a sketch of the proof is presented: First, consider a perturbation in the initial state x0 and show that for a small change in the initial state, the switching times vector make an accordingly small change in order to drive the final state to the origin. Secondly, consider an increment dp in p, and show that for a small change in the parameter vector, the switching times vector makes an according small adjustment to still drive the final state to the origin. In both steps, perturbation analysis is conducted, and integration of system equations and certain norm are applied, along with the Assumptions, especially AS9. In particular, for the case of the optimal state trajectory x∗ (x0 , tf ) and the opti mal final switching time t∗f , and the perturbed initial condition x0 of x0 , by properly  choosing 3 , one can obtain |tf − t∗f | < 1 ,and ||x(x0 , tf ) − x∗ (x0 , t∗f )|| < 2 .

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For the case of the optimal state trajectory x∗ (x0 , p, tf ), the optimal final  switching time t∗f , and the perturbed parameter vector p of p, by properly  choosing 6 , one can obtain that |tf −t∗f | < 4 ,and ||x(x0 , p , tf )−x∗ (x0 , p, t∗f )|| < 5 . This completes the proof. The above result indicates that the system dynamic behavior for an unknown parameter vector p can be closely approximated by those corresponding to a tessellation resulted parameter sub-region which is sufficiently small and contains the unknown parameter vector p. In addition, the theoretical results altogether presented in the paper clearly justify the proposed design methodology as to how the switch manifolds for fueloptimal control problems are identified using neural networks; and how the hierarchical neural network conducts the system control in a near optimal manner.

6

Conclusions

Different than a previous attempt to address adaptive, time-optimal control of uncertain nonlinear systems, this paper aims at achieving adaptive control of uncertain nonlinear systems in an approximately fuel-optimal control manner. The studied nonlinear system control problem is characterized by the nonlinear systems affine in both the control variables and uncertain parameters; the control variables physically restricted; and a cost functional to be minimize that is integral of a function that is linear in the absolute values of control variables over an applicable time horizon. Since the conventional adaptive control techniques can not be directly applied to solve the adaptive control of the studied systems, a neural network based control methodology is adopted. This novel control design allows for incorporation of control performance criteria, constraints on the control variables while achieve practical effectiveness of addressing the parameter uncertainty. The proposed hierarch neural controller consists of the lower level neural networks for fuel-optimal control for respective nominal cases and an upper level neural network for determining the relative contribution of each lower level neural controller. The control design procedures are presented with theoretical justifications which are practically convenient and useful in synthesizing robust, adaptive and fuel-optimal neural controllers.

References 1. Mohler, R.R.: Nonlinear Systems Volume I, Dynamics and Control. Prentice-Hall, Englewood Cliffs (1991) 2. Mohler, R.R.: Nonlinear Systems Volume II, Applications to Bilinear Control. Prentice Hall, Englewood Cliffs (1991) 3. Mohler, R.R.: Bilinear Control Processes. Academic Press, New York (1973) 4. Moon, S.F.: Optimal Control of Bilinear Systems and Systems Linear in Control, Ph.D. dissertation. The University of New Mexico (1969) 5. Lee, E.B., Markus, L.: Foundations of Optimal Control Theory. Wiley, New York (1967)

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6. Rugh, W.J.: Linear System Theory. Prentice-Hall, Englewood Cliffs (1993) 7. Chen, D., Mohler, R., Chen, L.: Neural-Network-Based Adaptive Control with Application to Power Systems. In: Proc. American Control Conf., San Diego, pp. 3236–3240 (1999) 8. Chen, D., Mohler, R.: Nonlinear Adaptive Control with Potential FACTS Applications. In: Proc. American Control Conf., San Diego, pp. 1077–1081 (1999) 9. Chen, D., Mohler, R.: The Properties of Latitudinal Neural Networks with Potential Power System Applications. In: Proc. American Control Conf., Philadelphia, pp. 980–984 (1998) 10. Chen, D., Mohler, R., Chen, L.: Synthesis of Neural Controller Applied to Power Systems. IEEE Trans. Circuits and Systems I 47, 376–388 (2000) 11. Chen, D.: Nonlinear Neural Control with Power Systems Applications. Ph.D. Dissertation, Oregon State University (1998) 12. Chen, D., Mohler, R., Shahrestani, S., Hill, D.: Neural-Net-Based Nonlinear Control for Prevention of Voltage Collapse. In: Proc. 38th IEEE Conference on Decision and Control, Phoenix, pp. 2156–2161 (1999) 13. Chen, D., Mohler, R.: Theoretical Aspects on Synthesis of Hierarchical Neural Controllers for Power Systems. In: Proc. 2000 American Control Conference, Chicago, pp. 3432–3436 (2000) 14. Sanner, R., Slotine, J.: Gaussian Networks for Direct Adaptive Control. IEEE Trans. Neural Networks 3, 837–863 (1992) 15. Yesidirek, A., Lewis, F.: Feedback Linearization Using Neural Network. Automatica 31, 1659–1664 (1995) 16. Chen, F., Liu, C.: Adaptively Controlling Nonlinear Continuous-Time Systems Using Multilayer Neural Networks. IEEE Trans. Automatic Control 39, 1306–1310 (1994) 17. Lewis, F., Yesidirek, A., Liu, K.: Neural Net Robot Controller with Guaranteed Tracking Performance. IEEE Trans. Neural Networks 6, 703–715 (1995) 18. Polycarpou, M.: Stable Adaptive Neural Control Scheme for Nonlinear Systems. IEEE Trans. Automatic Control 41, 447–451 (1996) 19. Zakrzewski, R.R., Mohler, R.R., Kolodziej, W.J.: Hierarchical Intelligent Control with Flexible AC Transmission System Application. IFAC J. Control Engineering Practice 2, 979–987 (1994) 20. Narendra, K., Mukhopadhyay, S.: Intelligent Control Using Neural Networks. IEEE Control Systems Magazine 12, 11–18 (1992) 21. Chen, D., Yang, J.: Robust Adaptive Neural Control Applied to a Class of Nonlinear Systems. In: Proc. 17th IMACS World Congress: Scientific Computation, Applied Mathematics and Simulation, Paris (2005) T5-I-01-0911 22. Chen, D., Yang, J., Mohler, R.: On Near Optimal Neural Control of a Class of Nonlinear Systems with Multiple Inputs. Neural Computing and Applications 2 (2007)

Three Global Exponential Convergence Results of the GPNN for Solving Generalized Linear Variational Inequalities Xiaolin Hu1 , Zhigang Zeng2 , and Bo Zhang1 1 State Key Laboratory of Intelligent Technology and Systems, Tsinghua National Laboratory for Information Science and Technology (TNList), Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China 2 School of Automation, Wuhan University of Technology, Wuhan 430070, China

Abstract. The general projection neural network (GPNN) is a versatile recurrent neural network model capable of solving a variety of optimization problems and variational inequalities. In a recent article [IEEE Trans. Neural Netw., 18(6), 1697-1708, 2007], the linear case of GPNN was studied extensively from the viewpoint of stability analysis, and it was utilized to solve the generalized linear variational inequality with various types of constraints. In the present paper we supplement three global exponential convergence results for the GPNN for solving these problems. The first one is different from those shown in the original article, and the other two are improved versions of two results in that article. The validity of the new results are demonstrated by numerical examples.

1

Introduction

The following problem is called the generalized linear variational inequality (GLVI): find x∗ ∈ m such that N x∗ + q ∈ X and (M x∗ + p)T (x − N x∗ − q) ≥ 0

∀x ∈ X,

(1)

where M, N ∈ m×m ; p, q ∈ m ; and X is a closed convex set in m . It has many scientific and engineering applications, e.g., linear programming and quadratic programming [1], extended linear programming [2] and extended linear-quadratic programming [2, 3]. If X is a box set, i.e., X = {x ∈ m |x ≤ x ≤ x}

(2)

where x and x are constants (without loss of generality, any component of x or −x can be −∞), a neurodyamic approach was proposed in [4] and [5] from different viewpoints for solving it. Moreover, in [5], the neurodynamic system was given a name, general projection neural network (GPNN). A general form of the system is as follows: dx = λW {−N x + PX ((N − αM )x + q − αp) − q}, dt F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 309–318, 2008. c Springer-Verlag Berlin Heidelberg 2008 

(3)

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where λ ∈ , W ∈ m×m and α ∈  are positive constants, and PX (x) = (PX1 (x1 ), · · · , PXm (xm ))T with ⎧ ⎨ xi , xi < xi , PXi (xi ) = xi , xi  xi  xi , (4) ⎩ xi , xi > xi . Recently, the stability of the above GPNN was studied extensively in [6]. Many global convergence and stability results were presented. In addition, when X in the GLVI (1) is not a box set, but a polyhedral set defined by inequalities and equalities, several specific GPNNs similar to (3) were formulated to solve the corresponding problems. Some particular stability results of those GPNNs were also discussed. In the present paper, we will give a few new stability results of the GPNNs, reflecting our up-to-date progress in studying this type of neural networks. Throughout the paper, x denotes the l2 norm of a vector x, I denotes the identity matrix with an appropriate dimension, and X ∗ stands for the solution set of GLVI (1), which is assumed to be nonempty. In addition, it is assumed that there exists at least one finite point in X ∗ . Define an operator D+ f (t) = lim suph→0+ (f (t + h) − f (t))/h, where f (t) is a function mapping from  → .

2 2.1

Main Results Box Set Constraint

First, we give a new stability result of the GPNN (3) for solving the GLVI with box-type constraint as described in (2). A useful lemma is introduced first [5, 4]. Lemma 1. Consider PX : m → X defined in (4). For any u, v ∈ m , we have PX (u) − PX (v) ≤ u − v. Theorem 1. Let N = {nij } and D = N − αM = {dij }. If nii >

m 

|nij | +

j=1,j=i

m 

|dij |,

∀i = 1, · · · , m,

(5)

j=1

then the GPNN (3) with W = I is globally exponentially stable. Proof. From (5) there exists θ > 0 such that nii ≥

m  j=1,j=i

|nij | +

m 

|dij | + θ,

∀i = 1, · · · , m.

(6)

j=1

Let x∗ be a finite point in X ∗ , L(t0 ) = max1≤i≤m |xi (t0 ) − x∗i | and zi (t) = |xi (t) − x∗i | − L(t0 )e−λθ(t−t0 ) . In the following we will show zi (t) ≤ 0 for any i = 1, · · · , m and all t ≥ t0 by contradiction. Actually, if this is not true, there

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311

must exists a sufficiently small  > 0, two time instants t1 and t2 satisfying t0 ≤ t1 < t2 , and at least one k ∈ {1, · · · , m}, such that zk (t1 ) = 0,

zk (t2 ) = 

D zk (t1 ) ≥ 0, +

zi (s) ≤ ,

(7)

D zk (t2 ) > 0 +

∀i = 1, · · · , m;

(8)

t0 ≤ s ≤ t2 .

(9)

From (3) we have dx/dt = λ{−N (x − x∗ ) + PX (Dx + q − αp) − PX (Dx∗ + q − αp)}

(10)

and from Lemma 1 we have |PXi (di x + qi − αpi ) − PXi (di x∗ + qi − αpi )| m  ≤|di (x − x∗ )| ≤ |dij ||xj − x∗j |, ∀i = 1, · · · , m,

(11)

j=1

where di ∈ 1×m denotes the ith row of D. Without loss of generality, we assume xk (t2 ) − x∗k > 0. (The case of xk (t2 ) − x∗k < 0 can be reasoned similarly.) It follows from (7), (9), (10) and (11) that xk (t2 ) − x∗k = L(t0 )e−λθ(t−t0 ) + , |xi (t2 ) − x∗i | ≤ L(t0 )e−λθ(t−t0 ) + , ∀i = 1, · · · , m, and D+ zk (t2 ) =D+ |xk (t2 ) − x∗k | + λθL(t0 )e−λθ(t−t0 ) m  ≤ − λnkk (xk (t2 ) − x∗k ) + λ |nkj ||xj (t2 ) − x∗j | j=1,j=k

+λ

m 

|dkj ||xj (t2 ) − x∗j | + λθL(t0 )e−λθ(t−t0 )

j=1 m 

≤ − λnkk (L(t0 )e−λθ(t−t0 ) + ) + λ

|nkj |(L(t0 )e−λθ(t−t0 ) + )

j=1,j=k

+λ ⎛

m 

|dkj |(L(t0 )e−λθ(t−t0 ) + ) + λθL(t0 )e−λθ(t−t0 )

j=1

=λ ⎝−nkk + ⎛

m 

|nkj | +

j=1,j=k

+ λ ⎝−nkk +

m 

m 

⎞ |dkj | + θ⎠ L(t0 )e−λθ(t−t0 )

j=1

|nkj | +

j=1,j=k

m 

⎞ |dkj |⎠ 

j=1

In view of (5) and (6), we have D+ zk (t2 ) < 0, which contradicts (8). Hence, |xi (t) − x∗i | ≤ L(t0 )e−λθ(t−t0 ) , The proof is completed.

∀i = 1, · · · , m;

t ≥ t0 .

(12)

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The above theorem is proved in the spirit of [7]. From the analysis it can be inferred that the convergence rate of (3) is at least λθ where θ is the difference between the left and right hand sides of (5). Different from most of the results in [6], the exponential convergence rate here is expressed in terms of every component of the state vector separately, which provides a more detailed estimation than the results obtained by the usual Lyapunov method. In the above proof, if we choose L(t0 ) = x(t) − x∗ 2 , following similar arguments we can arrive at the following condition which assures the global exponential stability results as well: the minimum eigenvalue of (N + N T )/2 is greater than D. Interestingly, this is a result stated in Corollary 1 of [6] where a different proof was given. 2.2

General Constraints

Consider the GLVI (1) with X defined as X = {x ∈ m |x ∈ Ωx , Ax ∈ Ωy , Bx = c},

(13)

where A ∈ h×m , B ∈ r×m , c ∈ r , and Ωx , Ωy are two box sets defined as {x ∈ m |x ≤ x ≤ x} and {y ∈ h |y ≤ y ≤ y}, respectively (cf. (2)). Let A˜ = (AT , B T )T and 







˜T p ˜ = M −A ˜ = N 0 , q˜ = q , M , p˜ = ,N ˜ ˜ 0 0 Aq AN 0 I T T T h+r T T T ˜ ˜ ˜y . Ωy = {y ∈  |(y , c ) ≤ y ≤ (y , c ) }, U = Ωx × Ω It was shown in [6] that the GLVI can be converted to another GLVI with a box ˜ only, and as a result, can be solved by using the following specific GPNN: set U du ˜ + P ˜ ((N ˜ − αM ˜ )u + q˜ − α˜ = λW {−Nu p) − q˜}, U dt

(14)

where λ > 0, α > 0, W ∈ (m+h+r)×(m+h+r) are constants, u = (xT , y T )T is the state vector, and PU˜ (·) is the activation function defined similarly as in (4). The output of the neural network is simply x(t), the first part of the state u(t). ˜ +αM ˜ )T , if M T N > 0 then the output In [6], it was proved that when W = (N trajectory x(t) of the neural network is globally convergent to the unique solution x∗ of the problem (1). In the following, we show that if this condition holds, the convergence rate can be exponential by choosing an appropriate scaling factor λ. The proof is inspired by [8]. ˜ + αM ˜ )T for solving the GLVI Theorem 2. Consider GPNN (14) with W = (N T with X defined in (13). If M N > 0 and λ is large enough, then the output trajectory x(t) of the neural network is globally exponentially convergent to the unique solution of the problem. Proof. It was shown in [6, Theorem 5] that the solution of the GLVI is unique, which corresponds to the first part of any equilibrium point of (14). Consider

Three Global Exponential Convergence Results of the GPNN

313

the function V (u(t)) = u(t) − u∗ 2 /2 where u∗ is a finite equilibrium point of (14). Following a similar analysis procedure to that of Corollary 4 in [5] we can derive dV (u(t)) ˜ TN ˜ (u−u∗ )−P ˜ ((N ˜ −αM ˜ )u+ q˜−α˜ ˜ u− q˜2 }. ≤ λ{−α(u−u∗ )T M p)− N U dt It follows that dV (u(t)) ˜ TN ˜ (u − u∗ )} = λα{−(x − x∗ )T M T N (x − x∗ )} ≤ λα{−(u − u∗ )T M dt ≤ λα{−βx − x∗ 2 }, where β > 0 denotes the minimum eigenvalue of (M T N + N T M )/2. Then

t

V (u(t)) ≤ V (u(t0 )) − λαβ

x(s) − x∗ 2 ds

t0

and x(t) − x∗ 2 ≤ 2V (u(t0 )) − 2λαβ



t

x(s) − x∗ 2 ds.

t0

Without loss of generality it is assumed x(t0 ) − x∗ 2 > 0 which implies V (u(t0 )) > 0. Then there exist τ > 0 and μ > 0 that depend on x(t0 ) only, so that

t0 +τ x(s) − x∗ 2 ds ≥ τ μ. If λ is large enough so that λ ≥ V (u(t0 ))/(αβτ μ), t0 we have V (u(t0 )) − λαβ

t0 +τ

x(s) − x∗ 2 ds ≤ 0.

t0

It follows that for any t > t1 ≥ t0 + τ x(t) − x∗ 2 ≤x(t1 ) − x∗ 2 + 2V (u(t0 )) − 2λαβ

t1

x(s) − x∗ 2 ds

t0 t

− 2λαβ

x(s) − x∗ 2 ds

t1

≤x(t1 ) − x∗ 2 + 2V (u(t0 )) − 2λαβ





t0 +τ

x(s) − x∗ 2 ds

t0 t

− 2λαβ

x(s) − x∗ 2 ds

t1 ∗ 2



t

≤x(t1 ) − x  − 2λαβ

x(s) − x∗ 2 ds.

t1

As a result, x(t) − x∗ 2 − x(t1 ) − x∗ 2 f (t) − f (t1 ) ≤ −2λαβ t − t1 t − t1

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where f (t) =

t t1

x(s) − x∗ 2 ds. Let t → t1 + 0, then we have dx(t) − x∗ 2 ≤ −2λαβx(t) − x∗ 2 . dt

Therefore x(t) − x∗  ≤ x(t1 ) − x∗ e−λαβ(t−t1 ) = c0 e−λαβ(t−t0 ) ,

∀t > t1

where c0 = x(t1 ) − x∗ eλαβ(t1 −t0 ) . Since dV (u(t))/dt ≤ 0, u(t) ∈ S = {u ∈ m |V (u(t)) ≤ V (u(t0 ))} for all t ≥ t0 . Moreover, V (u(t)) is radially unbounded, then S is bounded, which implies that x(t) − x∗  is bounded over t ≥ t0 . Let Δ = maxt0 ≤t≤t1 x(t) − x∗  and c1 = Δ/e−λαβ(t1 −t0 ) . We have x(t) − x∗  ≤ Δ = c1 e−λαβ(t1 −t0 ) ≤ c1 e−λαβ(t−t0 ) , Hence

x(t) − x∗  ≤ cm e−λαβ(t−t0 ) ,

∀t0 ≤ t ≤ t1 .

∀t ≥ t0 ,

where cm = max(c0 , c1 ). The proof is completed. 2.3

Inequality Constraints

Consider X in (13) with inequality constraints only; i.e., X = {x ∈ m |Ax ∈ Ωy },

(15)

where the notations are the same as in (13). Let ˆ = AN M −1 AT , qˆ = −AN M −1 p + Aq. N The following specific GPNN is proposed to solve the problem: – State equation du ˆ + PΩy ((N ˆ − αI)u + qˆ) − qˆ}; = λW {−Nu dt – Output equation

v = M −1 AT u − M −1 p,

(16a)

(16b)

where λ ∈ , α ∈ , λ > 0, α > 0 and W ∈ h×h . ˆ + αI)T , if M T N > 0 then the output In [6], it was proved that when W = (N trajectory v(t) of the neural network is globally convergent to the unique solution x∗ of the problem (1). In the following, we show that if this condition holds, the convergence rate can be exponential by choosing an appropriate λ. ˆ + αI)T for solving the GLVI Theorem 3. Consider GPNN (16) with W = (N T with X defined in (15). If M N > 0 and λ is large enough, then the output trajectory v(t) of the neural network is globally exponentially convergent to the unique solution of the problem.

Three Global Exponential Convergence Results of the GPNN

315

Proof. From [6, Theorem 6], the solution of the GLVI is unique, which is identical to v ∗ = M −1 AT u∗ − M −1 p where u∗ is any equilibrium point of (16a). Define a function 1 V (u(t)) = u(t) − u∗ 2 , t  t0 . 2 From (16b), we have v − v ∗ 2 = M −1 AT (u − u∗ )2  M −1 AT 2 u − u∗ 2 . ∗ 2

v−v  Thus V (u)  2M −1 AT 2 . Following a similar analysis to that of Corollary 4 in [5] we can deduce

dV (u(t)) ˆ (u − u∗ ) − PΩy ((N ˆ − αI)u + qˆ) − N ˆ u − qˆ2 }.  λ{−α(u − u∗ )T N dt It follows that dV (u(t))  λα{−(u − u∗ )T AN M −1 AT (u − u∗ )} dt = λα{−[M −1 AT (u − u∗ )]T M T N [M −1 AT (x − x∗ )]} = λα{−(v − v ∗ )T M T N (v − v ∗ )}  λα{−βv − v ∗ 2 }, where β > 0 denotes the minimum eigenvalue of (M T N + N T M )/2. Then t v(s) − v ∗ 2 ds V (u(t)) ≤ V (u(t0 )) − λαβ t0

and v(t) − v ∗ 2 ≤ 2γV (u(t0 )) − 2λαβγ



t

v(s) − v ∗ 2 ds,

t0

where γ = M −1 AT 2 . The rest of the proof is similar to the latter part of the analysis of Theorem 2, and is omitted for brevity.

3

Illustrative Examples

Example 1. Let’s first solve a GLVI (1) with a box set, where ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 4 2 −1 5 2 −1 −1 0 M = ⎝ 0 3 0 ⎠ , N = ⎝ 1 5 0 ⎠ , p = ⎝ 2 ⎠ , q = ⎝2⎠ , −1 3 6 −1 3 8 5 0 and X = {x ∈ 3 |(−4, 0, −4)T  x  (6, 6, 6)T }. Let α = 1, it is easy to verify that 3 the condition in Theorem 2 is satisfied. Actually, n11 − |n12 | − |n13 | − j=1 |d1j | =   1, n22 − |n21 | − |n23 | − 3j=1 |d2j | = 1, n33 − |n31 | − |n32 | − 3j=1 |d3j | = 2. Then the GPNN (3) is globally exponentially stable. All numerical simulations validated

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X. Hu, Z. Zeng, and B. Zhang

10

States

5

x 3 (t) x 1 (t) 0

x 2 (t)

−5

0

1

2

3

4

5

Time unit t

Fig. 1. State trajectories of the GPNN (3) in Example 1 with W = I, λ = α = 1 and x(0) = (10, 6, −5)T 4

ln x1 (t) − x∗1  ln x2 (t) − x∗2  ln x3 (t) − x∗3  L(t0 ) − λθt

2 0 −2 −4 −6 −8 −10 −12 −14

0

1

2

3

4

5

Time unit t

Fig. 2. Solution error of the GPNN (3) in Example 1. The estimated upper bound (dashed line) is also plotted.

this conclusion. Fig. 1 demonstrates the state trajectories started from the initial point x(0) = (10, 6, −5)T with λ = 1 (t0 is set to 0), which converge to the unit solution of the problem x∗ = (0.4265, −0.4853, −0.2647)T . To show their exponential convergence rates, we take the natural logarithm of both sides of (12), ln |xi (t) − x∗i | ≤ ln L(t0 ) − λθt,

∀i = 1, · · · , 3;

t ≥ 0.

and depict both sides of above inequality in Fig. 2. (It is evident that θ can be chosen as θ = 1). The right-hand-side quantity now becomes a straight line in the figure. It is seen that the error of the states are all upper bounded by this line.

Three Global Exponential Convergence Results of the GPNN

317

20 15 10

Outputs

5

v1 (t)

v3 (t)

0

v2 (t)

−5 −10 −15 −20

0

0.2

0.4

0.6

0.8

1

Time unit t

ˆ + αM ˆ )T , Fig. 3. Output trajectories of the GPNN (16) in Example 2 with W = (N λ = α = 1 and ten random initial points

4

ln v(t) − x∗ 

2

0

−2

−4

−6

−8

0

0.2

0.4

0.6

0.8

1

Time unit t

Fig. 4. Solution error of the GPNN (16) in Example 2. Because of numerical errors in simulations, when ln v(t) − x∗  ≤ −8, the trajectories become unstable, and thus are not shown here.

Example 2. Consider a GLVI with a polyhedron set X defined in (15). Let M=



    1 1 0 1 −1 −1 −1 0  , N = 0 −1 0  , p = −1 , q = 2 , A = −5 , 5 −1

1 −1 −1 −1 1 0 0 1 −1

0 3 −1

2

0

and Ωy = {y ∈ 2 | − 10  y  10}. It can be verified that M T N > 0. The ˆ + αM ˆ )T can be used to solve the problem acGPNN (16) with W = (N cording to Theorem 3. Simulation results showed that from any initial point this neural network globally converges to the unique equilibrium point u∗ = (−0.0074, −0.7556)T . Then, the solution of the GLVI is calculated as x∗ = (−0.4444, −3.2296, −1.9852)T . Fig. 3 displays the output trajectories of the neural network with λ = α = 1 and 10 different initial points, and Fig. 4 displays

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the solution error (in natural logarithm) along with these trajectories. It is seen that for any of the 10 curves in Fig. 4 there exits a straight line with negative slope above it, that is, the convergence rate is upper bounded by an exponential function of t which tends to zero as t → ∞.

4

Concluding Remarks

The general projection neural network (GPNN) has attracted much attention in recent years. The paper presents three sets of global exponential convergence conditions for it, which extend our recent results to some extent. Numerical examples illustrate the correctness of these new results. Acknowledgments. The work was supported by the National Natural Science Foundation of China under the grant No. 60621062 and 60605003, the National Key Foundation R&D Projects under the grant No. 2003CB317007, 2004CB318108 and 2007CB311003, and the Basic Research Foundation of Tsinghua National Laboratory for Information Science and Technology (TNList).

References 1. Hu, X., Wang, J.: Design of General Projection Neural Networks for Solving Monotone Linear Variational Inequalities and Linear and Quadratic Optimization Problems. IEEE Trans. Syst., Man, Cybern. B 37, 1414–1421 (2007) 2. He, B.: Solution and Applications of a Class of General Linear Variational Inequalties. Sci. China Ser. A-Math. 39, 395–404 (1996) 3. Hu, X.: Applications of the General Projection Neural Network in Solving Extended Linear-Quadratic Programming Problems with Linear Constraints. Neurocomputing (accepted) 4. Gao, X.B.: A Neural Network for a Class of Extended Linear Variational Inequalities. Chinese Jounral of Electronics 10, 471–475 (2001) 5. Xia, Y., Wang, J.: A General Projection Neural Network for Solving Monotone Variational Inequalities and Related Optimization Problems. IEEE Trans. Neural Netw. 15, 318–328 (2004) 6. Hu, X., Wang, J.: Solving Generally Constrained Generalized Linear Variational Inequalities Using the General Projection Neural Networks. IEEE Trans. Neural Netw. 18, 1697–1708 (2007) 7. Zeng, Z., Wang, J., Liao, X.: Global Exponential Stability of a General Class of Recurrent Neural Networks with Time-Varying Delays. IEEE Trans. Circuits Syst. I 50, 1353–1358 (2003) 8. Xia, Y., Feng, G., Kamel, M.: Development and Analysis of a Neural Dynamical Approach to Nonlinear Programming Problems. IEEE Trans. Automatic Control 52, 2154–2159 (2007)

Disturbance Attenuating Controller Design for a Class of Nonlinear Systems with Unknown Time-Delay Geng Ji School of Mathematics and Information Engineering, Taizhou University, Linhai 317000, P.R. China [email protected]

Abstract. An adaptive neural network control design approach is proposed for a class of nonlinear systems with unknown time delay. By constructing a proper Lyapunov-Krasoviskii functional, the uncertainty of unknown time-delay is compensated. In addition, the semiglobally input-to-state practically stable (ISpS) disturbance attenuation problem is solved by neural network technique. The feasibility of neural network approximation of unknown system functions is guaranteed over practical compact set. Finally, a numerical simulation is given to show the effectiveness of the approach. Keywords: Disturbance attenuation, input-to-state practically stable, adaptive neural network control, nonlinear time-delay systems.

1

Introduction

Control of nonlinear systems has received much attention and many analysis techniques and design methodologies have been developed [1-2]. In [2], from the geometric theory of nonlinear systems, under certain assumptions, a nonlinear system can be decomposed into two cascaded systems, one is a nonlinear system and the other is a linearizable system. A class of systems with this structure, called minimum phase nonlinear systems, has been discussed heavily [3-4]. In the past few years, disturbance attenuation and almost disturbance decoupling problems have been extensively studied for uncertain nonlinear dynamic systems. Many interesting results in this area have been obtained [5-7]. In [5], disturbance attenuation was studied for a class of nonlinear systems which only contain uncertain disturbance. The problem of almost disturbance decoupling was considered in [7]. In these works, the systems contain either known functions or bounded unknown parameters, and the robust control method was used. Nevertheless, when the systems contain unknown functions, they are hard to be dealt via robust method. In recent years, adaptive neural control schemes have been found to be useful for the control nonlinear uncertain systems with unknown smooth functions, and many significant developments have been achieved [8-10]. Direct adaptive neural network control was presented for a class of affine nonlinear systems in F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 319–329, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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the strict-feedback form with unknown nonlinearities by Ge and Wang [9]. The problem of semiglobally ISpS disturbance attenuation was investigated for a class of uncertain nonlinear minimum-phase systems in [10]. However, all these works study the nonlinear systems without time delay. It is well known that time delays are often appeared in practical systems. In general, the existence of time delays degrades the control performance and sometime makes the closed-loop stabilization difficult, especially for nonlinear systems. Stabilization of nonlinear systems with time delay has received considerable attention, and many approaches to this issue have been developed (see [11-13] ). In [11], adaptive neural control was presented for a class of strict-feedback nonlinear time-delay systems. The unknown time delays were compensated for through the use of appropriate Lyapunov-Krasovskii functionals. In this paper, we discuss the nonlinear systems with unknown time delay. Motivated by references [10,11], we will use the adaptive neural network control method to study the problem of semiglobally ISpS disturbance attenuation for a class of minimum phase nonlinear systems with some structure uncertainties which cannot be directly coped with by the existing robust control design method. An appropriate Lyapunov-Krasovskii functional is used to construct Lyapunov function candidate such that the uncertainty from unknown time delay is removed. The paper is organized as follows: The problem description and preliminary results are given in section 2. In section 3, an adaptive neural network control design scheme is presented. Simulation result is shown in section 4. Finally, conclusion is given in section 5.

2

Problem Description and Preliminary

Consider an uncertain nonlinear system with time-delay x(t) ˙ = f (x(t), ξ(t)) ˙ξ(t) = u(t) + d1 (x(t), ξ(t)) + d2 (ξ(t − τ )) + p (x(t), ξ(t)) ω

(1)

y(t) = h (x(t), ξ(t)) where x ∈ Rn−1 and ξ ∈ R are the state variables, u ∈ R is the control input, y ∈ Rp is the regulated output, ω ∈ Rq is an exogenous input (reference and/or noise). The function vectors f ( ·, · ), d1 ( ·, · ) and h( ·, · ) are unknown smooth function vectors satisfying f (0, 0) = 0, d1 (0, 0) = 0 and h(0, 0) = 0. d2 ( · ) is a unknown smooth function satisfying assumption 2.2. p( ·, · ) is a known smooth matrix with proper dimension. τ is a unknown time delay, which is bounded by a known constant, i.e. τ ≤ τmax . Since f (x, ξ), h(x, ξ) are smooth with f (0, 0) = 0, h(0, 0) = 0, it can be decomposed into f (x, ξ) = f0 (x) + f1 (x, ξ)ξ, h(x, ξ) = h0 (x) + h1 (x, ξ)ξ where f0 (x) = f (x, 0), h0 (x) = h(x, 0) with f0 (0) = 0, h0 (0) = 0.

(2)

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The following assumption for system (1), which will be used throughout the paper, is proposed. Assumption 2.1: For the x-subsystem of (1), there exist a radially unbounded positive definite differentiable function ψ(x) and a positive constant α satisfying ∂ψ(x) f0 (x) ≤ −αψ(x), ∀x ∈ Rn−1 ∂x

(3)

This assumption assumes that the x-subsystem is asymptotically stable with respect to ξ ≡ 0. ψ(x) is an unbounded positive definite differentiable function which is not needed to be known. Assumption 2.2: The unknown smooth function d2 (ξ(t)) satisfy the following inequality |d2 (ξ(t))| ≤ |ξ(t)| ρ (ξ(t)) where ρ (ξ(t)) is known smooth function. Assumption 2.3: Define Z = [xT , ξ T ]T ∈ ΩZ ⊂ Rn with ΩZ known compact set. Firstly, we will discuss the problem of disturbance attenuation for system (1) under the assumption that f ( ·, · ), d1 ( ·, · ), and h( ·, · ) are known. To obtain a good performance with asymptotic stability, a controller is designed to stabilize the system and to guarantee the closed-loop system has the good L2 performance for any given constant γ > 0, i.e.  t  t T 2 y(τ ) y(τ )dτ ≤ γ ω(τ )T ω(τ )dτ + β(x0 , ξ0 ) 0

0

where t ≥ 0, β : R × R → R with β(x0 , ξ0 ) ≥ 0. In assumption 2.1, since ψ(x) is radially unbounded and positive definite, denote s = ψ(x), there exists a class K-function k( · ) such that n−1

hT0 (x)h0 (x) ≤ k(s)

(4)

Based on the function k(s), a storage function V0 (s) can be constructed [15]  2s dk(t) V0 (s) = s sup + k(t)dt (5) 0≤t≤1 dt s And it satisfies the following inequalities V0 (s) ≥ k(s), s

d V0 (s) ≥ V0 (s) ds

Define the following Lyapunov function candidate  t 1 T 1 V = V0 (s) + ξ ξ + ξ 2 (τ )ρ2 (ξ(τ ))dτ 2 2 t−τ

(6)

(7)

The derivative of V along the trajectory of (1) is dV0 (s) ∂ψ V˙ = [f0 (x) + f1 (x, ξ)ξ] + ξ T ξ˙ ds ∂x 1 1 + ξ 2 (t)ρ2 (ξ(t)) − ξ 2 (t − τ )ρ2 (ξ(t − τ )) 2 2

(8)

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In viewing of Assumption 2.1, (9) is easily obtained from (1)(6) (8) dV0 (s) ∂ψ V˙ ≤ −αV0 (s) + f1 ξ + ξ T [u + d1 + pω] ds ∂x 1 1 +ξ T d2 (ξ(t − τ )) + ξ 2 (t)ρ2 (ξ(t)) − ξ 2 (t − τ )ρ2 (ξ(t − τ )) 2 2

(9)

Applying Assumption 2.2, we have dV0 (s) ∂ψ 1 V˙ ≤ −αV0 (s) + f1 ξ + ξ T [u + d1 + pω] + ξ 2 (t)ρ2 (ξ(t)) ds ∂x 2 1 2 2 + |ξ| · |ξ(t − τ )| · ρ (ξ(t − τ )) − ξ (t − τ )ρ (ξ(t − τ )) 2

(10)

By using Young’s Inequality |ξ| · |ξ(t − τ )| · ρ (ξ(t − τ )) ≤

1 2 1 2 ξ + ξ (t − τ )ρ2 (ξ(t − τ )) 2 2

(11)

Substituting (11) into (10), (10) becomes dV0 (s) ∂ψ 1 1 V˙ ≤ −αV0 (s) + f1 ξ + ξ T [u + d1 + pω] + ξ 2 + ξ 2 (t)ρ2 (ξ(t)) (12) ds ∂x 2 2 Now, choose α0 with 0 < α0 < α and considering (2) (4) (6) and (12), it yields V˙ + α0 y T y − α0 γ 2 ω T ω ≤ − (α − α0 ) V0 (s) − α0 γ 2 ω T ω + ξ T pω 1 1 +ξ T [u + ϕ(Z)] + ξ 2 + ξ 2 (t)ρ2 (ξ(t)) 2 2

(13)

where  ϕ(Z) =

dV0 (s) ∂ψ f1 ds ∂x

T + d1 + 2α0 hT1 h0 + α0 hT1 h1 ξ

T  with Z = xT , ξ T ∈ Rn . With the following inequality − α0 γ 2 ω T ω + ξ T pω ≤

1 ξ T ppT ξ 4α0 γ 2

(14)

And select the following controller u u = −cξ − ϕ(Z) −

1 1 1 ppT ξ − ξ − ξρ2 4α0 γ 2 2 2

where c > 0 is a constant scalar to be designed. Consequently, letting c ≥ 12 (α − α0 ), we have   1 T T 2 T ˙ V + α0 y y − α0 γ ω ω ≤ − (α − α0 ) V0 (s) + ξ ξ ≤ 0 2

(15)

(16)

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323

Therefore, by integrating both sides of (16) from 0 to t, and considering that V (x, ξ) ≥ 0, it is easily obtained 



t

y ydτ ≤ γ T

0

t

ω T ωdτ +

2 0

1 V (x0 , ξ0 ) , ∀t ≥ 0 α0

(17)

Then, the following result is obtained. Theorem 2.1: Consider the certain version of the nonlinear time-delay system (1) satisfying Assumption 2.1 and Assumption 2.2 with ψ(x) being known. For  T any γ > 0, given any initial state x(0)T , ξ(0)T ∈ Rn , and scalar α0 with 0 < α0 < α, there exists a feedback controller given by (15), which solves the problem of disturbance attenuation with globally asymptotic stability. Remark 1: In the above analysis, we get the theorem 2.1 under the assumption that f ( ·, · ), d1 ( ·, · ), and h( ·, · ) are known. If f ( ·, · ), d1 ( ·, · ), and h( ·, · ) are unknown smooth functions, the problem of disturbance attenuation cannot be easily solved. This motivates us to seek a new design method. A natural way to do this is to use neural networks method with which the unknown nonlinear functions in the systems can be approximated by certain neural networks. Refer to [14], we have Definition 1: The system x(t) ˙ = f (x, u) is input-to-state practically stable (ISpS) if there exist a function R1 of class KL, a function R2 of class K and a nonnegative constant δ such that, for any initial condition x(0) and each measurable essentially bounded control u(t) defined for all t ≥ 0, the associated solution exists x(t) for all t ≥ 0 and satisfies |x(t)| ≤ R1 (|x(0)| , t) + R2 ( ut ) + δ

(18)

where ut is the truncated function of u at t and · stands for the L∞ supremum norm. Definition 2: A C 1 function V is said to be an exp-ISpS Lyapunov function for system x(t) ˙ = f (x, u) if (1) there exist functions α1 , α2 of class K∞ such that α1 (|x|) ≤ V (x) ≤ α2 (|x|) , ∀x ∈ Rn

(19)

(2) there exist two constants k > 0, δ ≥ 0 and a class k∞ -function R3 such that ∂V f (x, u) ≤ −kV (x) + R3 (|u|) + δ ∂x

(20)

Proposition 1: For any control system x(t) ˙ = f (x, u), the following properties are equivalent: (i) It is ISpS. (ii) It has an exp-ISpS Lyapunov function

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Adaptive Neural Control Design

In this section, we assume that f ( ·, · ), d1 ( ·, · ) and h( ·, · ) are unknown. Thus, it is impossible to deal with the problem of disturbance attenuation by the controller (15) directly. An adaptive neural method will be proposed for the system (1) and the main result will be obtained in the latter. For the purpose of the practical controller design in the latter, let us define sets Ωcξ ⊂ ΩZ and ΩZ0 as follows: Ωcξ := {ξ| |ξ| < cξ } , ΩZ0 := ΩZ − Ωcξ

(21)

where cξ is a constant that can be chosen arbitrarily small and  − in (21) is used to denote the complement of set B in A set as A − B := {x|x ∈ A and x ∈ / B}. As the controller (16) contains the uncertainty vector field ϕ(Z), we employ RBF neural networks to approximate it. According to the main result state in [16], any real-valued continuous function can be arbitrarily closely approximated by a network of RBF type over a compact set. The compactness of set ΩZ0 is a must to guarantee the feasibility of neural networks approximation, which is shown in the following lemma. Lemma [11] Set ΩZ0 is a compact set. Based on the above lemma, given any ε > 0, by appropriately choosing μi ∈ Rn , i = 1, 2, · · · , l, for some sufficiently large integer l, one can see that the functions ϕ(Z) can be approximated by RBF neural networks on certain compact set ΩZ0 , i.e., ϕ(Z) = W ∗ T S(Z) + ε∗

(22)

where W ∗ is the ideal constant weights, and |ε∗ | ≤ ε is the approximation error with constant ε > 0. Consequently, the ideal controller u∗ is given by u∗ = −cξ − W ∗ T S(Z) − ε∗ −

1 1 1 ppT ξ − ξ − ξρ2 2 4α0 γ 2 2

ˆ be the estimate of W ∗ . Since W ∗ is unknown, let W Choose the practical controller as ˆ T S(Z) − u = −cξ − W

1 1 1 ppT ξ − ξ − ξρ2 4α0 γ 2 2 2

(23)

Consider the following Lyapunov function candidate 1 1 1 ˜ T −1 ˜ V = V0 (s) + ξ T ξ + W Γ W+ 2 2 2



t

ξ 2 (τ )ρ2 (ξ(τ ))dτ t−τ

˜ =W ˆ − W ∗. where Γ = Γ T > 0 is an adaptation gain matrix and W

(24)

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325

In light of Assumption 2.1 and Assumption 2.2, referring to (12), (13) and (14), it is easily obtained that V˙ + α0 y T y − α0 γ 2 ω T ω 1 1 ξ T ppT ξ + ξ 2 ≤ − (α − α0 ) V0 (s) + 2 4α0 γ 2 1 2 2 T ˆ˙ ˜ T Γ −1 W + ξ (t)ρ (ξ(t)) + ξ (u + ϕ(Z)) + W 2   ˆ˙ − Γ S(Z)ξ ˜ T Γ −1 W = − (α − α0 ) V0 (s) − cξ T ξ + ξ T ε∗ + W Consider the following adaptation law   ˆ˙ = Γ S(Z)ξ − σ W ˆ W

(25)

(26)

where σ > 0 is a small constant. t 0) 0) 2 2 For ξ ∈ ΩZ0 , choosing c = (α−α + c1 + (α−α 2 2ξ 2 t−τmax ξ (τ )ρ (τ )dτ , where c1 > 0. Since [t − τ, t] ⊂ [t − τmax , t], we have the inequality  t  t ξ 2 (τ )ρ2 (τ )dτ ≤ ξ 2 (τ )ρ2 (τ )dτ t−τ

t−τmax

Because of the following inequalities

2 ˜

 σ W σ W ∗ 2 T T ∗ ˆ = −σ W ˜ ˜ +W ≤− ˜ W W + −σ W 2 2 −c1 ξ T ξ + ξ T ε∗ ≤

ε∗ 2 ε2 ≤ 4c1 4c1

Hence V˙ + α0 y T y − α0 γ 2 ω T ω  t α − α0 T α − α0 ξ ξ− ≤ − (α − α0 ) V0 (s) − ξ 2 (τ )ρ2 (τ )dτ 2 2 t−τ 2 ˜ σ W σ W ∗ 2 ε2 − + + 2 2 4c1

(27)

 ∗ 2 ε2 Let δ = σW2  + 4c . If we choose σ and Γ such that σ ≥ (α − α0 ) λmax Γ −1 , 1 then from (27) we have the following inequality V˙ + α0 y T y − α0 γ 2 ω T ω α − α0 T ≤ − (α − α0 ) V0 (s) − ξ ξ 2  t α − α0 α − α0 ˜ T −1 ˜ W Γ W +δ − ξ 2 (τ )ρ2 (τ )dτ − 2 2 t−τ = − (α − α0 ) V + δ

(28)

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As α0 y T y ≥ 0, it following from (28) that V˙ ≤ − (α − α0 ) V + α0 γ 2 ω T ω + δ

(29)

Referring to definition 1, definition 2 and proposition 1, it is easy to obtain from (29) that the closed-loop system is input-to-state practically stable with respect to ω. Then, it is easily deduced from (18) that the state variables of closed-loop system are ultimately bounded if the states and NN weights are initiated in some compact set ΩZ0 with bounded ω. From inequality (28), we obtain that V˙ + α0 y T y − α0 γ 2 ω T ω ≤ δ Integrating both sides of (30), it yields  t  t 1 T 2 y ydτ ≤ γ ω T ωdτ + (V (0) + δt) α0 0 0

(30)

(31)

Theorem 3.1: Consider the uncertain nonlinear time-delay system (1) satisfying Assumption 2.1-2.3 with ω ∈ Lq2e [0, ∞). For any γ > 0 and compact set ΩZ0 , given ε > 0, σ > 0, α0 with 0 < α0 < α, there exist l, η, μi , Γ, c such that the solution of the closed-loop system is uniformly ultimately bounded and inequality (31) holds. Remark 2: Theorem 3.1 has essentially described the solvability of the problem of ISpS disturbance attenuation for the uncertain nonlinear time delay system (1). As the L2 -gain γ can be arbitrarily small, the disturbance’s affect on the output can almost be removed. Due to the fact that the system is ISpS, the energy of output in [0, ∞) is unbounded despite the fact that δ can be arbitrarily small, but it’s power is bounded by δ.

4

Simulation

Consider the following form of system (1) x˙ 1 (t) = −x1 (t) + x1 (t) · x22 (t) + ξ 2 (t) x˙ 2 (t) = x1 (t) · x2 (t) · ξ(t) + ξ(t) ˙ = u(t) + x1 (t) + ξ 2 (t) + sin (ξ(t − τ )) + p (x(t), ξ(t)) ω ξ(t) y(t) = (x1 (t), x2 (t), ξ(t))T

(32)

where ω = sin(t) is the disturbance input and p(x, ξ) = cos (x1 (t) · x2 (t) · ξ(t)). For simulation purpose, we assume that τ = 5. RBF neural network is employed to approximate ϕ(Z) and the practical controller u is given as follows: ˆ T S(Z) − u = −cξ − W ˆ is updated by (26). and W

1 1 1 ppT ξ − ξ − ξ 4α0 γ 2 2 2

(33)

Disturbance Attenuating Controller Design for a Nonlinear Systems −4

8

x 10

γ =0.1 γ =0.05

7 6

Output y(1)

5 4 3 2 1 0

0

10

20

30

40

50

Time(Sec)

Fig. 1. The output y(1) with different γ

0.07 γ =0.1 γ =0.05

0.06

Output y(2)

0.05

0.04

0.03

0.02

0.01

0

0

10

20

30

40

50

Time(Sec)

Fig. 2. The output y(2) with different γ

0.04 γ =0.1 γ =0.05

0.03 0.02

Output y(3)

0.01 0 −0.01 −0.02 −0.03 −0.04

0

10

20

30

40

50

Time(Sec)

Fig. 3. The output y(3) with different γ

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ˆ T S(Z) contain 27 nodes (i.e., l = 27), with centers μk (k = Neural networks W 1, 2, · · · , l) evenly spaced in [−2, 2] × [−2, 2] × [−2, 2], and widths η = 2. The designed parameters of the above controller are c = 7, α0 = 1, σ = 0.2, Γ = ˆ = [0, · · · , 0]T . The initial states diag {2.0, 2.0, · · · , 2.0}. The initial weight W T T [x1 (0) , x2 (0) , ξ(0)] = [0, 0, 0] . Figs 1-3 shows the simulation results of applying the controller (33) to system (32) with γ = 0.1 and γ = 0.05. From Figs 1-3, we can see that the output y is attenuated by smaller gain γ.

5

Conclusion

The paper has considered the problem of semiglobally ISpS disturbance attenuation for a class of nonlinear systems with unknown time delay. The time delay term is cancelled by using appropriate Lyapunov-Krasovskii functional. Based on neural network technique and Lyapunov theory, an adaptive controller has been designed. Simulation result is presented to show the effectiveness of the approach.

References 1. Krsti´c, M., Kanellakopoulos, I., Kokotovi´c, P.: Nonlinear and Adaptive Control Design. Wiley, New York (1995) 2. Isidori, A.: Nnonliear Control Systems, 3rd edn. Springer, New York (1995) 3. Xie, L.H., Su, W.Z.: Robust H∞ Control for a Class of Cascaded Nonlinear Systems. IEEE Transactions on Automatic Control 42, 1465–1469 (1997) 4. Byrnes, C.I., Isidori, A.: Asymptotic Stabilization of Minimum Phase Nonlinear Systems. IEEE Transactions on Automatic Control 36, 1122–1137 (1991) 5. Jiang, Z.P.: Global Output Feedback Control with Disturbance Attenuation for Minimum-phase Nonlinear Systems. System and Control Letter 39, 155–164 (2000) 6. Su, W.Z., Xie, L.H., Souza, C.E.: Global Robust Disturbance Attenuation and Almost Disturbance Decoupling for Uncertain Cascaded Nonlinear Systems. Automatica 35, 697–707 (1999) 7. Lin, Z.: Almost Disturbance Decoupling with Global Asymptotic Stability for Nonlinear Systems with Disturbance-affected Unstable Zero Dynamics. System and Control Letter 33, 163–169 (1998) 8. Tang, Y.G., Sun, F.C., Sun, Z.Q.: Neural Network Control of Flexible-link Manipulators Using Sliding Mode. Neurocomputing 70, 288–295 (2006) 9. Ge, S.S., Wang, C.: Direct Adaptive NN Control of a Class of Nonlinear Systems. IEEE Transactions on Neural Networks 13, 214–221 (2002) 10. Zhou, G.P., Su, W.Z., Wang, C.: Semiglobally ISpS Disturbance Attenuation via Adaptive Neural Design for a Class of Nonlinear Systems. In: Proceedings of the 6th World Congress on Control and Automation, pp. 2964–2968 (2006) 11. Ge, S.S., Hong, F., Lee, T.H.: Adaptive Neural Network Control of Nonlinear Systems with Unknown Time Delays. IEEE Transactions on Automatic Control 48, 2004–2010 (2003) 12. Zeng, Z.G., Wang, J.: Analysis and Design of Associative Memories Based on Recurrent Neural Networks with Linear Saturation Activation Functions and Timevarying Delays. Neural Computation 19, 2149–2182 (2007)

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13. Cao, J., Ren, F.: Exponential Stability of Discrete-time Genetic Regulatory Networks with Delays. IEEE Transactions on Neural Networks 19, 520–523 (2008) 14. Jiang, Z.P., Praly, L.: Design of Robust Adaptive Controllers for Nonlinear Systems with Dynamic Uncertainties. Automatica 34, 825–840 (1998) 15. Jiang, Z.P., Teel, A., Praly, L.: Small-gain Theorem for ISS Systems and Applications. Math. Contr., Signals Syst. 7, 104–130 (1994) 16. Hornik, K., Stinchcome, M., White, H.: Multilayer Feed-forward Networks are Universal Approximators. Neural networks 2, 359–366 (1989)

Stability Criteria with Less Variables for Neural Networks with Time-Varying Delay Tao Li, Xiaoling Ye, and Yingchao Zhang Department of Information and Communication, Nanjing University of Information Science and Technology, 210044 Nanjing, Jiangsu China [email protected]

Abstract. In this paper, new delay-dependent stability criterion for neural networks is derived by using a simple integral inequality. The result is in terms of linear matrix inequalities and turn out to be equivalent to the existing result but include the least number of variables. This implies that some redundant variables in the existing stability criterion can be removed while maintaining the efficiency of the stability conditions. With the present stability condition, the computational burden is largely reduced. A numerical example is given to verify the effectiveness of the proposed criterion. Keywords: Delay-dependent, Asymptotic stability, Neural networks, Linear matrix inequality (LMI).

1

Introduction

In recent years, neural networks (NNs) have attracted much attention in research and have found successful applications in many areas such as pattern recognition, image processing, association, optimization problems [1,2]. One of the important research topics is the globally asymptotic stability of the neural network models. However, in the implementation of artificial NNs, time delays are unavoidable due to the finite switching speed of amplifiers. It has been shown that the existence of time delays in NNs may lead to oscillation, divergence or instability. Recently, the stability issue of NNs with time delays has been extensively studied (see [3,4,5,6,7,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]). Among virous stability methods, a notable one is the free-weighting matrix method in [15,16], which is very effective to tackle the delay-dependent stability problem for timedelay NNs since neither bounding techniques on some cross-product terms nor model transformations are involved. However, the free weighting matrix method often needs to introduce many slack variables in obtaining LMI conditions and thus leads to a significant increase in the computational demand. One natural question is how to simplify existing stability results using matrix variables as less as possible while maintaining the effectiveness of the stability conditions. In this paper, simplified delay-dependent stability criterion for neural networks is obtained by using a simple integral inequality. The result is shown to F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 330–337, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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331

be equivalent to those in [16] but with much less variables. This implies that our result is more efficient as the computational burden is largely reduced.

2

Problem Formulation

Consider the following delayed neural networks: x(t) ˙ = −Cx + Ag(x(t)) + Bg(x(t − d(t))) + u .

(1)

where x(·) = [x1 (·), x2 (·), · · ·, xn (·)]τ ∈ Rn is the neuron state vector, g(x(·)) = [g1 (x1 (·)), g2 (x2 (·)), · · ·, gn (xn (·))]τ ∈ Rn denotes the neuron activation function, u = [u1 , u2 , · · ·, un ]τ ∈ Rn is a constant input vector. C = diag{c1 , c2 , · · ·, cn } is a diagonal matrix with ci > 0, i = 1, 2, ..., n. A and B are the connection weight matrix and the delayed connection weight matrix, respectively. The time delay, d(t), is a time-varying continuous function that satisfies ˙ ≤μ. 0 < d(t) < h, d(t)

(2)

where h > 0 and μ are constants. In the following, we assume that each neuron activation function in (1), gi (·), i = 1, 2, ..., n, satisfies the following condition: 0≤

gi (x) − gi (y) ≤ ki , ∀ x, y ∈ R, x = y, i = 1, 2, ..., n . x−y

(3)

where ki , i = 1, 2, ..., n are some constants. Assume x∗ = [x∗1 , x∗2 , ..., x∗n ]τ is an equilibrium of system (1). From (1), the transformation z(·) = x(·) − x∗ transforms system (1) into the following system: z(t) ˙ = −Cz(t) + Af (z(t)) + Bf (z(t − d(t))) .

(4)

where z(·) = [z1 (·), z2 (·), · · ·, zn (·)]τ is the state vector of the transformed system, f (z(·)) = [f1 (z1 (·)), f2 (z2 (·)), · · ·, fn (zn (·))]τ and fi (zi (·)) = gi (zi (·) + x∗i ) − gi (x∗i ), i = 1, 2, ..., n. Note that the functions f (z(t)) satisfy the following conditions:

0≤

fi (zi ) ≤ ki , fi (0) = 0, ∀zi = 0 . zi

which is equivalent to fi (zi )(fi (zi ) − ki zi ) ≤ 0, fi (0) = 0 . The purpose of this paper is to establish a simplified LMI condition with less slack variables such that NNs described by (4) is globally asymptotically stable while obtaining the allowable delay bound h as large as possible.

332

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Main Results

Theorem 1. For given scalars h > 0 and μ, the origin of system (4) with (2) is asymptotically stable if there exist positive matrices P, Qi (i = 1, 2, 3), Z, positive diagonal matrices Λ = diag{λ1 , λ2 , ...λn }, T1 = diag{t11 , t12 , ...t1n } and T2 = diag{t21 , t22 , ...t2n } such that the following LMI holds: ⎤ ⎡ 1 Γ11 − h1 Z Γ13 P B − hC τ ZB 0 hZ 1 ⎥ ⎢ ∗ −(1 − μ)Q1 − h2 Z 0 KT2 hZ ⎥ ⎢ τ ⎥ < 0 , (5) ⎢ ∗ ∗ Γ ΛB + hA ZB 0 Γ =⎢ 33 ⎥ ⎦ ⎣ ∗ ∗ ∗ Γ44 0 ∗ ∗ ∗ ∗ −Q3 − h1 Z where Γ11 = −P C − C τ P + Q1 + Q3 + hC τ ZC , Γ13 = P A − C τ Λ + KT1 − hC τ ZA , Γ33 = ΛA + Aτ Λ + Q2 − 2T1 + hAτ ZA , Γ44 = −(1 − μ)Q2 + hB τ ZB − 2T2 , K = diag{k1 , k2 , ..., kn } . and ∗ denotes the symmetric term in a symmetric matrix. Proof: Introduce the following Lyapunov-Krasovskii functional:  0  t  t τ τ V (z(t)) = z (t)P z(t) + z˙ (s)Z z(s)dsdθ ˙ + f τ (z(s))Q2 f (z(s))ds +2

n i=1

 λi

−h

t+θ



zj

t−d(t) t

fj (s)ds +

t

z τ (s)Q3 z(s)ds .

z (s)Q1 z(s)ds + t−d(t)

0



τ

t−h

where P > 0, Z > 0, Qi > 0 (i = 1, 2, 3). The time derivative of V (z(t)) along the trajectories of system (4) gives  t τ τ τ ˙ V (z(t)) ≤ 2z (t)P z(t) ˙ + 2f (z(t))Λz(t) ˙ + hz˙ (t)Z z(t) ˙ − z˙ τ (s)Z z(s)ds ˙ t−h

+z τ (t)(Q1 + Q3 )z(t) − (1 − μ)z τ (t − d(t))Q1 z(t − d(t)) −z τ (t − h)Q3 z(t − h) + f τ (z(t))Q2 f (z(t)) −(1 − μ)f τ (z(t − d(t)))Q2 f (z(t − d(t))) . From integral inequality [8], we have  t−d(t)   t z˙ τ (s)Z z(s)ds ˙ =− z˙ τ (s)Z z(s)ds ˙ − − t−h

t−h

1 ≤− h



z(s)ds ˙ t−h

z˙ τ (s)Z z(s)ds ˙ t−d(t)

τ

t−d(t)

t





t−d(t)

Z

z(s)ds ˙ t−h

Stability Criteria with Less Variables for NNs with Time-Varying Delay

1 − h

=



τ

t

z(s)ds ˙ t−d(t)





t

Z

333

z(s)ds ˙ t−d(t)

  

−1Z 1Z z(t − d(t)) h h z τ (t − d(t)) z τ (t − h) 1 Z −1Z z(t − h)   1h 1 h  

τ z(t) −hZ hZ + z (t) z τ (t − d(t)) (6) 1 1 z(t − d(t)) hZ −hZ

On the other hand, It is clear that fi (zi (t))(fi (zi ) − ki zi (t)) ≤ 0, i = 1, 2...n ,

(7)

fi (zi (t − d(t)))(fi (zi (t − d(t))) − ki zi (t − d(t))) ≤ 0, i = 1, 2...n .

(8)

Thus, for any Tj = diag{t1j , t2j , ..., tnj } ≥ 0, j = 1, 2, we have V˙ (z(t)) ≤ η τ (t)Θη(t) − 2

n

ti1 fi (zi (t))(fi (zi ) − ki zi (t))

i=1

−2

n

ti2 fi (zi (t − d(t)))(fi (zi (t − d(t))) − ki zi (t − d(t))) ,

(9)

i=1

where η(t) = [z τ (t) z τ (t − d(t)) f τ (z(t)) f τ (z − d(t)) z τ (t − h)]τ , ⎡ ⎤ 1 Γ13 P B − hC τ ZB 0 Γ11 − h1 Z hZ 1 ⎢ ⎥ ∗ Γ22 − h2 Z 0 0 hZ ⎢ ⎥ τ ⎥. ∗ ∗ Γ ΛB + hA ZB 0 Θ=⎢ 33 ⎢ ⎥ ⎣ ⎦ 0 ∗ ∗ ∗ Γ44 ∗ ∗ ∗ 0 −Q3 − h1 Z Applying the Schur complement to (5) gives V˙ (t) < 0 by (9). Hence, system (4) is asymptotically stable. Recently, a less conservative delay-dependent stability condition for delayed NNs was proposed in [16] by introducing some free-weighting matrices, which is restated as follows. Lemma 1. For given scalars h > 0 and μ, the origin of system (4) with (2) is asymptotically stable, if there exist positive matrices P, Qi (i = 1, 2, 3), Z, positive diagonal matrices Λ = diag{λ1 , λ2 , ...λn }, T1 = diag{t11 , t12 , ...t1n }, T2 = diag{t21 , t22 , ...t2n }, N = [N1τ N2τ N3τ N4τ N5τ ]τ , M = [M1τ M2τ M3τ M4τ M5τ ]τ , such that the following LMI hold: ⎡ ⎤ N5τ − M1 hN1 hM1 −hC τ Z Φ11 Φ12 Φ13 P B + N4τ ⎢ ∗ Φ22 Φ23 ⎥ Φ24 −N5τ + M5τ − M2 hN2 hM2 0 ⎢ ⎥ τ ⎢ ∗ ∗ Φ33 ⎥ ΛB −M hN hM hA Z 3 3 3 ⎢ ⎥ τ τ ⎢ ∗ ∗ ∗ ⎥ Φ −M hN hM hB Z 44 4 4 4 ⎢ ⎥ < 0 ,(10) Φ=⎢ τ ⎥ − M − M hN hM 0 ∗ ∗ ∗ ∗ −Q 3 5 5 5 5 ⎢ ⎥ ⎢ ∗ ∗ ∗ ⎥ ∗ ∗ −hZ 0 0 ⎢ ⎥ ⎣ ∗ ∗ ∗ ⎦ ∗ ∗ ∗ −hZ 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ −hZ

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where Φ11 = −P C − C τ P + Q1 + Q3 + N1 + N1τ , Φ12 = N2τ − N1 + M1 , Φ13 = P A − C τ Λ + KT1 + N3τ , Φ22 = −(1 − μ)Q1 − N2 − N2τ + M2 + M2τ , Φ23 = −N3τ + M3τ , Φ24 = KT2 − N4τ + M4τ , Φ33 = ΛA + Aτ Λ + Q2 − 2T1 , Φ44 = −(1 − μ)Q2 − 2T2 . Although Theorem 1 and Lemma 1 are obtained via different methods, they turned out to be equivalent. To show this, we give the following theorem. Theorem 2. Inequality Γ < 0 in Theorem 1 is feasible if and only if Φ < 0 in Lemma 1 is feasible. Proof: Note that Φ in Lemma 1 can be expressed as Φ = Γ1 + XW + W τ X τ < 0 , where⎡

⎤ Γ11 + N1 + N1τ N2τ − N1 + M1 Γ13 Γ14 −M1 −N1 −M1 ⎢ ∗ Φ22 0 L2 T2 −M2 −N1 −M2 ⎥ ⎢ ⎥ ⎢ ∗ ∗ Γ 0 0 0 ⎥ 33 Γ34 ⎢ ⎥ ∗ ∗ ∗ Γ44 0 0 0 ⎥ Γ1 = ⎢ ⎢ ⎥, ⎢ ⎥ ∗ ∗ ∗ ∗ −Q 0 0 3 ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ ∗ − h1 Z 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ − h1 Z     τ 0 0 N3τ N4τ N5τ 0 0 I −I 0 0 0 −I 0 X= , W = , 0 I 0 0 −I 0 −I 0 0 M3τ M4τ M5τ 0 0 and Γ11 , Γ13 , Γ14 , Φ22 Γ33 , Γ34 , Γ44 are defined in Theorem 1 and Lemma 1. For t  t−d(t) τ τ τ ˙ ( t−h z(s)ds) ˙ ] = 0, it is seen that W ξ(t) = 0. ξ(t) = [η τ (t) ( t−d(t) z(s)ds) According to Finsler’s Lemma, Φ < 0 holds if and only if the following inequality is true ξ τ (t)Γ1 ξ(t) < 0 , Then it yields that Γ1 < 0 is equivalent to Γ2 = ΠΓ1 Π τ < 0, where ⎡ ⎡ ⎤ −N1 − h1 Z −M1 I0000 I 0 ⎢ 0 I 0 0 0 −I I ⎥ ⎢ −N2 + 1 Z −M2 − 1 Z h h ⎢ ⎢ ⎥ ⎢0 0 I 0 0 0 0 ⎥ ⎢Γ 0 0 ⎢ ⎢ ⎥ ⎢ ⎥ 0 0 Π=⎢ ⎢ 0 0 0 I 0 0 0 ⎥ , Γ2 = ⎢ ⎢ 0 0 0 0 I 0 −I ⎥ ⎢ 0 0 ⎢ ⎢ ⎥ ⎣0 0 0 0 0 I 0 ⎦ ⎣ − h1 Z 0 0 00000 0 I 0 − h1 Z

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

Stability Criteria with Less Variables for NNs with Time-Varying Delay

335

It is obvious that Γ < 0 holds if Γ2 < 0 holds. Conversely, if Γ < 0 holds, then Γ2 < 0 is feasible by taking N1 = − h1 Z, M1 = 0, N2 = h1 Z, M2 = − h1 Z. Thus, Φ is also feasible. This completes the proof. Remark 1. From the proof of Theorem 2, it is clear that Theorem 1 is equivalent to Theorem 1 in [16]. This means that the free weighting matrices Ni , Mi (i = 1, ..., 5) in [16] can be removed while maintaining the effectiveness of the stability condition. Theorem 1 provides stability criterion for NNs with d(t) satisfying 0 < d(t) < ˙ h and d(t) ≤ μ. In many cases, μ is unknown. For this circumstance, a rateindependent criterion for the delay satisfying 0 < d(t) < h is derived as follows by choosing Q1 = Q2 = 0 in Theorem 1. Corollary 1. For given scalar h > 0, the origin of system (4) with delay d(t) satisfy 0 < d(t) < h is asymptotically stable if there exist positive matrices P, Q3 , Z, positive diagonal matrices Λ = diag{λ1 , λ2 , ...λn }, T1 = diag{t11 , t12 , ...t1n } and T2 = diag{t21 , t22 , ...t2n } such that the following LMI holds: ⎡ ⎤ 0 Υ11 h1 Z P A − C τ Λ + KT1 − hC τ ZA P B − hC τ ZB 1 ⎢ ∗ −2Z ⎥ 0 KT2 h hZ ⎢ ⎥ τ ⎢ ∗ ⎥<0, ΛB + hA ZB 0 ∗ Υ33 ⎢ ⎥ τ ⎣ ∗ ⎦ ∗ ∗ hB ZB − 2T2 0 1 ∗ ∗ ∗ ∗ −Q3 − h Z where Υ11 = −P C − C τ P + Q3 + hC τ ZC − Υ33 = ΛA + Aτ Λ + hAτ ZA − 2T1 ,

1 Z, h

and Γ13 , Γ14 and Γ55 are defined in Theorem 1. Remark 2. It is easy to show that Corollary 1 is equivalent to Corollary 1 in [16] by following a similar line to the proof of Theorem 2.

4

Numerical Examples

This section provides a numerical example to illustrate the effectiveness of our method and do comparisons with existing results. Example 1. Consider the delayed NN (4) with [16] ⎡ ⎤ ⎡ ⎤ 1.2769 0 0 0 −0.0373 0.4852 −0.3351 0.2336 ⎢ 0 0.6231 0 ⎢ ⎥ 0 ⎥ ⎥ , A = ⎢ −1.6033 0.5988 −0.3224 1.2352 ⎥ , C=⎢ ⎣ 0 ⎣ 0.3394 −0.0860 −0.3824 −0.5785 ⎦ 0 0.9230 0 ⎦ 0 0 0 0.4480 −0.1311 0.3253 −0.9534 −0.5015 ⎤ ⎡ ⎤ ⎡ 0.1137 0 0 0 0.8674 −1.2405 −0.5325 0.0220 ⎢ 0 0.1279 0 ⎢ 0.0474 −0.9164 0.0360 0.9816 ⎥ 0 ⎥ ⎥ ⎢ ⎥ . B=⎢ ⎣ 1.8495 2.6117 −0.3788 0.8428 ⎦ , K = ⎣ 0 0 0.7994 0 ⎦ −2.0413 0.5179 1.1734 −0.2775 0 0 0 0.2368

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Table 1 provides some comparisons of the maximum allowable delay bounds and the numbers of the variables involved in the results of this paper and [14,16]. From Table I, it is seen that our results contain the least numbers of computed variables while maintaining the effectiveness of the stability conditions. Table 1. Comparisons of maximum allowable delay bound h for Example 1 Method

μ = 0.5 μ = 0.9 Numbers of variables unknown μ Numbers of variables

[14]

2.1502 1.3164

248

1.2598

228

[16]

2.2245 1.5847

222

1.5444

202

Theorem 1 2.2245 1.5847

62

1.5444

42

5

Conclusions

This paper has established simplified delay-dependent stability criteria for neural networks with time-varying delay. The technique is based on a simple integral inequality. The results are shown to be equivalent to the latest results but include much less variables in the LMI conditions. Thus, the present method could largely reduce the computational burdens in solving LMIs. Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the present method. The results could be easily used for analysis of NNs with time-varying delays and parameter uncertainties.

Acknowledgement This work was supported by the National Science Foundation of China under Grant No. 60774013.

References 1. Cichocki, A., Unbehauen, R.: Neural Networks for Optimization and Signal Processing. Wiley, Chichester (1993) 2. Liu, G.P.: Nonlinear Identification and Control: A Neural Network Approach. Springer, New York (2001) 3. Zhang, B.Y., Xu, S., Li, Y.M., Chu, Y.M.: On Global Exponential Stability of High-order Neural Networks with Time-varying Delays. Phys. Lett. A. 366, 69–78 (2007) 4. Liao, X.F., Chen, G., Sanchez, E.N.: Delay-dependent Exponential Stability Analysis of Delayed Neural Networks: An LMI approach. Neural Netw. 15, 855–866 (2002) 5. Zeng, Z.G., Wang, J.: Improved Conditions for Global Exponential Stability of Recurrent Neural Networks with Time-varying Delays. IEEE Trans. Neural Netw. 17, 623–635 (2006) 6. Chen, T., Rong, L.: Delay-independent Stability Analysis of Cohen- Grossberg Neural Networks. Phys. Lett. A 317, 436–449 (2002)

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7. Zhang, Q., Wei, X., Xu, J.: Stability Analysis for Cellular Neural Networks with Variable Delays. Chaos, Solitons Fractals 28, 331–336 (2006) 8. Gu, K.: An Integral Inequality in the Stability Problem of Time-delay Systems. In: Proceedings of 39th IEEE conference on decision and control, pp. 2805–2810 (2000) 9. Singh, V.: Global Robust Stability of Delayed Neural Networks: An LMI approach. IEEE Trans. Circuits Syst. II, Exp. Briefs 52, 33–36 (2005) 10. Wang, Z., Shu, H., Liu, Y., Ho, D.W.C., Liu, X.: Robust Stability Analysis of Generalized Neural Networks with Discrete and Distributed Time Delays. Chaos, Solitons and Fractals 30, 886–896 (2006) 11. Wang, Z., Liu, Y., Li, M., Liu, X.: Stability Analysis for Stochastic CohenGrossberg Neural Networks with Mixed Time Delays. IEEE Trans. Neural Netw. 17, 814–820 (2006) 12. Xu, S., Lam, J., Ho, D.W.C., Zou, Y.: Novel Global Asymptotic Stability Criteria for Delayed Cellular Neural Networks. IEEE Trans. Circuits Syst. II, Exp. Briefs 52, 349–353 (2005) 13. Xu, S., Lam, J.: A new LMI Condition for Delay-dependent Asymptotic Stability of Delayed Hopfield Neural Networks. IEEE Trans. Circuits Syst. II, Exp. Briefs 53, 230–234 (2006) 14. Hua, C.C., Long, C.N., Guan, X.P.: New Results on Stability Analysis of Neural Networks with Time-varying Delays. Phys. Lett. A 352, 335–340 (2006) 15. He, Y., Wang, Q.G., Wu, M.: LMI-based Stability Criteria for Neural Networks with Multiple Time-varying Delays. Physica D 212, 126–136 (2005) 16. He, Y., Liu, G.P., Rees, D.: New Delay-dependent Stability Criteria for Neural Networks with Time-varying Delay. IEEE Trans. Neural Netw. 18, 310–314 (2007) 17. Park, J.H.: Robust Stability of Bidirectional Associative Memory Neural Networks with Time Delays. Phys. Lett. A 359, 494–499 (2006) 18. Yang, H., Chu, T., Zhang, C.: Exponential Stability of Neural Networks with Variable Delays via LMI Approach. Chaos, Solitons and Fractals 30, 133–139 (2006) 19. Cho, H.J., Park, J.H.: Novel Delay-dependent Robust Stability Criterion of Delayed Cellular Neural Networks. Chaos, Solitons and Fractals 32, 1194–1200 (2007) 20. Liu, Y., Wang, Z., Liu, X.: Global Exponential Stability of Generalized Recurrent Neural Networks with Discrete and Distributed Delays. Neural Networks 19, 667– 675 (2006) 21. Li, T., Guo, L., Sun, C.Y.: Further Result on Asymptotic Stability Criterion of Neural Networks with Time-varying Delays. Neurocomputing 71, 439–447 (2007) 22. Li, T., Guo, L., Sun, C.Y.: Robust Stability for Neural Networks with Time-varying Delays and Linear Fractional Uncertainties. Neurocomputing 71, 421–427 (2007) 23. Zhang, H.G., Wang, Z.S.: Global Asymptotic Stability of Delayed Cellular Neural Networks. IEEE Trans. Neural Netw. 18, 947–950 (2007) 24. Cao, J.D., Xiao, M.: Stability and Hopf Bifurcation in a Simplified BAM Neural Network with Two Time Delays. IEEE Trans. Neural Netw. 18, 416–430 (2007)

Robust Stability of Uncertain Neural Networks with Time-Varying Delays Wei Feng1,2, , Haixia Wu2 , and Wei Zhang2 1

2

College of Automation, Chongqing University, Chongqing 40044, China Department of Computer and Modern Education Technology, Chongqing Education College, Chongqing 400067, China [email protected]

Abstract. This paper is concerned the robust stability analysis problem for neural networks with time-varying delay and time-varying parametric uncertainties. By utilizing a Lyapunov-Krasovskii functional, we show that the addressed neural networks are robustly, asymptotically stable if a convex optimization problem is feasible. A stability criterion is derived and formulated by means of the feasibility of a linear matrix inequality (LMI), which can be effectively solved by some standard numerical packages. Two numerical examples are given to demonstrate the usefulness of the proposed robust stability criterion. Keywords: Robust stability, uncertain neural networks, linear matrix inequality, time-varying delays.

1

Introduction

During the last decade, there has been increasing interest in the potential applications of neural networks in many areas, including pattern recognition, image processing, intelligent controls and numerous other fields. Some of these applications require that the equilibrium points of the designed network be stable. Therefore, it is important to study the stability of neural networks. Actually, time-delay occurs in many real-world control systems, and it often results in instability and poor performance. Besides, it is difficult to obtain the exact value for the delay. Therefore, the problem of stability and stabilization of time-delay systems has been received considerable attention over the decades. At the same time, in biological and artificial neural networks, time delays often arise in the processing of information storage and transmission. In recent years, the stability of delayed neural networks (DNNs) has been extensively investigated by many researchers [1,2,3,4,5,6]. Recently, LMI-based techniques have been successfully used to tackle various stability problems for neural networks with time delays [7,8,9,10]. The advantage of the LMI-based approach is that resulting stability criterion can be solved 

This work described in this paper is supported by Scientific Research Fund of Chongqing Municipal Education Commission Grant No. kj081501.

F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 338–346, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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efficiently via existing numerical convex optimization algorithms such as the interior-point algorithms [11]. In practice, the connection weights of the neurons depend on certain resistance and capacitance values which include uncertainties. It is important and interesting to investigate the robust stability of neural networks with parametric uncertainties. In [12,13,14], the authors studied the robust stability of interval delayed neural networks. In [10], the author studied the robust stability of a delayed cellular neural networks with parametric uncertainties by using an LMI approach, but the uncertainties in this paper are time-invariant. Practically, uncertainties are usually time varying. With this motivation, we consider the robust stability of a general delayed neural network with time-varying parametric uncertainties and time-varying delay. Based on the Lyapunov-Krasovskii functional method, a stability criterion is derived in terms of LMI.

2

System Description

Consider the following uncertain DNNs model: u(t) ˙ = −(C + ΔC)u(t) + (A + ΔA)g(u(t)) +(B + ΔB)g(u(t − h(t))) + b,

(1)

where u(t) = [u1 (t), . . . , un (t)]T ∈ Rn is the neuron state vector, g(u(t)) = [g1 (u1 (t)), g2 (u( t)), . . . , gn (un (t))]T denotes the bounded neuron activation function with gi (0) = 0, b = [b1 , b2 , . . . , bn ]T is a constant input vector, the time ¯ with delays h(t) is bounded nonnegative function satisfying 0 ≤ h(t) ≤ h ˙h(t) < hd < 1, C = diag(c1 , c2 , . . . , cn ) is a positive diagonal matrix with ci > 0, i = 1, 2, . . . , n, A = (aij )n×n , B = (bij )n×n are the interconnection matrices representing the weight coefficients of the neurons, and ΔC, ΔA, ΔB are the uncertainties of system matrices of the form [ΔC

ΔA ΔB] = HF (t)[E

E0

E1 ],

(2)

where the time-varying nonlinear function F (t) satisfy F T (t)F (t) ≤ I,

∀t ∈ R.

(3)

In this paper,it is assumed that the activation function g(u) is bounded and globally Lipschitz; that is 0≤

gi (ξ1 ) − gi (ξ2 ) ≤ ki , ξ1 − ξ2

i = 1, 2, . . . , n.

(4)

Then, by using well-known Brouwer’s fixed point theorem [12], one can easily prove that there exists an equilibrium point for Eq.(1). Assume that u∗ = (u∗1 , u∗2 , . . . , u∗n ) is an equilibrium point of the system (1), then we will shift the equilibrium point u∗ to the origin, 0. The transformation x(·) = u(·) − u∗ puts system (1) into the following form: x(t) ˙ = −(C + ΔC)x(t) + (A + ΔA)f (x(t)) + (B + ΔB)f (x(t − h(t))), (5)

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where x(t) is the state vector of the transformed system, fj (xj ((t)) = gj (xj (t) + u∗j ) − gj (u∗j ) with fj (0) = 0 for j = 1, 2, . . . , n. It is noted that each activation function fi (·) satisfies the following sector condition: 0≤

fi (ξ1 ) − fi (ξ2 ) ≤ ki , ξ1 − ξ2

i = 1, 2, . . . , n.

(6)

Definition 1. The parametric uncertainties ΔC, ΔA, ΔB are said to be admissible if both (2) and (3) hold. Definition 2. The equilibrium point 0 is said to be globally robustly stable if for all admissible uncertainties ΔC, ΔA, ΔB, it is locally stable in the sense of Lyapunov and global attractive, where global attraction means that every trajectory tends to the equilibrium point as t → ∞. Lemma 1. Given any real matrices Σ1 , Σ2 , Σ3 with appropriate dimensions and a scalar ε > 0, such as that 0 < Σ3 = Σ3T , then the following inequality holds: Σ1T Σ2 + Σ2T Σ1 ≤ εΣ1T Σ3 Σ1 + ε−1 Σ2T Σ3−1 Σ2 .

(7)

Fact 1. [Schur complement] Given constant symmetric matrices Σ1 , Σ2 , Σ3 , where Σ1 = Σ1T , and 0 < Σ2 = Σ2T , then Σ1 + Σ3T Σ2−1 Σ3 < 0 if and only if 

3

 Σ1 Σ3T < 0, Σ3 −Σ2

 or

 Σ1 Σ3 < 0. Σ3T −Σ2

(8)

Main Result

Theorem 1. The equilibrium point of system(5) is globally robustly stable if there exist symmetrical and positive definite matrices P, Q1 , Q2 , Q3 , a positive diagonal matrix Λ = diag {λ1 , λ2 , . . . λn } and two positive scalars 1 , 2 , satisfying the following LMI: ⎡

Ξ11 ⎢

⎢ ⎢

⎢ Ξ =⎢ ⎢

⎢

⎢ ⎣



0 Ξ22







Ξ13 0 Ξ33





Ξ14 0 Ξ34 Ξ44





0 0 0 0 Ξ55



where Ξ11 = −2P C + Q1 + Q2 + (1 + 2 )E T E Ξ13 = P A − C T ΛT + (1 − 2 )E T E0 Ξ14 = P B + (1 − 2 )E T E1

Ξ16 0 0 0 0 Ξ66

⎤ 0 0 ⎥ ⎥ Ξ37 ⎥ ⎥ 0 ⎥ ⎥<0 0 ⎥ ⎥ Λ ⎦ Ξ77

(9)

Robust Stability of Uncertain Neural Networks with Time-Varying Delays

341

Ξ16 = P H Ξ22 = −(1 − hd )Q1 Ξ33 = 2ΛA + Q2 + (1 + 2 )E0T E0 Ξ34 = ΛB + (1 + 2 )E0T E1 Ξ37 = ΛH Ξ44 = −(1 − hd )Q2 + (1 + 2 )E1T E1 Ξ55 = −Q3 Ξ66 = −1 I Ξ77 = −2 I and denotes the symmetric terms in a symmetric matrix. Proof. To obtain the result, we will employ the following positive definite Lyapunov functional as follows: V (xt ) = V1 (xt ) + V2 (xt )

(10)

where V1 (xt ) = xT (t)P x(t) + 2 

n



xi

λi

i=1

fi (s)ds 0

t

[xT (s)Q1 x(s) + f T (x(s))Q2 f (x(s))]ds

V2 (xt ) = t−h(t)  t

+ ¯ t−h

xT (s)Q3 x(s)ds

The time derivative of Lyapunov-Krasovskii functional V (x(t)) along the trajectory of (5) is derived as following: V˙ (x(t)) = V˙ 1 (x(t)) + V˙ 2 (x(t)) ¯ ¯ (x(t)) V˙ 1 (x(t)) = 2xT (t)P [−Cx(t) + Af ¯ (x(t − h(t)))] +Bf ¯ ¯ (x(t)) +2f T (x(t))Λ[−Cx(t) + Af ¯ (x(t − h(t)))] +Bf ¯ ¯ (x(t)) = −2xT P Cx(t) + 2xT (t)P Af T ¯ +2x (t)P Bf (x(t − h(t))) ¯ ¯ (x(t)) −2f T (x(t))ΛCx(t) + 2f T x(t)ΛAf ¯ (x(t − h(t))) +2f T (x(t))ΛBf V˙ 2 (x(t)) = xT (t)(Q1 + Q3 )x(t) + f T x(t)Q2 f (x(t)) T ˙ (t − h(t))Q1 x(t − h(t)) −(1 − h(t))x T ˙ −(1 − h(t))f (t − h(t))Q2 f (x(t − h(t)) ¯ h)Q3 x(t − h) −xT (t − ¯ ≤ xT (t)(Q1 + Q3 )x(t) + f T x(t)Q2 f (x(t)) ¯ −xT (t − ¯ h)Q3 x(t − h)

(11)

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−(1 − hd )f T (x(t − h(t)))Q2 f (x(t − h(t))) −(1 − hd )xT (t − h(t))Q1 x(t − h(t)) ¯ = B + ΔB, C¯ = C + ΔC. where A¯ = A + ΔA, B T T ¯ T , we have Let ξ(t) = [x (t), x (t − h(t)), f T (x(t)), f T (x(t − h(t))), xT (t − h)] V˙ (x(t)) = ξ T (t)Ξ ∗ ξ(t) with

⎡

∗ Ξ11 ⎢

⎢ Ξ∗ = ⎢ ⎢

⎣



where ∗ = −2P C¯ + Q1 + Q2 Ξ11 ∗ Ξ13 = P A¯ − C¯ T ΛT ∗ ¯ Ξ14 = PB ∗ = −(1 − hd )Q1 Ξ22 ∗ Ξ33 = 2ΛA¯ + Q2 ∗ ¯ Ξ34 = ΛB ∗ Ξ44 = −(1 − hd )Q2 ∗ Ξ55 = −Q3 And let ⎡

Π11 ⎢

⎢ Π=⎢ ⎢

⎣



where Π11 Π13 Π14 Π33 Π34 Π44 Π55 and

0 ∗ Ξ22





0 Π22





(12)

∗ Ξ13 0 ∗ Ξ33



∗ Ξ14 0 ∗ Ξ34 ∗ Ξ44

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥<0 0 ⎦ ∗ Ξ55

(13)

Π13 0 Π33



Π14 0 Π34 Π44

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥<0 0 ⎦ Π55

(14)

= −2P C + Q1 + Q2 = P A − C T ΛT = P B Π22 = −(1 − hd )Q1 = 2ΛA + Q2 = ΛB = −(1 − hd )Q2 = −Q3 consider(2),then Ξ ∗ = Π + 2Ω1 F (t)Ω2 + 2Ω3 F (t)Ω4 < 0

where Ω1 = [P H

0

0 0

0]T

(15)

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343

Ω2 = [E 0 E0 E1 0] Ω3 = [0 0 ΛH 0 0]T Ω4 = [−E 0 E0 E1 0] By Lemma 1 and Eq.(15), we have −1 T T T T Ξ ∗ ≤ Π + −1 1 Ω1 Ω1 + 1 Ω2 Ω2 + 2 Ω3 Ω3 + 2 Ω4 Ω4 ≡ Π + Ω < 0,

with

⎡

Ω11 ⎢

⎢ Ω=⎢ ⎢

⎣



0 Ω13 0 0

Ω33





Ω14 0 Ω34 Ω44

⎤ 0 0⎥ ⎥ 0⎥ ⎥ < 0, 0⎦ 0

(16)

(17)

where T T Ω11 = ε−1 1 P HH P + (ε1 + ε2 )E E T Ω13 = (ε1 − ε2 )E E0 Ω14 = (ε1 − ε2 )E T E1 T Ω33 = (ε1 + ε2 )E0T E0 + ε−1 1 ΛHH Λ T Ω34 = (ε1 + ε2 )E0 E1 Ω44 = (ε1 + ε2 )E1T E1 So far ⎡

∗∗ Ξ11 ⎢

⎢ Ξ∗ = ⎢ ⎢

⎣



where ∗∗ Ξ11 ∗∗ Ξ13 ∗∗ Ξ14 ∗∗ Ξ22 ∗∗ Ξ33 ∗∗ Ξ34 ∗∗ Ξ44 ∗∗ Ξ55

0 ∗∗ Ξ22





∗∗ Ξ13 0 ∗∗ Ξ33



∗∗ Ξ14 0 ∗∗ Ξ34 ∗∗ Ξ44

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥<0 0 ⎦ ∗∗ Ξ55

(18)

T T = −2P C + Q1 + Q2 + ε−1 1 P HH P + (ε1 + ε2 )E E T T T = P A − C Λ + (ε1 − ε2 )E E0 = P B + (ε1 − ε2 )E T E1 = −(1 − hd )Q1 T = 2ΛA + Q2 + (ε1 + ε2 )E0T E0 + ε−1 1 ΛHH Λ = ΛB + (ε1 + ε2 )E0T E1 = −(1 − hd )Q2 + (ε1 + ε2 )E1T E1 = −Q3

If the matrix Ξ ∗ is negative definite, V˙ < 0. By Fact 1(Schur complement),the inequality Ξ ∗ < 0 is equivalent to the LMI(9). So, this completes the proof of Theorem 1.

4

Numerical Examples

In this section, two examples are given to illustrate the effectiveness of our theoretical results.

W. Feng, H. Wu, and W. Zhang

u2(t)

u1(t)

344

1.5 1 0.5 0

0

1

2

3

4

5 t

6

7

8

9

10

0

1

2

3

4

5 t

6

7

8

9

10

1 0 −1

Fig. 1. The time response curve of system(1) with time-varying delays

Example 1. Consider a two-neuron neural network (1), where       10 −1 0.5 0.2 0.1 C= ,A= ,B= , 01 0.5 −1.5 0 0.2       0.1 0.1 sin(t) 0 0.1 0 H= , F (t) = ,E= , 0 0.2 0 cos(t) 0 0.1     0.3 0 0.1 0 , E1 = , E0 = 0 0.2 0 0.2 u1 (θ) = 0.2, u2 (θ) = −0.5, hd = 0.6, b1 = b2 = 1, gi (x) = tanh(x), h(t) = 1.2 arctan(t). Now,the LMI (9) solutions 1 for this example are  of Theorem   451.1427 −11.9378 306.0134 −8.8818 P = , Q1 = , −11.9378 466.8521 −8.8818 300.4937     166.7071 −88.0047 287.7446 0 , Q3 = , Q2 = −88.0047 252.4010 0 287.7446   190.4515 0 Λ= , 1 = 293.9841, 2 = 291.0204. 0 190.4515 Hence, by Theorem 1, the unique equilibrium u∗ = [0.7880 0.6028]T of the system (1) in this case is globally robustly stable. It, s convergence dynamics is shown in Fig. 1. Example 2. Consider the neural network in example 1 without uncertainties, where       10 −1 0.5 0.2 0.1 C= ,A= ,B= , 01 0.5 −1.5 0 0.2 H = F (t) = E = E0 = E1 = 0, hd = 0.6, b1 = b2 = 1, u1 (θ) = 0.2, u2 (θ) = −0.5, gi (x) = tanh(x), h(t) = 1.2 arctan(t). It should be pointed out that, if we let β = 1 in Theorem 8 of [15], the conditions are not feasible when employing the LMI Toolbox in MATLAB, but using Theorem 1 in this paper, we can find that system (1) without uncertainties is also globally robustly stable, and the solutions of LMI (9) as follows:

Robust Stability of Uncertain Neural Networks with Time-Varying Delays

345

u1(t)

1.5 1 0.5

u2(t)

0

0

2

4

6

8

10 t

12

14

16

18

20

0

2

4

6

8

10 t

12

14

16

18

20

1 0 −1

Fig. 2. The time response curve of system(1) with time-varying delays without uncertainties



   484.5290 −9.1897 331.0123 −1.5010 , Q1 = , −9.1897 499.5615 −1.5010 330.2821     180.5504 −92.7723 304.1502 0 Q2 = , Q3 = , −92.7723 279.8001 0 304.1502   209.6066 0 Λ= , 1 = 304.1502, 2 = 304.1502. 0 209.6066 So, by Theorem 1, the unique equilibrium u∗ = [0.7989 0.6175]T of the system (1) without uncertainties in this case is globally robustly stable. It, s convergence dynamics is shown in Fig. 2. Therefore, our method is less conservative in some degree than that in [15]. P =

5

Conclusions

A robust stability criterion for general delayed neural networks with time-varying parametric uncertainties and time-varying delay has been presented. The stability criterion is given in terms of linear matrix inequality (LMI) which can be easily solved by some existing software packages. Two numerical examples have been provided to show the effectiveness of our theoretical results.

References 1. Wu, H., Mizukami, K.: Robust stability criteria for dynamical systems including delayedperturbations. IEEE Trans. Automat. Control 40, 487–490 (1995) 2. Yan, J., Tsai, J., Kung, F.: A new result on the robust stability of uncertain systems with time-varying delay. IEEE Trans. Circuits and Systems I 48, 914–916 (2001) 3. Xu, B.: A new result on the robust stability on uncertain systems with time-varying delay. Int. J. Syst. Sci. 33, 543–550 (2001) 4. Zhong, Q.: Robust stability analysis of simple systems controlled over communication networks. Automatica 19, 1309–1312 (2003) 5. Liu, P., Su, T.: Robust stability of interval time-delay systems with delaydependence-oscillation. Time Lages. Syst. Control Lett. 33, 231–239 (1998) 6. Jing, X., Tan, D., Wang, Y.: An LMI approach to stability of systems with severe time-delay. IEEE Trans. Automat. Control 49, 1192–1195 (2004)

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7. Zhang, H., Liao, X.: LMI-based robust stability analysis of neural networks with time-varying delay. Neurocomputing 67, 306–312 (2005) 8. Singh, V.: LMI approach to the global robust stability of a larger class of neural networks with delay. Chaos, Solitons and Fractals 32, 1927–1934 (2007) 9. Liao, X., Chen, G., Sanchez, E.: LMI-based approach for asymptotically stability analysis of delayed neural networks. IEEE Trans. Circuits and Systems I 49, 1033– 1039 (2002) 10. Singh, V.: Robust stability of cellular neural networks with delay: linear matrix inequality approach. IEE Proceedings- Control Theory and Applications 151, 125– 129 (2004) 11. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear matrix inequalities in system and control theory. SIAM, Philadelphia (1994) 12. Cao, J.: Global asymptotic stability of neural networks with transmission delays. Int. J. Syst. Sci. 31, 1313–1316 (2000) 13. Liao, X., Wong, K., Wu, Z., Chen, G.: Novel robust stability criteria for intervaldelayed hopfieldneural networks. IEEE Trans. Circuits and Systems I 48, 1355–1359 (2001) 14. Arik, S.: Global robust stability of delayed neural networks. IEEE Trans. Circuits and Systems I 50, 156–160 (2003) 15. Liao, X., Li, C.: An LMI approach to asymptotical stability of multi-delayed neural networks. Physica D 200, 139–155 (2005)

Novel Coupled Map Lattice Model for Prediction of EEG Signal Minfen Shen, Lanxin Lin, and Guoliang Chang College of Engineering, Shantou University, Guangdong 515063, China [email protected]

Abstract. Considering that electroencephalogram(EEG) signal is a typical chaotic signal induced from the nonlinear spatial temporal dynamic system, in this paper, we propose a new spatial temporal model combined couple map lattices(CML) with normalized radial basis function(NRBF), namely CMLNRBF model. NRBF neural network is employed to reconstruct the nonlinear map to obtain a more robust model with low sensitive for the selection of the basis function parameters. In particular, genetic algorithm (GA) is used to search for the optimal parameters of the proposed model, including the spatial coupling coefficients and the centers of NRBF network. The effectiveness of the proposed model is illustrated in terms of several experiments with real EEG by comparing the prediction and detection results of the presented model with the common RBF network. Keywords: Couple Map Lattices; Normalized Radial Basis Function; EEG.

1 Introduction Epilepsy contains various significant information of the brain function. The instantaneous detection or prediction of the occurrence of epilepsy in the background of normal EEG signal is a challenge mission for bio-signal processing, which can help us to observe and diagnose the state of the brain function [1]. However, most methods based on the conventional bio-signal processing are not valid for our purposes since the model establishment is from the linear assumption and cannot reflect the essential nonlinear dynamic characteristic of EEG, which results in the poor detection ability when predicting the occurrence of epilepsy. Nowadays, nonlinear dynamics has become an important tool for processing many practical signals generated by physical systems in life [2-4], such as EEG signals. People expect more useful information involved in the EEG signal can be extracted effectively and applied in clinical practice. More and more evidences shows that EEG is a chaotic signal [5], which can be modeled using the radial basis function (RBF) neural network and the phase-space reconstruction technique. According to the Taken’s time-delay reconstruction theory, the key point in the phase-space reconstruction means the choice of the embedding dimension and the time delay. In this paper, the mutual information method is adopted to get time delay and the embedding dimension is obtained through the combination of both false nearest neighbor (FNN) and Cao methods [6-8]. Moreover, normalized radial basis function F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 347–356, 2008. © Springer-Verlag Berlin Heidelberg 2008

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(NRBF) neural network are proposed to approximate the nonlinear map of CML by taking advantage of its universal approximation and generalization properties, which leads to the more robust and less sensitive to the selection of network parameters than RBF when modeling the EEG signal [9]. It is more complex that many physical systems, including nervous systems, are spatial-temporal nonlinear systems which reveal the nonlinearity not only in time dimension but also in the space dimension. In fact, the spatial effects are always ignored in the traditional nonlinear dynamic analysis [10-11]. For the purpose of exploiting the inherent spatial-temporal nonlinear dynamic characteristics of EEG signal, we propose a novel model in terms of coupled map lattice with normalized radial basis function (CML-NRBF) map in this paper to fit the underlying EEG signals. The established model is used as a predictor to detect the instantaneous occurrence of epilepsy involved in the EEG signal by evaluating the abrupt change from the prediction errors of the incoming signal. The incoming signal is more likely to be the epilepsy signal if the prediction error is large enough due to the obvious dynamic difference between the normal EEG and epilepsy signal. To train the model adequately and estimate the optimal model parameters, Genetic algorithm (GA) is proposed in this work to select the optimal coefficient of the model spatial coupling and the center of the NRBF network [12-13]. Several experimental results exhibit that GA procedure provides us an effective way for constructing the proposed spatialtemporal nonlinear model and significantly improves the prediction error with the optimal estimation of the spatial coupling coefficient of the model.

2 The CML-NRBF Scheme 2.1 CML-NRBF Model CML is widely used in studying the spatial-temporal chaotic phenomena. CML can be regarded as a dynamic system with discrete time and space and continuous state. In general, the CML has the form as: α

x(n + 1, i ) = f ( ∑ ε j x(n, i + j ))

(1)

j =−α

f denotes a differentiable mapping which can be approximated by neural network and n is the discrete-time step. i represents the lattice point and ε j is the spatial coupling coefficient. If α = 1 , three adjacent lattices are used for prediction, while five lattices will be involved if α = 2 .

where

A normalized RBF with M kernels, or hidden units, has the following overall response function: K

f (x(n)) = ∑ wkφk (x(n)) k =1

(2)

Novel Coupled Map Lattice Model for Prediction of EEG Signal

where

φi (x(n)) is a response function of the ith

kernel, and

349

wi denotes the weights

associated with the ith kernel. The common nonlinear kernel response function used in NRBF network is the Gaussian function given as:

φk (x(n)) = Where

exp[− x(n) − ck

∑

2

β]

K

exp[− x(n) − ck k =1

is the Euclidean distance measure,

2

(3)

β]

ci stands for the position of the ith

kernel’s center in the input space of NRBF network and

β

is known as the width of

the ith kernel. Then the CML-NRBF model can be defined as: K

xˆ (n + 1, i ) = ∑ wkφk (x ' (n, i ))

(4)

k =1

where

x' (n, i ) =

α

∑ ε x(n, i + j )

j =−α

(5)

j

The time-delay state vector in the reconstruction phase-space is given as:

x(n, i ) = ( x(n, i ), x(n − τ , i )," , x(n − (m − 1)τ , i ) )

(6)

xˆ (n + 1, i ) denotes the prediction of ith cell at time n+1 using the proposed CMLNRBF model. m is the embedding dimension while time-delay.

τ

represents the minimal

2.2 Phase-Space Reconstruction In order to model the nonlinear dynamic of EEG signal, it is necessary to make a time-delay reconstruction of a phase space from the observed time series. We use mutual information method to compute τ which is selected when the mutual information gets the minimum value. Fig.1 shows the mutual information corresponding the time delay. In our CML-NRBF model for EEG signal, τ =1 provides a good result for obtaining the small prediction error of the model. False nearest neighbor(FNN) and Cao method are combined to get the minimum embedding dimension. Fig.2 shows the results of Cao’s method with the solid line and dashed line, while the results of FNN are given with the star, plus and circle lines. From the Fig.2, obviously when m=20, the number of FNN is near to 0 and therefore the Cao’s two criterions are well satisfied. In detail in Fig.2, the solid line represents the mean space distance variation rate for the embedding dimension changes from m to m+1, and the dash line denotes another quantity which is useful to distinguish

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Fig. 1. the mutual information of time-delay

Fig. 2. Embedding dimension result of FNN and Cao combined method. The solid line and dashed line are the results of Cao’s method, while the star, plus, circle lines denote the results of three criterions in FNN respectively.

deterministic signals from stochastic signals. The star, plus and circle lines are the results of three criterions in FNN respectively. 2.3 Optimization of Model’s Parameters To train the CML-NRBF model in (4), we need to solve the following optimization problem:

min

α ,ε j ,ω k ,ck , K , β

x ( n, i ) − x ' ( n, i )

(7) '

where x (n, i ) denotes the actual signal at lattice i and time n, and x ( n, i ) is the predicted value. In order to obtain the optimal nonlinear parameters in (7), we adopt the genetic algorithm (GA) to determine several parameter of the model such as

Novel Coupled Map Lattice Model for Prediction of EEG Signal

ε j and ck . Parameter β

351

is chosen as the square of the maximum distance between

the selected centers divided by 2 K . The number of α and K is selected by trial and the is error based on the training result. Once the parameters are determined a priori, the weight vector w can be obtained by solving the following equations:

y = Φw where

(8)

Φ is a projection matrix with ( N − (m − 1)τ )× K given by:

Φ jk = ϕk (x' ( j + m − 1, i ))

(9)

The output of the proposed CML-NRBF model is given as:

y = ( x(m + 1, i )," , x( N , i )) where vector

N is the length of the observed time series. In order to obtain the weight w in (8), the least squares estimation is applied such as: w LS = Φ + y

where

(10)

(11)

Φ + is the pseudoinverse of the projection matrix Φ .

2.4 Genetic Algorithm In this paper, genetic algorithm is employed to train the proposed CML-NRBF model. First of all, the data are divided into three subsets (x1 , y1 ), ( x 2 , y 2 ), (x 3 , y 3 ) , which denotes the training, testing and validation sets. Then the algorithm starts with the selection of the following 4 parameters, including the size of the population L, the number of generation G, the probabilities of crossover ( pc ) and mutation ( p m ). These procedures are discussed in the following 4 subsections. 2.4.1 Initialization Firstly L matrices (chromosomes) including

c1 , c2 ,", c L with size (( K + 1) × m) are created. For each chromosome, the top left corner three ( 2α + 1) elements in the

first row is selected for encoding the spatial coupling coefficients. Three random values between 0 and 1 are generated to initial the spatial coupling coefficients and the middle one is chosen about ten times of the others two as it is more strong correlated with itself than the two adjacent lattices. The kernel’s centers are encoded in the left K rows, and the initial centers will be chosen from some successive embedded vectors of the input reconstruction state space. 2.4.2 Selection According to the CML-NRBF model, prediction error

E1 , E2 , " El of all

chromosomes can be computed. In order to give more chances of surviving to the

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chromosome associated with the smaller error values, the probability of selection

pl

of every chromosome is estimated via the following equation:

pl =

1 El L

∑1 E

(12) l

l =1

The cumulative probability is computed by the following formula: l

ql = ∑ pi

(13)

i =1

In addition, to protect the good chromosomes in current generation from crossover all mutation, the chromosome with minimum prediction error is selected firstly, and then a random number r between 0 and 1 is generated. If r < q1 , the first chromosome

c1 is selected. Otherwise, the chromosome cl is selected for

ql −1 < r < ql . It is expected that chromosomes with a higher probability of selection will get more copies in the new generation and the ones with lower probability of selection will be deleted. 2.4.3 Crossover As discussed above, a random number

rl between 0 and 1 is generated for every chromosome in the new generation. If pc > rl , the chromosome is selected for the

crossover operation and the number of crossover chromosomes should be even. Then they are grouped into pairs. A crossover position should be generated randomly for each pair before the crossover operation is undergone. 2.4.4 Mutation For every non-zero element in each chromosome, a random number rm between 0 and 1 is generated. if p m > rm , the element is replaced by a new value between 0 and 1 for spatial coupling coefficients and from -1 to 1 for NRBF centers. The mutation operation is very important since it can introduces some new values to the population and help the training procedure to avoid getting trapped in local minima. It is noted that high probabilities of mutation should not be adopted as it always leads to random search.

3 Experimental Results 3.1 Prediction Results To evaluate the performance of the proposed model, a segment of real spontaneous EEG signals with 3000 points are used for the purpose. The first 1000 points are used for training the CML-NRBF model, and the second 1000 points are used for testing the model with GA optimization. Finally, the last 1000 points are employed for

Novel Coupled Map Lattice Model for Prediction of EEG Signal

353

validation of the novel model. Genetic algorithm (GA) is developed to search for the model parameters such as the spatial coupling coefficients and the centers of NRBF. The GA parameters used here are shown as L = 50, pc = 0.5, p m = 0.2 , Fig.3 illustrates the development of the prediction mean absolute error (MAE) corresponding to the generation G for the CML-NRBF model. It can be seen that the MAE rapidly converge to the minimum value after a number of generations and then the optimal parameters will be obtained. In this paper the generation G = 500 was adopted to construct the model. Fig.4 shows the prediction result for 300 EEG data points using the presented CML-NRBF model. It can be seen from Fig.4 that the proposed CML-NRBF model tracks the EEG signal at all time steps with small prediction error. The prediction mean absolute error is estimated as 0.0073924 which is only about 1.48% of the real EEG data. Furthermore, the experiment result indicates that the proposed model can 6

5.8

preditive error

5.6

5.4

5.2

5

4.8

4.6

0

50

100

150

200

250 step

300

350

400

450

500

Fig. 3. the prediction MAE corresponding to the generation G for CML- NRBF model the real and preditive value of CML-NRBF model 0.8

0.6

0.4

EEG

0.2

0

-0.2

-0.4

-0.6

0

50

100

150

200

250

300

350

time

Fig. 4. Prediction result of CML-NRBF model, the solid line denote the real EEG data and the dot represent the predictive data

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9

x 10

8

CML-RBF

NRBF

2

3

RBF

CML-NRBF

7

prediction error

6 5 4 3 2 1 0

1

4

four models

Fig. 5. Prediction error of four models, CML-NRBF, CML-RBF, NRBF, RBF respectively

effectively capture the dynamic characteristics of the EEG signals. To compare the performances of modeling EEG with different schemes, 4 kinds of model, CMLNRBF, CML-RBF, NRBF, and RBF, were employed to the EEG and estimated the prediction error, as shown in Fig.5. From the results in Fig.5, it can be found that the CML-NRBF model provides the best predictor. Both CML-NRBF and CML-RBF can fit the nonlinear spatial-temporal dynamic of EEG well by comparing with the NRBF and the RBF network since the spatial factor was included in the CML model. The experimental results indicate that the normalized procedure provides the robust tracking ability and the low sensitive to the selection of the model’s parameters. 3.2 Detection Results Normal EEG signal and epilepsy signal are used here to evaluate the detection performance of the proposed CML-NRBF model. We randomly choose an epilepsy detection results of CML-NRBF CML-RBF NRBF and RBF model 0.09 0.08 0.07

prediction MSE

0.06 0.05 0.04 0.03 0.02 0.01 0

0

2

4

6

8

10

12

14

16

18

time

Fig. 6. Prediction MSE of four models, the solid lines represents the CML-NRBF model, the dashed lines represents the CML-RBF model, the dotted lines represents the NRBF model, the dashdot lines represent the RBF

Novel Coupled Map Lattice Model for Prediction of EEG Signal

355

signal segment contains 10 successive points and then insert it into the normal EEG. Then four models CML-NRBF, CML-RBF, NRBF, RBF are used to predict the mixed signal, and the prediction MSE (mean square error) is 0.70519, 0.68509, 0.6916, 0.6477 respectively, in order to differentiate the detection results clearly, the prediction MSE are preprocessed and then figured in Fig.6. It can be seen from Fig.6 that the proposed CML-NRBF model has the largest prediction error and RBF model has the minimum one, which tells us that the proposed model has the best detection ability and CML-RBF model can get the comparable result with NRBF when the RBF model is the worst one.

4 Conclusion This paper presents a nonlinear spatial temporal prediction based on CML-NRBF model for prediction of EEG signals and detection of epilepsy in normal EEG signal, which can capture the inherent nonlinear characteristic of EEG effectively. NRBF neural network is used here because it is more robust and less sensitive to the selection of parameters, GA is used to search for the optimal parameters including spatial coupling coefficients and the centers of NRBF network. The superiority of the proposed model is evaluated by experiments, and the CML-NRBF is shown to consistently outperform the CML-RBF and the conventional NRBF and RBF model when they are used to predict spontaneous EEG or detect the epilepsy occurrence in normal EEG signal. Acknowledgements. This research is supported by the Natural Science Foundation of China ( Grant No.60571066 ).

References 1. Babloyantz, A., Destexhe, A.: Low-dimensional Chaos in an Instance of Epilepsy. Proceeding of the National Academy of Sciences of the United States of American 83, 3513–3517 (1986) 2. Dudkin, A.O., Sbitnev, V.I.: Coupled Map Lattice Simulation of Epileptogenesis in Hippocampal Slice. Biological Cybernetics 78, 479–486 (1998) 3. Leung, H., Hennessey, G., Drosopoulos, A.: Signal Detection Using the Radial Basis Function Coupled Map Lattice. IEEE Transaction on Neural Networks 11, 1133–1151 (2000) 4. Oketani, N., Ushio, T.: Chaotic Synchronization of Globally Coupled Maps with an Application in Communication. International Journal of Bifurcation and Chaos 6, 2145– 2152 (1996) 5. Soong, A.C.K.: Evidence of Chaotic Dynamics Underlying the Human Alpha-Rhythm Electroencephalogram. Biological Cybernetics 62, 55–62 (1989) 6. Matthew, B.K., Reggie, B., Henry, D.I.A.: Determining Embedding Dimension for Phasespace Reconstruction using a Geometrical Construction. Physical Review A 45, 3403– 3411 (1992) 7. Matthew, B.K., Henry, D.I.A.: False Neighbors and False Strands: A Reliable Minimum Embedding Dimension Algorithm. Physical Review E 66, 026209 (2002)

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8. Cao, L.Y.: Practical Method for Determining the Minimum Embedding Dimension of a Scalar Time Series. Physica D 110, 43–50 (1997) 9. Cowper, M.R., Mulgrew, B.C., Unsworth, P.: Nonliear Prediction of Chaotic Signal using a Normalized Radial Basis Function Network. Signal Processing 82, 775–789 (2002) 10. Lorenz, E.N.: Dimension0 of Weather and Climate Attractors. Nature 353, 241–244 (1991) 11. Lachaux, J.P.: Spatial Extension of Brain Activity Fools the Single-Channel Reconstruction of EEG Dynamics. Human Brain Mapping 5, 26–47 (1997) 12. Haralambos, S., Alex, A., Stefanos, M., George, B.: A New Algorithm for Developing Dynamics Radial Basis Function Neural Network Models based on Genetic Algorithms. Computers and Chemical Engineering 28, 209–217 (2004) 13. Nan, X., Leung, H.: Reconstruction of Piecewise Chaotic Dynamics Using a Genetic Algorithm Multiple Model Approach. IEEE Transaction on Circuits and Systems 51, 1210–1222 (2004)

Adaptive Synchronization of Delayed Chaotic Systems Lidan Wang and Shukai Duan School of Electronics & Information Engineering, Southwest University, Chongqing 400715, PR China {ldwang,duansk}@swu.edu.cn

Abstract. In this paper the adaptive synchronization between two delayed chaotic systems with uncertain parameters is investigated. The sufficient condition for parameter identification is derived based on the Lyapunov stability theory. Through choosing the appropriate adaptive controller, parameter identification and adaptive synchronization can be achieved simultaneously. Finally, numerical results illustrate the effectiveness of the theorems. Keywords: Chaos, Adaptive synchronization, Delay.

1

Introduction

Recently, many researchers have directed their attention to the study of chaos because of its distinct features such as continuous broadband frequency spectrum, complex and unpredictable behavior, and sensitive dependence on the initial conditions [1-2]. Undoubtedly, these interesting features make it have potential applications in secure communication, especially since Pecora and Carroll’s pioneer work [3]. It has been noticed that most chaos synchronization schemes are based on identical chaos systems depicted by ordinary differential equations (ODEs). However, uncertainties and parameters mismatch will affect the stability of chaos synchronization in the synchronous process. As we know, most of the synchronization methods belong to drive-response type and the behavior of the response subsystem is dependent on the behavior of the driving subsystem. In many cases, determining some system parameters in advance may be difficult and some parametrical values are characterized by uncertainties related to experimental conditions that can destroy or even break the synchronization. These problems can be tackled to a certain extent through adaptive synchronization [4-6]. Adaptive synchronization schemes between two chaotic systems based on delayed differential equations (DDEs) have seldom been reported. Until recently, Yu and Cao [7] proposed adaptive synchronization and lag synchronization schemes of uncertain dynamical system with time delay. 

Corresponding author.

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The present work addresses a delayed chaotic model and focuses on investigating an adaptive synchronization of the delayed chaotic systems based on parameter identification. An adaptive synchronization controller and a parameter estimation updating law are considered, it can make the states of two delayed chaotic systems globally asymptotically synchronized in the presence of system’s unknown constant parameters. The remaining materials are organized in this manner: Section 2 gives the system description necessary for our study; Section 3 presents and proves the main result of this paper: several sufficient conditions are presented for the adaptive synchronization of dynamical systems. In Section 4, a series of computer simulations is demonstrated the effectiveness of the proposed scheme. The conclusions are finally drawn in Section 5.

2

Problem Formulation

Consider the following time-delay differential equation: x(t) ˙ = −ax(t − τ ) + bx(t) + cf (x(t − τ ))

(1)

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Adaptive Synchronization

We assume that there are two linear coupled delayed systems in which the drive system with the subscript 1 drives the response system having identical equations denoted by the subscript 2. To facilitate statement, the drive and response systems are defined as following delayed differential equations.

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y(t) ˙ = −ay(t − τ ) + by(t) + cf (y(t − τ ))

(2)

x(t) ˙ = −ax(t − τ ) + bx(t) + c˜f (x(t − τ )) + u(t)

(3)

where u(t) is controller for synchronizing two delayed chaotic systems with the unknown parameter in spite of the differences in initial conditions.in (3) and described as u(t) = k[y(t) − x(t)] (4) c˜ is an unknown parameter. To obtain simple parameter estimation update law, by the way, the sensitive parameter c is only analyzed. In order to estimate the control functions, we subtract (2) from (3), yields error system between drive and response systems (2) and (3) are defined as following equations. e(t) = y(t) − x(t) (5) Then, our target is shifted to make error system (5) approach asymptotically to zero when t approaches to infinite. System (5) is governed by the following dynamical system e(t) ˙ = −ae(t − τ ) + be(t) + cf [y(t − τ )] − c˜f [x(t − τ )] − u(t) = −ae(t − τ ) + be(t) + cf [y(t − τ )] − c˜f [x(t − τ )] − ke(t). (6) The goal is to find a controller and a parameter estimation update law for (6) such that the states of response system (3) and the states of drive system (2) are globally synchronized asymptotically, this translates to, lim e(t) = 0, ∀a, b ∈ R

t→∞

(7)

Theorem 1. The drive system (2) synchronizes with the response system (3) if we choose (1)The controller u(t) = k[y(t) − x(t)] (8) where k > [cN 4−a] + 1 + b. N is the boundary of the first continuous derivative of f (·). There exists a positive constant N that satisfies the inequality 2

f  (ξ) ≤ N (2) Parameter estimation update law is employed as 1 c˜˙ = f (x(t − τ ))e(t) μ

(9)

where c˜ is independent of c. Proof. Let we take a Lyapunov function candidate for (6) as following  0 1 2 1 2 V = e (t) + μ(c − c˜) + e2 (t + θ)dθ 2 2 −τ

(10)

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For facility, e(t) and e(t − τ ) can be denoted as e and eτ . Differentiating V with respect to time along the trajectory of the error system (6) yields

V˙ = ee˙ − μ(c − c˜)c˜˙ + e2 − e2τ = e{−aeτ + be + c[f (yτ ) − f (xτ )] + (c − c˜)f (xτ ) −ke} − μ(c − c˜)c˜˙ + e2 − e2τ

(11)

According to mean value theorem, we obtain f (yτ ) − f (xτ ) = f  (ξ)(yτ − xτ ) = f  (ξ)eτ

(12)

So, Eq. (11) becomes V˙ = [cf  (ξ) − a]eeτ + (c − c˜)[f (xτ )e − μc˜˙ ] + (1 − k + b)e2 − e2τ

(13)

Let parameter updating rule is 1 c˜˙ = f (xτ )e μ

(14)

Then Eq. (13) is [cf  (ξ) − a] 2 [cf  (ξ) − a]2 V˙ = −{eτ − e} + { + (1 − k + b)}e2 2 4 [cf  (ξ) − a]2 + (1 − k + b)}e2 . ≤{ 4

(15)

Obviously, when the following inequality k>

[cf  (ξ) − a]2 +1+b 4

is satisfied, we can obtain V˙ ≤ 0. According to Lyapunov second method, if V ≥ 0 and V˙ ≤ 0, this translates to, lim e(t) = 0, ∀b ∈ R

t→∞

lim ˜b − b = 0, ∀˜b, b ∈ R

t→∞

is satisfied. e = 0 is the full and necessary condition for V = 0. Hence, the synchronization of two delayed chaotic systems (2) and (3) with unknown parameters is achieved under the adaptive controller (8) and parameters estimation update law (9). The inductive proof of the claim is completed.

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Theorem 2. For continuous delayed chaotic systems (2) and (3), if f : Rn → Rn is continuous and differentiable in its definition domain, its unknown parameter can be identified by following equation, c˜˙ ∝ e ∗ f (x) c − c = 0, ∀˜ c, c ∈ R, under the controller u(t), the response Proof. If limt→∞ ˜ system can synchronize with the drive system , i.e. e2 will trend to zero. Hence, for searching the form of unknown parameter of synchronization systems, we can employ state variable to identify unknown parameter. By applying the steepest descent method, we can obtain ∂e2 c˜˙ ∝ ∂˜ c i.e. c˜˙ ∝ e ∗ f (x) So, we can choose c˜˙ =

4

1 μ f (x(t

− τ ))e(t) in Theorem 1.

Numerical Simulation

To support the analysis made in section 3, a numerical simulation example is constructed to show the effectiveness. Example. Consider following drive-response systems. y(t) ˙ = −0.2y(t − τ ) + 0.04y(t) + f (y(t − τ ))

(16)

x(t) ˙ = −0.2x(t − τ ) + 0.04x(t) + c˜f (x(t − τ )) + k[y(t) − x(t)]

(17)

By using Theorem 1 and 2, we obtain 1 c˜˙ = f (x(t − τ ))e(t) μ

(18)

here function f (x) = 1/[1 + e−10x ] − 0.5, f  (x) = 10e−10x /[1 + e−10x ]2 , constant N ≥ |f  (x)| = 2.5. According to Theorem 1 and 2, while k > 2.4 and μ = 0.2, adaptive synchronization can be reached. The initial values of the drive-response system states are set as y(0) = 0.1 and x(0) = −0.9, respectively. Initial value of estimate for ”unknown” parameters is taken as c˜(0) = 0.8 and time delay is set as τ = 10. The time response of the error system and procedure of adjusting parameter c˜ can be seen in Fig. 3 and Fig. 4. Obviously, with the time evolvement, adaptive synchronization is achieved and c − c˜ approaches to zero immediately. In fact, adaptive synchronization can be achieved while weakly coupling factor is adopted, for example, it can be achieved while k = 1.5 (shown in Fig. 5 and Fig. 6). Of course, with the decreasing of the coupling factor, the time while synchronization achieves will spend much longer, as shown in Fig. 7 and Fig. 8. These numerical results demonstrated the effectiveness of the proposed adaptive synchronization scheme.

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Conclusions

Based on Lyapunov theory, we investigated two coupled chaotic systems with delays, derived some simple generic theorems, and proposed an adaptive synchronization controller and a parameter update law, which can make the states of two delayed chaotic systems globally asymptotically synchronized in the presence of system’s unknown constant parameters. The simulations demonstrated the effectiveness of the proposed method. Acknowledgments. This research was supported by the Natural Science Foundation of Chongqing CSTC under Grant 2007BB2331, the Doctorate Foundation of the Southwest University under Grant SWUB2007008.

References 1. Chen, G.R., Ueta, T.: Chaos in Circuits and Systems. World Scientific, Singapore (2002) 2. Thangavel, P., Murali, K., Lakshmanan, M.: Bifurcation and Controlling of Chaotic Delayed Cellular Neural Networks. International Journal of Bifurcation and Chaos 8, 2481–2492 (1998) 3. Pecora, L.M., Carroll, T.L.: Synchronization in Chaotic Systems. Physical Review Letters 64, 821–824 (1990) 4. Li, S., Alvarez, G., Chen, G.: Breaking a Chaos-based Secure Communication Scheme Designed by an Improved Modulation Method. Chaos, Solitons & Fractals 25, 109–120 (2005) 5. Liao, T., Tsai, S.: Adaptive Synchronization of Chaotic Systems and Its Application to Secure Communications. Chaos, Solitons & Fractals 11, 1387–1396 (2000) 6. Wang, C., Su, J.: A New Adaptive Variable Structure Control for Chaotic Synchronization and Secure Communication. Chaos, Solitons & Fractals 20, 967–977 (2004) 7. Yu, W.W., Cao, J.D.: Adaptive Synchronization and Lag Synchronization of Uncertain Dynamical System with Time Delay Based on Parameter Identification. Physica A 375, 467–482 (2007)

Research on Fish Intelligence for Fish Trajectory Prediction Based on Neural Network Yanmin Xue1, Hongzhao Liu1, Xiaohui Zhang1, and Mamoru Minami2 1

Xi’an University of Technology, Xi’an, China Tel.: 86-29-82066424; Fax:86-29-62690260 [email protected], [email protected], [email protected] 2 Fukui University, Fukui, Japan [email protected]

Abstract. This paper researches the behavior modes of some intelligent creature in some environment. The gained modes are used as movement models to construct NN to predict the moving trajectory and then catch it. Firstly the behavior patterns of fish that kept trying to escape from the net attached at robot’s hand were studied through lots of experiments. The patterns were divided into five sorts and the learning procedures were divided into three stages. Based on this, the position, orientation and speed of each time were used as the input of multi layer perceptron (MLP) neural networks (NN), and the positions of the fish at next time were the outputs. The NN adopted extended delta-bar-delta (DBD) algorithm as learning method. Thus the NNs were constructed to study the moving regulations of fish in every pattern to predict the moving trajectory. The simulation results shows that the BP NN constructed here have the advantage of faster learning rate, higher identifying precision and can predict the fish trajectory successfully. The research is significant for visual servo in robotic system. Keywords: Visual servo, Intelligent robot, Neural network, Predicting trajectory, Genetic algorithm.

1 Introduction In recent 20 years progress in visual control of robots has been a marked increase in scientific research and industrial application such as prototyped span manufacturing, teleoperation, missile tracking cameras and fruit picking as well as robotic ping-pong, juggling, balancing, car steering and even aircraft landing [1]. In the so many application fields, target tracking is one typical application of visual servoing technology [2]. It is still a difficult task to track high speed target with current visual servo system [3]. There have been intensive efforts in this aspect [4,5,6] and have gained some progress. In recent years, Artificial Intelligence (AI) including NN,GA,fuzzy logic and so on are more and more widely used in many scientific applications and researches[7].And it is also applied to this field. Some good jobs were done in recent F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 364 – 373, 2008. © Springer-Verlag Berlin Heidelberg 2008

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years. [8] describes an additional neural mechanism that enhances the ability of a visual system to track curved trajectories, while the linear trajectories can be tracked by existing NN models. [9] designed a Neural Location Predictor to predict some mobility model. In this paper, a new tracking object, fish, is encountered. And NN is chosen here for predicting fish trajectory. The reason to choose NN is that the behaviour mode is too irregular to form some formulations to learn, while NN can be a function approximator for us. In this paper, we novelly summarized some typical movement patterns of swimming fishes in some pool. And some multi-layer network architectures to predict the near future location of the fishes were also designed. The NNs can be called as Neural Location Predictor (NLP) too. The transformation for moving model into NLP was based on lots of experiments to observe the fish movements. And then five kinds of movement patterns that were seemingly random were summarized. And each one pattern was corresponding to one MLP NN. The computational simulation results were presented that show how the neural mechanism could learn to track the location of objects and how the network operated to predict the trajectories of the movement of an intelligent creature. Thus the simulation results well demonstrated the ability of the system to track some moving creature. A whole visual servo system should include image processing, camera calibration, position and orientation estimation of robot, control system and so on. The job this paper done was help to robot control system. And the environment this paper based on was a position-based robot visual servo system with single camera, and ‘eye in hand’. The image processing procedure and robot control system adopted the techniques in [10,11,12]. As a result these were not described in this paper. The proposed intelligent prediction scheme promises a more appropriate approach and can predict a future location. Accurate and speedy location prediction can help to enhance the efficiency of the whole visual servo system. The remainder of this paper is organized as follows. In the beginning, the intelligent behaviours observed in experiments to catch fish are studied. Then the MLP neural network is constructed to predict the trajectory of the fish, the simulations show the performance of the predicting method. Finally, the conclusion of the work is presented. All of these are in order to help to realize an intelligent robot to catch fish.

2 Intelligent Behaviours of Fish 2.1 Sequence Catching Experiment of a Fish We performed a tracking and catching experiment of a swimming fish using the proposed method. The photograph of our experimental system is shown in Fig. 1. In this figure, the camera-to-fish distance is 450 mm. The size of the water pool is 400×350×100 mm and the net is 80×100 mm. Catching the fish is executed by pulling up the net when the fish is within an area of 40×50 mm at the center of the net. In the 2-D servo experiment, the camera-to-fish distance is fixed.

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Fig. 1. Photograph of our experimental system

Using this experimental system, we performed the catching experiment of fish for almost an hour continuously each time to observe the behavior of the fish. We released one fishe (length is about 40 mm) in the pool, and the fish caught by the net was returned to the pool at once. The result of this experiment is shown in Fig. 2. In this figure, the vertical axis represents the number of catching times, and the horizontal axis represents the time. As you see in the figure, the fish was caught continuously. Catching Number of the Fish

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We had expected that catching of the fish becomes easier in the latter half of experiments, because we had expected that the fish would be made to be tired to escape by the persistent tracking of the net for long time. However, in the actual result of this experiment, the number of fish caught by the net in ten minutes as shown in Fig. 2 is decreasing smaller and smaller as time goes on. We consider that a learning of the fish causes the result. The results also show intelligence of the fish to survive, and we understood even a small fish, it is obviously more intelligent than a simple machine having an eye, hand, net and infinite physical strength. In order to realize the intelligent catching robot system, we should study the intelligent behaviours observed in experiments to catch fish.

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2.2 Movement Patterns of Fish From the experiment, we found that the movement patterns of fish were finite, and the intelligent of fish was shown in the decision process selecting a certain movement pattern, so we studied the movement patterns of fish firstly. The movement patterns observed of fish are shown in Fig. 3. The rectangle area in Fig. 3 is a pool, and the size of it is 400×350 mm. There are five movement patterns founded, they are 1) Linear Movement (LM): The track is shown in Fig. 3-1. The fish most selected it as escape method from the net and swam very quickly in initial stage of taking the fish into the pool.

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Fig. 3. Movement patterns of the fish

2) Movement with Changing Orientation (MCO): The track is shown in Fig. 3-2. The fish swam with different speeds, but the frequency of changing orientation was high and the changing angle was large. 3) Movement with Arc Track (MAT): The track is shown in Fig. 3-3. The movement track likes an arc. The movement patterns from 1) to 3) have different speeds. 4) Rest State (RS): The track is shown in Fig. 3-4. The fish continued to stay and have a rest in the clearance between the side of the pool and the net position. Here, the net cannot move to the side of the pool to catch the fish, because we fixed a motion area of the net to avoid hitting against the wall of the pool by the net. 5) Movement Around the Net (MAN): The track is shown in Fig. 3-5. The fish kept a close distance to the net and swam slowly, but the net couldn’t catch it, because the fish learnt how to run away around the net, the opportunity of catching also was known by fish. It is very stable with a period of about 3 seconds.

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We performed a tracking and catching experiment of a swimming fish using the proposed method. The photograph of our experimental system is shown in Fig. 1. In this figure, the camera-to-fish distance is 450 mm. The size of the water pool is 400×350×100 mm and the net is 80×100 mm. Catching the fish is executed by pulling up the net when the fish is within an area of 40×50 mm at the center of the net. In the 2-D servo experiment, the camera-to-fish distance is fixed. Besides above five movement patterns of the fish, from the experiment, we also found that the factors influencing movement of fish includes: position, orientation and speed of the fish; position, orientation and speed of the net; the bound of the pool; the durative time of the experiment (It shows the physical force, dread, curiousness and mood of the fish.); the interval from current time to last caught. 2.3 Learning Process of the Fish From section B we know that the fish is intelligent, here, we consider the learning process of fish during the experiment, the learning process of fish is divided into three stages, they are as follows. 1) Dread Stage (DS): The stage is in initial stage of cat-ching fish, and it is about ten minutes. During the stage, the fish had high degree of dread, and no curiousness. The movement patterns selected by fish were LM and MCO with high speed. At end of the stage, the fish knew some basic knowledge about dangerous area and safe area (Safe area is some places that the net can’t reach, and dangerous area is opposition). The times of fish caught was the most in the stage, but the most caught was at the end of the stage, because the fish swam very quickly at the beginning, it was difficult to catch it, but at the end of the stage, the physical force of the fish decreased very quickly, then catching it was comparatively easy. 2) Curious Stage (CS): It is about twenty minutes. During the stage, the degree of dread of fish was decreased, and the degree of curiousness was increased. The times of caught were less than the first stage. The physical force of fish was decreased continuously. The movement patterns selected by fish were LM, MCO, MAT, RS and MAN with different speeds. In the end, the fish distinctly knew the bound of the dangerous area and safe area, and had a basic knowledge about the opportunity of catching. 3) Adaptive Stage (AS): The remainder time is AS. During the stage, the degree of dread of fish was decreased to the lowest level, and the grade of curiousness also was weakened. The times of success catching were small because the fish had known anything about the catching. Physical force of fish was decreased, but its decreasing speed was low for the movement of fish was not acutely like before. The movement patterns selected by fish were MCO, MAT, RS and MAN with lower speed than last stage. Finally, the fish distinctly knew the bound of dangerous area and safe area, and the opportunity of catching. In addition, sometimes the fish was worried. From above, we know the learning process of the fish clearly. At the same time, the physical force (PF), dread (D), curiousness (C) and mood (M) of fish is also changed during the experiment, the changes of them are shown in Fig. 4. There are some peak values in curve of D shown in Fig. 4 when the fish was caught once. A fish is an intelligent biology with the ability of learning from the surrounding.

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3 Predicting Trajectory of Fish We want to realize a set of intelligent robot system to catch fish. The word “intelligent” means the robot doesn’t only follow the fish, but it can estimate the trajectory of fish and decide to take actions just like human beings, so we should make the robot predict the trajectory of the fish firstly. It is impractically that using a certain mathematic model to describe the total movement process of fish. In practice, we often use the neural network as a function approximator, but when the function changes a little, the neural network will spend much time to adjust the relative parameters to adapt to its variety, because of the low convergence speed. Here, we customarily hope the neural network can adjust its relative parameters online to fit the learning of the fish. The learning speed of fish is very quickly, and the fish also is an intelligent biology, not a function, so we think it is difficult to predict the movement of the fish only depending on the ability of adjusting online of the neural network during the whole process of catching fish. From the experiment, we can know that the movement patterns of a fish are finite, the fish can finally select some correct movement patterns to escape from the pursuing of net, but every movement pattern is simple, so we possibly find a neural network model to predict the trajectory of every movement pattern off-line. The decision process of the fish to select which kind of movement pattern in different surrounding shows its learning intelligence. So we only need to find a decision model to learn the decision process of the fish (Here, we can reference the Reinforcement Learning methods in machine learning, there are some kind of successful methods about Markov Decision Process (MDP) decision; in addition, we also can use some expert strategies aiming at every movement patterns). Using above method, the request of computing and the degree of complexity of the problem will be decreased, so it is possible to predict the movement of fish. In this paper, we use MLP to predict the trajectory of the fish in a pool. 3.1 Predicting with Neural Network There are five movement patterns of the fish from above, but the RS is a special pattern because of the moving limit of the net, it can be solved by a certain special strategy. We only need to predict four patterns, and they are LM, MCO, MAT and MAN.

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We adopt a MLP as predictive model because it has achieved many successful applications in engineering field, the theory of MLP and BP algorithm is well known, here, we use extended DBD(delta-bar-delta) algorithm [13] as a learning method of MLP neural network model because of its rapidly convergent speed. The related parameters adjusting methods are described as follows in mathematics. Δ ω ij ( k ) = − η ij ( k )

∂J + μ ij ( k ) Δ ω ij ( k − 1) ∂ ω ij ( k )

η ij ( k + 1) = min [η max η ij ( k ) + Δ η ij ( k ) ]

(1)

μ ij ( k + 1) = min [μ max μ ij ( k ) + Δ μ ij ( k ) ]  D 1 exp(  J 1 G ij ( k ) )  G ij ( k  1) G ij ( k ) ! 0 ' K ij ( k )

 ) 1 K ij ( k )   G ij ( k  1) G ij ( k )  0

 0 

(2)

 Other

 D m exp(  J m G ij ( k ) )  G ij ( k  1) G ij ( k ) ! 0 ' P ij ( k )

 ) m P ij ( k )   G ij ( k  1) G ij ( k )  0

 0 

(3)

 Other

δ ij ( k ) = ∂ J / ∂ ω ij ( k ) δ ij ( k ) = (1 − θ ) δ ij ( k ) + θ δ ij ( k − 1

(4)

ω is the connection weight value of NN, η is the learning rate, μ is the momentum coefficient, α 1 , γ 1 , Φ 1 , α m , γ m , Φ m , θ are parameters set.

Where

3.2 Simulation The data from the experiment were divided into five sets by five movement patterns defined before. The neural net-works predicted the next position of the fish by the data of the current time, the last time and the time before last time, the prediction interval is 480 ms. The NN architecture implemented in this paper has three layers (input, hidden, and output). In the authors’ application, there were four neural networks, the input layer of every neural network contained twenty six neurons, representing the position (12), orientation (6), speed (6) of fish and net in the three times, and the durative time of experiment (1) and the interval to last fish caught (1) of current time. Two output neurons of them represented the prediction position of fish next time. The number of hidden-layer neurons of them was different because their different movement rules. The architectures are shown in Table 1. The input neurons were fully connected to hidden-layer neurons, which were fully interconnected to output neurons. A sigmoid transfer function was used for the hidden layer, and a linear function was used for the output layer. All weights were initialized to random values between –0.1 and 0.1. The learning rate and momentum relate the variation of the weights to the gradient of the error function. They were set to 0.5 before training and then optimized on a trial basis during the training procedure. RMS was used as the error function in this study. It is defined as

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n

RMS = ∑ || d i − o i || 2 / n

(5)

i =1

Where d i = [ xi , y i ]T is output position calculated, o i = [ x ' i , y ' i ]T is practical position from the experiment, xi , x ' i is x coordinates, and y i , y ' i is y coordinates. || d i − o i || is the distance from prediction position to practical position. n is the total number of patterns presented to the network in the learning procedure. The performance of NN was tested on a predicting set (a subset of the data set) and monitored during the learning procedure to determine when the learning process had to be stopped. An early stopping strategy was adopted in this work to avoid overtraining: When the descending rate of training error was small enough and the testing error began to increase, the learning procedure stopped. Table 1. Architectures of Neural Networks Pattern

Input

Hidden

Output

LM MCO

26 26

7 9

2 2

MAT MAN

26 26

14 16

2 2

Computer simulation of the networks was carried out with special software, developed by the authors. It is programmed using the Visual C++ language and run on an IBM-686-compatible PC under a Windows XP environment. This software package includes a NN error back-propagation algorithm for developing neural networks. The extended DBD algorithm used in this study worked efficiently. The results of the simulation are shown in Fig. 5 and Fig. 6. Fig. 5 is the learning error curve chart of neural networks about four movement patterns, where the solid curves are learning samples error curve and the dash ones are test sample error curve, Learning error Test error

0.8

0.3 Error

Error

0.6 0.4 0.2 0

0.2 0.1

100 200

300

Times

400 500

0

600

100 200 300 400 500 600 Times

LM

0.4

MCO

0.3

0.3 Error

Error

0.2

0.1

0.1 0

0.2

100

200

300

Times

400

500 550 MAT

0

100

200

300

Times

400

Fig. 5. Learning error curve of neural networks

500 MAN

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0

200

100

100 y

0

0

y

-150 -100 -50

0 x

50

-200

100 150

-150 -100 -50

0 x

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0

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y

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0 -----Fish -----NN

-100

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0 x

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0 x

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MAN

Fig. 6. Contrast trajectory chart of NN prediction and fish

the RMS is 0.002. Fig. 6 is the contrast trajectory chart of NN predictions and practical movement patterns of fish, where the triangle is practical position of fish and the circle is predicting position by neural networks. From the simulations we can know that four MLP neural networks can successfully predict movement patterns of fish.

3 Conclusion This paper has introduced a new method to construct NNs using the movement patterns studied by lots of experiments. First of all, in order to realize a set of intelligent robot system to catch fish, we have performed the catching experiment of fish using the same experimental system for almost an hour continuously to observe the behaviors of the fish. As the result of the experiments, we have studied the interesting strategy of the fish, which is an intelligence of the fish to survive, and divided the movement of the fish into five patterns. Then, we used four MLP neural networks to predict the trajectory of four movement patterns, and it has gotten a satisfactory result by the simulations. The prediction of the trajectory of the fish only is the first step of realizing the intelligent robot system to catch fish, there are more works need to be done in the next steps. The following step should be real experiments to justify the proposed method.

References 1. Hulchinson, S., Haner, G., Corke, P.: A Tutorial Introduction to Visual Servo Control. IEEE Trans. on Robotics and Automation 12, 651–677 (1996) 2. Tian, M.Q., Luo, X.H., Wei, Y.: Research on Visual Servoing Robot to Manipulate Moving Objects. Robot. 25, 548–553 (2003)

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3. Wang, G., Zhang, S.F., Hong, Y., Li, X.Q., Qi, H.: Spatial Trajectory Prediction of Visual Servoing. Chinese Journal of Mechanical Engineering 12, 7–10 (2003) 4. Musso, Cardillo, A., Cosentino, O., Memmo, A.: A Balloon Trajectory Prediction System. Advances in Space Research 33, 1722–1726 (2004) 5. Zhao, J.B., Wang, J.Z., Zhong, Q.H.: Adaptive Track Predicting Control for Target Tracking Control Systems. Journal of Beijing Institute of Technology 15, 62–65 (2006) 6. Xi, Y.G., Chen, W.D., Fan, C.H., Dong, S.L.: Study of a New Method for Vision Based Robot Target-Tracking Problem. High Technology Letters 7, 54–57 (2001) 7. Fukuda, T., Shimojima, K.: Intelligent Control for Robotics. Computational Intelligence, 202–215 (1995) 8. Marshall, J.A., Srikanth, V.: Curved Trajectory Prediction Using a Self-Organizing Neural Network. International Journal of Neural Systems 10, 59–70 (2000) 9. Liou, S.C.: An Application of Trajectory Prediction in Mobile Networks. Journal of Leader University 1, 49–65 (2003) 10. Minami, M., Agubanhan, J., Asakura, T.: Manipulator Visual Servoing and Tracking of Fish using Genetic Algorithm. Int. J. of Industrial Robot. 29, 278–289 (1999) 11. Minami, M., Agbanhan, J., Asakura, T.: GA Pattern Matching Based Manipulator Control System for Real-Time Visual Servoing. Advanced Robotics 12, 711–734 (1999) 12. Minami, M., Suzuki, H., Agbanhan, J., Asakura, T.: Visual Servoing to Fish and Catching Using Global/Local GA Search. In: Proc. of Int. Conf. on Advanced Intelligent Mechatronics, pp. 183–188 (2001) 13. Minai, A., Williams, R.: Back-Propagation Heuristics: A Study of Extended Delta-BarDelta Algorithm. IJCNN 2, 595 (1990)

A Hybrid MCDM Method for Route Selection of Multimodal Transportation Network Lili Qu and Yan Chen Dalian Maritime University 116026 Dalian, China [email protected]

Abstract. Multimodal movements are those in which two or more different transportation modes are linked end-to-end in order to move freight and/or people from point of origin to point of destination. Computational efficiency required to select the path will be drastically deteriorated by increasing the number of traffic modes and terminals. With different goals, the minimization of cost, time, risk and unreliability, it is usually assumed that the route selection of multimodal transportation is a multicriteria decision making (MCDM) problem. On the basis of the Fuzzy Analytic Hierarchy Process (AHP) and artificial neural network (ANN) theory, we set up a hybrid MCDM model. The proposed method makes the route selection system in multimodal transportation more comprehensive, scientific, fair and accurate. Keywords: Hybrid multicriteria decision making (MCDM), Multimodal transportation, Route selection, Fuzzy Analytic Hierarchy Process (AHP), Artificial neural network (ANN).

1 Introduction Multimodal transportation is defined by the European Conference of Ministers of Transport (ECMT) as the carriage of goods by at least two different modes of transport in the same loading unit without stuffing or stripping operations when changing modes[1][2]. For example, multimodal transportation can combine the advantages of rail and road, rail for long distances and large quantities, road for collecting and distributing over short or medium distances [3]. Multimodal transportation offers a full range of transportation modes and routing options, allowing them to coordinate supply, production, storage, finance, and distribution functions to achieve efficient relationships [4][5]. While multimodal operations are increasing, little has been done to study the properties of multimodal networks and the complexities of the algorithms to model and optimize operations on them [6]. Multimodal networks are characterized by dynamically changing conditions and multiple modes of transportation operating simultaneously on them. Goals of relevant interest, like the minimization of cost, time, risk and unreliability, are conflict. Therefore, in general, there is no single optimal solution, but rather a set of nondominated or noninferior solutions, from which the decision F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 374 – 383, 2008. © Springer-Verlag Berlin Heidelberg 2008

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maker must select the most preferred one, or the best compromise solution. It is usually assumed that the route selection of multimodal transportation is a multiple criteria decision making (MCDM) problem. Many experts solve these problems based on the mathematic programming methods. Crainic and Rousseau encountered the problem of computing optimum paths on a multimodal freight networks as a component of a more general freight network design problem [7]. In this paper, considering the kinds of objects and requirements in the transportation process comprehensively, we adopt a hybrid MCDM method to find the best route in the multimodal transportation. Computational efficiency required to select the path will be drastically deteriorated by increasing the number of modes, terminals and time intervals. Artificial neural networks (ANN) have an important role when these relationships are unknown or non-linear. Provided there are enough observations with flexible form and universal approximation property [8], ANN can learn from experience and generalize, estimate, predict. These attributes have made the ANN approach fairly efficient for MCDM problem solving [9]. In order to search the “best” way to go from a certain origin to a specific destination through a multimodal transportation network in efficient and effective way, this paper is to demonstrate the potential role of ANN for MCDM and propose a new hybrid MCDM method.

2 MCDM Base on ANN 2.1 MCDM Notation [10] In general, an MCDM problem p=(X, f) with q objective functions can be formulated as (1), with X⊂ Rn being the set of feasible alternatives and f: Rn → Rq representing q objective functions to be maximized. Such a problem is usually called a multiple objectives decision making (MODM) problem. max f (x)

(1)

s.t. x∈X Another case is when X is discrete. A set of alternatives is a finite set, X={x1, x2,…, xp}, an MCDM problem can be defined in a simple way. The objective or criteria values for each feasible alternative can be represented by a decision matrix (2), where zij is the value of objective j for alternative xi, zij =fj(xi). Such a problem is usually called a multiple attributes decision making (MADM) problem.

Z = ( zij ) i ∈ {1, 2,... p}, j ∈ {1, 2,...q}

(2)

Present MCDM in a mathematical sense, usually the set of efficient alternatives is considered as the solutions of an MCDM problem. This set is defined as (3), where f(y) ≥ f(x) means that fj(y) ≥ fj(x) for all j∈{1,2,…,q} and fj(y) > fj(x) for at least one j∈{1,2,…,q}. For x, y ∈ X with f(y) ≥ f(x), we also write y ; x.

Eff ( x, f ) = {x ∈ X : do not exist y ∈ X such that f ( x) ≥ f ( y)}

(3)

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If the decision makers (DM) preferences can be represented by a function F(f(x)), the definition of efficiency Eff(x, f) implies that F(f(x)) is a non-decreasing function with respect to each objective fj(x). General MCDM methodology involves four steps: 1) Identify the set of alternatives, 2) Identify and assess criteria, 3) Choose a method for ranking alternatives, 4) Apply the method for ranking alternatives and choose the best alternative. 2.2 FANN and MCDM In solving MCDM problems [11][12], we demonstrate that feedforward ANN (FANN) can provide an approach to represent the DM preferences quantitatively through flexible and versatile nonlinear mapping function. We use FANN to model the objective function f(x), f(x) = (f1(x), f2(x),…,fq(x)). One objective function or criteria is an input point to the FANN, so in this case, it is assumed that the neural network gets q inputs, hence the objective values of an alternative x to be aggregated to a result F (f(x)) ∈R. To train the FANN to solve MCDM is best accomplished if F (f(x))> F (f(y)). In general, the neural network function F: Rq→R, depends on weights and other parameters of the neural network, is supported by following propositions. Proposition 1: an MCDM problem p=(X, f) with q objective functions, then utility function of alternatives F(f(x)) can be approximated by a three layers FANN with back- propagation learning algorithm. In this case, the FANN would have one output and q inputs where q is the number of criteria. Appling ANN to solve MCDM, the monotonous increasing characteristic of BP FANN is the base to model utility function F(f(x)). But in the ANN establishment process, we do not make sure the monotone feature through analysis the network topology. But according to the proposition 2 and 3, we can utilize the network connection weights to realize this condition. Proposition 2: Given that all node functions in an FANN are non-decreasing, an assessed FANN function is a non-decreasing function with respect to its inputs f(x), if its weights on the links are required to be non-negative. Proposition 3: Given that the node function in a FANN is a non-increasing function, an assessed FANN function is a non-increasing function with respect to its inputs f(x), if its weights on the links are required to be non-negative. In order to solve MCDM problems by FANN, we can require that all weights of FANN be non-negative while FANN is being trained. If a weight wij is going to be negative, let wij be equal to zero or a small positive number.

3 Hybrid MCDM We develop some methods to improve the ANN performance, including determine the input-output structure, standardize input nodes value, provide enough observations in universal flexible form, and determine the appropriate initial input weights. The

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Fig. 1. The framework of applying improved FANN to solve MCDM

procedure (as shown in Fig. 1) of hybrid MCDM method consists of 10 steps, described as follows: (1) Identify the set of alternatives. Alternatives refer to objects to be compared with. (2) Identify criteria by reviewing the related references and interviewing the domain experts to evaluate the alternatives. (3) Determine the ANN input-output nodes, utilizing AHP multicriteria hierarchical structure. The input layer nodes should be equal to number of the evaluation criteria and the comprehensive evaluation result is the output, so the ANN output layer has only one node. (4) Conduct a questionnaire to investigate the criteria weights. The pairwise comparison is recorded by triangular fuzzy number, with peak (or center) m, left width l and right width u, defined by a triplet (l, m, u) [13]. (5) Determine the appropriate ANN initial input weights utilizing Fuzzy AHP. The most popular activation function is sigmoid or logistic, nearly linear in the central part, which are suitable for MCDM application for mild regularity conditions: continuous, bounded, differentiable and monotonic increasing. But the initial weights will affect training time, whether the training trap into the local minimum or converge. That is to say, if the initial weights is bigger, the weighted input-hidden valued will go to the saturation area of sigmoid function and make the derivative small, then the modified weights tend to be 0 and the training process will stop. In order to

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improve the convergence capability in ANN training process, we apply Fuzzy AHP method to calculate the criteria weights [14] and use them as the ANN initial weights with practical definition. Triangular fuzzy number Mij, (i,j=1,2,…,n) expresses the pairwise comparison between ith criteria and jth criteria. With the fuzzy set theory, the comprehensive fuzzy grade of the ith criteria (Si) is: Si =

n

∑M j =1

n

ij

⊗ [∑

i =1

n

∑M j =1

ij

]−1

(4)

Si is used to compute the criteria rank. S1= (l1, m1, u1) and S2= (l2, m2, u2). The probability of S1≥ S2 , denoted by V(S1≥ S2) is defined as follows:

V (S1 ≥ S2 ) = 1 when m1 > m2 l2 − u1 ⎧ , l2 ≤ u1 ⎪ V (S1 ≥ S2 ) = ⎨ (m1 − u1 ) − (m2 − l2 ) when m1 ≤ m2 ⎪ 0 , other ⎩

(5)

Let Ai presents the ith criteria, then ith criteria fuzzy AHP weight w’(Ai) is:

w ′( Ai ) = min V ( S i ≥ S k ) ( k = 1, 2, ..., n , k ≠ i )

(6)

Use function (7) to normalize w’(Ai) (i=1,2,…,n)

w( Ai ) =

w′( Ai )

n

∑ w ′( A j )

(i = 1, 2,..., n )

(7)

j =1

(6) Collect performance data. For quantitative criteria, data are collected through some government publications and actual investigation; for qualitative criteria, a questionnaire survey is conducted to determine the performance value of alternatives to the qualitative criteria with five points method (1 -very bad, 2- bad, 3-medium, 4good, 5-very good). (7) Normalize the performance values as input nodes for ANN learning and test. The performance values should be transformed into dimensionless units to solve: 1) criteria are conflict and non-commensurability, 2) when MCDM criteria are used as the ANN input nodes, the small performance value to one criterion may be submerged by the great performance value to another criterion, but in fact, the first criterion may be more important, with the higher weight than the second one, and 3) when the input value is between [0,1], the Sigmoid like activation functions will be more flexible and efficient. The following normalization method is suggested: use (8.1) for benefit criteria (the higher the better), use (8.2) for cost criteria (the lower the better), use (8.3) for moderation criteria (the more moderate the better).

rij =

xij max( xij ) 1≤i ≤q

(8.1)

rij =

min(xij ) 1≤i ≤q

xij

(8.2)

(8)

A Hybrid MCDM Method for Route Selection of Multimodal Transportation Network

rij = rij =

xij − min( xij ) avg ( xij ) − min( xij ) max( xij ) − xij max( xij ) − avg ( xij )

379

when rij ∈[min( xij ), avg( xij )] (8.3)

when rij ∈[avg ( xij ),max( xij )]

(8) Increase the data samples for training ANN If there are enough observations with flexible form and universal approximation property used in the training set, then in the ANN test part, the network output will make smaller error with respect to the real value and we can make conclusion that this ANN has a good generalization property. But in general, there may be only finite discrete alternatives in MCDM problem. In order to make ANN memory more training patterns, we should produce some reasonable training pairs for ANN training and improve the ANN generalization performance [15]. Supposing the best alternative should have the highest criteria value and the worst alternative should have the lowest ones, recommend by TOPSIS concept [16], we develop the function (9) (10) to produce the relative highest performance value r+ and lowest one r-, respectively. Then, use the arithmetic average function (11) to compute the middle value r*. Repeat this process many times, we can produce suitable samples and promise monotonous increasing characteristic as input-output pairs for training ANN.

r + = { r1+ , r2+ , ..., rq+ }

r j+ = max rij

(9)

r − = {r1− , r2− ,..., rq− } r j− = min rij

(10)

i =1,... p

i =1... p

−

rq+ + rq r+ + r r+ + r } r = {r , r ,..., r } = { 1 1 , 2 2 ,..., 2 2 2 −

*

* 1

* 2

* m

−

(11)

(9) Train the ANN and test generalization performance 1) Divide data samples (the practical and produced data) into two subsets randomly: training and test set. 2) Select ANN architecture and set training parameters. 2.1) Determine the number of network layers: Kolmogrov theory has proved that an ANN with one hidden layer can compute any continuous approximation, so in this paper, we establish an ANN with three layers, one input, one hidden and one output layer. 2.2) The number of input and output nodes is determined by the practical application referred in step 3. 2.3) The initial input weights are obtained in step 5. 2.4) Set learning rate and momentum, maximum allowed error and maximum training epochs. 3) When reach maximum allowed error or maximum training epochs, stop training. 4) Repeat these steps using different architectures and training parameters to select the best ANN network architecture. In the training process, we should set all weights non-negative to guarantee the solution efficient. 5) Use the test part data to test ANN generalization performance. (10) Establish the best ANN model for MCDM and apply this ANN for future decision making and choose the best new alternative.

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4 Application – Select Path in Multimodal Transportation In order to select the “best” way from a certain origin to a specific destination through a multimodal transportation network, we make 15 criteria clustered into 6 groups by reviewing the related references and interviewing the domain experts. The hierarchical MCDM system is shown as Fig 2. The first level is the selection of the optimal path in multimodal transportation. The following layer consists of the decision factors for evaluating the potential route. After comprehensive consideration, six main aspects are extracted, the total transport cost, the total transport time, the transport quality, the transport tools, the service in the transport, the social benefit. The sub-factors in the level followed consist 15 sub-criteria.

Fig. 2. Hierarchical MCDM structure for the optimal path selection in multimodal transportation Table 1. The weights of criteria Si SA SB SC SD SE SF

li 0.024 0.137 0.058 0.069 0.182 0.044

mi 0.081 0.253 0.115 0.154 0.311 0.086

ui 0.197 0.498 0.240 0.278 0.548 0.185

w’ 0.06 0.84 0.23 0.38 1.00 0.02

w 0.024 0.332 0.091 0.150 0.395 0.008

Use the same method, we can calculate the sub-criteria (A1-F2) weights, and further get the synthesis weights (0.303, 0.050, 0.014, 0.193, 0.064, 0.034, 0.035, 0.123, 0.034, 0.086, 0.020, 0.007, 0.007, 0.027, 0.005). We can use the A-F as the ANN input nodes and their weights as the input weights correspondingly. Also, if we want to use 15 sub criteria as the ANN input nodes, we should use the sub-criteria weights as the ANN initial weights. This paper uses the six criteria (A-F) in first layer to form a simple ANN structure. The performance values to the six criteria are cumulated by the multiplication of the sub- criteria weight with normalized performance value to sub- criteria correspondingly.

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We put out a practical problem in the multimodal transportation. The multimodal network is shown as Fig. 3. There are six regions/ terminal (t1 to t6) in this network and the transport modals include air, water, rail, and road.

Fig. 3. The multimodal network (The binary real line denotes road mode, the single real line denotes water mode, the binary dashed line denotes air mode and the single dashed line denotes rail mode.)

From t1 to t6, 36 paths can be selected at present (Table 2.). Table 2. Remark 36 paths Rmn (m= {r,o,i} and n={1,2,...,12})(The first subscript denotes three connections from t1 to t2, r=rail, o=road, i=air, and the second subscript denotes the 12 paths from t2 to t6) r o i 1 2

t1-(rail)- t2 t1-(road)- t2 t1-(air)- t2 t2- t4 -(rail)- t6 t2- t4-( road)- t6

3 4 5 6 7

t2- t4-(air)- t6 t2- t3- t4-(rail)- t6 t2- t3- t4-( road)- t6 t2- t3- t4-(air)- t6 t2- t5 -(rail)- t6

8 9 10 11 12

t2- t5 -( road)- t6 t2- t5 -(air)- t6 t2- t3- t5 -( rail)- t6 t2- t3- t4-(road)- t6 t2- t3- t4-(air)- t6

The normalized criteria data (source: [1]) of these paths are shown in Table 3. Table 3. The normalized criteria data NO.

Paths

A

B

C

D

E

F

results

1 2 3 4 5 … 36

Rr,1 Rr,2 Rr,3 Rr,4 Rr,5 … Ri,12

1 0.11 0.66 0.29 0.33 … 1

1 0.05 0.49 0.49 0.75 … 0.75

1 0.11 0.67 0.89 0.89 … 0.89

1 0.07 0.66 0.92 0.82 … 0.82

1 0.12 0.6 0.4 0.8 … 0.8

0.87 0.07 0.67 0.87 0.8 … 0.8

0.96 0.303 0.69 0.562 0.79 … 0.83

Use the functions (9)-(11) to increase the data samples for ANN training. In Table 4, we list some input-output data pairs produced repeatedly. We divide the data into two parts, the first 20 alternatives results listed in Tab. 3 and some pairs produced in Tab. 4 as the ANN training data pairs, the second 16 data pairs as test part. The ANN has 3 layers and 6 input nodes and one output nodes, use sigmoid function as the input-hidden activation function and linear function as the hidden-output

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A 1 … 0.7775 … 0.555

B 1 … 0.7625 … 0.525

Input nodes C D 1 1 … … 0.7775 0.7675 … … 0.555 0.535

E 1 … 0.78 … 0.56

F 1 … 0.7675 … 0.535

Output results 1 … 0.82575 … 0.6515

… 0.3325 … 0.11

… 0.2875 … 0.05

… 0.3325 … 0.11

… 0.34 … 0.12

… 0.3025 … 0.07

… 0.47725 … 0.303

… 0.3025 … 0.07

activation function, back propagation as learning algorithm. Set learning rate and momentum. Determine the training stop condition: set maximum sum-squared error and the maximum training epochs. Then calculate the mean square errors (MSE) of ANN performance with the test data, when MSE in the train and test parts are all satisfied, this FANN model can be applied to choose the best alternative in MCDM problem. Since the best ANN system is established for path selection in multimodal transportation, which presents the nonlinear relation of six criteria with MCDM solution, we put all the 36 possible paths into the ANN (listed in Table 3.) and find the t1(rail)- t2-(water)- t3-(water)- t4-(rail)- t6 is the optimal route, as the same as the practical condition. This ANN can be applied for future decision making and choose the best new alternative.

5 Conclusion According to the characteristic and the basic principles of the multimodal transportation system, the path selection is a MCDM problem, which is associated with a set of conflicting and non-commensurable criteria. Combining the Fuzzy AHP and ANN, we propose a hybrid MCDM method. In order to make the proposed ANN more efficient, we not only use Fuzzy AHP to determine the appropriate initial input-hidden weights, but also some other improvement methods are developed. The improved ANN with error backpropagation is applied to study the relationship between the criteria and the alternatives performance. This hybrid MCDM framework, which is an accurate, flexible and efficient system, can be used in many other transportation applications.

Reference 1. Su, Y., Li, T.Z.: A Route Choice Method for International Multi-modal Transportation. Journal of Transportation Systems Engineering and Information Technology 16, 91–94 (2006) (in Chinese) 2. Rondinelli, D., Berry, M.: Multimodal Transportation, Logistics, and the Environment: Managing Interactions in a Global Economy. European Management Journal 18, 398–410 (2000)

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3. Southworth, F., Peterson, B.E.: Intermodal and International Freight Network Modeling. Transportation Research Part C l.8, 147–166 (2000) 4. Arnold, P., Peeters, D., Thomas, I.: Modeling a Rail/Road Intermodal Transportation System. Transportation Research Part E 40, 255–270 (2004) 5. Nijkamp, P., Reggiani, A., Tsang, W.F.: Comparative Modeling of Interregional Transport Flows: Applications to Multimodal European Freight Transport. European Journal of Operational Research 155, 584–602 (2004) 6. Xie, F., Zhang, N., Ji, S.W.: Study on Multi- transportation Path Selection Based on AHP. Logistic technology 29, 13–17 (2006) (in Chinese) 7. Crainic, T.G., Rousseau, J.M.: Multicommodity, Multimode Freight Transportation: a General Modeling and Algorithmic Framework for the Service Network Design Problem. Transportation Research B 3, 225–242 (1986) 8. Sun, M., Stam, A., Steuer, R.E.: Interactive Multiple Objective Programming Using Tchebycheff Programs and Artificial Neural Networks. Computers & Operations Research 27, 601–620 (2000) 9. Zhou, Y., Malakooti, B.: An Adaptive Feedforward Artificial Neural Network with the Applications to Multiple Criteria Decision Making. In: Proceedings of IEEE International Conference on Systems, Man, Cybernetics, Los Angeles, CA, pp. 164–169 (1990) 10. Gal, T., Hanne, T.: Nonessential Objectives within Network Approaches for MCDM. European Journal of Operational Research 168, 584–592 (2006) 11. Sun, X.D., Li, Z.B., Chen, F.M.: esearch on Multiple Attribute. Synthectical Evaluation Methods based on Artificial Neural Network. Journal of Zhengzhou institute of light industry 18, 11–14 (2003) (in Chinese) 12. Qu, L.L., Chen, Y.: An Interactive Integrated MCDM based on FANN and Application in the Selection of Logistic Center Location. In: 2007 International Conference of Management Science and Engineering (14th), Harbin, pp. 162–167 (2007) 13. Chang, D.Y.: Applications of the Extent Analysis Method on Fuzzy AHP. European Journal of Operational Research 95, 649–655 (1996) 14. Qu, L.L., Chen, Y.: Evaluating the Selection of Logistics Centre Location Using Fuzzy MCDM Model Based on Entropy Weight. In: The 6th World Congress on Intelligent control Automation, Dalian, pp. 7128–7132 (2006) 15. Qiu, C.G., Liu, Y.S.: Neural Network Based Multiple Criteria Decision Making. Journal of Beijing Institute of Technology 20, 65–68 (2000) (in Chinese) 16. Shih, H.S., Shyur, H.J., Lee, E.S.: An Extension of TOPSIS for Group Decision Making. Mathematical and Computer Modeling 45, 801–813 (2007)

Function Approximation by Neural Networks Fengjun Li School of Mathematics and Computer Science, Ningxia University, 750021 Yinchuan, People’s Republic of China [email protected] [email protected] [email protected]

Abstract. Neural networks are widely used to approximating continuous functions. In order to study its approximation ability, we discuss the constructive approximation on the whole real lines by an radial basis function (RBF) neural network with a fixed weight. Using the convolution method, we present a family of RBF neural networks with fixed weights that are able to uniformly approximate continuous functions on a compact interval. Our method of proof is constructive. And this work provides a method for function approximation. Keywords: approximation, radial basis function, neural networks, convolution.

1

Introduction

The use of radial basis function (RBF) neural networks to approximate functions has been the focus of much research. Recent results ([1-8]) have shown that feedforward multilayered RBF neural networks with one hidden layer can serve as a universal approximator to arbitrary continuous functions. Binfan and Jennie ([8]) proved that an RBF to approximate any C 2 function with support on the unit hypercube Im = [0, 1]m in the sense of best approximation. Li ([9]) showed a constructive method that any multivariate function and all its existing derivatives can be simultaneously approximated by an RBF neural network, and proved that the error of the approximation was inversely proportional to the number of hidden neurons. Duy and Cong ([10]) presented of a numerical approach, based on RBF neural networks, for the approximation of a function and its derivatives. In previous papers ([1-11]) the weight in RBF neural networks vary and so the results are very difficult to be applied in engineering and other fields. Naturally, we put forward the following question: can we fix the weight in RBF neural networks to make them facilitate in practical application? In this paper, we answer this question and prove that an RBF neural network with a fixed weight can approximate any function in C0 on R, where C0 denotes the collection of all continuous functions that converge to 0 as |x| → ∞. Using convolution method, we show that any continuous functions in a compact interval F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 384–390, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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can be approximated by an RBF neural network with a fixed weight. Our method of proof is constructive. The paper is organized as follows. In Section 2, we obtain an approximation theorem in C0 by using a constructive method. And then, in Section 3, we introduce the convolution and prove a theorem by using it. We also prove another approximation theorem about RBF neural networks on a bounded closed interval [a, b] on R. Finally, in Section 4, we briefly summarize our current research.

2

Constructive Function-Approximation by RBF Neural Networks with a Fixed Weight in C0

In general, it is impossible to approximate an arbitrary continuous function on R by an RBF neural network. For example, f (x) = x is a continuous function on R and |f (x)| → ∞ as |x| → ∞. In other words, f can not be approximated by a neural network with an radial basis activation function on R with respect to the sup norm. Thus, if we want to f can be approximated by a neural network with an radial basis activation function according to given accuracy, then we need some limits on the target function. From this view, the following theorem is obtained. Theorem 1. Let arbitrary  > 0 be given and let σ be an RBF on R. If f ∈ C0 , then there exists constants ci , θi ∈ R and positive integers W, M such that M      ci σ(W x + θi ) < . f (x) −

(1)

i=1

Proof. Since f ∈ C0 , there exists a positive integer K such that |f (x)| < /2 for |x| ≥ K. From mathematic analysis, we know that f is uniformly continuous on R. Therefore, there exists δ > 0, if any x, y ∈ R with |x − y| < δ, then |f (x) − f (y)| < /(2M 2 ), where M = max{K, [1/δ] + 1}, and “[x]” denotes the integer part of number x. Divide the interval [−M, M ] into 2M 2 equal segments, each with length of 1/M . Let (2) −M = x0 < x1 < · · · < x2M 2 = M and θi = (xi + xi+1 )/2, (0 ≤ i ≤ 2M 2 − 1).

(3)

Since σ is an RBF, there exists a positive integer constant L such that |σ(x)| < 1/(2M ) for |x| ≥ L. Choose a positive integer W such that W/(2M ) > L. Now, we construct a neural network as follows N (x) =

2 2M −1

(f (xi+1 ) − f (xi ))σ(W (x − θi )).

i=0

(4)

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If |x| ≥ M , then |W (x − θi )| ≥ L, and hence, |σ(W (x − θi ))| < 1/(2M ) for i = 0, 1, · · · , 2M 2 − 1. Therefore, we obtain |f (x) − N (x)| ≤ |f (x)| + |N (x)| 2M   ≤ + |f (xi+1 ) − f (xi )||σ(W (x − θi ))| 2 i=0   ≤ + 2 2M ≤ 2

If |x| ≤ M , then there must exist a nonnegative integer j (0 ≤ j ≤ 2M 2 − 1) such that x ∈ [xj , xj+1 ]. So we can see that W (x − θi ) ≥ L for i = 0, 1, · · · , j − 1 and W (x − θi ) ≤ −L for i = j + 1, j + 2, · · · , 2M 2 − 1. In addition, we can easily see that j−1  (f (xi+1 ) − f (xi ))σ(W (x − θi )) i=0 j−1  = (f (xi+1 ) − f (xi ))(σ(W (x − θi )) − 1) + f (xj ) − f (x0 ) (5) i=0

From the fact that σ is a bounded function, we have |f (x) − N (x)| = |f (x) −

j−1 

(f (xi+1 ) − f (xi ))σ(W (x − θi ))

i=0

− (f (xj+1 ) − f (xj ))σ(W (x − θj )) +

2 2M −1

|f (xi+1 ) − f (xi )||σ(W (x − θi ))|

i=j+1

≤ |f (x) − f (xj )| + |f (x0 )| +

j−1 

|(f (xi+1 ) − f (xi ))||σ(W (x − θi )) − 1|

i=0

+ |(f (xj+1 ) − f (xj ))||σ(W (x − θj ))| +

2 2M −1

|f (xi+1 ) − f (xi )||σ(W (x − θi ))|

i=j+1

1 (|σ(W (x − θi )) + 1|) M2 ≤ A, ≤

where A is a positive constant. This finishes the proof of Theorem 1.

3



Convolution and RBF Neural Networks Approximation

By using of the Theorem 1 and the property of convolution, we can study the universal approximation of some special type of RBF neural networks.

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3.1

387

Convolution and Its Property of Convergence

For two continuous functions f and g defined on R, their convolution is defined by  (f ∗ g)(x) = f (y)g(x − y)dy. (6) R

For any x ∈ R, we define a function  1 − 1−x 2 , if |x| < 1, F (x) = de 0, if |x| ≥ 1,

(7)

 where d is chosen such that R F (x)dx = 1. Then, F ∈ C0∞ , where C0∞ denotes the class of functions in C0 and they have any order derivatives. For any positive integer k, we define (8) Fk (x) = kF (kx).  Then, we obtain R Fk (x)dx = 1 and Fk (x) ∈ C0∞ . By using the property of convolution, we obtain the following result. Theorem 2. If f is a uniformly continuous function on R, then Fk ∗f uniformly converges to f on R. Proof. Since f is uniformly continuous, then, for any  be given, there exists a positive integer M such that |f (x)−f (y)| <  for any x, y ∈ R with |x−y| < 1/M . For any x ∈ R and any integer k(k ≥ M ), we obtain        (Fk ∗ f )(x) − f (x) =  Fk (y)f (x − y)dy − Fk (y)f (x)dy   R   R    kF (ky)(f (x − y) − f (x))dy  =  R  z   ≤ F (z)||f (x − ) − f (x)dz k R  1   z |F (z)|f (x − ) − f (x)dz = k −1  1 ≤ |F (z)|dz = .

−1

This finishes the proof of Theorem 2. 3.2



Approximation by an RBF Neural Network with a Fixed Weight on a Compact Interval

In this subsection, by using the above conclusions, we prove a Theorem on the universal approximation of an RBF neural network with a fixed weight. The

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obtained result will qualitatively characterize approximation capability of the RBF neural networks to any continuous functions on a bounded closed interval. The corresponding result is as follows. Theorem 3. Let any  > 0 be given and let f be a continuous function on a bounded closed interval [a, b] of R. If σ is a measurable RBF on R, then there exists constants ci , θi ∈ R and positive integer W, M such that M      (x) − ci σ(W x + θi ) < . f

(9)

i=1

Proof. We construct a uniformly continuous function f˜ on R as follows: ⎧ f (a)x + (a − 1)f (a), if x ∈ [a − 1, a], ⎪ ⎪ ⎨ f (x), if x ∈ [a, b], f˜(x) = −f (b)x + (b + 1)f (b), if x ∈ [b, b + 1], ⎪ ⎪ ⎩ 0, if x ∈ (−∞, a − 1] ∪ [b + 1, ∞).

(10)

˜ From Theorem 2, we can see that Fk ∗f˜ uniformly  converges to f on R. Therefore, ˜ ˜ Fk ∗ f uniformly converges to f on [a, b]. Since R Fk (x − y)f (y)dy < ∞ for each positive integer k, we know that the convolution (Fk ∗ f˜)(x) is approximated by an Riemann sum. Hence, for any positive integer k, there exists a positive integer Mk and constants yi , ci for i = 1, 2, · · · , Mk such that Mk       ci Fk (x − yi )f˜(yi ) < , (Fk ∗ f )(x) − 3 i=1

where yi ∈ R for i = 1, 2, · · · , Mk . From Eqs.(7) and (8), we can see that Fk ∈ C0 . Let

M k

˜

ci f (yi ) i=1 αj,k , βj,k ∈ R

(11)

= B. By using

Theorem 1, we know that there exists constants and a positive integer K such that     Fk (x − yi ) − . (12) βj,k σ(K(x − yi ) + αj,k ) < 3B j,k

As Fk ∗ f˜ uniformly converges to f on [a, b], so we can choose a positive integer k such that  (13) |f (x) − (Fk ∗ f )(x)| < . 3 From Eqs. (11), (12) and (13), we have   M  k ˜   ci f (yi ) βj,k σ(K(x − yi ) + αj,k ) f (x) − i=1

j,k

Mk      ≤ |f (x) − (Fk ∗ f )(x)| + (Fk ∗ f )(x) − ci Fk (x − yi )f˜(yi ) i=1

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Mk Mk       + ci Fk (x − yi )f˜(yi ) − ci f˜(yi ) βj,k σ(K(x − yi ) + αj,k ) i=1

< .

i=1

j,k



Theorem 3 shows that an radial basis activation function with a fixed weight is sufficient for the neural network approximation of a continuous function on a compact subset of R.

4

Conclusion

In approximation by neural networks, we have two main problems—“density” and “complexity”. In this paper, we study the density problem. We introduce convolution and prove that any RBF neural networks with a suitable fixed weight can universally approximate any continuous functions in C0 or over a compact interval if these neural networks have sufficient 2M 2 hidden neurons. The significance of these results lies in that we show potential approximation of neural networks computation in functions approximation. Our thought leads us to study the complexity problem by neural networks with a fixed weight in the future.

Acknowledgement This work was supported by NSFC projects under contract Nos. 70531030 and 10726040.

References 1. Franco, S., Ah, C.T.: Universal Approximation Using Feedforward Neural Networks: A Survey of Some Existing Methods And Some New Resluts. Neural Networks 1, 15–37 (1998) 2. Vera, K.: Approximation of Functions by Perceptron Neural Networks with Bounded Number of Hidden Units. Neural Networks 5, 745–750 (1995) 3. Nahmwoo, H.: An Approximation by Neural Networks with a Fixed Weight. Neural Computers and Mathematics with Applications 47, 1897–1903 (2004) 4. Park, J., Sandberg, I.W.: Universal Approximation Using Radial-Basis-Function Network. Neural Comput. 3, 246–257 (1991) 5. Leonard, J.A., Kramer, M.A., Ungar, L.H.: Using Radial Basis Functions to Approximate a Function And Its Error Bounds. IEEE Trans. Neural networks 3, 622–630 (1992) 6. Chen, T.P., Chen, H.: Approximation Capability to Function of Several Variables, Nonlinear Functionals, And Operators by Radial Basis Function Neural Networks. IEEE Trans. Neural Networks 6, 904–910 (1995) 7. Mhaskar, H.N., Micchelli, C.A.: Approximation by Supersition of Sigmoidal And Radial Basis Functions. Advances Applied Mathematics 13, 350–373 (1992) 8. Binfan, L., Jennie, S.: The Best Approximation to C 2 Functions And Its Error Bounds Using Regular Center Gaussian Networks. IEEE Trans. Neural networks 5, 845–847 (1994)

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9. Li, X.: On Simultaneous Approximations by Radial Basis Function Neural Networks. Applied Mathematics and Computation 95, 75–89 (1998) 10. Duy, N.M., Cong, T.T.: Approximation of Function And Its Derivatives Using Radial Basis Function Networks. Applied Mathematical Modelling 27, 197–220 (2003) 11. Hartman, E.J., Keeler, J.D., Kowalski, J.M.: Layered Neural Networks with Gaussian Hidden Units as Universal Approximators. Neural Comput. 2, 210–215 (1990) 12. Mulgrew, B.: Applying Radial Basis Functions. IEEE Signal Processing, 50–65 (May 13, 1996) 13. Sanner, R., Sloitine, J.: Gaussian Networks for Direct Adaptive Contral. IEEE Trans. Neural Networks 3, 837–863 (1992) 14. Orr, M.: Regularization in the Selection of Radial Basis Function Centers. Neural Comput. 7, 606–623 (1995)

Robot Navigation Based on Fuzzy RL Algorithm Yong Duan, Baoxia Cui, and Huaiqing Yang School of Information Science and Engineering, Shenyang University of Technology, Shenyang 110178, China [email protected]

Abstract. This paper focused on the problem of the autonomous mobile robot navigation under the unknown and changing environment. The reinforcement learning (RL) is applied to learn behaviors of reactive robot. T-S fuzzy neural network and RL are integrated. T-S network is used to implement the mapping from the state space to Q values corresponding with action space of RL. The problem of continuous, infinite states and actions in RL is able to be solved through the function approximation of proposed method. Finally, the method of this paper is applied to learn behaviors for the reactive robot. The experiment shows that the algorithm can effectively solve the problem of navigation in a complicated unknown environment. Keywords: Reinforcement learning, Robot navigation, T-S fuzzy neural network, Q-learning.

1 Introduction The mobile robot navigation has become a hot research subject in robotics and artificial intelligence field, recently. The robot navigation issue is that the robot can choose the correct action and accomplish the task of reaching the destination without collision, according to the environment information perceived by the sensor of robot. The traditional research methods for the reactive robot navigation usually depend on the environment model and expert knowledge. However, they are not easy to obtain when robot is running in unknown and dynamic environment. Reinforcement learning (RL) is an unsupervised learning method. It is applicable where no model of the environment is available [1,2]. RL allows robots to learn mapping from their states to their actions by reward (reinforcement signal) obtained through interacting with their environment. RL requires the agent to obtain the mapping from state to action. The aim is to maximize the future accumulated rewards received from the environment. In the case of application, the state and action space of RL are often large, which brings on the search space of training is overly large. Therefore, the agent is difficult to visit each state-action pair. To cope with the problem, some generalization approaches are used to approximate or quantize the state space, which aims to reduce the complexity of the search space. Neural network (NN) and fuzzy inference system (FIS) have widely approximate characteristic. So they can implement arbitrarily non-linear mapping F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 391 – 399, 2008. © Springer-Verlag Berlin Heidelberg 2008

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from inputs to outputs [3]. Recently, multi-layer feedforward neural networks have been applied to perform RL algorithms [2,4]. Furthermore, Glorennec and Jouffe proposed Q-learning algorithm based on FIS [5]. In this paper, a reinforcement learning algorithm based on Takagi-Sugeno (T-S) model fuzzy neural network is proposed. T-S fuzzy neural network effectively integrates the FIS and NN and has the advantage of them. The presented method can solve the “curse of dimensionality” problem of RL. Moreover, this method is used to learning the robot navigation behaviors. The experiment results can denote the validity and self- adaptability. It can make robot accomplish the navigation tasks in complicated environment.

2 RL Based on T-S Fuzzy Neural Network 2.1 Q-Learning Algorithm In Markov decision process (MDP), the agent is able to perceive the state set S = {si | si ∈ S } of the environment, and there is the corresponding action set A = {ai | ai ∈ A} . At every time step t , the agent senses the current state st and chooses an action at . Through implementing this action, the agent can receive the feedback reward rt and achieve the new state st +1 . The aim of RL is to achieve an optimum control scheme π : S → A , which makes the sequence <state, action> maximize cumulative rewards. Q-learning is an important algorithm of RL [1]. In Q-learning, the idea is to directly optimize Q-function, which can be calculated recursively. The function of Q( s, a ) is to represent the evaluation of state-action. Q-learning is given by [6]:

Qˆ ( st , at ) = Q( st , at ) + η ⋅ [rt + γ ⋅ max Q( st +1 , at +1 ) − Q ( st , at )] . a

(1)

Where, the value of η is a learning rate. γ denotes the discount factor. rt is the reinforcement signal. This means only perform iterative learning of current Q-function the global optimal sequence actions can be selected. The equation 1 updates the current Q-function based on the evaluation value of next state. When the convergence of Q-function is achieved, the optimal policy can be confirmed 2.2 Architecture of Fuzzy RL System

The state variables of RL algorithms serve as the inputs of T-S fuzzy neural network. The consequent variables of T-S network are the linear combination of input variables. They are evaluation value Q corresponding with the action variables of the RL algorithms. T-S fuzzy neural network is used to perform the mapping from state space to action space of RL. The reinforcement signal (reward) function from the environment constitutes the error cost function of the learning system. The RL system based on T-S network includes the antecedent network and consequent network. The antecedent network has four layers. Nodes at layer one are input

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nodes which represent input linguistic variables. The nodes in this layer transmit input state variables of RL directly to the next layer. Each node at layer two performs a membership function. The output value of this node is the degree of membership of the state variable. We use bell-shaped function as the membership function of the fuzzy rules. The output variable μi j of the jth term of the ith input linguistic variable xi is calculated as follows:

μi j = exp[ fij(2) ] = exp[−( xi − cij )2 / δ ij2 ] .

(2)

Where cij and δ ij are, respectively, the mean and the variance of the bell-shaped function. A layer-three node is a rule node which represents one fuzzy logic rule. The output variables α j are antecedent matching of fuzzy rules. The links at layer four perform the normalization operation to integrate the fired rules which have the same consequent. The output variable of the node is given by: m

f j(4) = α j = α j / ∑ α i .

(3)

i =1

The consequent network part has l paratactic sub-networks with the same structure. Each sub-network produces an output variable which denotes the Q value of RL. The first layer of sub-network is input layer. The input value of the first node in this layer is 1. It offers the constant item in consequent of fuzzy rules. Each node at layer two expresses one fuzzy rule, which is used to calculate the consequent of each rule, that is: n

qij = ∑ p ijk sk = p ij 0 + pij1 s1 + " + pijn sn k =0

.

(4)

The third layer of sub-network produces the output of learning system. Consequently, the Q value of each candidate action of RL, that is: m

Qi = ∑ α j qij j =1

(5) .

2.3 Error Back- Propagation Learning Algorithms

According to architecture of learning system, the tuned parameters are c ij and σ ij of fuzzy membership function at the second layer of the antecedent network, and link weight p kji of the consequent network. In the learning of the function approximation system, how to calculate the gradient of the function approximation error is the major problem. The direct gradient algorithms can retain the fast learning speed, but fail to converge to an answer. The residual gradient algorithms are shown to always converge, but it may converge very

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slowly in some cases. Contrasting to above two algorithms, the residual algorithms can both guarantee convergence and converge quickly. We use the mean squared Bellman residual as the error cost function, which is defined to be [7]:

Et =

1 2 ∑ [rt + γ ⋅ max Qi (st +1 , at +1 ) − Qi (st , at )] . a ∈ A 2 i

(6)

The weight of the residual algorithms is updated as follows: ∂Q (st +1 , at +1 ) ∂Q (st , at ) ∂E = ϑt ⋅ [φγ − ]. ∂w ∂w ∂w

Δw = Where ϑt = −

∂Et = rt + γ ⋅ max Q(st +1, at +1 ) − Q(st , at ) . a∈A ∂Q( st )

(7)

β is the learning gain. The

real number φ lies in the range [0,1] . Firstly, the learning method of the weight parameters p kji of the consequent network is discussed. ∂E ∂E ∂Qk ∂qkj = −[rt + γ ⋅ max Q (st +1 , at +1 ) − Q (st , at )]α j si . = k a∈ A ∂p ji ∂Qk ∂qkj ∂p kji

(8)

Hence the update formula is as follows: p kji (t + 1) = p kji (t ) − β =

p kji (t ) +

∂E ∂p kji

(9)

β ⋅ [rt + γ ⋅ max Q(st +1 , at +1 ) − Q(st , at )]α j si . a∈ A

Now parameters p kji can be regard as fixedness. By back propagating error, the intermediate variables are given by:

δ i(5) = − δ (4) =− j

∂E = rt + γ ⋅ max Qi (st +1 , at +1 ) − Qi (st , at ) . a∈ A ∂Qi ( s ) ∂Qi ( s ) ∂f j(4)

l

= ∑ δ i(5) qij . i =1

N

N

k =1

l =1,l ≠ j

2 (4) δ (3) ∑ αi − j = [1/( ∑ α k ) ] ⋅ (δ i

N

δ ij(2) = ∑ δ k(3) μi j . k =1

N

∑

m =1, m ≠ j

δ m(4)α i ) .

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( 2)

f ij ∂Q(s) ∂Q( s) f ij . . = = −δ ij( 2) . ( 2) ∂w ∂ w ∂w ∂f ij

(10)

According to the equation 10, we have 2[ si (t ) − cij (t )] ∂Q(st , at ) = −δ ij(2) ⋅ . ∂cij (t ) σ ij2 (t )

(11)

2[ si (t ) − cij (t )] ∂Q(st , at ) . = −δ ij2 . ∂σ ij (t ) σ ij3 (t )

(12)

2

By implementing the output action a ( st ) of RL system, the state st transform into the new state st +1 . We also can calculate

∂Q(st +1 , at +1 ) ∂Q(st +1 , at +1 ) and . ∂cij (t ) ∂σ ij (t )

According to equation 10, the mean and the variance of the bell-shaped function are updated by: cij (t + 1) = cij (t ) − β ⋅

∂Et . ∂cij (t )

(13)

σ ij (t + 1) = σ ij (t ) − β ⋅

∂Et . ∂σ ij (t )

(14)

According to above algorithms, the parameter learning and tuning of RL based on T-S fuzzy network can be implemented. After learning, agent senses the environment state information. Q value corresponding with each action is obtained by network. Then the action with the maximal Q value is selected. Thereby agent implements this action.

3 RL Based on Embedding Expert Knowledge RL method searches optimal strategy without antecedent knowledge. When learning tasks are complex or working environment dynamic changes, the convergence speed of learning is too slow. In the case of practical application, some expert knowledge can be obtained in advance. Therefore, RL system is built on the effective antecedent knowledge, the learning speed can be improved. The key problem of improving RL speed based on embedding expert knowledge is how to denote knowledge form and integrate knowledge and RL. The proposed RL method of this paper is used T-S fuzzy network perform learning process of RL. FIS can effectively make use of limited expert knowledge. So some fuzzy rules can be designed by expert knowledge. This available expert knowledge can not only greatly

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decrease learning complexity, but also direct interaction of learning agent and environment. Hence, learning speed and efficiency can be improved. The fuzzy rules of above learning system are composed of two parts: one part is designed through expert knowledge; other part is obtained through RL. Agent senses the environment state information, firstly, searching rules in expert knowledge base. If the rule is defined by expert knowledge, the action is outputted based on fuzzy rule. The output action acted on environment and got reward. If the reward is encouragement, the fuzzy rule is testified reasonable. So it need not modify. Contrarily, agent obtain punish, this rule is not sound. This rule will be removed to RL system and optimized. For the input state variables, the corresponding rule is not find in fuzzy rule base of expert knowledge. Then the fuzzy rule is learned through RL system. Therefore, the proposed RL based on T-S fuzzy neural network can make the best of existing expert knowledge to improve the learning efficiency.

4 Robot Navigation Based on RL In the research field of autonomous mobile robots, navigation is the core technology [8,9]. The robot model adopts a two wheel differential drive at its geometric center (see Fig.1). The control actions can be indicated by a traversal speed vc and rotational speed ωc .The sensors of robot are divided into three groups according to their overlay areas. Respectively, the distance of obstacles to the right, at the front, and to the left of the robot sensed. In every group, the distance between the robot and obstacles is the minimum of the sensor data, i.e. D min = min( d i ) . Input variables are the obstacle distances Dl , Dc , Dr , and the target orientation TD .

Fig. 1. Perceptive model of robot. θ is the angle between the moving direction of the robot and the line connecting the robot center with the target . L is the distance between robot and target.

The traversal speed vc and rotational speed ωc can be discretized into crisp set. The traversal speed vc can be denoted the three velocity values which are restricted the

maximal mobile velocity of the robot. The rotational speed ωc can be described the five

variables ωc = {ωc | −600 , −300 , 0,300 , 600 } . The various combinations of traversal

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speed v c and rotational speed ω form the action space of RL. The input variables Dl , Dc and Dr are described by three linguistic labels: NEAR, MED and FAR. The variable TD is expressed using five linguistic terms: NB, NS, ZE, PS and PB. The target orientation TD is defined in a range of (−180 0 ,+180 0 ] . The environment state information captured by sensors of the robot is fuzzified to the premise of the fuzzy rules base. The actions set of RL is composed of the different combinations of v c and ωc , which corresponding Q values are the output of T-S fuzzy network. The behaviors of robot include avoiding obstacles and approaching the target. To avoidance obstacles, it is natural to wish the robot is far away from obstacles. If the robot is close to obstacle, it will receive the punishment (negative reinforcement signal); On the contrary, the robot receives the bonus (positive reinforcement signal). To the behavior of approaching the target, when the distance of the robot and target is diminishing, the robot receives the bonus. Otherwise, it receives punishment. Synthesizing two kinds of behavior, the reinforcement signal is defined as follows:

rt

1.0 ­ °W ˜ ( D  d ) ° A t ® 0.5 ° °¯ 0.0

dt d DS DS  dt d DA dt ! DA and dT 1 d dT 0 RWKHUV

.

(15)

Where rt is instant reinforcement signal at the time step t . d t denotes the minimum obstacle distance of triple directions around the robot, i.e., dt = min{Dl , Dc , Dr } . The parameter τ is proportional gain. D S represents the safe distance threshold. When the distance of robot and obstacles is less than D S , the robot is considered collision. D A is the distance threshold of avoidance obstacles. In this range, the robot is able to effectively avoid obstacles. d T 0 and dT 1 are the distances of robot and target in successive time steps.

5 Experiments In order to demonstrate the effectiveness of the proposed RL method, the experiments are performed with the simulation and the real mobile robot Pioneer II. According to the previous section, the ultrasonic sensors of Pioneer II are divided into three groups. Each group sensors can measure the distance between the robot and the obstacles in the different directions. In order to increase the learning speed of the RL method and decrease the exhaustion of the real robot, we apply the proposed method to the robot that tries to learn the behaviors in the simulation environment. Then the learned results are tested with the real robot Pioneer II. The simulated robot is located in the complicated unknown environment to train the obstacle-avoidance and approaching target behavior. According to the equation 15, the robot receives the reinforcement signal. If the robot collides with obstacles,

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Fig. 2. Navigation trajectories in simulation. Black blocks are the obstacle. Red curves are robot’s trajectories.

Fig. 3. Navigation behavior of Pioneer II robot. It shows robot avoid the obstacles.

reaches the target or completes the trials, it will return the start state and perform a new learning stage. After learning, the mapping strategy from state space to action space of RL can serve as the controller of robot navigation. Figure 2 denotes the navigation trajectories of the robot in the unknown simulation environment. Figure 3 shows that the robot Pioneer II performs the wandering behavior in the real environment. The effectiveness of the proposed method is demonstrated through simulator and the real robot experiments. The robot with controller designed by proposed method can explore the environment without collision.

6 Conclusions In this paper, the fuzzy neural network based on T-S model and RL are integrated. The T-S network is used to implement Q-learning and perform the mapping from the state space of RL to Q values corresponding with the action variables of RL. The navigation behavior learning of the mobile robot based on proposed method is researched. The robot can master the behavior ability through interact with environment. The experimental results indicate the proposed method with the reasonable reinforcement signal function can complete effectively the learned tasks.

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References 1. Erfu, Y., Gu, D.B.: Multiagent Reinforcement Learning for Multi-robot Systems: A Survey. University of Essex Technical Report CSM-404, Department of Computer Science (2004) 2. Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction. MIT Press, Cambridge (1998) 3. Sun, Z.Q.: Intelligent Control Theory and Technology. Tsinghua University Press, Beijing (2000) 4. Claude, F.T.: Neural Reinforcement Learning for Behaviour Synthesis. Robotics and Autonomous Systems 22(3-4), 251–281 (1997) 5. Jouffe, L.: Fuzzy Inference System Learning by Reinforcement Methods. IEEE Trans. on Systems, Man, and Cybernetics 28(3), 338–355 (1998) 6. Watkins, C.J., Dayan, P.: Q-learning. Machine Learning 8(3), 279–292 (1992) 7. Baird, L.C.: Residual Algorithms: Reinforcement Learning with Function Approximation. In: Proceeding of the 12th International Conference on Machine Learning, pp. 9–12. IEEE Press, San Francisco (1995) 8. Shi, H.Y., Sun, C.Z., Chen, D.Y., An, Y.J.: Research on Navigation Complexity for Autonomous Mobile Robots under Dynamic Environments. Journal of Shenyang University of Technology 28(5), 534–538 (2006) 9. Xu, W.L., Sun, M.X.: Robust Control of Omni-direction Wheeled Mobile Robot. Journal of Shenyang University of Technology 29(3), 512–516 (2007)

Nuclear Reactor Reactivity Prediction Using Feed Forward Artificial Neural Networks Shan Jiang1 , Christopher C. Pain1 , Jonathan N. Carter1 , Ahmet K. Ziver2 , Matthew D. Eaton1 , Anthony J. H. Goddard1 , Simon J. Franklin3 , and Heather J. Phillips3 1

Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College London, U.K [email protected] 2 RM Consultant, Abingdon, U.K 3 Imperial College Reactor Centre, Silwood Park, Berkshire, U.K

Abstract. In this paper, a feed forward artificial neural network (ANN) is used to predict the effective multiplication factor (kef f ), an indication of the reactivity of a nuclear reactor, given a fuel Loading Pattern (LP). In nuclear engineering, the kef f is normally calculated by running computer models, e.g. Monte Carlo model and finite element model, which can be very computationally expensive. In case that a large number of reactor simulations is required, e.g. searching for the optimal LP that maximizes the kef f in a solution space of 1010 to 10100 , the computational time may not be practical. A feed forward ANN is then trained to perform fast and accurate kef f prediction, by using the known LPs and corresponding kef f s. The experiments results show that the proposed ANN provides accurate, fast and robust kef f predictions. Keywords: Feed forward neural network, Nuclear reactor, kef f prediction.

1

Nuclear Reactor Reactivity Estimation

Nuclear Reactors are loaded with fission fuel elements to generate energy for electricity or neutrons for medical/physics research. A crucial part of nuclear reactor operations is to decided how to load the fuel, in another word, to design the core Loading Pattern (LP), to improve the performance of the reactor, indicated by an estimation of the core reactivity and/or some other key measures, e.g. power and temperature distributions. A reactor core normally contains a number of loadable positions (or fuel channels), from tens to hundred of thousands, to be filled with designated fuel elements. An LP can be regarded as an ordering assignment of n items to m positions. In this case, n is the number of fuel elements, and m is number of fuel channels. When the size of the reactor is large, the number of possible LPs need to be examined before loading the reactor can be very large. Given a designed LP, it is desired that some key parameters of the reactor can be estimated. Such calculations are done by building a computer model and F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 400–409, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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simulating it. One of the most important measures is the effective multiplication factor (kef f ), an indication of the core reactivity. The kef f is defined as the ratio of the neutrons produced by fission in one generation to the number of neutrons lost through absorption and leakage in the preceding generation. kef f =

neurons generated from fission reaction neurons absorbed and leaked

(1)

The value of kef f is one, when the number of neutrons keeps consistent. Under this condition, the chain reaction is self-sustaining. It is also known as the critical point. To keep consistent energy generation or neutron emission, kef f is required to be over one. To operate the reactor efficiently, engineers need to estimate/optimize the kef f when loading the reactor. This process is also referred to as core design or LP design. To calculate the kef f by running the full computer model can be very time consuming and may not be acceptable, particularly when the number of possible LPs is large, e.g., 1010 to 10100 . Because the number of LPs to be examined is large and the computational cost to evaluate the kef f for an LP is expensive, a fast kef f estimation method is required in engineering practice. In this paper, we present a fast and reliable method using a feed forward Artificial Neural Network (ANN), to predict the kef f from a given LP. The rest of the paper is organized as follows: Firstly, the CONSORT reactor is briefly introduced in section 2. Secondly, the ANN methodology is introduced and adapted to solve the reactor kef f prediction problem in section 3. In section 4, experiments based on the CONSORT reactor are set up and the results are shown, followed by conclusions drawn in section 5.

2 2.1

The CONSORT Reactor Core A Brief Introduction to the CONSORT Reactor

The Imperial College Consort Reactor (CONSORT) was first constructed in 1965, then expanded in 1971, and has been in continuous safe operation since 1965 at the Imperial College Reactor Centre. It is the only civil research reactor in the UK. The CONSORT reactor is designated as a low power research reactor and to provide the neutral particles, or neutrons, which result from nuclear reactions inside the reactor core, for research and engineering applications. The reactor centre provides facilities for the university and other educational institutions to be used for teaching and research in many fields of nuclear science and technology such as reactor physics, reactor engineering, neutron physics, solid state physics, radiochemistry and activation analysis. The reactor centre also provides radioisotopes for use in other laboratories. A picture of the CONSORT reactor is shown in figure 1.

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Fig. 1. A birds’ eye view of the CONSORT Reactor

Table 1. k∞ of five fuel types the Imperial College CONSORT Reactor, calculated using the WIMS code, with the assumption that all fuels are fresh Fuel Name MARK I A MARK I B MARK I C MARK II MARK III Fuel Type 1 2 3 4 5 1.66197 1.16471 0.79238 1.77473 1.78583 k∞

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Fuel Inventory and Core Map of the CONSORT Reactor

There are five types of fuel assemblies in the CONOSRT reactor, which contain highly enriched Uranium fuel plates in an Uranium/Aluminium alloy ([4], [14]). The five fuel types are referred to as MARK I A, MARK I B, MARK I C, MARK II and MARK III, or fuel type ‘1’ to ‘5’. Instead of representing the fuel elements with an integer coded fuel type (i.e. 1 to 5), another parameter is used, which is the infinite multiplication factor (k∞ ). The value of k∞ can be used as a measure of the reactivity of individual fuel elements. The initial k∞ of each fuel type is calculated by using WIMS8A (wims8a99) code and summarized in table 2.2. The CONSORT core consists of 24 fuel elements and four control rods, i.e. rod no.1, 2, 3, and 4. Note that rods no. 3 and no. 4 are not included in our modelling because they act as shut down rods to ensure the safety. Normally they are fully withdrawn. A 2D core plan is illustrated in figure 2. 2.3

Modelling the CONSORT Reactor

A three-dimensional reactor core model has been constructed using the finite element method based radiation transport code EVENT ([7]). The WIMS8A ([5]) code was used to generate group constants for EVENT. The kef f can be obtained by running an EVENT simulation. Visualised 3D model is shown in figure 3.

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Fig. 2. A 2D view of the CONSORT reactor (Drawing is not to scale)

3 3.1

Predicting the keff by an ANN The ANN Methodology

The ANN has been widely applied to various engineering applications for parameter prediction, pattern recognition, classification and others ([6]). In nuclear engineering, ANNs have been utilized to solve a number of real-world problems. For examples, in [8] and [13], modified recurrent neural network is used to optimise the in-core fuel arrangement. Similarly, a feed forward ANN is used to optimise the LP in [11]. In [12], [15], [10], feed forward ANNs are used to predict key parameters of reactors. A mathematical description of a feed forward Multi-Layer Perceptron (MLP) ANN is stated in equation 2. n m   h h wi ϕ ( wij xj )) y = ϕ( i=1

(2)

j=1

in which y is the output prediction; xj is the input data and j is the dimension of the input data; ws are the weights associated with each input variables or internal variables (hidden neurons), and i is the index of the hidden neurons; ϕ and ϕh are the same non-linear function (transfer function). A frequently used transfer function is the sigmoid function: ϕ(v) =

1 1 + e−v

(3)

We choose feed forward ANN for this work, because it is well established in theory and have been successfully applied to applications such as parameter prediction and function approximation ([6]). In this work, the ANN is used to find the mapping between a given LP (a set of ordered fuel elements) to the corresponding kef f . This is achieved by training the ANN with a set of known LPs and their kef f s. Once trained, the correlation is stored in the weights, w. The resulting ANN will be able to estimate the kef f given any LP.

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Fig. 3. A 3D model of the CONSORT reactor by GEM and EVENT

3.2

The Input and Output Data for the ANN

In this work, the inputs for our ANN are LPs. Since there are twenty four loadable fuel channels, an LP is a vector consisting of twenty four fuel elements. For example, a sub-LP X from channel 1 to channel 6 (refer to figure 2): X = [5, 1, 3, 5, 2, 4]

(4)

means channel 1, 2, 3, 4, 5 and 6 are loaded with MARK III, MARK I A, MARK I C, MARK III, MARK I B and MARK II, respectively. However, the fuel type identifiers, 1 to 5, do not contain any physical property of a fuel element. The fuel type IDs can be replaced by their reactivity, or the k∞ in table 2.2. The mapping between fuel elements’ reactivity and the kef f is expected to be easier to calculate than using the integer fuel type IDs. A further improvement is to combine the k∞ and position-wise information to represent the LP more precisely. Some pre-calculation is required. A certain type of fuel i is inserted in a particular fuel channel j, with other fuel channel filled with a generic fuel element, e.g. fuel type 1. The kef f is calculated by the EVENT simulation. The resulting ‘kef f ’ is called the stand-alone kef f for fuel i at channel j. This process is repeated for each fuel type at all the channels. The result is a 24 × 5 matrix. Entry [i, j] is a realistic indication of the core reactivity, when fuel i is inserted into fuel channel j. For this ANN, instead of using the fuel element identifiers in integer form, we use the square of the normalised stand alone kef f with fuel coupling. For a MARK III fuel element (fuel type ‘5’ )in channel 1, the input to the ANN will

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Table 2. The calculated stand-alone kef f ’s using EVENT for the five different fuel types in the CONSORT reactor. The S.D. is the fuel-channel-wise standard deviation. Channel No. MARK I A MARK I B MARK I C 1 1.20460 1.19972 1.19744 2 1.20460 1.19919 1.19654 3 1.20460 1.19919 1.19654 4 1.20460 1.20164 1.20016 5 1.20460 1.20164 1.20016 6 1.20460 1.18920 1.18184 7 1.20460 1.19186 1.18581 8 1.20460 1.19186 1.18581 9 1.20460 1.19833 1.19525 10 1.20460 1.19833 1.19525 11 1.20460 1.19652 1.19252 12 1.20460 1.19652 1.19252 13 1.20460 1.18736 1.17857 14 1.20460 1.18736 1.17857 15 1.20460 1.18168 1.16935 16 1.20460 1.19797 1.19469 17 1.20460 1.19797 1.19469 18 1.20460 1.19216 1.18589 19 1.20460 1.19216 1.18589 20 1.20460 1.19092 1.18435 21 1.20460 1.20156 1.20004 22 1.20460 1.20156 1.20004 23 1.20460 1.19907 1.19636 24 1.20460 1.19907 1.19636 S.D. 0.00000 0.00545 0.00827

MARK II MARK III 1.20690 1.20883 1.20717 1.20930 1.20717 1.20930 1.20611 1.20735 1.20611 1.20735 1.21091 1.21600 1.21004 1.21448 1.21004 1.21448 1.20754 1.20995 1.20754 1.20995 1.20831 1.21133 1.20831 1.21133 1.21153 1.21700 1.21153 1.21700 1.21312 1.21972 1.20778 1.21038 1.20778 1.21038 1.20989 1.21412 1.20989 1.21412 1.21043 1.21515 1.20615 1.20742 1.20615 1.20742 1.20728 1.20949 1.20728 1.20949 0.00200 0.00356

be given by the square of the corresponding entry [1, 5] in the stand alone kef f table 2, which is 1.20883. Applying the square of stand alone kef f , LP X, as in equation 4 can be transformed to: X  = [1.208832, 1.204622, 1.1965422, 1.2073522, 1.2016422, 1.2109122]

(5)

By doing this, the variance is reasonably rescaled and therefore helps the ANN to recognize different loading patterns more precisely. Then all the entries in X  are normalized between [0.2, 0.8], using the maximum and minimum value in table 2 as upper and lower boundaries, respectively: X  = [0.6683, 0.6174, 0.5208, 0.6504, 0.5818, 0.6934]

(6)

The corresponding kef f s are normalized between [0.2, 0.8] which are the expected output of the ANN. We use [0.2, 0.8] instead of [0, 1] to ensure the training data is not too close to the output boundary of the ANN, which is [0, 1].

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Experiments and Results

We set up two experiments to validate the proposed ANN method. The first test case is a simple fresh core with the basic five fuel types, while in the second test case, we use a different fuel inventory and modified core model to test the ANN in under more complicated and realistic conditions. 4.1

Test Case 1

Fuel Inventory The five basic fuel types are used in this case. Please refer to table 2.2 for the fuel inventory. The corresponding stand alone kef f table is summarized in table 2. Core State In this test case, we have removed the aluminium clad on and below the fuel elements and the control rods (figure 3) to form a fresh and simpler core state. ANN Design For test case 1, a three layer feed forward ANN is constructed, which has 24 input nodes representing the fuel channels, 65 hidden nodes and 1 output node. The number of hidden neurons is chosen by ‘trial and error’ experiments. The transfer function for hidden neurons and output neurons are both sigmoid function given in equation 3. A total of 1535 different LPs are generated randomly, their Kef f are calculated by EVENT. A subset of 767 of LPs were used for training, 460 for validation and 308 for testing. The ANN is generated and trained using the Stuttgart Neural Network Simulator (SNNS, [3]) package. The training algorithm chosen is the scaled conjugate gradient method ([2]). Results The EVENT model with fine mesh takes approximately 10 minutes or more to calculate the kef f of an LP. Using a trained ANN developed here, the kef f can be predicted within a fraction of a second on the same computer with very little accuracy lost. The average prediction error, the largest prediction error, number of samples of which error ranges within 1% and 0.5% are listed in table 3 and figure 4. The error measurement is defined as below: Error =

4.2

ANN Prediction - EVENT Calculation EVENT Calculation

(7)

Test Case 2

Fuel Inventory In this test case, we set up a fuel inventory consists of 35 fuel elements (not necessarily the real fuel inventory in the CONSORT reactor). They still belong

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Table 3. The testing result of the ANN predicting the Kef f for test case 1 - the fresh core LPs Set Total No.LPs Error < 0.5% Error < 1% Average Error Training Set 767 762 767 0.12% Unseen Set 308 290 307 0.19%

keff calculated by EVENT

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to the five fuel types, but we have modified the fission material’s atom density by estimating the natural decay. Therefore the 35 fuel elements are treated individually. The corresponding stand alone kef f are computed using the method described in section 3. In this case, the result is a 24 × 35 matrix. The fuel inventory and tje stand alone kef f table are not included in the main text. Core State To set up a realistic case, we used the full 3D model of the CONSORT reactor as shown in figure 3. The aluminium clad is included in the core model and control rods 1 and 2 are fully inserted in. ANN Design A three layer feed forward ANN is built, which has 24 inputs nodes representing 24 fuel channels, 44 hidden neurons, and 1 output. It uses the sigmoid function given in equation 3 as the transfer function for hidden and output neurons. A total number of 2025 LP are generated randomly and their kef f are computed by EVENT. A subset of 1620 LPs are used as the training set, 202 for validation and 203 for testing. The ANN is generated and trained using SNNS package. The training algorithm is the scaled conjugate gradient method. Results Similar to test case1, the ANN can predict the kef f accurately compared to the EVENT model. The computational time is reduced by a factor of 103 . The results of test case 1 is shown in table 4 and figure 5.

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Table 4. The testing result of the ANN predicting the Kef f in test case 2: a realistic core state LPs Set Total No.LPs Error < 0.5% Error < 1% Average Error Training Set 1620 1611 1620 0.13% Unseen Set 203 203 203 0.13%

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Conclusions

The experiments based on a research nuclear reactor show that the ANN proposed for the kef f prediction is very accurate compared to a well established finite element model. In both cases, running EVENT for one LP costs 10 minutes or more. For our trained ANN, it only takes a fraction of a second. The computational time is reduced significantly by a factor of 103 or more. The robustness of the ANN is also demonstrated by testing it on two different reactor models. The proposed ANN method can be used in manual reactor core loading pattern design and/or in an automatic optimal LPs searching routine ([15],[12] and [10]), as a fast reactor core reactivity (kef f ) estimation tool. Acknowledgements. This work is partially funded by Imperial College Reactor Centre as part of the Reactor Future Options project. The authors would also like to thank Prof. C.R.E. de Oliveira for the use of the EVENT code.

References 1. Kim, H.G., Chang, S.H., Lee, B.H.: Pressurized water reactor core parameter prediction using an artificial neural network. Nuclear Science and Engineering 113, 70–76 (1993) 2. Moller, M.F.: A scaled conjugate gradient algorithm for fast supervised learning. Neural Networks 6, 525–533 (1993)

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3. Zell, A., Mauier, G., Vogt, M., Mache, N.: SNNS: Stuttgart Neural Network Simulator user manual, Institute of Parallel and Distributed High Performance System (IPVR), University of Stuttgart, Version 4.1 (1995) 4. Franklin, S.J., Goddard, A.J.H., O’Connell, J.S.: Research reactor facilities and recent development at Imperial College London. Research Reactor Fuel Management 1998: European Nuclear Society, Bruges, Belgium (1998) 5. SERCO-ANSWERS: User guide for version 8, ANSWERS/WIMS (1999) 6. Haykin, S.S.: Neural Networks: A comprehensive foundation. Prentice-Hall, Englewood Cliffs (1999) 7. de Oliveira, C.R.E., Eaton, M.D., Umpleby, A.P., Pain, C.C.: Finite element spherical harmonics solutions of the 3D Kobayashi benchmarks with ray tracing void treatment. Prog. Nuclear Energy 261.39, 243–261 (2001) 8. Sadighi, M., Setayeshi, S., Salhi, A.A.: PWR fuel management optimization using neural networks. Annals of Nuclear Energy 29, 41–51 (2002) 9. Yamamoto, A.: Application of neural network for loading pattern screening of incore optimisation calculations. Nuclear Technology 144, 63–75 (2003) 10. Erdogan, A., Geckinli, M.: A PWR reload optimisation code (Xcore) using artificial neural networks and genetic algorithms. Annals of Nuclear Energy 30, 35–53 (2003) 11. Faria, E.F., Pereira, C.: Nuclear fuel loading pattern optimisation using a neural network. Annals of Nuclear Energy 30, 603–613 (2003) 12. Ziver, A.K., Pain, C.C., Carter, J.N., de Oliveira, C.R.E., Goddard, A.J.H., Overton, R.S.: Genetic algorithms and artificial neural networks for loading pattern optimisation of advanced gas-cooled reactors. Annals of Nuclear Energy 31, 431– 457 (2004) 13. Ortiz, J.J., Requena, I.: Using a multi-state recurrent neural network to optimize loading patterns in BWRs. Annals of Nuclear Energy 31, 789–803 (2004) 14. Franklin, S.J., Gardner, D., Mumford, J., Lea, R., Knight, J.: Business operations and decommissioning strategy for Imperial College London research reactor ’CONSORT’ - A financial risk management approach. Research Reactor Fuel Management 2005, European Nuclear Society, Budapest, Hungary (2005) 15. Jiang, S., Ziver, A.K., Carter, J.N., Pain, C.C., Goddard, A.J.H., Franklin, S.J., Phillips, H.: Estimation of distribution algorithms for nuclear reactor fuel management optimisation. Annals of Nuclear Energy 33, 1039–1057 (2006)

Active Noise Control Using a Feedforward Network with Online Sequential Extreme Learning Machine Qizhi Zhang and Yali Zhou School of Automation, Beijing Information Science & technology University, P.O. Box 2865, Beijing, 100192, China [email protected]

Abstract. In practical active noise control (ANC) systems, the primary path and the secondary path may be nonlinear. The actuators of an ANC system often have non-minimum phase response. A linear controller under such situations yields poor performance. Neural networks using Filtered-x back-propagation (FX-BP) algorithm are often used as a controller for the nonlinear ANC systems. But FX-BP algorithm often converges slowly and may converge to a local minimum. A novel feedforward network-based ANC algorithm is proposed in this paper. The Online Sequential Extreme Learning Machine(OS-ELM) is generalized to meet the requirements of the nonlinear ANC systems. Computer simulations have been carried out to demonstrate that the proposed algorithm outperforms the FX-BP algorithm when the primary path is nonlinear. Keywords: ANC, Neural networks, Extreme Learning Machine.

1 Introduction The exponential increase of noise pollution and ineffectiveness of passive techniques for noise attenuation have led to the development of active noise control (ANC) system [1-6]. This technique has been successfully applied to heating, ventilating, and air conditioning systems [4-5], exhaust and motor noise [4][7], headsets [4][6], and airplanes [4]. ANC, in general, is based on the principle of destructive interference between an original primary noise source and a secondary source, whose acoustic output is governed by a controller. The controller has to be adaptive because of changes in the environment, degradation of system components (e.g., loudspeaker), and alterations of the noise source [4]. Most of the conventional ANC systems are linear in nature and they employ the standard filtered-x least mean square (FXLMS) algorithm [1-2]. Even though this algorithm involves less computational complexity for active control of broadband and narrowband noise, it may not perform well in the cases where the ANC system exhibits nonlinear response characteristics [4]. Nonlinearities may be present in the primary path when the noise is propagating with high sound pressure [8] and the nonlinearity of the air is taken into account. To deal with these effects, different structures for nonlinear controllers have been recently proposed in the literature [3-4], [7]. In [3], a nonlinear controller based on a filtered-error BP neural network was proposed to offer computational advantage over F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 410–416, 2008. © Springer-Verlag Berlin Heidelberg 2008

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the FX-BP neural network algorithm. In [4], a nonlinear controller based on a normalized Gaussian radial basis neural network was proposed to compensate the nonminimum phase secondary path transfer function and control the nonlinear noise process. In [7], adaptive Volterra filters have been introduced for feedforward ANC system. All the parameters of the BP neural network need to be tuned and the gradient descent-based method has been used in learning algorithms, which are generally very slow due to improper learning steps or may easily converge to local minima[10]. Moreover, the learning algorithm of updating weights in hidden layer is a too large computing load to DSP implement in the ANC system [3]. In [10-12], Extreme learning machine (ELM) is proposed to training a single-hidden layer feedforward neural network (SLFN) with arbitrarily chosen input weights, and the output weights of SLFNs can be analytically determined through simple generalized inverse operation of the hidden layer output matrices. The learning speed of ELM can be thousands of times faster than traditional feedforward network learning algorithms like backpropagation algorithm, in the meantime, it can obtain better generalization performance. In [11], On-line learning ELM (OS-ELM) algorithm is proposed, which can learn data one-by-one or chunk-by-chunk with fixed or varying chunk size. The ELM algorithm cannot be used to directly adapt weights of the neural networks controller in the ANC system. Because the desire outputs of the neural networks can not be obtained directly, and only the error, i.e., the sum of primary noise and secondary noise, can be measured. In this paper the OS-ELM is generalized to meet the requirements of the nonlinear ANC systems. The organization of the paper is as follows. Section II gives a brief review of the OS-ELM. Section III presents the derivation of OS-ELM based nonlinear ANC algorithm. The simulation results using the proposed algorithm and other conventional algorithms are shown in Section IV. Finally, conclusions are presented in Section V.

2 OS-ELM This section briefly reviews the OS-ELM developed by Liang and Huang et al. [11] to provide the necessary background for the development of OS-ELM algorithm for the ANC system. Only one output node is considered for convenience. The output of an SLFN with M sigmoid additive hidden nodes can be represented by M

f ( x ) = ∑ β i g (ai ⋅ x + bi ) .

(1)

i =1

where ai and bi are the learning parameters of hidden nodes and βi is the weight connecting the ith hidden node to the output node. g(ai⋅x +bi) is the output of the ith hidden node with respect to the input x. In supervised batch learning, the learning algorithms use a finite number of input– output samples for training. For N arbitrary distinct samples (xi, ti)∈Rn×R. Here, xi is a n×1input vector and ti is the target value. If an SLFN with M hidden nodes can approximate these samples with zero error, it then implies that there exist βi ,ai and bi, such that

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f ( x j ) = ∑ βi g (ai ⋅ x j + bi ) = t j ,

j=1,…,N.

i=1

(2)

Eq. (2) can be written compactly as

Hβ = T .

(3)

⎡ t1 ⎤ ⎡ β1 ⎤ ⎡ g(a1 ⋅ x1 + b1 ) " g(aM ⋅ x1 + bM ) ⎤ ⎢ # ⎥ T =⎢# ⎥ . ⎥ H = ⎢⎢ # " " β = ⎢ ⎥ ⎢ ⎥ ⎥ ⎢⎣t N ⎥⎦ N×1 ⎢⎣βM ⎥⎦ M×1 ⎢⎣ g(a1 ⋅ xN + b1 ) " g(aM ⋅ xN + bM )⎥⎦ N×M

(4)

where

The output weights β are estimated as

βˆ = H +T .

(5)

Where H + = ( H T H )−1 H T is the Moore–Penrose generalized inverse of the hidden layer output matrix H. When the training data is received one-by-one, we can obtain the following simple on-line estimating format of β

Pk +1 = Pk − Where hk +1 =

Pk hk +1hkT+1 Pk , ( k+1) β = β (k ) + Pk +1hk+1 (tk +1 − hkT+1 β (k ) ) . 1 + hkT+1 Phk +1

[g(a1⋅xk +1 + b1)

training data. Let P0=I, β cursive Eq. (6).

(0)

(6)

" g(aM ⋅xk +1 + bM )] , and (xk+1, tk+1) is the (k+1)th

as a random victor, β can be estimated on-line by the re-

3 OS-ELM for ANC System The block diagram of a feedforward ANC system using the OS-ELM-based NN control algorithm is shown in Fig.1. The SLFN is selected as a NN controller. The primary path P(z) is from the noise source to the error microphone, and the secondary path S(z) is from the canceling loudspeaker to the error microphone. The OS-ELMbased NN controller is used to generate an anti-noise signal u(k). At any time instant k, defining the input signal vector xk as

xk = [ x(k ) x(k − 1) " x(k − n + 1)]T .

(7)

Where n is the length of the input vector xk, [.]T denotes transpose of a vector. The output of the NN controller is calculated as follows M

u ( k ) = ∑ β i g (ai ⋅ x k + bi ) . i =1

(8)

ANC Using a Feedforward Network with OS-ELM x(N)

d(k)

P(z)

413

e(k) + y(k)

NN controller

u(k)

S(z) S(Z)

OS-ELM

Fig. 1. The block diagram of an ANC system

The output of the secondary path, y(k), and the error signal, e(k), are given by m −1

y ( k ) = ∑ s ( i ) u ( k − i ) =s ⋅ u k .

(9)

e( k ) = d ( k ) + y ( k )

(10)

i=0

Where s =[s(0) s(1) … s(m-1)]T is the FIR of the secondary path ,m is the length of the FIR. uk =[u(k) u(k-1) … u(k-m+1)]T is the output vector of the NN controller. It is obvious that Eq. (10) is different from Eq. (2). First, the primary noise signal d(k) can not be measured in the ANC system, and we can only measure the error signal e(k). Second, the FIR of secondary path appears in Eqs.(8-9). So we can not obtain the similar formula to Eq.(3) to apply the OS-ELM algorithm directly. From Eqs. (8-9), we can obtain m−1

M

M

m−1

j =0

i =1

i =1

j =0

y(k ) = ∑ s( j)∑ βi g (ai ⋅ xk − j + bi ) = ∑ βi ∑ s( j) g (ai ⋅ xk − j + bi ) .

(11)

And the similar formula to Equation (3) is given as follows

hTk β + d (k ) = e(k ) .

(12)

Where

⎡m−1 hk = ⎢∑s( j)g(a1 ⋅ xk − j + b1) " ⎣ j =0

T

⎤ s( j)g(aM ⋅ xk − j + bM )⎥ . ∑ j =0 ⎦ m−1

(13)

In an ANC system, the objective is to cancel the error signal e(k). It means that β should be estimated by y(k) = -d(k), but the primary noise signal d(k) can not be measured in the ANC system. Supposing the impulse response of the secondary path can be estimated off-line by a FIR filter sˆ , and it is invariant when the ANC system is running. The primary noise signal d(k) can be estimated recursively, and Eq. (12) can be changed as follows

hTk β = tk = hkT β (k ) − e(k ) .

(14)

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From Eq.(14), the β can be estimated by the recursive Eq. (6) on-line. The OS-ELM algorithm for the ANC system is as follows. Step 1: Initialization Phase a) Assign random input weights ai, bias bi and output weights β(k) b) Set k=0, P0=I Step 2: Sequential Learning Phase a) Send out the control signal u(k) as shown in Eq.(8) b) Calculate the filtered hidden layer output vector

⎡m−1 hk+1 = ⎢∑sˆ( j)g(a1 ⋅ xk− j+1 + b1) " ⎣ j=0 c) Input error signal e(k) and Set tk +1

T

⎤ sˆ( j)g(aM ⋅ xk− j+1 + bM )⎥ ∑ j =0 ⎦ m−1

(15)

= hkT β ( k ) − e(k ) .

d) Calculate the output weight

Pk +1 = Pk −

Pk hk +1hkT+1 Pk , ( k+1) β = β (k ) + Pk +1hk+1 (tk +1 − hkT+1 β (k ) ) T 1 + hk +1 Phk +1

(16)

e) Set k=k+1, Go to Step 2

4 Simulation Examples To demonstrate the effectiveness of the proposed algorithm, computer simulations are performed on a nonlinear ANC system. A comparison between the proposed OSELM based algorithm and the FX-BP algorithm is made. The SLFN with sigmoid activation function is selected as a NN controller in the proposed OS-ELM based ANC system and the FX-BP based ANC system. The number of neurons in the OS-ELM algorithm and the FX-BP algorithm is set as 6-16-1. i.e., 6 input nodes, 16 hidden nodes and 1output node. A 50Hz sinusoidal signal corrupted with Gaussian noise is used to generate the primary disturbance signal and also used as the reference signal to the control algorithm. The sampling frequency used is 1 kHz, and the total simulation duration is 6 second. The learning rate for the FX-BP algorithm is set as 0.04. The model used in this simulation has the following expressions with nonlinear terms: The primary disturbance d(k) is expressed as d(k)= 0.8x(k-3)+0.6x(k-4)-0.2x(k-5)-0.5x(k-6)-0.1x(k-7)+0.4 x(k-9) -0.05x (k-9) +0.9x2(k-3)

(17)

The secondary path transfer function is assumed to be time-invariant, and the control signal y(k) is expressed as y(k)= u(k-2)+1.6u(k-3)+0.1u(k-4)-0.1u(k-5)-0.005u(k-6)

(18)

ANC Using a Feedforward Network with OS-ELM

Fig. 2. The error signal spectrum (OS-ELM)

415

Fig. 3. The error signal spectrum (FX-BP)

Fig.2 and Fig.3 present the simulation results of the canceling error in the frequency domain. From Fig.2, it may be seen that the OS-ELM algorithm performs well in the case where the primary path transfer function is a nonlinear model, the major disturbance frequency and the broadband noise is attenuated by approximately 10 dB, and the harmonic noise signal caused by nonlinearity is attenuated by approximately 7 dB also. From Fig.3, it may be seen that the FX-BP algorithm does not work well in the case where the primary path transfer function is a nonlinear model, the major disturbance frequency is attenuated by approximately 20 dB, but the broadband noise and the harmonic noise signal caused by nonlinearity is not attenuated.

5 Conclusions In this paper, a novel OS-ELM-based algorithm is developed for use in a feedforward ANC system for situations where the primary path is nonlinear. The proposed algorithm is derived using the OS-ELM algorithm as the basic structure. This approach assigns random input weights ai, bias bi, which are not needed to adapt when the ANC system is running. Only the output weights are adaptive with OS-ELM algorithm online. Therefore, the presented ANC algorithm can fast converge. Computer simulations have been carried out to assess the performance of the proposed algorithm as a candidate for nonlinear ANC. Its performance in terms of error power spectrum has been compared to that of FX-BP algorithm. It is shown that for the nonlinear control, in terms of the error power spectrum, the proposed algorithm outperforms the FX-BP algorithm.

Acknowledgments This research is supported by Training Funds for Elitist of Beijing (20061D0500600164,20051A0500603).and Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality(PXM2008_014215_055942) .

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References 1. Kuo, S.M., Morgan, D.R.: Active Noise Control Systems—Algorithms and DSP Implementations. Wiley, New York (1996) 2. Nelson, P.A., Elliott, S.J.: Active Sound Control. Academic Press, London (1991) 3. Zhou, Y.L., Zhang, Q.Z., Li, X.D., Gan, W.S.: Analysis and DSP Implementation of an ANC System using a Filtered-Error Neural Network. Journal of Sound and Vibration 285(1), 1–25 (2005) 4. Strauch, P., Mulgrew, B.: Active Control of Nonlinear Noise Processes in a Linear Duct. IEEE Trans. Signal Processing 46(9), 2404–2412 (1998) 5. Pelton, H., Wise, K.S., Sims, W.S.: Active HVAC Noise Control System Provide Acoustical Comfort. Sound Vibr. 28(1), 14–18 (1994) 6. Salloway, A.J., Millar, C.E.: Active Vibration and Noise Control. GEC Rev. 11(3), 138– 145 (1996) 7. Tan, L., Jiang, J.: Adaptive Volterra Filter for Active Control of Nonlinear Noise processes. IEEE Trans. Signal Processing 49(8), 1667–1676 (2001) 8. Klippel, W.: Active Attenuation of Nonlinear Sound. Patent, U. S (1999) 9. Zhou, Y.L., Zhang, Q.Z., Li, X.D., Gan, W.S.: Model-Free Control of a Nonlinear ANC System with a SPSA-based Neural Network Controller. In: Wang, J., Yi, Z., Żurada, J.M., Lu, B.-L., Yin, H. (eds.) ISNN 2006. LNCS, vol. 3972, pp. 1033–1038. Springer, Heidelberg (2006) 10. Huang, G.B., Zhu, Q.Y., Siew, C.K.: Extreme Learning Machine: Theory and Applications. Neurocomputing 70, 489–501 (2006) 11. Liang, N.Y., Huang, G.B., Saratchandran, P., Sundararajan, N.: A Fast and Accurate Online Sequential Learning Algorithm for Feedforward Networks. IEEE Trans. On Neural Networks 17(6), 1411–1422 (2006) 12. Huang, G.B., Chen, L., Siew, C.K.: Universal Approximation Using Incremental Constructive Feedforward Networks With Random Hidden Nodes. IEEE Trans. On Neural Networks 17(4), 879–892 (2006)

A Probabilistic Method to Estimate Life Expectancy of Application Software Shengzhong Yuan1 and Hong He2 1

Department of Modern Education Technology, Shandong University at Weihai, Weihai 264209, China 2 School of Information Engineering, Shandong University at Weihai, Weihai 264209, China [email protected], [email protected]

Abstract. The security of network applications is more important in Internet days. A kind of randomized protocol to find out application software security bug is raised. We discuss the security testing method with randomized protocol; put forward an attack-proof theorem which shows a way of computing average number of attack times easily with a Weibull probability map before a software system having concealed bugs becomes invalid. It is also proved a fairly good method to estimate life expectancy of application software on Internet under various random attacks. Keywords: Network application, Randomized protocol, Probabilistic method, Life expectancy, Weibull probability map.

1 Introduction In the face of endless malicious attacks and viruses, people have made the firewall and intrusion detection increasingly complicated[1,3], the virus database growingly bigger, and the rate of false alarms higher; accordingly, maintenance and management is more complicated and hence difficult to carry out. Meanwhile, the efficiency of information system is greatly decreased. This situation has opened our eyes to the fact that we have stepped into misunderstandings in terms of security. The basis of system security lies in software security. Application software which supports Internet functions is where the highest security risks lie. The two most fundamental types of software security risks are implementation error and architectural weakness in code realization. With the inherent complexity of software increasing continuously, errors in code realization can usually be detected with the help of mature experience and powerful automated security testing tools [2,11]. However, the automatic analysis technology of software security on the construction layer lags far behind that of the implementation tools. Therefore, finding weaknesses in the software construction remains a hard job and it should be regarded as an important component of software testing to be completed by professionals. Software lifetime, which will be mentioned in the following discussion, means the actual survival period of the application software, or the survival period counted from a particular time during which it works normally. The software which cannot function normally is called “breakdown” or “invalid”. F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 417–421, 2008. © Springer-Verlag Berlin Heidelberg 2008

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The second part presents a randomized protocol in a systematical way which is applicable to all phases of the tests to find security bugs, discusses the application scope of the method, and elaborates attack-proof theorem of enterprise portal system which can be used to estimate how many random attacks fail a loosely-connected enterprise portal system possessed of security bugs. Data acquired from the security testings with randomized protocol can not only be used to construct a threat mode [3], but can also be applied in the experiments and analysis of software’s lifetime estimation. Following is the conclusion.

2 Randomized Protocol—An Adversary Pattern to Find Out Security Bugs in Internet Application 2.1 Security Testing by Randomized Protocol Randomized protocol is derived from random algorithm to find security bugs of software. Random algorithm, which is meant to make random choice in the implementation process, has two advantages: simple and fast. We intend to make use of the advantages to deal with software security testing and software’s lifetime estimation, for the two issues are of much uncertainty and complication. We only studies the common attack types which use security bugs of others’ information system, intrude into it through network commands and special software, and attack the other’s application software to achieve the purpose of undermining its network system. The theoretical basis of randomized protocol’s ability to find security bugs is that a system with security weakness may “defeat ” some definite attacks, but it can’t always “defeat” attacks selected randomly[4]. The designing notion of randomized protocol comes from experienced security testingers’ intuitions: if you want to know the security performance of the application software, let the attackers try from the network. For a certain system to be tested, when the security analysis on its absolute action is meaningless, we can find an adversary to attack it to make the system security perform worst. In fact, after the application software is released, it is exposed to thousands of people’s attacks at the same time. It is unrealistic to try to protect the system from all types of attacks through security design [5]. But if the security testingers are independent of the development team, and besides the normal software testing channels, they launch random attacks that are intensive and concentrated on the software; it will be able to find the security bugs earlier, more effectively and more veraciously than intruders. A testing pattern, once confirmed to be a randomized protocol, mainly covers the following steps: collection of goal information, construction of attacks, intensity control, implementation of tests, and testing feedback. As it is impossible to find all the bugs in the software, security testing needs a stop sign. When every part of the system is acceptable to the risk threshold value, the risk treatment can be stopped. 2.2 Application of Randomized Protocol Note the degree of an attack on system security in a numeral value called attack value, which is defined on the interval (0,1) according to the meaning of probability.

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Suppose that the attack value of each attack obeys distribution G , independent with each other. When the attack with the value x arrived, each running module becomes invalid instantaneously at the possibility of x . What we research is the distribution of N which is the number of attack times. Theorem 1. In the randomized protocol, the average number of attacks N to get all n ⎛ n ⎞ (−1) i +1 E[ N ] = ∑ ⎜⎜ ⎟⎟ i =1 ⎝ i ⎠ 1 − p i

the modules of software system valid is number of software system modules,

，where n is the

p j is the possibility when all the j modules

designated are good after a random attack. Proof

：To calculate P{N > k} , we define E (i = 1,2, i

, n) as the event that

part i is still good after k attacks, then n

n

P{N > k } = P(∪ Ei ) = ∑ P( Ei ) −

∑ P( E E ) + i

i
Let

l

i =1

i =1

+ (−1)

n +1

(1)

P( E1 E 2

En )

p j be the possibility that all j parts designated are still good after an attack.

p j = ∫ P{all j parts designated are good Attack value x}dG( x) = ∫ (1 − x) j dG( x)

P( Ei ) = p1k P( Ei El ) = p 2k , , P( E1 E 2 E n ) = p nk ⎛ n⎞ ⎛ n⎞ P{N > k} = np1k − ⎜⎜ ⎟⎟ p 2k + ⎜⎜ ⎟⎟ p3k − + (−1) n +1 p nk ⎝ 2⎠ ⎝ 3⎠ The average of N can be calculated as follows [8] E[N ] =

∞

∑

P{ N > k } =

k =0

=

n

∑

i =1

⎛n ⎜⎜ ⎝ i

∞ ⎞ ⎟⎟ ( − 1 ) i + 1 ∑ p ik = k =0 ⎠

∞

⎛n⎞ ⎜⎜ ⎟⎟ ( − 1 ) i + 1 p ik ⎝ i ⎠ ⎛ n ⎞ ( − 1) i + 1 ⎜⎜ ⎟⎟ ⎝ i ⎠ 1 − pi

So

(2)

n

∑∑

k = 0 i =1 n

∑

i =1

(3)

2.3 Experiment Launch security testings on a network application P in randomized protocol has been carried for 6 times. The lifetime of P is recorded whenever P becomes invalid under

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the attacks by its adversary and a detailed security testing report is formed to be submitted to the system testing group. Then the designers make corresponding improvement. Repeat the above steps. The lifetime of P in weeks during the 6 experiments: 3, 4, 5, 6, 8, 10. Draw the points (t (i ),100(i − 0.5) / n) i = 1, ,6 in Weibull probability map. Then draw a straight line on it in eyes as shown in figure 1.

Fig. 1. Weibull probability map of 6 times security testing on P

63.2 percentile points is approximately equal to 6.5 weeks, which is the estimated value of 1 / λ . So the estimated value of λ is 0.154. Draw a straight line parallel to the fitting straight lines from the starting point on the top. The intersection which is approximately equal to 2.75 at the left auxiliary linear calibration is the estimated value. Then the average lifetime of P can be estimated using Webull distribution with these values:

μ=

Γ(1 + 1 / γ )

, So

λ

μ = Γ(1.364) / 0.154 = 0.8897/ 0.154 = 5.78

weeks.

Γ(γ ) is the famous Γ function defined by Abramowitz and Stegun (1964) [10]. Lastly, Calculate the potential lifetime of P with the lifetime μ = 5.78 weeks es-

Here

timated under the circumstances of the laboratory using the following formula:

LP = μ ×

f adversary f general

，where

f adversary is the frequency of attacks on P during the

security testing in randomized protocol,

f general is the frequency of attacks on P in

general network state. The frequency of attacks on P given by testing group is 120 times a week. The frequency of attacks on P in general network state is about 5 day/week × 1 time/day × (the possibility of P under attack)50%=2.5 time/week. Then the potential lifetime of P is

LP = μ ×

f adversary f general

= 5.78×

（

）

120 = 277.44 weeks ≈ 5.34 2.5

（years）

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3 Conclusions In the analysis above, the effect of module varies with the different issues under discussion and different methods employed; the complexity of module and the relevant statistical methods also vary according to different applications[11]; an application software is likely to be affected by any one of the reasons leading to “invalidity”; etc.. Therefore, in the cases which involve software’s lifetime, the process of data production is often difficult to clarify, which is one of the factors making things more complicated. It is due to such complexity that we have chosen not to subdivide the different effects exerted by various kinds of network attacks on the software’s lifetime; instead, on a more abstract and higher level, we have put all attacks and their countless new variants into one attack environment to study.

References 1. Li, T.: An Immune Based Model for Network Monitoring. Chinese Journal of Computers (2006) 2. Tang, Y., Lu, X.C., Hu, H.P., Zhu, P.D.: Automatic Generation of Attack Signatures Based on Multi-Sequence Alignment. Chinese Journal of Computers (2006) 3. Michael, H., David, L.B.: Writing Secure Code. Microsoft Press (2002) 4. Rajeev, M., Prabhakar, R.: Randomized Algorithms. Cambridge University Press, Cambridge (1995) 5. James, W., Herbet, H.T.: How to Break Software Security: Effective Techniques for Security Testing. Addison Wesley, Reading (2003) 6. Herbet, H.T., Scott, G.C.: The Red TeamProgram Testing, Software Research and Programmin (2004) 7. Gary, M.G., John, V.: Ten Principles of Software Security, http://www.nap.edu/catalog/6161.html 8. Ross, S.M.: Stochastic Processes. John Wiley, Chichester (1983) 9. Foster, I., Kesselman, C. (eds.): The Grid: Blueprint for a New Computing Infrastructure, ch. 3. Morgan Kaufmann, San Francisco (1998) 10. Lawless, J.F.: Statistical Models and Methods for Lifetime Data. John Wiley, Chichester (1983) 11. Li, A.G., Hong, B.R., Wang, S.: An Approach for Identifying Software Vulnerabilities Based on Error Propagation Analysis. Chinese Journal of Computers (2007)

Particle Filter with Improved Proposal Distribution for Vehicle Tracking Huaping Liu and Fuchun Sun Department of Computer Science and Technology, Tsinghua University, P.R. China State Key Laboratory of Intelligent Technology and Systems, Beijing, P.R. China {hpliu,fcsun}@tsinghua.edu.cn

Abstract. Symmetry is an important characteristic of vehicles and has been frequently used for detection tasks by many researchers. However, existing results of vehicle tracking seldom used symmetry property. In this paper, we will utilize the detected symmetry feature to design a proposal distribution of particle filter for vehicle tracking. The resulting proposal distribution can be closer to the true posterior distribution. Experimental results show that the use of symmetry information will obtain better tracking performance than the conventional color histogram-based particle filters. Keywords: Visual tracking, particle filter.

1

Introduction

Symmetry is a characteristic that is common to most vehicles. Some researchers have already used symmetry to detect vehicles [10][25]. They have proposed varied approaches to find symmetry on images: using edges, pixel intensity, and other features. Especially, [25] pointed out that the mirror symmetry with respect to a vertical axis is one of the most striking generic shape features available for object recognition in a vehicle-following situation. The obtained vertical axis of symmetry is an excellent feature for measuring the leading vehicle’s relative lateral displacement in consecutive images because it is invariant under vertical nodding movements of the camera and under changes of object size. However, though the symmetry property is a strong feature of vehicles, it is up to now mainly used in the field of vehicle detection. For vehicle tracking, only [1] and [19] recently tried to used the symmetry features, but their results were based on the combination information of vision and radar. As to the tracking algorithm, earlier results were based on the famous Kalman filtering[5], which can obtain optimal solution in the case of linear dynamics and Gaussian noise. Unfortunately, very few practical visual tracking problems belong to this case. For nonlinear or non-Gaussian problems, it is impossible to evaluate the probability distribution analytically and many algorithms have been proposed to approximate them. The particle filter, also known as sequential Monte Carlo[6], or Condensation, is the most popular approach which recursively F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 422–431, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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constructs the posterior probability distribution function of the state space using Monte Carlo integration[7][9][13]. On the other hand, the color histogram is robust against noise and partial occlusion, but suffers from the presence of the confusing colors in the background. Recently, [1] and [19] incorporated the symmetry information into the particle filter framework to track vehicles, which show excellent performance with the aid of radar information. In this paper, we will use detected symmetry feature to design the proposal distribution of particle filter for vehicle tracking. Experimental results show that the use of symmetry information will strongly increase the tracking performance than the conventional color histogram-based particle filters. In addition, the calculation of symmetry and the combination way of different features are very different from the algorithms in [1] and [19]. In fact, in [1] and [19] , the symmetry feature is combined with other features to design the observation likelihood and therefore the symmetry feature must be calculated many times for each candidate particle. Since the calculation of symmetry feature is rather time-consuming, it will spend a lot online computational time using the approaches of [1] and [19]. In this paper, the symmetry feature is just used to the design the proposal distribution in one region and it is calculated only once. This will strongly reduce the computational burden and even obtain more accurate tracking results. This paper focuses on the vision-based vehicle tracking problem and the remainder of this paper is organized as follows. In Section II, the basic color histogram-based particle filter algorithm is introduced. In Section III, the new proposal distribution is presented. Section IV gives some representative experimental results. Finally, some conclusions are presented in Section V.

2

Color Histogram-Based Particle Filter

The task of tracking is to use the available measurement information to estimate the hidden state variables. Given the available observations z1:k−1 = z1 , z2 , · · · , zk−1 up to time instant k − 1, the prediction stage utilizes the probabilistic system transition model p(xk |xk−1 ) to predict the posterior at time instant k as  p(xk |z1:k−1 ) = p(xk |xk−1 )p(xk−1 |z1:k−1 )dxk−1 (1) At time instant k, the observation zk is available, the state can be updated using Bayes s rule p(zk |xk )p(xk |z1:k−1 ) p(xk |z1:k ) = (2) p(zk |z1:k−1 ) where p(zk |xk ) is described by the observation equation. In general, the integrals in (1) and (2) are analytically intractable. To solve this problem, the particle filter approaches are proposed [6]. The kernel of particle filter is to recursively approximate the posterior distribution using a finite set of weighted samples. Each sample xik represents one hypothetical state of the

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object, with a corresponding discrete sampling probability ωki , which satisfies N i i=1 ωk = 1. The posterior p(xk |z1:k ) then can be approximated as p(xk |z1:k ) ≈

N 

ωki δ(xk − xik )

(3)

i=1

where δ(·) is Dirac function. The candidate samples {xik }i=1,2,···,N are drawn from a proposal distribution q(xk |x1:k−1 , z1:k ) and the weight of the samples are i ωki = ωk−1

p(zk |xik )p(xik |xik−1 ) q(xk |x1:k−1 , z1:k )

(4)

The samples are re-sampled to generated an unweighed particle set according to their importance weights to avoid degeneracy. In the case of the bootstrap filter[6], q(xk |x1:k−1 , z1:k ) = p(xk |xk−1 ) and the weights become the observation likelihood p(zk |xk ). In visual tracking, the color histogram is an extensively used feature[16]. Color distributions are used as object models as they achieve robustness against nonrigidity, rotation and partial occlusion. In our experiments, the histograms are typically calculated in the RGB space using 8×8×8 bins. The resulting complete histogram is thus composed of Nh = 512 bins. The color-similarity measure is based on the similarity between the color histogram of a reference region and that of the image region in frame k represented by a sample xik . To estimate the proper weight for this sample during the measurement update step, we need the observation model p(zk |xk = xik ). This model can be obtained by the following equation p(zk |xk = xik ) ∝ exp{−λD2 (q∗ , qk (xik ))}

(5)

where λ is an experimentally determined constant and q∗ and qk (xik ) are the color histograms of the reference region and the region defined by xik , respectively. The distance measure D(·, ·) is derived from the Bhattacharyya similarity coefficient and is defined as D2 (q∗ , qk (xik )) = {1 −

Nh   q ∗ (n)qk (n; xik )}1/2

(6)

n=1

More details can be found in [16]. Although color histogram is a robust feature, it also present some disadvantages.

3

Improved Proposal Distribution

The proposed particle filter introduces an important extension: It uses detected symmetry axis in the construction of the proposal distribution q(xk |x1:k−1 , z1:k ). This improves the robustness of the algorithm substantially. It is widely accepted

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(i)

a

b (ii)

(iii)

d

c

Fig. 1. Detection of symmetry axis

that proposal distributions that incorporate the recent observations (in our case, through the symmetry detections) outperform naive transition prior proposals considerably[15]. In the following we will describe the algorithm details (see Fig.1). We assume that the concerned state is xk = (xk , yk , sk ), where (xk , yk ) is the detection box center and sk is the scale. Fig.1(a) represents the previous frame, where the white dashed box is the tracking results, we denote it as xk−1 = (xk−1 , yk−1 , sk−1 ). After obtaining this result, we enlarge the corresponding detection box to obtain a searching region Rk for the current frame, which is used for searching the vehicle in the current frame. This region can be obtained as Rk = (xk−1 , yk−1 , γsk−1 )), where γ is a coefficient which can be set by the designer (Here we set it to be 1.5). In Fig.1(a), the region Rk is represented as a yellow box. Also, we propagate it into the current frame (see Fig.1(b)). Next, we will search the axis which achieves the strongest symmetry in region Rk . For computing the symmetry, the intensity distribution of each row of the rectangle can be regarded as a one-dimensional function. Denote the left and right border of the region Rk are lef t and right respectively. For each column i ∈ [lef t + 1, right − 1], we can compute its symmetry strength S¯i . The computation procedure is presented as follows:

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(i) El (u)

 G(l,i+u)+G(l,i−u) =

2

0  G(l,i+u)−G(l,i−u)

(i)

Ol (u) =

2

0

if − w(i) ≤ u ≤ w(i) otherwise

(7)

if − w(i) ≤ u ≤ w(i) otherwise

(8)

where w(i) = min{i−lef t, right−i}, G(·, ·) is the corresponding gray scale image (i) (i) of current frame. It is obvious that El (u) is an even function and Ol (u) is (i) an odd function. For fair comparison purpose, the function El (u) should be normalized as  w(i) 1 (i) (i) (i) ˆ El (u) = El (u) − (i) El (v)dv (9) w 0 (i) ˆ (i) (u) are odd functions, we can define Since both Ol (u) and E l

 w(i) (i) Sl

= (i)

0  w(i) 0

(i)

 w(i)

(i)

 w(i)

ˆ (u)|du − |E l ˆ (u)|du + |E l

(i)

0

|Ol (u)|du

0

|Ol (u)|du

(10)

(i)

(i)

In general, the value Sl will locate in the interval [−1, 1]. Sl = 1 represents (i) ideal symmetry and Sl = −1 represents ideal antisymmetry. Therefore we can use this value to measure the symmetry property of the i-th column in the region Rk . Finally, assume this region has H rows, then we have to compute this value H (i) row by row S¯(i) = H1 l=1 Sl . One demo of the symmetry function can be found in Fig.1(c). We can easily find one peak in this figure and then we can locate the x-coordinate of the vehicle in this current frame (see Fig.1(d)). Remark 1. Different from existing results, such as [25], the symmetry function in (10) are defined by using the absolute value rather than square operator. This will reduce computation time and shows better performance in our practice. After obtaining the symmetry axis, we can use it to lead the particles to cluster this high likelihood region. Certainly, we notice that this can only obtain the estimate of x-coordinate. It is clear from the symmetry detection results that they could be improved if we considered the motion models of the vehicles. For this reason, the expression for the proposal distribution is given by the following mixture. q(xk |x1:k−1 , z1:k ) = αqsym (xk |xk−1 , zk ) + (1 − α)p(xk |xk−1 ) where qsym is a Gaussian distribution. The parameter α can be set dynamically without affecting the convergence of the particle filter[15]. When α = 0, the proposed algorithm reduces to the conventional color histogram-based particle filter. By increasing α we place more importance on the symmetry detections. We can adapt the value of α depending on tracking situations.

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Experimental Results

In this section, the proposed particle filter is used to track a vehicle in the road. We have done a lot of experiments but can only include some representative results due to the page limitation. The dynamics of the objects are assumed to be a random walking model, which can be represented as xk = xk−1 + vk , where vk is a multivariate zero-mean Gaussian random variable. Its variances are set by [σx , σy , σs ] = [10, 10, 0.1]. We can initialize the particle filters and the reference color histogram with a detector algorithm or a manfully specified image patch in the first frame. The image sizes of all of the sequence are 640 × 480. For fair comparison, we also run a conventional color histogram-based particle filter (CCH-PF). All of the particle filters for one sequence are started with same initial detection results. We assign 50 particles for our particle filter and assign 50 and 200 particles for CCH-PF. The running time for each frame is shown in Fig.2. It can be seen that the proposed approach will spend more time than CCH-PF with the same number of particles. Figs.3-8 give some representative tracking results, where left column corresponds to the CCH-PF with 200 particles, and the right column corresponds to the proposed particle filter with 50 particles. It can be shown that CCHPF will produce obvious deflection during frames 115-145, even lose the vehicle

Conventional approach (50 particles) Conventional approach (200 particles) Proposed approach (50 particles) 1.2

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entirely from frame 155. However, the proposed approach, which uses only 50 particles, will give rather accurate tracking results. In addition, our approach

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Fig. 6. Frame 145

Fig. 7. Frame 155

Fig. 8. Frame 165

will be more fast than CCH-PF with 200 particles (see Fig.2). If we assign 50 particles for CCH-PF, its performance will be more worse and we omit it in this paper.

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Conclusions

In this paper, we utilized the symmetry feature to design a proposal distribution of particle filter for vehicle tracking. Experimental results are given to show the advantages of the proposed approach.

Acknowledgements This work was jointly supported by National Science Fund for Distinguished Young Scholars (Grant No: 60625304), National Natural Science Foundation of China (Grants No: 60504003, 60474025), National Key Project for Basic Research of China (Grant No: 2007CB311003), and National High-tech Research and Development Plan (Grant No: 2007AA04Z232).

References 1. Alefs, B., Schreiber, D., Clabian, M.: Hypothesis based vehicle detection for increased simplicity in multi sensor ACC. In: Proc. of IEEE Intelligent Vehicles Symposium, pp. 261–266 (2005) 2. Avidan, S.: Support vector tracking. IEEE Trans. Pattern Analysis and Machine Intelligence 26(8), 1064–1072 (2004) 3. Broggi, A., Cerri, P., Antonello, P.C.: Multi-resolution vehicle detection using artificial vision. In: Proc. of IEEE Intelligent Vehicles Symposium, pp. 310–314 (2004) 4. Collado, J.M., Hilario, C., de la Escalera, A., Armingol, J.M.: Model based vehicle detection for intelligent vehicles. In: Proc. of IEEE Intelligent Vehicles Symposium, pp. 572–577 (2004) 5. Dellaert, F., Thorpe, C.: Robust car tracking using Kalman filtering and Bayesian templates. In: SPIE Conference on Intelligent Transportation Systems, pp. 72–83 (1997) 6. Doucet, A., de Freitas, N., Gordon, N.: Sequential Monte Carlo Methods in Practice. Springer, New York (2001) 7. Du, M., Guan, L.: Monocular human motion tracking with the DE-MC particle filter. In: Proc. of Int. Conf. on Acoustics, Speech, and Signal Processing, pp. 205– 208 (2006) 8. Fairfield, N., Kantor, G., Wettergreen, D.: Towards particle filter SLAM with three dimensional evidence grids in a flooded subterranean environment. In: Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 3575–3580 (2006) 9. Hamlaoui, S., Davoine, F.: Facial action tracking using an AAM-based condensation approach. In: Proc. of Int. Conf. on Acoustics, Speech, and Signal Processing, pp. 701–704 (2005) 10. Hilario, C., Collado, J.M., Armingol, J.M., de la Escalera, A.: Pyramidal image analysis for vehicle detection. In: Proc. of IEEE Intelligent Vehicles Symposium, pp. 88–93 (2005) 11. Isard, M., Blake, A.: Condensation - conditional desity propagation for visual tracking. Int. J. of Computer Vision 29(1), 5–28 (1998) 12. Khan, Z., Balch, T., Dellaert, F.: MCMC-based particle filtering for tracking a variable number of interacting targets. IEEE Trans. on Pattern Analysis and Machine Intelligence 27(11), 1805–1819 (2005)

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13. Maggio, E., Cavallaro, A.: Hybrid particle filter and mean shift tracker with adaptive transition model. In: Proc. of Int. Conf. on Acoustics, Speech, and Signal Processing, pp. 221–224 (2005) 14. Nummiaro, K., Meierb, E.K., Gool, L.V.: An adaptive color-based particle filter. Image and Vision Computing 21, 99–110 (2003) 15. Okuma, K., Taleghani, A., De Freitas, N., Little, J.J., Lowe, D.G.: A boosted particle filter: multitarget detection and tracking. In: Proc. of European Conf. on Computer Vision, pp. 28–39 (2004) 16. Perez, P., Hue, C., Vermaak, J., Gangnet, M.: Color-based probabilistic tracking. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2350, pp. 661–675. Springer, Heidelberg (2002) 17. Ranganathan, A., Dellaert, F.: A Rao-Blackwellized particle filter for topological mapping. In: Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 810–817 (2006) 18. Sabbi, A.S., Huber, M.: Particle filter based object tracking in a stereo vision system. In: Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 2409–2415 (2006) 19. Schweiger, R., Neumann, H., Ritter, W.: Multiple-cue data fusion with particle filters for vehicle detection in night view automative applications. In: Proc. of IEEE Intelligent Vehicles Symposium, pp. 753–758 (2005) 20. Sun, Z., Bebis, G., Miller, R.: On-road vehicle detection: A review. IEEE Trans. Pattern Analysis and Machine Intelligence 28(5), 694–711 (2006) 21. Viola, P., Jones, M.J.: Robust real-time face detection. Int. J. Computer Vision 52(2), 137–154 (2004) 22. Xiong, T., Debrunner, C.: Stochastic car tracking with line- and color-based features. IEEE Trans. on Intelligent Transportation Systems 5(4), 324–328 (2004) 23. Yang, C., Duraiswami, R., Davis, L.: Fast multiple object tracking via hierarchical particle filter. In: Proc. of the 10th IEEE Int. Conf. on Computer Vision (ICCV 2005), pp. 212–219 (2005) 24. Zhou, S., Chellappa, R., Moghaddam, B.: Visual tracking and recognition using appearance-adaptive models in particle filter. IEEE Trans. on Image Processing 13(11), 1491–1506 (2004) 25. Zielke, T., Brauckmann, M., Seelen, W.V.: CARTRACK: Computer vision-based car-following. In: Proc. of IEEE Workshop on Applications of Computer Vision, pp. 156–163 (1992)

Cluster Selection Based on Coupling for Gaussian Mean Fields Yarui Chen and Shizhong Liao School of Computer Science and Technology Tianjin University, Tianjin 300072, P.R. China {yrchen,szliao}@tju.edu.cn

Abstract. Gaussian mean field is an important paradigm of clusterbased variational inference, and its cluster selection is critical to the tradeoff between the variational accuracy and the computational complexity of cluster-based variational inference. In this paper, we explore a coupling based cluster selection method for Gaussian mean fields. First, we propose the model coupling and the quasi-coupling concepts on Gaussian Markov random field, and prove the coupling-accuracy theorem for Gaussian mean fields, which regards the quasi-coupling as a cluster selection criterion. Then we design a normalized cluster selection algorithm based on the criterion for Gaussian mean fields. Finally, we design numerical experiments to demonstrate the validity and efficiency of the cluster selection method and algorithm. Keywords: Variational Probabilistic Inference, Gaussian Mean Field, Gaussian Markov Random Field, Cluster Selection.

1

Introduction

Gaussian Markov random field (GMRF) is a graphical model with multivariate Gaussian distribution and conditional independence assumptions [1]. The model provides a natural way to describe real systems, and has wide application in spatiotemporal statistics, image analysis, time-series analysis and semi-parametric statistics [2]. Gaussian mean field is a basic cluster-based variational inference method that manipulates on disjoint clusters on the GMRF model. It plays an important role in huge models for its concise variational formulation and low computational complexity. Cluster selection is critical for cluster-based variational inference, and concerns the tradeoff between variational accuracy and computational complexity. Saul, Jordan, and Wiegerinck put forward the cluster selection problem, and provide cluster reduction methods for manual selection respectively [3,4]. Since manual cluster selection is unreliable for large-scale models and new applications, automated cluster selections are desirable. Taking free energy change as a selection criterion, Welling addresses the automated cluster selection issue with region pursuit algorithm for generalized belief propagation, but the candidate clusters and cluster number should be appointed beforehand [5]. Later, he improves the region pursuit algorithm, and makes the choice of the number of F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 432–441, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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clusters automatic [6]. Xing proposes cluster partition strategies with weights as partition criteria for generalized mean field, and the equal-cardinality constraint condition is added for partition to avoid unbalance computational complexity among clusters [7]. However, current automated cluster selection methods only concern the structures of graphical models, paying little attention to the interferences among variables. Besides, these selection algorithms require critical initialization, such as the candidate cluster of the pursuit algorithm [6] and the equal-cardinality constraint on cluster partition [7], to maintain balanced clusters. In this paper, we propose coupling conception to measure the dependence among variable clusters, establish the quasi-coupling as a natural cluster selection criterion, and present an automated cluster selection algorithm for Gaussian mean fields, which avoids unbalanced clusters through cluster normalization. The remainder of this paper is organized as follows. Section 2 introduces the GMRF model and Gaussian mean field method. Section 3 establishes the coupling theory that can be used to guide cluster selection, and designs the normalized cluster selection algorithm based on quasi-coupling. Section 4 presents and analyzes numerical experiment results. Finally section 5 is conclusions.

2

Backgrounds

In this section, we introduce the GMRF model and Gaussian mean field method. 2.1

Gaussian Markov Random Field

The GMRF model G = (V, E) is formed by a vertex set V and an undirected edge set E. Associated with V is a Gaussian random vector x = {x1 , . . . , xn }. Let A, B denote the inner product of symmetric matrixes A and B, that is A, B = trace(AB), and suppose all integrals are Lebesgue integrals, the exponential distribution p(x; θ) of G is [8]   p(x; θ) = exp Ux (θ), X − A(θ) , (1)  A(θ) = log exp{Ux (θ), X}dx. (2) D

Here, 1. A(θ) denotes the log partition function of exponential distribution which ensures that p(x; θ) is properly normalized. 2. D denotes the value spaces of Gaussian vector x. 3. Symmetric matrix ⎡ ⎤ 1 x1 · · · xn ⎢ x1 x21 · · · x1 xn ⎥   1 ⎢ ⎥ 1 xT = X=⎢ . ⎥ . . . .. .. .. ⎦ x ⎣ .. xn xn x1 · · · x2n denotes the sufficient statistic matrix of G.

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4. Symmetric matrix

⎡

0 ⎢ θ1 ⎢ Ux (θ) = ⎢ . ⎣ ..

θ1 θ11 .. .

θn θn1

⎤ · · · θn  · · · θ1n ⎥ 0 z T (θ) ⎥ .. .. ⎥ = z (θ) Zx (θ) x x . . ⎦ · · · θnn

denotes the exponential parameter matrix associated with X. Here, (a) zx (θ) denotes the n-vector [θ1 , . . . , θn ]T . (b) Zx (θ) denotes the n × n matrix [θst ]n×n . The Markov property is represented by constraint condition {θst = 0 | ∀(s, t) ∈ / E}. 5. The exponential parameter vector θ belongs to the set   Θ = θ ∈ Rd |Zx (θ) ≺ 0 , where d = 2n + |E|. 2.2

Gaussian Mean Field

Given probability distribution p(x), variable set x is divided into two types, visible variable set xV , and hidden variable set xH . Probabilistic inference is to compute the likelihood distribution p(xV ) by summing over all 2|xH | configurations of hidden variables. The variational inference is to transform the log likelihood log p(xV ) into the extremum of a functional by Jensen’s inequality, and then solve the optimization problem, obtain a low bound of log p(xV ) [9]. This is done as follows:  p(xH , xV )dx log p(xV ) = log xH  p(xH , xV ) dx q(xH ) = log q(xH ) xH  ≥ (q(xH ) log p(xH , xV ) − q(xH ) log q(xH )) dx, (3) xH

where q(xH ) is an arbitrary distribution (free distribution) on hidden variables used to construct the functional. For model G, the key problem of probabilistic inference is to compute the log partition function A(θ), which involves the sum of all variables. Based on the variational principle (3) and exponential distribution (1),(2), A(θ) has the variational formula [8]   (4) A(θ) = sup Ux (θ), Wx (μ) − A∗ (μ) . μ∈MGauss

Here, 1. μ denotes the mean parameter vector of the free probability distribution q(x; θ(μ)). 2. A∗ (μ) denotes the negative entropy of q(x; θ(μ)) A∗ (μ) = −H(q(x; θ(μ))).

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3. Wx (μ) is the expectation of X based on q(x; θ(μ)) ⎤ ⎡ 1 μ1 · · · μn ⎢ μ1 μ11 · · · μn1 ⎥  1 zxT (μ) ⎥ ⎢ Wx (μ) = Eq {X} = ⎢ . . . . ⎥ = . zx (μ) Zx (μ) ⎣ .. .. .. .. ⎦ μn μn1 · · · μnn 4. MGauss denotes the constraint set of μ of all the free probability distribution q(x; θ(μ))   MGauss = μ ∈ Rd | Wx (μ)  0, [Wx (μ)]11 = 1 . The Gaussian mean field method restricts the optimization problem (4) to a tractable constraint set MTract , where the free distribution q(x; θ(μ)) satisfies the following factorial form based on disjoint variable clusters {xc1 , . . . , xcm } [10,11] m  qi (xci ; θ(μ)). q(x; θ(μ)) = i=1

The elements of MTract satisfy the conditions: μi = Eq [xi ], i ∈ cg , g ∈ {1, . . . , m}; μij = Eq [xi xj ], i, j ∈ cg , g ∈ {1, . . . , m}; μij = Eq [xi xj ] = μi μj , i ∈ cl , j ∈ ch , l, h ∈ {1, . . . , m}, l = h. Hence MTract has the form MTract = {μ ∈ Rd|Wx (μ)  0, μij = μi μj , i ∈ cl , j ∈ ch , l, h ∈ {1, . . . , m}, l = h}. And the entropy function H(q(x; θ(μ))) can be expressed as 1 n log det Wxci (μ) + log 2πe, 2 i=1 2 m

H(q(x; θ(μ))) =

where Wxci (μ) is the expectation of Xci based on distribution qi (xci ; θ(μ)), ⎤ ⎡ 1 μ1 · · · μ|ci | ⎢ μ1 μ11 · · · μ1|ci | ⎥  1 zxTci (μ) ⎥ ⎢ = . Wxci (μ) = Eqi {Xci } = ⎢ . .. .. .. ⎥ zxci (μ) Zxci (μ) ⎣ .. . . . ⎦ μ|ci | μ|ci |1 · · · μ|ci ||ci | Obviously, MTract ⊆ MGauss , and we can obtain a optimal lower bound of A(θ) on MTract . Hence, the Gaussian mean field variational formula is A(θ) ≥ sup μ∈MTract

 m i=1

Uxci (θ), Wxci (μ)

 m 1 n + log det Wxci (μ) + log 2πe . (5) 2 i=1 2

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Cluster Selection

In this section, we first propose the model coupling and the quasi-coupling concepts to measure the dependance among clusters of GMRF model, and prove the coupling-accuracy theorem that can guide cluster selection. Then we design the normalized cluster selection algorithm based on quasi-coupling for Gaussian mean fields. 3.1

Couplings

Definition 1. Let G be a GMRF model with m disjoint variable clusters {xc1 , xc2 , . . . , xcm } and corresponding parameter matrixes {Zxc1 (θ), Zxc2 (θ), . . . , Zxcm (θ)} of variable set x, where Zxci (θ) is the |xci |-th order principal submatrix of Zx (θ). Then k =1−

|Zx (θ)| |Zxc1 (θ)| · · · |Zxcm (θ)|

is defined as the model coupling of G. Definition 2. Let G be a GMRF model with m disjoint variable clusters {xc1 , xc2 , . . . , xcm } of variable set x. Then M=

2  θij θii θjj i,j

is defined as the quasi-coupling of G, where i ∈ cl , j ∈ ch , l, h ∈ {1, . . . , m}, l = h. Since k and M measure the same change trend of the dependance degree among variable clusters, we have: Property 1. k is positive correlated with M , denoted as k  M. Theorem 1 (Coupling-Accuracy Theorem). For Gaussian mean field method, let A (θ) denote the optimal lower bound with variable clusters {xc1 , . . . , xcm }, and d the variational accuracy of A (θ). Then d=

1 1 log . 2 1−k

Proof. Taking derivatives of the right side of (4) with respect to zx (μ) and Zx (μ), we can get the true values of mean parameters z0 (μ), Z0 (μ), ⎧ =−[Zx (θ)]−1 zx (θ), ⎨ z0 (μ) 1 ⎩ Z0 (μ) − z0 (μ)z0T (μ) =− [Zx (θ)]−1 . 2

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Substituting the results into formula (4), we have A(θ) = 2zx (θ), z0 (μ) + Zx (θ), z0 (μ)z0T (μ) +

a0 1 log( ) + c0 , 2 |Zx (θ)|

(6)

and

n 1 n a0 = (− )n , c0 = log 2πe − . 2 2 2 The Gaussian mean field method is convergent. Taking derivatives of the right side of (5) with respect to zx (μ) and Zx (μ), we can obtain m iterative updates, where the iterative updates for the variable cluster xci is ⎧ ⎡ ⎤  θ1 + j∈N (xc ) μj θ1j ⎪ ⎪ i ⎪ ⎢ ⎥ ⎪ ⎪ .. ⎪ ⎥, ← −Zx−1 (θ) ⎢ ⎨ zxci (μ) . ci ⎣ ⎦  + μ θ θ k |ci | |ci |k k∈N (xci ) ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎩ Zxc (μ) ← zxc (μ)zxT (μ) − Zx−1 (θ). ci i i 2 ci Then A (θ) = 2zx (θ), zx (μ) + Zx (θ), zx (μ)zxT (μ)+ a0 1 log( ) + c0 . 2 |Zxc1 (θ)|...|Zxcm (θ)|

(7)

Since the first order mean parameters zx (μ) converge to the exact values, that is zx (μ) → z0 (μ), from formulas (6) and (7), we have d = A(θ) − A (θ) =

1 1 log . 2 1−k



From the coupling-accuracy Theorem 1 and coupling Property 1, we get the following corollary. Corollary 1 d  M. The quasi-coupling is more computationally efficient than the model coupling, so it can be used as a cluster selection criterion for Gaussian mean fields. 3.2

Algorithm

From above analysis, we can see that the cluster selection problem for Gaussian mean field is equivalent to the optimization problem of min M . Following is a formal description of the assertion. Let en denote the n × 1 matrix of all ones; B the n × n similarity matrix 2 of Gaussian vector x, where Bij = θij /θii θjj ∀i = j and Bii = 0 ∀i; D the n × n degree matrix of x with D = Ben ; and S the n × m cluster selection

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matrix corresponding to variable clusters {xc1 , . . . , xcm }, where the j-th column S·j = (S1j , . . . , Snj )T is the indicator vector for the cluster xcj ,  1 xi ∈ xcj , ∀i, Sij = 0 xi ∈ / xcj . Then the optimization problem for the m-way cluster selection (CS) is  1 m T minimize M (S) = S (D − B)S·i , CS 2 i=1 ·i subject to Sem = en . To avoid unbalance computational complexities among clusters, we normalize the clusters with cluster degrees to penalize large clusters [12]. The normalized cluster selection (NCS) formula of the CS optimization problem is ⎧ m S T (D − B)S·i ⎨ minimize , (8) N (S) = i=1 ·i T NCS S·i DS·i ⎩ subject to Sem = en . (9) As this optimization problem is NP-hard even for m = 2, we present an approx1 imate global optimal solution. First, let T = S(S T BS)− 2 be the normalized cluster selection matrix, and ignore the discrete constraint condition (9), the discrete optimization problem (8) transforms into a tractable continuous optimization problem  maximize N (T ) = Tr(T T BT ), (10) subject to

T T DT = Im ,

(11)

where Tr(·) denotes the matrix trace. This continuous optimization problem can 1 1 be solved by the eigendecomposition of symmetric matrix D− 2 BD− 2  1 1 D− 2 BD− 2 V = V Λ, V TV = In , where Λ = Diag(λ) denotes the eigenvalue matrix with nonincreasingly order: λ1 ≥ . . . ≥ λn ; V = [V1 , . . . , Vn ] the corresponding eigenvector matrix. Obviously, the global solution T ∗ for formulae (10) and (11) is obtained by the first m largest eigenvalues ⎧ ∗ ⎪ = Diag[λ1 , . . . , λm ], ⎨Λ 1 ∗ T = D− 2 × [V1 , . . . , Vm ], ⎪ ⎩ N (T ∗ ) = Tr(Λ∗ ). And the optimal continuous cluster selection matrix S˜∗ can be characterized by 1 S˜∗ = Diag(diag 2 (T ∗ T ∗T ))T ∗ .

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Then, we calculate the approximate optimal cluster selection matrix S ∗ by clustering method. Treat each row S˜i·∗ as a point in Rm , cluster them into m clusters using the standard K-means algorithm, and return a n × 1 index matrix ID containing the cluster indices of S˜i·∗ . Based on the clustering results, S ∗ can be ∗ computed by Sij = 1 iff S˜i·∗ belongs to the cluster j. The formal description of the algorithm is shown in Algorithm 1.

Algorithm 1. Normalized cluster selection based on quasi-coupling Data: Graphical model G with n variables, cluster number m. Result: Cluster selection matrix S ∗ . begin 2 B ← [θij /θii θjj ]n×n − In D ← Ben 1 1 (V, Λ) ← Eigendecomposition(D− 2 BD− 2 ) ∗ −1 T ← D 2 [V1 , . . . , Vm ] 1 S˜∗ ← Diag(diag− 2 (T ∗ T ∗T ))T ∗ ID ← K-means(S˜∗ , m) for i ← 1 to n do for j ← 1 to m do if ID(i) = j then ∗ Sij ←1 else ∗ Sij ←0 end end end end

4

Experiments

In this section, we compare three cluster selection algorithms — the NCS al2 /θii θjj ] based on coupling theory (NCS-B1 ), the NCS gorithm with B1 = [θij algorithm with B2 = [|θij |] based on Xing’s theory (NCS-B2 ), and the random cluster selection algorithm (RCS) that selects the clusters randomly with the same cardinalities as in NCS. We generate the GMRF models through specifying exponential parameters zx (θ) and Zx (θ) randomly, but ensuring that Zx (θ) ≺ 0. Here, we create three models G1 , G2 , G3 with the variable number n = 16, 32, 64 separately. We execute the Gaussian mean field variational formula (5) based on the clusters determined by the three algorithms, and calculate the corresponding optimal lower bound A (θ). The experiment results are showed in Fig. 1 and Table 1. In Fig. 1, the top dotted line indicates the exact value A(θ), and the bottom one represents the naive optimal lower bound A (θ). From Fig. 1, we can draw that (a) the lower bounds of A(θ) based on NCS-B1 are tighter than that based on NCS-B2 , which shows that the quasi-coupling presents superior guidance

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Fig. 1. Cluster selection comparison, where m denotes cluster number, n variable number, A (θ) optimal lower bound Table 1. Accuracy comparison, where m denotes cluster number, n variable number, A (θ) optimal lower bound m n = 16

n = 32

n = 64

2 4 8 2 8 16 2 16 32 48

NCS-B1 90.0801 89.9055 89.6168 136.2399 134.6604 134.1896 164.6212 157.3659 156.6051 156.2517

A (θ) NCS-B2 89.9845 89.5723 89.4298 135.7963 134.4977 134.1135 160.7819 156.9741 156.5891 156.2422

RCS 89.5756 89.3505 89.2883 135.009 133.8552 133.6846 159.8708 156.2664 156.0954 156.0311

Accuracy Increase Rate 12.20% 27.87% 13.98% 18.58% 04.41% 01.87% 68.80% 04.17% 00.16% 00.01%

efficiency for Gaussian mean field cluster selection; (b) the accuracy based on NCS-B2 is higher than that on RCS, which indicates that Xing’s theory has certain guidance to cluster selection for Gaussian mean field. Table 1 is a detailed accuracy comparisons for three cluster selection algorithms, where the accuracy increase rate denotes the accuracy increase ratio of A (θ) based on NCS-B1 to that on NCS-B2 . It shows that the accuracy of A (θ) based on model coupling theory increases 15% than the value under Xing’s theory on average. Furthermore, the smaller the clusters number m is, the higher the accuracy increasing rate is.

5

Conclusions

In this paper, we have proposed the model coupling and the quasi-coupling concepts to measure the dependance among clusters of GMRF model, proved the coupling-accuracy theorem for Gaussian mean fields, which establishes the quasicoupling as a selection criterion, and designed the normalized cluster selection algorithm based on the quasi-coupling for Gaussian mean fields. Theoretical

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analysis and experimental results elucidate that the quasi-coupling is an efficient and superior cluster selection criterion, and that the coupling-based approach to cluster selection for Gaussian mean fields is valid and convenient. Acknowledgement. This work is supported in part by Natural Science Foundation of China under Grant No. 60678049 and Natural Science Foundation of Tianjin under Grant No. 07JCYBJC14600.

References 1. Speed, T., Kiiveri, H.: Gaussian Markov Distributions over Finite Graphs. Annals of Statistics 14, 138–150 (1986) 2. Rue, H., Held, L.: Gaussian Markov Random Fields: Theory and Applications. Chapman & Hall, London (2005) 3. Saul, L., Jordan, M.: Exploiting Tractable Substructures in Intractable Networks. In: Advances in Neural Information Processing Systems, vol. 8, pp. 486–492. MIT Press, Cambridge (1996) 4. Wiegerinck, W.: Variational Approximations between Mean Field Theory and the Junction Tree Algorithm. In: 16th Conference on Uncertainty in Artificial Intelligence, pp. 626–633. Morgan Kaufmann, San Francisco (2000) 5. Welling, M.: On the Choice of Regions for Generalized Belief Propagation. In: 20th Conference on Uncertainty in Artificial Intelligence, pp. 585–592. AUAI Press, Arlington (2004) 6. Welling, M., Minka, T.P., Teh, Y.W.: Structured Region Graphs: Morphing EP into GBP. In: 21th Conference on Uncertainty in Artificial Intelligence, pp. 609–619. AUAI Press, Arlington (2005) 7. Xing, E., Jordan, M., Russell, S.: Graph Partition Strategies for Generalized Mean Field Inference. In: 20th Conference on Uncertainty in Artificial Intelligence, pp. 602–610. AUAI Press, Arlington (2004) 8. Wainwright, M., Jordan, M.: Graphical Models, Exponential Families, and Variational Inference. Technical Report 649, Department of Statistics, University of California, Berkeley (2003) 9. Saul, L., Jaakkola, T., Jordan, M.: Mean Field Theory for Sigmoid Belief Networks. Journal of Artificial Intelligence Research 4, 61–76 (1996) 10. Ghahramani, Z., Jordan, M.: Factorial Hidden Markov Models. Machine learning 29, 245–273 (1997) 11. Xing, E., Jordan, M., Russell, S.: A Generalized Mean Field Algorithm for Variational Inference in Exponential Families. In: 19th Conference on Uncertainty in Artificial Intelligence, pp. 583–591. Morgan Kaufmann, San Francisco (2003) 12. Yu, S.X., Shi, J.: Multiclass Spectral Clustering. In: 9th IEEE International Conference on Computer Vision, pp. 313–319. IEEE Computer Society, Washington (2003)

Multiresolution Image Fusion Algorithm Based on Block Modeling and Probabilistic Model Chenglin Wen1,2 and Jingli Gao2,3 1

Institute of Information and Control, Hangzhou Dianzi University, 310018 Hangzhou, China 2 Department of Computer and Information Engineering, Henan University, 475001 Kaifeng, China 3 Department of Software Engineering, Pingdingshan University, 467000 Pingdingshan, China {wencl,gaojingli}@hdu.edu.cn

Abstract. One new multi-resolution image fusion algorithm based on probabilistic model, block modeling, and wavelet transform is presented here. First, the multisensor images are decomposed into multiple subimages which form a multiresolution pyramid via wavelet transform. Second, the obtained multiple subimages are modeled by block-stack modeling method, then the E-M algorithm is used to get the model parameters. Third, in each level the block fusion estimates of subimages can be obtained according to maximum a posterior method along with the above model and constructed to form subimages. Finally, the image that fused all images is achieved by taking the inverse wavelet transform of all subimages in each level. Experimental results illustrate the feasibility and effectiveness of the algorithm. Keywords: Block modeling, E-M algorithm, Wavelet transform, Maximum a posterior estimate.

1 Introduction Image fusion is to integrate the complementary information from multi-sensor images to generate one composite image that contains a more accurate description of the true scene than any one of the individual sensor images. The goal of image is to decrease the uncertainty about the true scene. With the advent of new imaging and computer technology, the image fusion is widely used in fields of navigation guidance, medical imaging, object detection and recognition, computer vision, robot intelligence, network security, remote sensing and so on [1-6]. There are lots of image fusion techniques and approaches such as intensity-huesaturation (HIS) transform fusion, principal component analysis (PCA) fusion, multiresolution decomposition[1-5]. However, until now, the researches mainly deal with the images with the same resolution. There are fewer methods appropriate to the images with different resolutions. Moreover, the acquisitions from different sensors with F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 442–451, 2008. © Springer-Verlag Berlin Heidelberg 2008

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different imaging mechanisms actually have different resolutions. So, the new schemes about multiple resolutions are needed to resolve the problem. This paper generalizes the probabilistic model based fusion method[6] to fuse the images with different resolutions. Suppose that the resolution proportion between different images is any positive integer. Based on probabilistic model Block modeling, and wavelet transform, an effective multiresolution fusion algorithm is presented.

，

2 Wavelet Transform Wavelet transform can provide a hierarchical decomposion of an image by recursive application of lowpass and highpass filters in each direction of the input image. The wavelet transform can overcome the instability of Laplacian pyramid scheme, because wavelet transform can be used to get rid of correlations between the information contained in subimages of two adjacent layers. Moreover, wavelet transform also offers other advantages such as directional information, computational efficiency and etc[7], [8]. Wavelet transform decomposes the source images into various frequency-domain and spatial subimages which adequately reflect the localized behavior of the original images. It is widely applied to image fusion. Assume that the Q sensor images Z j , j = 1 , 2 , " , Q of an actual scene X have been obtained and the Q images have different resolutions. The wavelet based block image fusion can be described as follows. First, we perform wavelet decomposition on the Q sensor images respectively, decompose the sensor image Z j ( j = 1,2,", Q) into one approximated subimage by lowpass filter and three detailed subimages by highpass filters, and repeat to decompose the lowpass subimages until stopping at a same proper level f to get the following subimages d 11j , d 12j , d 13j , " , d fj 1 , d fj 2 , d fj 3 , c fj 4 ( j = 1,2,", Q) ; Second, at each level, supposing

image Z Q is the basis image, we do blocking operation on all the subimages with different resolutions obtained from above two steps to get different blocks. We can conclude that the size proportionality between the corresponding blocks is positive integer. Then we fuse the corresponding blocks by the proposing method until all the corresponding blocks are fused and integrated into new subimages. Finally, we do the same operation on the final lowpass subimages and then take the inverse wavelet transform the subimages at each level and get the fused image Xˆ of the true scene X .

3 Multiresolution Image Modeling and Fusion 3.1 Block-Based Modeling in Multiscale Domain Suppose that there exists a true scene X that is being imaged by multiple sensors. Usually, the true scene X can not be acquired directly, but the observing image can be gotten under conditions of uniform lighting, unlimited visibility. Because different subimages obtained by multiscale transform have different sizes, the pixel-based

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modeling method can not be used in such a situation. So, the blocking method is applied to present a block-stack modeling method [8-9]. Taking subimages Z 1m,r and Z 2m,r ( r =1,2,3,4 and m is the decomposion level), for example, the dimensions of which are m1 × n1 and m 2 × n 2 respectively, where m

m1 ≥ m2 , n1 ≥ n 2 , m12 = m1 /m2 and n12 = n1 / n2 . We write matrix Z 1, r as the form of

block matrices, that is, matrix Z 1m,r can be divided into m12 × n12 blocks and each block has a size of m12 × n12 , i.e., Z 1m, r = ( Z 1m, r ( g , q )) , where Z 1m, r ( g , q ) is a matrix composed of rows from m 12 ( g − 1 ) + 1 to m 12 g , columns from n 12 ( q − 1 ) + 1 to n 12 q of matrix Z 1m, r , that is, Z 1m, r ( g , q ) = Z 1m, r ( m 12 ( g − 1 ) + 1 : m 12 g , n 12 ( q − 1 ) + 1 : n 12 q )

Let Z 1m, r

= ( Z 1m, r ( g , q ))

.

(1)

be arranged in row sequence, namely

Z1m,r (1,1), Z1m,r (1,2),", Z1m,r (1, n2 ), Z1m,r (2,1), Z1m,r (2,2),", Z1m,r (2, n2 ),", Z1m,r (m2 ,1), Z1m,r (m2 ,2),", Z1m,r (m2 , n2 ) ,

and denoted as Z1m,r ( k ) for k = 1,2,", m2 n2 . Similarly, taking the above blocking operation on the sensor bias sor noise ε1m,r , and then denoting matrices

α

m 1 ,r

α

α

( g , q ) , ε1m,r ( g , q ) by

m 1 ,r

m 1,r

and the sen-

( k ) and ε1m,r ( k )

respectively for k = 1, 2 , " , m 2 n 2 . After above blocking manipulations, the block measurement equations can be written by[8], [9].

α

m m,r Z mj,r ( k ) = C1m,r ,j ( k ) X r ( k )C 2 ,j (k ) +

m j,r ( k ) +

ε mj,r ( k ) , j = 1, 2; k = 1, 2, " , m 2 n 2 .

(2)

m21 ×m12 n12 × n 21 m12 × m12 n12 ×n12 , C 2m,r , C1m,r , C1m,r . where C1m,r ,2 ( k ) ∈ R ,2 ( k ) ∈ R ,1 ( k ) ∈ R ,2 ( k ) ∈ R

Using the definition of Kronecker product, equation(4) can be rewritten as z mj,r ( k ) = C

where

m j,r

( k ) x rm ( k ) + α

m j,r

(k ) + ε

m j,r

( k ) , j = 1, 2 ; k = 1 , 2 , " , m 2 n 2 .

x rm ( k ) = vec ( X rm ( k )) ∈ R m12 n12 ×1

α

α1m,r (k ) = vec(

m 1,r ( k ))

,

z1m,r (k ) = vec( Z1m,r (k ))

,

m 12 n12 × m 12 n12 m,r T C 1m,r ( k ) = C 1m,r ,1 ( k ) ⊗ ( C 2 ,1 ( k )) ∈ R

1× m12 n12 m,r T ; C 2m,r ( k ) = C1m,r ,2 ( k ) ⊗ (C 2 ,2 ( k )) ∈ R

ε 1m, r ( k ) = vec ( ε1m,r ( k )) ∈ R m12 n12 ×1

α

z 2m,r ( k ) = vec ( Z 2m,r ( k )) , α 2m,r (k ) = vec (

(3) , , ,

, ε 2m,r ( k ) = vec( ε 2m,r ( k )) ∈ F 1×1 .

m 2 ,r ( k ))

“ R ”stands for real domain, and,

[

]

T vec( X ) = X1 X2 " Xm1 .

X l is the lth row vector X .

(4)

Multiresolution Image Fusion Algorithm Based on Block Modeling

445

m,r T C 1m,r ( k ) = C 1m,r ∈ R m 12 n12 × m 12 n12 is the Kronecker product of ma,1 ( k ) ⊗ ( C 2 ,1 ( k ))

trix

and

C1m,1,r ( k )

trix A = ( a le ) g1 × g 2 ∈ R (g1 ⋅ g

2

)× (g

3

⋅ g

4

)

.

( C 2m,1, r ( k )) T g1 × g 2

The

Kronecker

and B = (ble ) g 3 × g 4 ∈ R

g3 × g4

product

of

max-

is a matrix of dimension

, and is written by. ⎡ a11 B ⎢a B 21 A ⊗ B : =⎢ ⎢ # ⎢ ⎢⎣ a g1 1 B Δ

a12 B a 22 B

" " %

a g1 2 B

"

a1 g 2 B ⎤ a 2 g 2 B ⎥⎥ . # ⎥ ⎥ a g 1 g 2 B ⎥⎦

(5)

Dividing all the measurement subimages in a manner similar to the above two subimage case, and considering the image with the lowest resolution as the basis image, we can obtain the following measurement equations.

z mj,r ( k ) = C mj,r ( k ) x rm ( k ) + α mj,r ( k ) + ε mj,r ( k ) , j = 1, 2, " , Q ; k = 1, 2, " , m Q nQ . where m jQ = m j / mQ , n jQ = n j / nQ and m jQ , n jQ are z mj,r ( k ) ∈ R

m jQ n jQ ×1

C mj,r ( k ) ∈ R

m jQ n jQ × m1Q n1Q

integers.

is the observation vector located at k from the jth sensor, is gain located at k from the jth sensor, xrm (k ) ∈ R

vector located at k from the true scene, α the jth sensor, ε

positive

(6)

m j,r

(k ) ∈ R

m

jQ

n

jQ

×1

m j,r

(k ) ∈ R

m jQ n jQ ×1

m1Q n1Q ×1

is the

is bias located at k from

is noise vector located at k from the jth sensor,

where k , j , m and r are all discrete integers. The image formation parameters and sensor noise distribution were assumed to vary slowly from one spatial location to another. Taking some stacking manipulations of equations (6), we write the measurement equation as[8].

z rm ( k ) = C rm ( k ) x rm ( k ) + α rm ( k ) + ε rm ( k ) .

(7)

where, ⎡ z1m,r ( k ) ⎤ ⎢ m ⎥ z (k ) ⎥ m z rm ( k ) = ⎢ 2 ,r C (k ) = ⎢ # ⎥ r ⎢ m ⎥ ⎢⎣ z Q,r ( k ) ⎥⎦

Let m jQ n jQ = u j ,

∑u

j

⎡ C 1m,r ( k ) ⎤ ⎡ ε 1m,r ( k ) ⎤ ⎡ α 1m,r ( k ) ⎤ ⎢ m ⎥ ⎢ m ⎥ ⎢ m ⎥ ⎢ C 2 ,r ( k ) ⎥ α m ( k ) = ⎢ α 2 ,r ( k ) ⎥ ε m ( k ) = ⎢ ε 2 ,r ( k ) ⎥ ⎢ # ⎥ r ⎢ # ⎥ ⎢ ⎥ r # ⎢ m ⎥ ⎢ m ⎥ ⎢ m ⎥ ⎢⎣ C Q,r ( k ) ⎥⎦ ⎢⎣ε Q,r ( k ) ⎥⎦ ⎢⎣α Q,r ( k ) ⎥⎦

.

(8)

= u , u1 = v ,then zrm (k) ∈ Ru×1 , xrm (k ) ∈ Rv×1 , Crm (k ) ∈ Ru×v ,

Q

α

m m u×1 .where r (k ), r (k ) ∈ R

ε

block-vector model.

Q is the number of sensors, the model (7) was called

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3.3 Multiresolution Block Fusion Algorithm

Consider next how to obtain the estimate Xˆ of the true scene X by using of the corresponding observations along with model (7). Hypothesis 1 1) the a priori probability density function of x rm (k ) obeys Gaussian distribution N [0, Σ x m (k ) ] r

(

)

p x rm ( k ) =

1

(2π )v / 2 Σ x

m r

(k )

⎡ 1 ⎤ exp ⎢ − x rm ( k ) T Σ −x 1m ( k ) x rm ( k ) ⎥ r ⎣ 2 ⎦

.

(9)

2) the noise ε mj,r (k ) of the ith sensor satisfies

ε mj,r (k ) ~ N [0,σ ε2m ( k ) I ε m ( k ) ] . j,r

(10)

j,r

with the following statistical behaviors ⎧⎪σ 2m I u × u E ε i,rm ( k )ε mj,r ( k ) T = ⎨ ε j,r ( k ) j j 0 u i ×u j ⎪⎩

{

{

}

i= j , E s ( k )ε Tj ( k ) = 0 .

}

i≠ j

(11)

where, ε i ( k ) ∈ R ui ×1 , ε j ( k ) ∈ R u j ×1 , i, j ∈ {1,2," , Q} , 0ui ×u j are matrices of dimension ui × u j , the elements of 0ui ×u j are all zeroes, I u j ×u j is unit matrix of dimension u × u . σ ε2m (k ) is the noise covariance matrix of the ith sensor at index k .

j

j

j,r

Property 1. From(10) and (11), the vector series ε rm (k ) constituted by all ε mj,r (k ) satis-

fies N [0, Σ ε m (k ) ] r

(

)

p ε rm ( k ) =

where, Σ ε

m r

(k )

1

(2π )

Σ ε m (k )

u/2

= diag [σ ε2 m

.

(12)

r

I

1 ,r

⎡ 1 ⎤ exp ⎢ − ε rm ( k ) T Σ ε−1m ( k ) ε rm ( k ) ⎥ r 2 ⎣ ⎦

( k ) ε 1m,r ( k )

, σ ε2 m

2 ,r

I

( k ) ε 2m,r ( k )

, " , σ ε2 m

Q,r

] , σ ε mj,r ( k ) I ε mj,r ( k ) is the 2

I

m ( k ) ε Q,r (k )

noise covariance matrix of the ith sensor, as in (11). Theorem 1. Under the Hypothesis 1, the posterior density function of xrm (k ) , given

the observed sensor data z rm (k )

(

)

p x rm ( k ) | z rm ( k ) =

，is

1 ( 2π )

v/2

M

-1

1/ 2

(

⎡ 1 m x r ( k ) − μ x|z exp ⎢ − ⎣ 2

)

T

(

)

⎤ M x rm ( k ) − μ x | z ⎥ ⎦

.

(13)

Multiresolution Image Fusion Algorithm Based on Block Modeling

447

where M -1 = cov( x rm ( k ) | z rm ( k ))

.

= E {[ x rm ( k ) − μ x | z ] 2 | z rm ( k )}

(14)

= ( C rm ( k ) T Σ ε− 1( k ) C rm ( k ) + Σ x m ( k ) ) − 1 r

is the posterior covariance, and μ x | z = E {x rm ( k ) | z rm ( k ) }

(

)

= M -1 { C rm ( k ) T Σ ε− 1m ( k ) z rm ( k ) − α rm ( k ) } r

.

(15)

is the conditional mean. The operator E { • } denotes expectation operation. If the maximum a posterior estimate of xrm (k ) is xˆrm ( k ) MAP , then Inference 1. Using Theorem 1, the maximum a posterior estimator xˆrm ( k ) MAP of the block at the corresponding location of each level is xˆ rm ( k ) MAP = μ

[

x|z

= C rm ( k ) T Σ ε− 1m ( k ) C rm ( k ) + Σ r

] {C −1

x rm

(k )

m r

(

( k ) T Σ ε− 1m ( k ) z rm ( k ) − α r

.

m r

(k )

)}

(16)

Next, the main task is: First, how to estimate the parameters Crm (k ) and α rm (k ) in model (7)

； Second,

obtain the maximum a posterior estimate xˆrm ( k ) MAP of

xrm (k ) using Inference 1 after getting the estimator of the parameters; Third, construct every xˆrm ( k ) MAP , k = 1,2,", mQ nQ by using the definition of equation (4), and then obtain the maximum a posterior estimate of sub-image r ( r =1,2,3,4) at scale m ; Finally, obtain the global estimator of the fused image by performing the wavelet reconstruction on the fused sub-images. 3.4 E-M Algorithm Multiresolution Block Fusion Algorithm

E-M algorithm is a parameter estimating technique for un-complete data, which estimate parameters by assuming the presence of the latent variables [10-13]. In order to derive the E-M algorithm for estimating parameters, the following hypothesis is given: C rm (k ) , α rm (k ) , Σ ε m (k ) and Σ x m (k ) are constant over the regions r

r

R k , denoted as C , α , Σ ε and Σ x . Fig. 1 shows the regions R k consists of local analysis window defined by 6*6 pixels, 3*3 pixels, or other similar forms; consider the latent variables {x n } to be ‘missing’ data, and the ‘complete’ data to comprise the block observations {z n } along with these latent variables, where the definition of z n is the same as z rm (k ) in (8), n = 1 , " , N , and N is the number of total blocks in local analysis window R k . Next, derive the E-M algorithm for the above block-based model.

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C. Wen and J. Gao

Rk

k

Fig. 1. Local analysis window

From the above hypotheses, the complete-data log-likelihood over region R written as N

∑ ln {p ( z

L =

n

, x n )} .

k

is

(17)

n =1

where, from the definition of section C, we have p (z n , x n ) = ⋅

⎡ 1 exp ⎢ − (z n − Cx n − α ⎣ 2

1

(2 π )u / 2

1/ 2

Σε

⎤ − Cx n − α )⎥ ⎦

.

(18)

⎡ 1 ⎤ exp ⎢ − x nT Σ −x 1 x n ⎥ ⎣ 2 ⎦

1

(2 π )v / 2

)T Σ ε− 1 (z n

Σx

In the E-step, we take the expectation with respect to the distributions p ( x n | z n , C , Σ ε , Σ x ) E

(L ) =

−

N

∑

n =1

1 (z 2 1 (z − 2 +

⎧ 1 ln Σ ⎨ ⎩ 2

ε

−1

n

− α )T Σ

ε

n

− α )T Σ

ε

−1

(z

n

−1 x

E

(x

[

T

Σ

ε

1 tr Σ 2

− α ) +

1 tr C 2

x

(x n ) −

CE

[

+

1 ln Σ 2

+

1 E 2

( x n )C

T

−1

T

n

x

)]

T n

(x

CE Σ

−1

ε

n

(z

x n

T n

)]

.

(19)

⎫ − α )⎬ ⎭

where we omit terms independent of the model parameters, and E

(x n ) =

E (x n x nT

)=

M

−1

M

C

−1

T

Σ

−1

ε

.

(20)

(x n ) .

(21)

(zn − α )

+ E ( x n )E

T

where M = C T Σ ε−1C + Σ x as before. Note that these statistics are calculated using the current values of the parameters. In the M-step, E ( L ) is maximized with respect to C , Σ ε and Σ x , by differentiating equation (19) and setting the derivatives to zero. After some manipulations, we have new parameter estimates ⎡ N ~ C = ⎢∑ ( zn − α )E ⎣ n =1

( x n )⎤⎥ ⎡⎢ ∑ N

T

⎦⎣

n =1

⎤ E (x n x nT )⎥ ⎦

−1

.

(22)

1 ~ Σε = N

∑ {( z N

n =1

n

Multiresolution Image Fusion Algorithm Based on Block Modeling

449

}

(23)

~ ~ ~ ~ − α )( z n − α ) T + C E (x n x nT )C T − ( z n − α ) E T (x n )C T − C E ( x n )( z n − α ) T

Σ

α =

1 N

x

=

N

∑

1 N

∑ E (x N

zn −

n =1

n

n =1

1 N

x nT

).

N

∑

CE ( x n )

(24) .

(25)

n =1

To maximize the likelihood then, the sufficient statistics of the posterior distributions are computed from equations (20) and (21), after which revised estimates for the parameters are obtained from equations (22), (23), (24) and (25). These four equations are iterated in sequence until the likelihood is already at a local maximum. The estimator of α is given by the mean of the data, in which case S is the sample covariance matrix of the observations {z n } . Note that C~ in (23) is the new gain matrix, while C is the old gain matrix. Now, we substitute the estimator of parameters obtained by the above E-M algorithm into (16) to get fusion image.

4 Experiments and Results Simulation of the proposed algorithm was performed on Matlab. The wavelet employed here is Daubechies wavelet. The proposed fusion method is compared with the wavelet based weighted average scheme which averages the low resolution image and the approximation image obtained by the wavelet decomposition of the high resolution image. The input images along with the fused images are shown in Fig. 2, where (a) has a size of 360*240 pixels, (b) has a size of 180*120 pixels, and (c) has a size of 120*80 pixels. (d) shows the fusion result of (a) and (b) by the wavelet-based weighted average method. (e) shows the fusion result of (a) and (c) with our method. (f) shows the fusion result of (a) and (b) with our method. In comparison with (d), (f) shows the invisible person in (a) and the invisible boskage in (b) more clearly. We can find that both the wavelet-based average method and our method are effective. Comparing (f) with (e), we find that (f) shows the invisible person in (a) more clearly than (e), because of the higher the resolution, the more information the original image contains. It accords with the actual fact. There are several methods to compare the performance of the image fusion schemes. Here we use the Entropy, Standard Error, and Clarity rule to evaluate the fused images[14]. The results of Entropy, Standard Error, Clarity of two approaches are shown in Table 1, where the second row shows the performance value of the wavelet-based weighted average method, and the third row and the fourth row show the performance data of fusion images obtained by using our method to fuse (a) and (c), and (a) and (b) respectively. The resolution ratio between (a) and (c) is 1:3, and the resolution ration between (a) and (b) is 1:2. Considering the visual judgment and the objective evaluation, we can find that our algorithm is effective to multiresolution image fusion.

450

C. Wen and J. Gao Table 1. Quantitative analysis of the two algorithms

Fusion method Wavelet-based weighted average Our algorithm (fusion of a and c) Our algorithm (fusion of a and b)

Entropy 5.9911 6.5169 6.7083

Standard error 0.0060 0.0124 0.0144

Clarity 0.0216 0.0341 0.0459

Fig. 2. (a)360*240 high resolution visible image (b)180*120 low resolution infrared image (c)120*80 low resolution infrared image (d)result of fusing (a) and (b) with the wavelet-based weighted average method (e)result of fusing (a) and (c) with our method (f)result of fusing (a) and (b)with our method

5 Conclusions and Future Work By using of wavelet transform, block modeling, and probabilistic model, we have presented a new approach to multiresolution image fusion. The new approach is established under the assumption that the relationship between those noise vectors from different sensor image is statistically independent, and the proportion of resolutions between different images is the positive integer. Actually, it can be improved and applied to other proportions in time domain. If the proportion of resolutions between different images is 1:1, then the proposed is degenerated into the block image fusion with respect to the same resolution, and if the size of the blocks is 1*1, the algorithm will be degenerated into the case of [6], so it can be considered to be the generation of [6]. Our work can be regarded as a helpful exploration to muliresolution image fusion techniques.

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Acknowledgments. This work is supported the National Nature Science Foundation of China (No.60434020, No.60572051). The authors are grateful for the anonymous reviewers who made constructive comments.

References 1. Tan, Z., Fu, F.L., Li, A.G., Yang, B., Gong, Y.G.: Digital Image Fusion. Xi’an Jiaotong University Press, Xi’an (2004) 2. Mao, S.Y., Zhao, W.: Comments on Multisensor Image Fusion Techniques. Journal of Beijing University of Aeronautics and Astronautics 28, 512–518 (2002) 3. Amolins, K., Zhang, Y., Dare, P.: Applications of Wavelet Transforms in Image Fusion. In: Urban Remote Sensing Joint Event, pp. 1–7. IEEE Press, New York (2007) 4. Wu, J., Huang, H.L., Liu, J., Tian, J.W.: Remote Sensing Image Data Fusion based on HIS and Local Deviation of Wavelet Transformation. In: IEEE International Conference on Robotics and Biomimetics, pp. 564–568. IEEE Press, New York (2007) 5. Xia, Y.S., Kamel, M.S.: Novel Cooperative Neural Fusion Algorithms for Image Restoration and Image Fusion. IEEE Transactions on Image Processing 16, 367–381 (2007) 6. Wen, C.L., Guo, C., Wen, C.B.: Multiresolution Image Fusion Algorithm based on Probabilistic Model. In: 6th World Congress on Intelligent Control and Automation, vol. 2, pp. 10398–10402. IEEE Press, New York (2006) 7. Boggess, A., Narcowich, F.J.: A First Course in Wavelets with Fourier Analysis. Publishing House of Electronics Industry, Beijing (2002) 8. Wen, C.L., Zhou, D.H.: Multiresolution Estimate Theory and Application. Qinghua University Publishing House, Beijing (2002) 9. Yan, L.P.: Study on Multirate Sensor based State Fusion Estimation and Multiresolution Image Fusion Algorithms. Ph. D. Dissertation, Tsinghua University (2006) 10. Yang, J., Blum, R.S.: A Statistical Signal Processing Approach to Image Fusion for Concealed Weapon Detection. In: 2002 International Conference on Image Processing, vol. 1, pp. 513–516. IEEE Press, New York (2002) 11. Tipping, M.E., Bishop, C.M.: Probabilistic Principal Component Analysis. Journal of the Royal Statistical Society 61, 611–612 (1999) 12. Sun, D.F., Chen, Z.G., Liu, W.J.: Discussion of Maximum Likelihood Parameter Estimation based on EM Algorithm. Journal of Henan University (Natural Science) 32, 35–41 (2002) 13. Xue, W.X., Liu, D.Y., Zhang, H.: Learning with a Bayesian Networks a Set of Conditional Probability Tables. Acta Electronica Sinica 31, 1686–1689 (2003) 14. Li, W.: Research on Pixel-Level Image Fusion and its Application. Ph. D. Dissertation, South China University of Technology (2006)

An Evolutionary Approach for Vector Quantization Codebook Optimization Carlos R.B. Azevedo1, , Esdras L. Bispo Junior1 , Tiago A. E. Ferreira2 , Francisco Madeiro1 , and Marcelo S. Alencar3 1

2

Catholic University of Pernambuco, Center for Science and Technology, Brazil {azevedo,bispojr,madeiro}@dei.unicap.br Federal Rural University of Pernambuco, Department of Statistics and Informatics 3 Federal University of Campina Grande, Department of Electrical Engineering

Abstract. This paper proposes a hybrid evolutionary algorithm based on an accelerated version of K-means integrated with a modified genetic algorithm (GA) for vector quantization (VQ) codebook optimization. From simulation results involving image compression based on VQ, it is observed that the proposed method leads to better codebooks when compared with the conventional one (GA + standard K-means), in the sense that the former leads to higher peak signal-to-noise ratio (PSNR) results for the reconstructed images. Additionally, it is observed that the proposed method requires fewer GA generations (up to 40%) to achieve the best PSNR results produced by the conventional method. Keywords: codebook optimization, genetic algorithms, accelerated Kmeans, hybrid systems, vector quantization, image compression.

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Introduction

The fundamental purpose of signal compression techniques is to reduce the number of bits required to represent a signal (speech, image, video, audio), while maintaining an acceptable signal quality. Signal compression is essential for applications which require minimization of the storage capacity and/or transmission rate, such as multimedia systems, video-conference, voice response systems, high definition television (HDTV), mobile telephony, storage of medical images, and transmission of remote sensing images obtained from satellites. Vector quantization (VQ) [1] plays an important role in many image coding systems, leading to high compression rates. VQ can be defined as a mapping Q from a vector x in K-dimensional Euclidean space, RK , into a finite subset W of RK containing N distinct reproduction vectors. Thus, Q : RK → W . Codebook W = {wi }N i=1 is a set of codevectors (reconstruction vectors), K is the dimension of the codevectors and N is the codebook size. The mapping Q leads to a partitioning of RK in N cells Si , i = 1, 2, . . . , N , such that 

Author thanks the National Council of Technological and Scientific Development (CNPq) for the financial support.

F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 452–461, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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Si = RK and Si ∩ Sj = ∅ for i = j,

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in which each cell Si is defined as Si = {x : Q(x) = w i } = {x : d(x, wi ) ≤ d(x, wj ), ∀j = i},

(2)

in which d(·,·) is a distortion measure. Codevector wi is the representative vector of all input vectors belonging to the corresponding cell Si . The K-means algorithm, also known as Linde-Buzo-Gray (LBG) algorithm [2], is the most used technique for codebook design. When applied to vector quantizer optimization, the genetic algorithm (GA) is fed with codebooks, in order to generate a better codebook by means of successive crossover and mutation operations in an evolutionary process. In this paper, a modified GA is applied to codebooks designed by the K-means method. However, the codebooks obtained at the end of each GA iteration are updated according to the method proposed by Lee et al. [3], which corresponds to an accelerated version of K-means. The computational complexity of the proposed method is compared to that one of the conventional methodology, which consists on using GA with standard K-means. Simulation results regarding image VQ show that the proposed method outperforms the conventional approach in the sense that it leads to better codebooks, which lead to reconstructed images with higher peak signal-to-noise ratio (PSNR), in earlier stages of the evolutionary process.

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K-Means Algorithm

Let the current iteration of the K-means algorithm be denoted by n. Given K, N and a distortion threshold  > 0, the K-means algorithm [2] consists of the following steps: Step 1 (initialization). Given an initial codebook W0 and a training set X = {xm ; m = 1, 2, . . . , M }, set n = 0 and D−1 = ∞. Step 2 (partitioning). Given Wn (codebook at the n-th iteration), assign each training vector (input vector) to the corresponding class (Voronoi cell) according to the nearest neighbor rule; determine the distortion Dn =

N  

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(3)

i=1 xm ∈Si

Step 3 (convergence test). If (Dn−1 − Dn )/Dn−1 ≤  then stop, with Wn representing the final codebook (designed codebook); else, continue. Step 4 (codebook updating). Calculate the new codevectors as = C(V(wni )), wn+1 i C(V(w ni ))

in which is the centroid of the partition n ← n + 1 and go to Step 2.

(4) V(w ni );

set Wn+1 ← Wn ; set

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The distortion decreases monotonically in the algorithm, since the codebook is iteratively updated to satisfy the centroid and nearest neighbor conditions. The convergence speed of the algorithm and the final distortion are sensitive to the initial codebook.

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Accelerated K-Means Algorithm

The algorithm proposed by Lee et al. [3] corresponds a modification introduced on K-means and generally results, for a given initialization, in a better codebook. The modification proposed by [3] is introduced in Step 4 of K-means. In this approach, the new codevector will be chosen according to = w ni + s(C(V(w ni )) − wni ). wn+1 i

(5)

Thus, the proposed algorithm selects a point between the old codevector and its reflected point as the new reference vector. This method may be seen as a look ahead approach aiming at improving convergence, while reaching a smaller value of average distortion. In the experiments reported in [3], when the value of the scale s is about 1.8, the algorithm generally achieves an adequate performance. It should be noted that a scale value of s = 1.0 corresponds to the standard K-means method.

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Genetic Algorithms

Genetic algorithms are biologically inspired search techniques commonly applied for solving constrained optimization problems. The canonical GA, proposed by Holland [4], arranges different combinations of the input space as a set of candidate solutions called population. With the abilities of distinguishing between bad and good solutions (also known as individuals) and of appropriately encoding them, a GA can evolve the population by operating on pairs of individuals (parents), yielding (possibly) better solutions. The genetic operators crossover and mutation are responsible for producing new individuals for the population. The first acts on the parents’ representation, exchanging information encoded on them, while the second acts solely on the newly generated solutions called offspring. An iteration of GA is known as a generation and comprises three operations over the individuals in the population: a probabilistic selection of the parents for reproduction; application of genetic operators on parents for producing the next offspring and a replacement strategy for substituting inadequate existing solutions by the newly generated ones. It should be noted that a GA is a global search technique and, thus, GA’s are notable on the task of escaping from local minima (provided the necessary diversity in population), as opposed to hill climbing techniques [5]. Diversity can be achieved by either raising the population’s size or by the introduction of stochastic elements on both genetic operators and selection mechanism. Typically, there is a tradeoff between the ability of evaluating distinct regions of

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the search space and the evolution’s convergence, i.e. how fast can the GA find a suitable solution. An individual in the population is (ideally) uniquely identified by means of its encoding of the problem’s domain. In that sense, each search parameter composes a gene. The set of genes composes the individual’s genotype. Typically, similar genotypes imply similar problem’s solutions. Moreover, the characterization of the individual’s viability is done following a heuristic estimation of the solution’s quality, called fitness function – the higher is the individual’s fitness, the higher it is adapted to the environment (the problem’s domain). This is usually the objective function which should be optimized.

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Hybrid Evolutionary Clustering

The K-means algorithm can be regarded as a hill climbing clustering strategy, inasmuch as it is a monotonic algorithm [6]. Thus, it is supposed to achieve local optima, i.e. suboptimal partitions. Much of this limitation is due to the update scheme which highly depends on the initial codevectors. Hence, stochastic approaches such as GA’s can be used to find a globally optimal partition [6]. Considering the population of a GA as composed of various codebooks, Fr¨ anti et al. [7] stated that the attempts of integrating GA’s and K-means fall into one of three categories: 1. Apply K-means for the output of GA (GA + K-means); 2. Apply GA for the output of K-means (K-means + GA); 3. Apply K-means for every codebook in the GA population. The first category (GA + K-means) is immediate: whichever is the output of GA, K-means can be used to continuously improve the best individual. The second strategy (K-means + GA) introduces a high complexity on the genetic operators, since GA should be designed to, hopefully, improve the K-means output. However, in this scheme and without mutation, according to simulations reported in [8], a GA was incapable of improving the quality of the K-means designed input codebooks. Finally, the last category yields too much computational cost, since K-means is applied to all individuals at each generation. It is also possible to design GA’s for VQ without using any integrated hill climbing technique such as K-means, but it has been shown that such GA’s perform worse than if a clustering algorithm is used altogether [7]. Since K-means algorithm attempts to satisfy VQ optimally conditions [1], its integration with GA is beneficial.

6

The Proposed Method

The method proposed in the present paper is derived from a GA + K-means strategy, as explained in Section 5, in which an individual is a codebook composed by N genes, that is, N reference vectors. However, some differences may be pointed out when comparing with those methods described by Fr¨anti et al. [7]:

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1. The GA replacement strategy has been modified to accept the offsprings which represent better solutions than those already known. Another difference from the conventional approach is the fact that the proposed method accepts, with probability pacc , solutions which are not better than the worst one present in the population. When accepted, an offspring will always replace the most inadequate individual. 2. The standard K-means algorithm was replaced by the accelerated version proposed by Lee et al [3]. 3. Instead of applying the accelerated K-means for the best individual at each generation, it is applied for the newly offspring. Fig. 1 illustrates the overall process realized by the hybrid evolutionary K-means optimizer. All modifications are explained in the following: the first one depends on the pacc parameter – for pacc = 0, this strategy is equivalent to a greedy genetic search [9]; for pacc = 1, the strategy becomes the one implemented on the canonical GA [4] (accepting all offspring). This additional parameter gives a fine control of diversity levels in the GA, while guarantees that good solutions will not be rejected. This improvement was benchmarked in Leung et al. [10] and it was used for training multilayer perceptron neural networks, as reported in that paper.

Fig. 1. The proposed method for VQ codebook optimization. The size of the population is denoted by psize and CBj is the j-th input codebook.

The second improvement constitutes the main contribution of this paper. It is expected that this modification will lead to better results in terms of peak signalto-noise ratio (PSNR) measure, as shown in the results’ section. Finally, the third modification is justified by the sake of maintaining diversity and as an attempt to achieve better results in earlier stages of the evolutionary process (when compared with the conventional approach): in any GA scheme, the common way of controlling diversity is by raising the mutation rate, i.e. the frequency that mutation will be applied to an offspring. In this sense, the accelerated K-means integration with GA can be regarded as an additional local optimization module, which acts on the newly candidate solution. Hence, the accelerated

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K-means module improves the newly offspring. The offspring is expected to have a higher probability of being better than its parents. Note that a similar strategy was successfully utilized in [6] for VQ codebook design. Other mechanisms of the proposed GA are now explained: Initialization – The initial population is the set {CBj }psize j=1 of initial codebooks −3 trained with K-means supervised algorithm with  = 10 , in which psize denotes the population size. Selection for reproduction – A fitness proportional selection (FPS) strategy commonly referred to as roulette wheel is adopted: the population is ordered by the individuals’ fitness and two parents are sampled with probability proportional to their fitness values, within a stochastic simulation scheme. Genetic operators – For parents’ recombination, a generalized crossover operator is implemented in the following way: first, a random integer segsize is sampled from the set {1, . . . , N/2}, in which segsize denotes the segment size. Then, for each block composed of segsize genes, one chooses, with equal probability, the corresponding block from one of the parents for composing the genes of the offspring. It can be seen that this algorithm is equivalent to uniform crossover [6] when segsize = 1 and to one-point crossover when segsize = N/2 with the locus point on the half of the chromosome. For mutation, let μ be a random uniformly distributed variable on the range [0.8, 1.2], i.e. μ ∼ U nif orm(0.8, 1.2). Then, a scalar product is applied on a random chosen gene with probability pmut.

7

Complexity Analysis

In a recent paper [11], analytical expressions were obtained for the number of arithmetic and comparison operations performed by K-means as well as by a competitive learning algorithm applied to VQ codebook design. Based on the expressions derived in [11], the second column of Table 1 summarizes the number of operations performed by one iteration of K-means, in terms of dimension (K), codebook size (N ) and number of training vectors (M ). Now the corresponding number of operations will be derived for the modified K-means [3]. For codebook updating, the modified K-means differs from the standard K-means by the fact that the former performs Equation 5, which requires KN additional multiplications, subtractions and additions. The third column of Table 1 summarizes the number of operations performed by one iteration of the modified K-means. Table 1. Analytical expressions for the number of computational operations performed by one K-means iteration in terms of K, N and M Op. Standard K-means Accelerated K-means + (M − 1)(1 + KN ) + M (K − N + 1) (M − 1)(1 + KN ) + M (K−N + 1) + KN 1 + KN M (1 + KN M ) + KN − KN M KN (M + 1) × 1 + KN 1 + KN / 1 + (N 1)M 1 + (N 1)M comp

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According to Table 1, it is observed that the modified K-means leads to an inKN 1 crease of KN M 100% = M 100% in terms of multiplications when compared to the standard K-means. As an example, for the purpose of image VQ with a training set with M = 4, 096 vectors, the increase is 0.0244%. The modified K-means KN 1 KN leads to an increase of 1+KN M 100% ≈ M 100% and (1+KN )(M−1)+(K−N +1)M 100% in terms of subtractions and additions respectively. For practical values of K and N , in the scenario of image VQ, the computational complexity of the modified K-means approaches that of the standard K-means.

8

Results

The coding performance of all optimization methods was evaluated on 256 × 256 monochrome 8 bpp images: Boat, Barbara, Clock, Elaine, Goldhill, Lena, Mandrill, Peppers and Tiffany (Fig. 2). For the simulation, given a target image to be compressed and a fixed N , the same input set of K-means designed codebooks was used for composing the initial population for each set of trails (50 runs). The distortion threshold  = 10−3 was assumed for designing the codebooks with the standard K-means. In addition, the following parameters were assumed for both GA + standard K-means and the proposed hybrid evolutionary method (modified GA + accelerated K-means): psize = 20, pacc = 0.1, n = 1 (number of either standard K-means and Lee et al. iterations performed within both GA versions), mutation rate of 0.2 and a maximum of 500 generations for both methods. Also note that PSNR was assumed as the fitness function.

Fig. 2. Image data set used on experiments

The peak signal-to-noise ratio (PSNR) [1] was also used to assess the objective quality of the reconstructed images. Vector quantization with dimension K=16 was considered, corresponding to the use of image blocks of 4 × 4 pixels. Codebook sizes of N =32, 64, 128, 256 and 512 were considered, corresponding to coding rates of 0.3125, 0.375, 0.4375, 0.5 and 0.5625 bpp. Table 2 summarizes the experimental results obtained after 50 runnings of the GA + standard K-means approach and the proposed method (modified GA + accelerated K-means with s = 1.5) for Lena image. From Table 2, it can be noted that the proposed method outperforms the conventional techniques investigated. Also, as the codebook size (N ) raises, the gain achieved with the proposed GA increases. It should be noted that the K-means column on Table 2 refers to the PSNR values derived from the population (K-means designed codebooks) which fed both GA’s versions.

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Table 2. Average PSNR for Lena image obtained in 50 experiments N 32 64 128 256 512

K-means 26.59 27.71 28.80 29.90 31.11

GA + K-means 26.69 27.91 29.07 30.55 32.33

Proposed Method 26.70 27.91 29.11 30.73 32.62

An experimental study was also conducted for adjusting the scale value (s) used on simulations. For this task, scale values from the set {1.1, 1.2, . . . , 2.0} were tested for each combination of image and codebook size (N ). The gains obtained by the proposed approach over both K-means and GA + K-means for the best values of s, considering images encoded at 0.5625 bpp, are reported in Table 3. The best results were achieved for Clock, Lena, Peppers, Tiffany, Boat and Elaine images, with gains from 1.10 dB up to 1.58 dB in comparison with the K-means input codebooks and gains from 0.20 dB up to 0.33 dB when compared to the GA + standard K-means approach. Fig. 3 (a) shows an example of the evolution performed by the proposed method in comparison with GA + standard K-means strategy for Lena with coding rate of 0.5625 bpp. For s ≤ 1.6, the simulations have shown that, on average (after 50 trials), the proposed method leads to PSNR results better than the ones obtained by using the conventional approach. For Lena image, by the end of 500 generations, the proposed method leads to a PSNR value of the recontructed image at around 32.62 dB, while the GA + standard K-means leads to a final PSNR value around 32.33 dB. Fig. 3 (a) also shows that, after about 257 generations, the PSNR value obtained by the GA + standard K-means approach stabilizes. The proposed method, by its turn, requires only 161 generations to achieve the same 32.33 dB PSNR result. In almost all experiments (taking into account 50 trials, five codebook sizes, and nine different images), it was observed that the proposed method Table 3. Average improvements of the proposed method over both K-means and GA + K-means for the best scale values (s) on images quantized at 0.5625 bpp Image Barbara (s = 1.2) Boat (s = 1.4) Clock (s = 1.1) Elaine (s = 1.3) Goldhill (s = 1.2) Lena (s = 1.5) Mandrill (s = 1.1) Peppers (s = 1.2) Tiffany (s = 1.5)

Improvement (dB) K-means GA + K-means 0.62 0.15 1.12 0.21 1.58 0.33 1.10 0.20 0.50 0.00 1.50 0.29 0.39 0.05 1.49 0.26 1.47 0.27

Improvement (%) K-means GA + K-means 1.74 0.50 3.85 0.72 5.12 1.03 3.36 0.58 1.62 0.01 4.86 0.90 1.49 0.19 4.82 0.81 4.39 0.79

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Fig. 3. (a)Average evolution of the peak signal-to-noise ratio (PSNR) for Lena image with N = 512 after 50 trials. (b) Effect of scale variation in the peak signal-to-noise ratio (PSNR) for Peppers image with N = 512.

achieves the best results obtained by the conventional technique in earlier stages of the evolutionary process (in a smaller number of generations). Fig. 3 (b) shows the effect of varying the scale parameter for Peppers image encoded at 0.5625 bpp. As it can be seen, scale values of s > 1.6 does not garantees any improvements over the conventional approach, but still achieves significant improvements over the initial K-means designed codebooks. Therefore, it can be asserted that the proposed method leads to better codebooks in fewer generations when compared to the conventional method.

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Conclusion

This paper proposed a hybrid evolutionary approach for codebook optimization with the use of Lee et al. algorithm instead of the standard K-means as a new local optimization module on a modified GA. Results have shown that the method is promising to improve the codebooks designed with standard clustering algorithms, yielding (in comparison with the GA + standard K-means method) better PSNR results for image compression applications in fewer generations. Further studies should consider the investigation of the effect of raising the number of Lee et al. iterations and of designing the initial population with other clustering algorithms. Also, auto-adaptive methods, which could automatically adjust the scale parameter during the global search, are of interest. In this sense, authors’ current works also include the investigation of a Terrain-Based Genetic Algorithm (TBGA) [12] for deriving heuristics for the adjustment of the scale parameter during the evolution process.

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References 1. Gersho, A., Gray, R.M.: Vector Quantization and Signal Compression. Kluwer Academic Publishers, Boston (1992) 2. Linde, Y., Buzo, A., Gray, R.M.: An Algorithm for Vector Quantizer Design. IEEE Trans. on Communications 28, 84–95 (1980) 3. Lee, D., Baek, S., Sung, K.: Modified K-means Algorithm for Vector Quantizer Design. IEEE Signal Processing Lett. 4, 2–4 (1997) 4. Holland, J.H.: Adaptation in Natural and Artificial Systems: An Introductory Analysis with Application to Biology, Control and Artificial Intelligence. MIT Press, Cambridge (1992) 5. Mitchell, M., Holland, J.H.: When Will a Genetic Algorithm Outperform Hill Climbing? In: Cowan, J.D., Tesauro, G., Alspector, J. (eds.) Advances in Neural Information Processing Systems, pp. 51–58. Morgan Kaufmann, San Mateo (1994) 6. Krishna, K., Murty, M.: Genetic K-means Algorithm. IEEE Trans. on Systems, Man and Cybernetics 29, 433–439 (1999) 7. Fr¨ anti, P.: Genetic Algorithm with Deterministic Crossover for Vector Quantization. Pattern Recog. Lett. 21, 61–68 (2000) 8. Fr¨ anti, P., Kivij¨ arvi, J., Kaukoranta, T., Nevalainen, O.: Genetic Algorithm for Codebook Generation in Vector Quantization. In: 3rd Nordic Workshop on Genetic Algorithms, Helsinki, pp. 207–222 (1997) 9. Smith, J.: On Replacement Strategies in Steady State Evolutionary Algorithms. Evol. Comput. 15, 29–59 (2007) 10. Leung, F., Lam, H., Ling, S., Tam, P.: Tunning of the Structure and Parameters of a Neural Network Using and Improved Genetic Algorithm. IEEE Trans. on Neural Networks 14, 79–88 (2003) 11. Madeiro, F., Lopes, W.T.A., Aguiar Neto, B.G., Alencar, M.S.: Complexidade Computacional de um Algoritmo Competitivo Aplicado ao Projeto de Quantizadores Vetoriais. Learning and Nonlinear Models 2, 34–48 (2004) 12. Gordon, V.S., Pirie, R., Wachter, A., Sharp, S.: Terrain-Based Genetic Algorithm (TBGA): Modeling Parameter Space as Terrain. In: Genetic and Evolutionary Computation Conference (GECCO), pp. 229–235. Morgan Kaufmann, Orlando (1999)

Kernel-Based Text Classification on Statistical Manifold Shibin Zhou, Shidong Feng, and Yushu Liu School of Computer Science and Technology Beijing Institute of Technology Beijing, 100081, P.R. China {guoguos.zhou, sdfenghx, liuyushu}@bit.edu.cn

Abstract. In the text literature, a variety of useful kernel methods have been developed by many researchers. However, embedding text data into Euclidean space is the key characteristic of common kernels-based text categorization. In this paper, we focus on representation text vectors as points on Riemann manifold and use kernels to integrate discriminative and generative model. And then, we present diffuse kernel based on Dirichlet Compound Multinomial manifold (DCM manifold) which is a space about Dirichlet Compound Multinomial model combining inverse document frequency and information gain. More specifically, as demonstrated by our experimental results on various real-world text datasets, we show that the kernel based on this DCM manifold is more desirable than Euclidean space for text categorization. And our kernel method provides much better computational accuracy than some current stateof-the-art methods. Keywords: Kernel method, Support vector machine, Statistical manifold.

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Introduction

Because the increased availability of documents in digital formats, there is a growing need for finding, filtering, and managing these resources. Text categorization is a major solution to these kinds of problems, which is a text contentbased classification technique that assigns texts to some predefined categories. In the past, Support Vector Machine (SVM) have been established as a most powerful and promising machine learning method for text categorization [1] and kernel methods related with SVM has been heavily developed. Especially, the Fisher kernel [2] and probability product kernel [3] examined the relation of discriminative and generative model in kernel framework. These methods explore a contact point between discriminative learning (support vector machines and kernels) and generative learning (graphical models). Discriminative learning directly create superplane or distance metric for a given classification task, while generative learning provides a rich probability analysis for exploring models of classification. Then, the Fisher kernel and probability product kernel try to integrate the generative models within a discriminative learning paradigm. F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 462–471, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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Meanwhile, from a very different angle, the diffuse kernel [4] has been proposed which are based on the heat equation of the graph and can be regarded as the discretization of the familiar Gaussian kernel of Eculidean space. Lafferty and Lebanon extend this work to define the diffuse kernel [5] on the Riemann manifold in continuous situation rather than on the graph in the discrete situation. The compelling feature of the diffuse kernel which arise from the statistical manifold of multinomial distributions, is that it is straightforward and efficient to compute over a wide range multinomial distributions and generative models. Furthermore, Zhang et al. [6] prove kernels on the multinomial manifold is conditionally positive definite on the TF-IDF schema. In our opinion, the kernels which embed text data into Euclidean space was not effective representation for text because the common text representation space is not linear such that adding text vectors does not make sense. So we think it makes more sense to view document feature vectors as points in a Riemannian manifold rather than in a Euclidean geometry. In this paper, We follow the Lafferty and Lebanon’s work [2] to relate discriminative and generative model in the framework of text statistical manifold. We investigate Dirichlet Compound Multinomial distribution which handle word burstiness phenomenon very well according to [7] and regard this distribution as the based generative model in our work. So, the key idea in our approach is to begin with a statistical model which embed vectors in high dimensional space called Dirichlet compound multinomial manifold (DCM manifold) which is a space about Dirichlet compound multinomial model combining inverse document frequency and information gain, and to consider the diffuse kernel or heat kernel on the manifold equipped with the Fisher information metric as geodesic distance. This paper is organized as follows. In Section 2, the statistical manifold theory is introduced briefly. We present our approaches about how to and classify documents with diffuse kernel on Dirichlet compound multinomial manifold in section 3. In section 4, we evaluate accuracy of our schema on 20newsgroup and WebKB datasets. Finally we conclude the paper with a summary in section 5.

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Relevant Concepts about Statistical Manifold

In this section, we introduces the concept of the statistical manifold and the trick to compute geodesic distances on it[5][8]. These are base techniques of our proposed kernel. 2.1

Statistical Manifold and Geodesic Distance

Let M = {p (·|θ) , θ ∈ Θ ⊂ Rn } be a n-dimensional statistical space on a set X . For each x ∈ X assume the mapping θ → p (x|θ) is C ∞ (infinitely differentiable mapping space) at each point in the interior of Θ. ∂ and θ (x) = logp (x|θ). The Fisher information matrix [gij (θ)] Let ∂i = ∂θ i is given by

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gij (θ) = Eθ [∂i θ ∂j θ ]  = ∂i logp(x|θ)∂j logp(x|θ)p(x|θ)dx X

or equivalently as  gij (θ) = 4

X

∂i

  logp(x|θ) ∂j logp(x|θ) dx

In coordinates θi , gij (θ) defines a Riemannian metric on Θ, giving M the structure of a n-dimensional Riemannian manifold. The Riemannian metric g assign for each point of the manifold x ∈ M an inner product on the tangent space Tx M. The metric is required to satisfy the usual inner product properties and to be C ∞  in x and allows us to measure lengths of tangent vectors v ∈ Tx M as vx = gx (v, v), leading to the definition of a length of a curve on the manifold b c : [a, b] → M as a c(t)dt. ˙ The geodesic distance function d(x, y) for x, y ∈ M is defined as the length of the shortest curve connecting x and y and turns the manifold into a metric space. 2.2

Text Multinomial Manifold and Metric Learning

Suppose we have m documents, d1 , d2 , . . . , dm , containing words form corpus D who has a vocabulary of size n. The corpus of text documents is summarized in a m by n co-occurrence table, where tft,d stores the number of occurrences of a word wt in document d. Then, the generative model of the corpus denoted by multinomial distribution is n  (n tft,d )!  (tf ) nt=1 p (D) = θt t,d t=1 tft,d ! t=1 d∈D

n where θ = {θ1 , θ2 , . . . , θm } should be on the (n − 1)-simplex defined by t=1 θt = 1. Therefore, the multinomial manifold is the parameter space of the multinomial distribution

n  n Pn−1 = θ ∈ R : ∀i θi > 0, θi = 1 i=1

equipped with the Fisher information metric: Jθ (u, v) =

n  ui vi i=1

θi

θ ∈ Pn−1

u, v ∈ Tθ Pn−1

It is a well known fact that the multinomial manifold is isometric to the positive n-sphere of radius 2.

n  n−1 n 2 θi = 4 S+ = θ ∈ R : ∀i θi > 0, i=1

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with the metric inherited from the embedding  n−1 Euclidean space. We can define the diffeomorphism F : (Pn−1 , J ) → S+ ,δ ,

√ √ √  F (θ1 , θ2 , . . . , θn ) −→ 2 θ1 , 2 θ2 , . . . , 2 θ n  n−1 and its inverse mapping F −1 : S+ , δ → (Pn−1 , J ) is F −1 (θ1 , θ2 , . . . , θn ) −→



θ2 θ12 θ22 , ,..., n 4 4 4



Associate with F is the push-forward map F∗ that maps v ∈ Tx Pn−1 to v  ∈ n−1 . It is defined as TF (x) S+ v (h ◦ F ) = (F∗ v) h

 n−1 ∀h ∈ C ∞ S+ ,R

 n−1 n−1 Where C ∞ S+ , R denotes infinitely differentiable function space on S+ . Intuitively, the push forward maps velocity vectors of curves to velocity vectors n−1 of the transformed curves. As we show above, the Riemmanian metric on S+ ∗ is δ, and we can obtain a metric F δ on Pn−1 called the pullback metric F ∗ δx (u, v) = δF (x) (F∗ u, F∗ v)

u, v ∈ Tθ Pn−1

The importance of this map is that it turns F (as well as F −1 ) into an isometry; that is,    dF ∗ δ θ, θ = dδ F (θ) , F θ θ, θ ∈ Pn−1 Therefore the geodesic distance between θ, θ ∈ Pn−1 can be derived as the  n−1 geodesic distance between F (θ) , F (θ) ∈ S+ . In other words, the geodesic distance is the short perimeter segment of great circle connecting F (θ) and F (θ) . Specifically, the geodesic distance between θ, θ ∈ Pn−1 is given by    dg θ, θ = 2 arccos F (θ) , F (θ)   n   θt θt = 2 arccos t=1

3

Diffuse Kernel Based on Dirichlet Compound Multinomial Manifold

In this section, we derive the multinomial simplex of Dirichlet compound multinomial (DCM) model, inverse document frequency and information gain function which can be represent exponential function of information gain at first. And then, Dirichlet Compound Multinomial manifold can be derived by embedding in the manifolds of these multinomial simplex. Finally, we propose diffuse kernel on the Dirichlet Compound Multinomial manifold. We present our method as follows.

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Three Multinomial Manifold

Given a document d = {w1 , w2 , . . . , wn }, it is a natural way to associate to the document d with a parameter vector θ(d) in the multinomial model. So, under the Dirichlet Compound Multinomial model, this amounts to the mapping which ˆ sends a document d to its maximum likelihood model θ(d) as   tfw1 ,d + α1 tfwn ,d + αn ˆ , . . . , n θ(d) = n t=1 tfwt ,d + αt t=1 tfwt ,d + αt and the Dirichlet prior parameter vector α is a convergent fixed-point iteration [9] that we adopt as:  tfwk ,d αnew = αk  k

d∈D tfwk ,d −1+αk  k tfwk ,d   d∈D k tfwk ,d −1+ k αk

where tfwk ,d denote the numbers of the k-th word in the document d. So, a multinomial manifold about Dirichlet Compound Multinomial model can be derived as:

n  model n θi = 1 (1) Pn−1 = θ ∈ R : ∀i θi > 0, i=1

The inverse document frequency(IDF) of the t-th word in the corpus D = {d1 , d2 , . . . , dm } is weighted by idf(wt ) as m idf(wt ) = df(wt ) Let

idf(wi ) βi =  j idf(wj )

Then, we can derive the inverse document frequency multinomial manifold about inverse document frequency (idf) as:

n  idf n Pn−1 = β ∈ R : ∀i βi > 0, βi = 1 (2) i=1

Let igk denote information gain of the k-th word in the corpus D and we can define a function of this information gain as f (igk ) = exp(igk ) Let

exp(igt ) γt = n j=1 exp(igj )

Then, we can derive the information gain function multinomial manifold as:

n  ig n Pn−1 = γ ∈ R : ∀i γi > 0, γi = 1 (3) i=1

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467

Embedding

In this paper, The embedding can be interpreted as a mapping from one multinomial manifold to another multinomial manifold through the following family of differomorphisms gλ : intPn−1 → intPn−1   xn λn x1 λ1 ,..., gλ (x) = λ ∈ intPn−1 x·λ x·λ n Where x · λ is the scalar product i=1 xi λi . The family gλ is a Lie group of transformations under compostition that is isomorphic to intPn−1 . idf Firstly consider the family of differomorphisms Gγ : intPn−1 → intPn−1   βn γn β1 γ1 ig ,..., Gγ (β) = γ ∈ intPn−1 β·γ β·γ which is mapping from multinomial manifold described by Eq.(2) to a new multinomial manifold. model dcm Secondly, consider the family of differomorphisms Fλ : intPn−1 → intPn−1   θ1 λ1 θn λn ,..., Fλ (θ) = (4) λ ∈ Gγ (β) θ·λ θ·λ which is mapping from multinomial manifold described by Eq.(1) to Dirichlet compound multinomial manifold or DCM manifold as we define. 3.3

DCM Manifold

With regard to Eq.(4), we can derive a multinomial simplex as:   θ1 λ1 θn λn ,..., φ= = (φ1 , . . . , φn ) θ·λ θ·λ As a result, we can describe our Dirichlet Compound Multinomial manifold (DCM manifold) of above multinomial simplex as follows.

n  dcm n φi = 1 (5) Pn−1 = φ ∈ R : ∀i φi > 0, i=1

In this manifold, the fisher information metric J can be defined as Riemann metric. But the fisher information was hardly derived, so as to geodesic distance dJ derived hardly too. n−1 Fortunately, we can define the diffeomorphism f : (Pn−1 , J ) → (S+ , δ), 

   φ1 , φ2 , . . . , φn f (φ1 , φ2 , . . . , φn ) → n−1 which is an isometry between (Pn−1 , J ) and (S+ , δ), and noticing that dδ (η, ζ) is givenby the length of the great circle connecting the two points dδ (η, ζ) = arccos( ηi ζi ). It then follows that  n   φi ϕi dJ (φ, ϕ) = dJ (f (φ), f (ϕ)) = arccos i=1

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The Kernel

The motivation of this paper is to exploit the intrinsic geometric structure of text data to design a kernel that can better capture document similarity or dissimilarity. Standard text categorization usually rely on the similarity measure defined by the dot product of two document vectors in a Euclidean space. The geometric interpretation of the dot product is that it computes the cosine of the angle between tow vectors provided they are normalized to unit length. When turning to the Riemannian geometry, this similarity measure is no longer available on general manifolds, because the concept of dot product is only defined locally on the tangent space but not globally on the manifold itself. However, there exists a natural dissimilarity measure on general manifold: geodesic distance. We can use the diffuse kernel by geodesic distance dJ of Dirichlet Compound dcm as Multinomial manifold Pn−1  n+1    1  −n 2  κ φ, φ = (4πt) 2 exp − arccos φi φi t i=1 which can be reduced to  n+1     φi φi κ φ, φ = exp −γ arccos2 i=1

4

Experiments and Results

In order to evaluate the properties of our kernel method, we have conducted experiments on two real-world datasets, WebKB and 20Newsgroups, to evaluate the effectiveness of our proposed model for text categorization. The WebKB1 dataset contains manually classified Web pages that were collected from the computer science departments of four university(Cornell, Texas, Washington and Wisconsin) and some other university. The pages are divided into seven categories: student, faculty, staff, course, project, department and other. In this paper, we use the four most populous entity-representing categories: student, faculty, course, and project, which all together contain 4199 pages. We called this selected WebKB dataset as WebKB top-4 dataset. Like handling 20Newsgroup dataset, We randomly select 70% of the documents to be used for training and the remaining 30% for testing. The 20Newsgroups(20NG)2 dataset is a collection of approximately 20,000 documents that were collected from 20 different newsgroups with about 1000 messages from each newsgroup. This collection consists of 19,974 non-empty documents distributed evenly across 20 newsgroups and we selected 19,946 nonempty documents which are all the same after feature selection. We use the 1 2

http://people.cs.cmu.edu/afs/cs.cmu.edu/project/theo-20/www/data http://people.csail.mit.edu/jrennie/20newsgroups

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Table 1. Experimental results on the WebKB top-4

macro-averaging precision macro-averaging recall macro-averaging F1 micro-averaging accuracy

poly 0.753 0.703 0.720 0.733

RBF 0.823 0.700 0.718 0.797

diffuse 0.876 0.726 0.734 0.835

Table 2. Experimental results on the 20NG dataset

macro-averaging precision macro-averaging recall macro-averaging F1 micro-averaging accuracy

poly 0.817 0.801 0.805 0.798

RBF 0.801 0.748 0.761 0.744

diffuse 0.855 0.855 0.854 0.852

Table 3. The comparison of performance (F1) on WebKB subset Category student faculty course project macroaveraged F1

poly 0.878 0.817 0.446 0.741 0.720

RBF 0.881 0.852 0.378 0.763 0.718

diffuse 0.938 0.874 0.295 0.827 0.734

newsgroups to form categories, and randomly select 70% of the documents to be used for training and the remaining 30% for testing. For these two datasets, we performed stop word removal, stemming, and caseconversion to lower case before feature selection was applied on the training set. Furthermore, We apply Information Gain feature selecting method to the documents of both WebKB and 20NG datasets with threshold −0.0436 to WebKB and 0.055 to 20NG. We exployed LIBSVM [10] as the implement of SVM which uses the “ones vs ones” method for multi-category classification because of its effectiveness and efficiency. We set all the parameters to their default values except few parameters were set special values for various datasets. Especially, we adapted LIBSVM with implementing diffuse kernel of DCM manifold to cooperate with the “Gerneral Text Toolkit” developing by our laboratory. We have tried diffuse kernel on the DCM manifold, polynomial kernel and RBF kernel respectively on the Euclidean space of TF-IDF weighted schema. The results of macro-averged and micro-averaged to WebKB and 20NG datasets are shown in Tables 1 and 2 for polynomial, RBF and diffuse kernel respectively. We also showed the comparison of performance F1 on WebKB and 20NG subsets in Tables 3 and 4. Specially, All results are averaged across 5 random runs for WebKB and 20NG dataset. According experimental results,

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S. Zhou, S. Feng, and Y. Liu Table 4. The comparison of performance (F1) on 20NG subset Category alt.atheism comp.graphics comp.os.ms-windows.misc comp.sys.ibm.pc.hardware comp.sys.mac.hardware comp.windows.x misc.forsale rec.autos rec.motorcycles rec.sport.baseball rec.sport.hockey sci.crypt sci.electronics sci.med sci.space soc.religion.Christian talk.politics.guns talk.politics.mideast talk.politics.misc talk.religion.misc macroaveraged F1

poly 0.708 0.736 0.806 0.753 0.815 0.830 0.607 0.877 0.895 0.934 0.948 0.884 0.766 0.897 0.926 0.875 0.803 0.893 0.659 0.480 0.805

RBF 0.700 0.707 0.761 0.760 0.818 0.783 0.554 0.834 0.896 0.919 0.924 0.851 0.743 0.828 0.869 0.811 0.796 0.533 0.661 0.467 0.761

diffuse 0.761 0.807 0.827 0.805 0.869 0.872 0.820 0.898 0.944 0.965 0.974 0.931 0.823 0.931 0.953 0.922 0.852 0.918 0.699 0.510 0.854

diffuse kernel on DCM manifold outperform polynomial kernel and RBF kernel of TF-IDF schema.

5

Conclusion

In this paper we derived three multinomial simplex and pullback mapping between multinomial manifold and semi-sphere manifold at first, and proposed diffuse kernel on Dirichlet Compound Multinomial (DCM) manifold which is a novel space model of kernel related the generative and discriminative schema as we known. We have demonstrated experiements use LIBSVM [10] with kernels on WebKB and 20 Newsgroups datasets. According to the experimental results, diffuse kernel on Dirichlet Compound Multinomial manifold which we proposed is more desirable than polynomial kernel and RBF kernel. Acknowledgements. We would like to thank the anonymous reviewers for their valuable comments and suggestions. We are grateful for Zhao Cao’s helpful discussion and advice. Many thanks also give to Jian Cao, Jinghua Bai, Xu Zhang and Yingfan Gao for their suggestions regarding this paper. This work was supported by the Pre-Research Project of the ’Eleventh Five-Year-Plan’ of China under grant No.200504123.

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References 1. Joachims, T.: Text Categorization with Support Vector Machines: Learning with Many Relevant Features. In: N´edellec, C., Rouveirol, C. (eds.) ECML 1998. LNCS, vol. 1398, pp. 137–142. Springer, Heidelberg (1998) 2. Jaakkola, T.S., Haussler, D.: Exploiting Generative Models in Discriminative Classifier. In: Proceedings of the 1998 Conference on Advances in Neural Information Processing Systems, Denver, Colorado, USA, vol. 11, pp. 487–493. MIT Press, Cambridge (1999) 3. Jebara, T., Kondor, R., Howard, A.: Probability Product Kernels. The Journal of Machine Learning Research 5, 819–844 (2004) 4. Kondor, R., Lafferty, J.: Diffusion Kernels on Graphs and Other Discrete Input Spaces. In: Proceedings of the Nineteenth International Conference on Machine Learning, San Mateo, CA, USA, pp. 315–322. Morgan Kaufmann, San Francisco (2002) 5. Lafferty, J., Lebanon, G.: Diffusion Kernels on Statistical Manifolds. Journal of Machine Learning Research(JMLR) 6, 129–163 (2005) 6. Zhang, D., Chen, X., Lee, W.S.: Text Classification with Kernels on the Multinomial Manifold. In: Proceedings of the 28th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR 2005), Salvador, Brazil, pp. 266–273. ACM Press, New York (2005) 7. Madsen, R.E., Kauchak, D., Elkan, C.: Modeling Word Burstiness Using the Dirichlet Distribution. In: Proceedings of the 22nd International Conference on Machine Learning, New York, NY, USA, pp. 545–552. Morgan Kaufmann, San Francisco (2005) 8. Lebanon, G.: Metric Learning for Text Documents. IEEE Transactions on Pattern Analysis and Machine Intelligence 28, 497–508 (2006) 9. Minka, T.: Estimating a Dirichlet Distribution (unpublished Paper, 2003) http://research.microsoft.com/∼ minka 10. Chang, C.-C., Lin, C.-J.: LIBSVM: A Library for Support Vector Machines (2001), http://www.csie.ntu.edu.tw/∼ cjlin/libsvm

A Boost Voting Strategy for Knowledge Integration and Decision Making Haibo He1 , Yuan Cao1 , Jinyu Wen2 , and Shijie Cheng2 1

2

Department of Electrical and Computer Engineering, Stevens Institute of Technology, Hoboken, NJ 07030 USA {hhe,ycao}@stevens.edu College of Electrical and Electronics Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China {jinyu.wen,sjcheng}@mail.hust.edu.cn

Abstract. This paper proposes a voting strategy for knowledge integration and decision making systems with information uncertainty. As ensemble learning methods have recently attracted growing attention from both academia and industry, it is critical to understand the fundamental problem of voting strategy for such learning methodologies. Motivated by the signal to noise ratio (SNR) concept, we propose a method that can vote optimally according to the knowledge level of each hypothesis. The mathematical framework based on gradient analysis is used to find the optimal weights, and a voting algorithm, BoostVote, is presented in detail in this paper. Simulation analyses based on synthetic data and real-world data sets with comparison to the existing voting rules demonstrate the effectiveness of this method. Keywords: Voting strategy, Ensemble learning, Signal-to-noise ratio, Classification.

1

Introduction

Voting strategy is a fundamental and critical issue for the ensemble learning systems. Generally speaking, voting strategy provides a mechanism to integrate the knowledge from multiple and diversified voting hypotheses to potentially improve the final decision making process. In this paper, we focus on the understanding of this problem for the ensemble classification systems. Our objective is to find an optimal voting weight for each individual hypothesis for improved performance over a given goal. We start the definition of the problem discussed in this paper explicitly: Definition: Given an ensemble learning system with multiple hypotheses over a classification target function Y = {1, ..., C} (C is the number of class labels): H = {hj } , j = 1, ..., L, each is developed by a learning method, Φ : {ξ, π}, based on the training data set, Dtr , find the optimal voting strategy, Ψ , for an improved final decision: P (Y |xt ), where xt is a testing data instance drawn from the testing data distribution Dte . F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 472–481, 2008. c Springer-Verlag Berlin Heidelberg 2008 

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The learning procedure Φ includes two subsystems: base algorithm ξ and learning procedure π. Generally speaking, any kind of learning algorithms can be used as the base algorithm to develop the decision boundary from the training data, such as neural networks, decision tree, and others. A common scenario in the machine learning community is to use weak learning algorithms (WeakLearn) to develop a strong ensemble learning system [1-4]. For instance, theoretical analysis of the weak learnability is discussed in detail in [1]. General bounds of the complexity of probably approximately correct (PAC) learning, and important proofs on the equivalent of “strongly learnable” and “weakly learnable” have been presented. The π represents a learning procedure used to obtain multiple hypotheses H. For instance, bootstrap sampling method can be used to sample in the instance space to train different hypotheses. Bagging [2] and adaptive boosting (AdaBoost) [3,4] are two representative works in this domain. While bagging uses a uniformly distributed sampling function (bootstrap with replacement) across all training instances, AdaBoost adopts an adaptive iteration learning procedure to automatically shift the decision boundary to be more focused on those difficult instances: examples tend to be misclassified (“hard instances”) receive higher weights compared to those tend to be correctly classified (“easy instances”) at each iteration. Subspace learning is another major category to build an ensemble of multiple hypotheses, such as the random subspace [5], random forest [6], ranked subspace [7], and rotation forests [8]. Other major work of ensemble learning includes stacked generalization [9] and mixture of experts [10]. A fundamental problem in the ensemble learning system is the voting strategy since multiple hypotheses are developed from different views of the training data. To this end, hypothesis diversity is an important criterion to assess the effectiveness of ensemble learning system, which raises the essential problem of voting strategy. As each hypothesis carries different knowledge level for the target function, it is natural to use different weights for different hypotheses to potentially improve the final decision: highly confident hypothesis should carry higher weights. However, in real-world applications, it is very difficult to evaluate the confidence level of each hypothesis over future testing instances [12]. In this paper, we propose a novel understanding to address this fundamental issue. Analogous to the signal and noise concept, we transform the decision making process to be an optimization problem targeting to find the optimal weights for each hypothesis to maximize the combined knowledge level of the ensemble system. In this way, the final decision is boosted from individual hypotheses for knowledge integration and accumulation. Mathematical formulation of the problem followed by a voting algorithm, BoostVote, is presented in detail in this paper. To our best knowledge, this is the first time for this idea to be presented in the community. We believe that this idea provides new insights to this fundamental problem and may motivate future theoretical and practical research developments in the society. The rest of this paper is organized as follows. Section 2 briefly reviews the major voting strategies in this field. In section 3, the detailed mathematical

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foundation and the BoostVote algorithm is presented. Section 4 presents the simulation analyses of the proposed method to synthetic data as well as real-world machine learning data sets. Comparative studies of the classification accuracy of the proposed method with those of the existing methods are used to illustrate the effectiveness of this method. Finally, a conclusion and a brief discussion of future research work is discussed in section 5.

2

Related Works

Considering the problem definition in the section 1, we represent the training data as Dtr with m instances {xq , yq }, where q = 1, ..., m, xq is an instance in the n dimensional feature space X, and yq is the class label associated with xq . Followed by the learning method Φ : {ξ, π}, a total number of L hypothesis is obtained: H = {hj } , j = 1, ..., L. Therefore, for each testing instance xt ∈ Dte , each hypothesis hj can produce an estimate of a posteriori probability across all the potential class labels: Pj (Yi |xt ), where Yi = 1, ..., C. In this way, the voting strategy can be defined to find a mapping function Ψ for an improved estimate of a final posteriori probability P (Yi |xt ) from the individual Pj (Yi |xt ): xt → Yi satisf y max P (Yi |xt )) Yi   = Ψ Pj (Yi |xt )|L j=1 , xt , ω

(1)

Where ω is a set of parameters used to adjust the contributions of each hypothesis. Based on the information theoretic criteria and Bayesian rationale [11,12], various voting strategies have been used in the research community. The most commonly adopted voting rules include geometric average (GA), arithmetic average (AA), median value (MV), and majority voting (MajV) rule. In this research, we will compare the proposed BoostVote strategy to all these existing voting methods. GA rule: xt → Yi satisf y max Yi

AA rule:

L 

Pj (Yi |xt ).

(2)

j=1

1 Pj (Yi |xt ). L j=1 L

xt → Yi satisf y max Yi

(3)

MV rule: xt → Yi satisf y max{median(Pj (Yi |xt ))}. Yi

(4)

MajV rule: xt → Yi satisf y max Yi

L  j=1

Δj (Yi |xt ),

(5)

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where Δj (Yi |xt ) =

3

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1; if hj (xt ) = Yi ; 0; otherwise.

The Proposed Method

Fig. 1 shows the general scenario of knowledge integration and decision making for an ensemble of classifier learning system. We assume each hypothesis, hj , is associated with a signal, sj , and noise, nj , defined as a measurement related to the posteriori probability Pj (Yi |xt ): sj = |Pj (Yi |xt ) − 0.5|

(6)

nj = 0.5 − |Pj (Yi |xt ) − 0.5|

(7)

In a two-class classification problem, Pj (Yi |xt ) = 0.5 represents the lowest certainty, meaning that each of the two classes is equally likely. On the other hand, Pj (Yi |xt ) = 1 or Pj (Yi |xt ) = 0 represents full knowledge, meaning that the hypothesis, hj , is certain about the class label. For multi-class classification problems, considering a class label Yi , the classification of any given testing instance xt can be represented in a Boolean type: xt ∈ Yi or xt ∈ Y¯i , where Y¯i = {Yk , k = i}. That is to say, xt either belongs to Yi or Y¯i , where Y¯i represents all other possible class labels in Y except Yi . In this way, the multi-class classification problem can also be transformed analogous to a two-class problem. Therefore, Equations (6) and (7) provide a uniform way to represent the signal and noise concept. In order to maximize the signal level in the ensemble system as shown in Fig. 1, we define the combined signal and noise as [13]: L  ωk sk )2 sˆ2 = (ω1 s1 + ω2 s2 + . . . + ωL sL )2 = (

(8)

k=1

2 2 n ˆ 2 = ω12 n21 + ω22 n22 + . . . + ωL nL =

L 

ωk2 n2k

k=1

Fig. 1. Boost voting strategy for knowledge integration and decision making

(9)

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Therefore, the combined signal to noise ratio (SNR) in the knowledge integration process can be defined as: sˆ2 Γ (ωj ) = 2 (10) n ˆ To find the maximal value, we take the gradient of Equation (10) with respect to ωj . ∂ˆ s2 2 ∂ n ˆ2 2 n ˆ − sˆ ∂ωj ∂ωj ∇Γ (ωj ) = n ˆ4 L L L    ωk2 n2k ωk sk − 2ωj n2j (ωk sk )2 2sj =

k=1

k=1

k=1

n ˆ4 (11)

By setting ∇Γ (ωj ) = 0, one can get: L 

(ωk nk )2 ωj n2j = k=1L sj  ω k sk

(12)

k=1

This lead to the conclusion of the following condition: ωj ωj = = constant 2 sj /nj βj

(13)

sj . Equation (13) means from the signal to noise ratio point of n2j view, each hypothesis hj should vote proportionally to βj in order to maximize the knowledge level in the ensemble system. Based on this analysis, we now present the proposed BoostVote algorithm as follows: where βj =

[Algorithm: BoostVote] − 1. Apply the testing sample xt to each hypothesis, hj , and return the decision profile Pd (Yi |xt ). − 2. Calculate the signal and noise for each class label: SYi = |Pd (Yi |xt ) − 0.5|

(14)

NYi = 0.5 − SYi

(15)

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− 3. Calculate the P¯Yi and PˆYi for each potential class label: P¯Yi = sign(Pd (Yi |xt ) − 0.5) · SYi

(16)

P¯Yi PˆYi = NYi

(17)

ˆYi − 4. Calculate βYi and β SYi (NYi )2

(18)

1 − 0.5 1 + e−αβYi

(19)

βYi = ˆYi = β

− 5. Calculate Pˆout (Yi ) and P¯out (Yi ) L 

βˆYi (k)P¯Yi (k)

Pˆout (Yi ) =  k=1  L  (βˆY (k))2 (NY (k))2 i i

(20)

k=1

P¯out (Yi ) =

Pˆout (Yi ) 2(1 + |Pˆout (Yi )|)

(21)

− 6. Calculate the final voting probability P (Yi |xt ) P (Yi |xt ) = P¯out (Yi ) + 0.5

(22)

Output: voting strategy: mapping function Ψ xt → Yi satisf y max P (Yi |xt ) Yi

(23)

In the BoostVote algorithm, a decision profile Pd (Yi |xt ) is defined, which provides a voting probability for each testing instance across all potential class labels. Such a decision profile can either be obtained directly from most of the off-the-shelf base learning algorithms or by slight modifications. For instance, for those base learning algorithms that can provide soft-type outputs (continuous values), such as neural networks, one can directly use a scaled output value to obtain the decision profile information to calculate the signal and noise value. Fig. 2 illustrates this idea for a neural network model with C output neurons; each represents a class identity label. In this case, the decision profile element

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Fig. 2. Decision profile calculation based on neural network model

Fig. 3. An example of BoostVote algorithm

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can be decided by the normalized output value from each corresponding output neuron. On the other hand, for hard-type (output discrete class labels only) base learning algorithms, one can obtain the decision profile information based on the cross-validation method. We also want to point out that for many off-the-shelf base learning algorithms, it is generally very straightforward to transform from the hard-type output to the soft-type output [3]. In the BoostVote algorithm, a modified logistic function is introduced in Equation (19) to adjust the voting sensitivity level from each hypothesis, and the value of α parameter can be decided by cross-validation method. In this research, we set α = 0.1 for all simulations. Fig. 3 shows an example of the BoostVote algorithm for a three-class classification problem with three hypotheses. From Fig. 3 one can see, BoostVote algorithm will vote the testing example xt as a class 3 label in this case. Considering the decision profile information at Step 1, if majority voting rule is used in this case, it will randomly select a class label for this testing instance since each class receives the same number of votes (one vote for each class). This indicates that the proposed algorithm can potentially boost the final decision making process from different knowledgeable voting hypotheses.

4

Simulation Analysis

To see how the proposed BoostVote algorithm can boost the knowledge level in the ensemble system, we present our first experiment on synthetic data set. Assume a two-class (positive and negative) ensemble classification system including 25 hypotheses (L = 25) and 50 testing instances. After a training procedure, assume each hypothesis, hj , will vote each testing example with a posteriori probability, Pj (Yi |xt ), according to a uniform distribution in [0, 1]. Fig. 4 shows the final combined posteriori probability P (Yi |xt ) for the positive class, where x-mark represents AA rule and circle represents BoostVote method. From Fig. 4 one can see, BoostVote algorithm can boost the final combined knowledge level to be more deterministic, therefore increase the separation margin to facilitate the final decision making process. We now illustrate the application of BoostVote to real-world data benches from the UCI machine learning repository [14]. Table 1 summarizes the Table 1. Data set characteristics used in this paper Data set # # # Name Examples Classes Attributes ecoli 336 8 7 shuttle 59000 2 9 spectf 267 2 44 wdbc 569 2 30 wine 178 3 13 yeast 1484 10 8

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Fig. 4. Final posteriori probability based on BoostVote and AA rule Table 2. Testing error performance(in percentage) Voting method ecoli shuttle spectf wdbc wine yeast Winning times

GA 13.21 7.38 21.92 8.2 20.28 39.88 1

AA 13.17 7.46 22.25 8.22 22.9 39.89 1

MV 13.2 8.85 23.4 8.48 28.31 39.87 0

MajV 13.28 8.94 23.4 8.48 29.97 39.89 0

BoostVote 13.29 7.35 21.74 8.22 14.73 39.86 4

characteristic of the data sets used in this research. For each data set, we randomly select half of the data for training and use the remaining half for testing purpose. Neural networks with multilayer perceptron (MLP) is used as the base learning algorithm in our current study. The number of hidden layer neuron is set to 10, and the number of input and output neurons are set to the number of features and classes for each data set, respectively. Sigmoid function is used as the activation function, learning rate is set to 0.05 and the number of learning iteration is 100. Bagging [2] is used to create the ensemble system, and 25 bootstrap sampling (with replacement) iterations are used as suggested in [15] for general use of bagging method. Table 2 presents the testing error performance based on the average of 100 random runs. Here we compare the performance of BoostVote with those of four popular voting strategies as discussed in section 2. For each data set, the winning strategy is also highlighted by underline. In addition, the total winning times for each method across all these data sets are also summarized in Table 2. These numerical results indicate that BoostVote can provide competitive voting results when multiple hypotheses are involved in a voting system.

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Conclusion and Future Work

In this paper, a novel voting strategy is proposed for knowledge integration and decision making systems. By using the concept similar to signal and noise ratio, the proposed method enables each hypothesis vote optimally according to their knowledge level for the target function; therefore boost the performance of the final decision. Mathematical analysis is presented in detail in this paper, and simulation analysis on various data sets is used to demonstrate the effectiveness of this method. There are various interesting directions can be further developed. For instance, large scale empirical studies and assessment metrics development will be useful to fully justify the effectives of this method in different applications. In addition, it would be interesting to analyze the performance of this voting method under the skewed data distributions (the imbalanced data learning problem). Since voting strategy plays a critical role in many machine learning methods, we hope the proposed research provides new insights to this fundamental problem, and can potentially be a powerful method for a wide range of application domains.

References 1. Schapire, R.E.: The Strength of Weak Learnability. Machine Learning 5(2), 197– 227 (1990) 2. Breiman, L.: Bagging Predictors. Machine Learning 24(2), 123–140 (1996) 3. Freund, Y., Schapire, R.E.: Experiments With a New Boosting Algorithm. In: International Conference on Machine Learning, pp. 148–156 (1996) 4. Freund, Y.: An Adaptive Version of the Boost by Majority Algorithm. Machine Learning 43(3), 293–318 (2001) 5. Ho, T.K.: Random Subspace Method for Constructing Decision Forests. IEEE Transactions on Pattern Analysis and Machine Intelligence 20(8), 832–844 (1998) 6. Breiman, L.: Random Forests. Machine Learning 45(1), 5–32 (2001) 7. He, H., Shen, X.: A Ranked Subspace Learning Method for Gene Expression Data Classification. In: International conference on Artificial Intelligence, pp. 358–364 (2007) 8. Rodr´ıguez, J.J., Kuncheva, L.I., Alonso, C.J.: Rotation Forest: A New Classifier Ensemble Method. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(10), 1619–1630 (2006) 9. Wolpert, D.H.: Stacked Generalization. Neural Network 5(2), 241–259 (1992) 10. Jacobs, R.A., Jordan, M.I., Nowlan, S.J., Hinton, G.E.: Adaptive Mixtures of Local Experts. Neural Computation 3(1), 79–87 (1991) 11. Kittler, J., Hatel, M., Duin, R.P.W., Matas, J.: On Combining Classifiers. IEEE Transactions on Pattern Analysis and Machine Intelligence 20(3), 226–239 (1998) 12. Theodoridis, S., Koutroumbas, K.: Pattern Recognition, 3rd edn. Elsevier, Academic Press (2006) 13. Starzyk, J.A., Ding, M., He, H.: Optimized Interconnections in Probabilistic SelfOrganizing Learning. In: IASTED International Conference on Artificial Intelligence and Applications, pp. 14–16 (2005) 14. UCI Machine Learning Repository, http://mlean.ics.uci.edu/MLRepository.html 15. Opitz, D., Maclin, R.: Popular Ensemble Methods: An Empirical Study. J. Artificial Intelligence Research 11, 169–198 (1999)

A New Strategy for Pridicting Eukaryotic Promoter Based on Feature Boosting Shuanhu Wu, Qingshang Zeng,, Yinbin Song, LihongWang, and Yanjie Zhang School of Computer Science and Technoloy, Yantai University, Yantai 264005, China [email protected]

Abstract. Computational prediction of eukaryotic promoter is one of most elusive problems in DNA sequence analysis. Although considerable efforts have been devoted to this study and a number of algorithms have been developed in the last few years, their performances still need to further improve. In this work, we developed a new algorithm called PPFB for promoter prediction base on following hypothesis: promoter is determined by some motifs or word patterns and different promoters are determined by different motifs. We select most potential motifs (i.e. features) by divergence distance between two classes and constructed a classifier by feature boosting. Different from other classifier, we adopted a different training and classifying strategy. Computational results on large genomic sequences and comparisons with the several excellent algorithms showed that our method is efficient with better sensitivity and specificity. Keywords: DNA sequence analysis, promoter prediction, motifs, feature boosting.

1 Introduction The eukaryotic promoter prediction is one of most important problems in DNA sequence analysis. It is possible to use the prediction of promoter sequences and transcriptional start point as a signal to deduce the approximate start of the transcript and delineate one end of some gene. Over the last several years, although a lot of efforts have been devoted to this problem and several excellent algorithms, such as PromoterInspector[1], Dragon Promoter Finder[2], FirstEF[3], Eponine[4] and the algorithm proposed by Wu [5] et al., have been developed with moderate success, but it is still need to further improve. The existing algorithms for promoter region recognition are based on the fact that the properties of the promoter regions are different from the properties of other functional regions. Almost all existing algorithms can be subdivided into three main categories: (1) search by signal, (2) search by content and (3) search by CpG island. “Search by signals” techniques are based on the identification of putative transcriptional patterns such as TATA-box and CAAT-box in DNA sequences, but these patterns cannot be the only determinants of the promoter function. For instance, in one study it was found that applying Buchers TATA-box weight matrix to a set of mammalian non promoter DNA F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 482–490, 2008. © Springer-Verlag Berlin Heidelberg 2008

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sequences resulted in an average of one predicted TATA-box every 120 bp [6]. That means that the application of some known transcriptional motifs to the prediction of promoters introduces many false positives. “Search by content” techniques are often based on the difference in the local base and local word composition between regulatory and non-regulatory DNA regions. This class of algorithms assumes that the difference is caused by the presence of transcriptional signals, such as the binding motifs for transcriptional regulators in the promoter regions. This concept was explored by analyzing the most frequent pentamers or hexamers [2, 7], other variant–length motifs and short words [1]. “Search by CpG island” techniques is based on the fact that most human promoters are correlated with CpG islands and many genes are recognized and validated successfully by using CpG island as gene markers. CpG islands are found around gene starts in approximately half mammalian promoters and are estimated to be associated with ~60% of human promoters [8]. Therefore, it is a good indicator for the presence of promoters. Algorithms such as the CpG-promoter [9], CpGProd [10] and FirstEF [3] make use of the information of CpG islands. Nevertheless, we must bear in mind that not all the human promoters are related to CpG islands, and from this point of view, correct predictions are limited and cannot exceed 60%, if the prediction is based on CpG islands alone. In this paper, we developed a new algorithm for human promoter regions recognition. The new algorithm is also word based and it belongs to the category of “search by content”, but we adopted a different strategy from others for constructing an efficient classifier. Our algorithm is base on following hypothesis: promoter is determined by some motifs or word patterns and different promoters are determined by different motifs. We select most potential motifs (i.e. features) by divergence between two classes and constructed a classifier by feature boosting. Different from other classifier, we also adopted a different training and classifying strategy. We tested our algorithm on large genomic sequences and compared the results with those from PromoterInspector and Dragon Promoter Finder. The experiments show that our algorithm is efficient with higher sensitivity and specificity in predicting promoter regions.

2 Classifier Constructing by Feature Boosting 2.1 Ideas for Classifier Constructing In this section, we introduce the ideas for our classifier according to the hypothesis: promoter is determined by some motifs and different promoters are determined by different motifs. Suppose that we are given a set of training or labeled samples including positive samples (promoter sequences) and negative samples (non-promoter sequences) {xi, yi|i=1,2,…N} where xi is training sequence and y i ∈ {−1,1} , and are asked to give a decision y to any sequence x (known or unkown). Usually, it is convenient for us to obtain some statistics from training samples for some potential features and we can get these features by a mapping function φ i () . In our study, φ i () is a mapping function from training sequence to some pentamers (word with length of 5 letters) that is potential feature for distinguishing promoters and non-promoters.

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Once we get a set of potential features by the mapping function, according to the ideas that promoters are determined by some upstream binding motifs, our classification function can be formed as follows: k

F ( x) = sign[ ∑ α i f i (φ i ( x))]

(1)

i =1

Where f i () is the classifier for the feature φ i (x) , α i is the ith weight or infection factor of ith feature for constructing a promoter, and k is the number of features. With each feature, we may obtained the occurring probabilities of positive (promoters) and negative samples (non-promoters) p + (φ i ( x)) and p − (φ i ( x)) . For an input sequence x, if p + (φ i ( x)) > p − (φ i ( x)) , it will be more likely from this evidence that x is a positive sample. Therefore, we can use f i () = log( p + () / p − ()) as a feature classifier for each feature, which is proved in [11] to maximize the margin. This way, the classification function becomes k

p + (φ i ( x))

i =1

p − (φ i ( x))

F ( x) = sign[∑ α i log

]

(2)

It is clear that classification function F(x) is constructed by weighting each feature classifier. 2.2 Feature Selection by Divergence Distance There are two terms to learn in the classification function (3), one is the feature set P = { φ i (x) , i=1,2,…k} and the other is combining weights { α i , i=1,2,…k }. To get the most potential and minimum feature set, we use divergence [12] and take a greed strategy to gradually add the most discriminating feature to the feature set one at a time. For the labeled positive samples (promoter sequences) and negative samples (non-promoter sequences), divergence distance is defined as follows N

D = ∑ ( p + (φ I ( x k )) − p − (φ I ( x k ))) log k =1

p + (φ I ( x k )) p − (φ I ( x k ))

(3)

Note that each term in (3) is positive, so the most discriminability features can be selected by first sorting ( p + (φ I ( x k )) − p − (φ I ( x k ))) log( p + (φ I ( x k )) / p − (φ I ( x k ))) in descending order and then feed to the classifier in turn one at one time until classification precision is satisfied. Our promoter prediction system consists of two classifiers, so it needs two different groups of features (pentamers) to distinguish promoter regions from exons and introns regions, these two groups features can be obtained by labeled training samples, we will discuss it in detail in the experimental part. 2.3 Parameter Learning by Boosting Strategy After a set of features has been obtained, next step is to tune the weights { α i , i=1,2,…k } that can best combine each feature classifier f i () = log( p + () / p − ()) to

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form a efficient classification function. Finding the optimal { α i , i=1,2,…k } is a difficult task when there are a large number of features. Optimal parameters { α i , i=1,2,…k } can be obtained by minimizing the recognition error rate at the kth step

{α i }ik=1 = arg min ε k {α i }

(4)

where the recognition error is

εk =

1 N ∑ δ ( y i ≠ F ( x i )) N i =1

(5)

Where δ ( x ) = 1 , if x is true, otherwise δ ( x ) = 0 . But finding these optimal { α i , i=1,2,…k } is a difficult task when there are a large number of features. In our work we adopt a greed algorithm to find better weights by assuming the last optimal weights { α i , i=1,2,…k -1} known. In the beginning, we fix the first (k-1) α i as the last optimal one and set α k = 0 . By this method, it can guarantees the recognition error to be no more than that of the last optimal weights, i.e. ε k ≤ ε k −1 . In this way, we can gradually find all the approximately optimal weights. We implemented the above procedure by boosting strategy by adding one feature classifier at each time. Through increasing the weights of misrecognized samples at each step, a corresponding α can be obtained by the similarly reasoning in [11]. Assume that at (k-1)th step, the weights of of positive and negative samples are D k −1 ( x i+ ) and D k −1 ( x i− ) , respectively. Then at step k we can re-weight the samples by

D k ( x i+ ) =

p + (φ k −1 ( x i+ )) 1 + + D x α y ( ) exp( − log( )) k −1 i k −1 i Z+ p − (φ k −1 ( x i+ ))

p + (φ ( x − )) 1 D k ( x ) = − D k −1 ( x i− ) exp(−α k −1 y i− log( − k −1 i− )) Z p (φ k −1 ( x i ))

(6)

− i

where Z + and Z − are normalization factors for D k ( x i+ ) and D k ( x i− ) , respectively, and α k −1 can be obtained by D k −1 ( x i+ ) and D k −1 ( x i− ) at (k-1)th step by the similar method used in [11] as follows:

α k −1 =

1 2 + r+ + r− log( ) 2 2 − r+ − r−

(7)

where N+

r + = ∑ D k −1 ( x i+ ) y i+ log( i =1

N−

p + (φ k −1 ( x i+ )) ) , p − (φ k −1 ( x i+ ))

and r − = ∑ D k −1 ( x i− ) y i− log( i =1

p + (φ k −1 ( x i− )) ) , N + and N − are the number of positive − − p (φ k −1 ( x i ))

and negative samples. Note that in [11], it need each feature classifier

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| log( p + () / p − ()) |≤ 1 , since each feature classifier is a constant and can be obtained by labeled training samples, so this can be simply achieved by following operation:

log

p + (φi ( x)) p − (φi ( x))

⇐

log( p + (φi ( x) / p − (φi ( x)) max{log( p + (φi ( x) / p − (φi ( x))}

(8)

i

Fig. 1 shows the complete flowchart of our feature boosting algorithm.

Given: {xi, yi|i=1,2,…N}, xi’s are labeled training sequence with fixed length, y i  {1,1}

1 , D1 ( x i )  N H k  H k 1 While k < Kmax or z0

Initialize: D1 ( x i )

1 , i = 1, Kmax N

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zSelect ith most discriminating feature and corresponding classifier via (3) and (8) zWeights computation: learning D i via (7) zCompute recognition error H k via (5) zUpdate D k 1 ( x i ) and D k 1 ( x i ) via (6) zk = k + 1 End Output classifier: F ( x)

k

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Fig. 1. The flowchart of feature boosting

2.4 Training and Classifying Strategy In practical training and classifying application, each input sequences have limited length. In our study, the length of each training and input sequence is 300 letters. For some input sequence x, we can’t guarantee each feature obtained by training sequences exists and corresponding classifier does work. Therefore in the training and classifying procedure, we process this situation by following restriction:

⎧ p + (φi ( x)) if φi x exists ⎪log − =⎨ p (φi ( x)) log − p (φi ( x)) ⎪ if φi x doesn' t exists ⎩0 p + (φi ( x))

(9)

This is just about our hypothesis that promoter is determined by some motifs and different promoters are determined by different motifs. By above strategy, we can obtain two classifiers that constructed our promoter prediction system: Promoter-Exon classifier and Promoter-Intron classifier. The prediction

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system assigns a sequence to the promoter calss only if all two classifiers decide that the sequence belongs to this class. For further control sensitivity and specificity, one can apply a threshold T > 0 to each classifier by the following: k

p + (φ i ( x))

i =1

p − (φ i ( x))

F ( x) = sign[(∑ α i log

) −T]

(10)

In the experiment part, we represent these two thresholds by symbols tpe and tpi for Promoter-Exon classifier and Promoter-Intron classifier, respectively.

3 Experimental Results and Discussions 3.1 Training Sequence Sets Our training sequence sets were downloaded from website: http://www.fruitfly.org /seq_tools/datasets/Human. All the training sequences are of 300 bp length and each promoter sequence is taken from 250 bp upstream to 50 bp downstream of the TSS. Three training sets for promoter, exon and intron include 565, 890 and 4345 sequences, respectively. We use these sequence sets to train our classifiers by setting Kmax=45=1024 and the algorithm stopped at k=641. For convenience, our promoter prediction system is called as PPFB in the following discussion. 3.2 Large Genomic Sequence Analysis and Comparisons Our promoter prediction system recognizes promoter regions in large genomic sequences by a sliding window. A window is moved over sequences and its content is classified. The window length is set as 300 bp and step length set to 1 bp in our system. A promoter region is obtained by clustering the prediction outputs with a gap tolerance 1 kb. To evaluate the performance of our algorithm, we compared our system with four other promoter prediction systems: PromoterInspector [1], Dragon Promoter Finder (DPF) [2], Eponine [4] and FirstEF [3]. These four methods can be accessible via the Internet and are currently the best four prediction systems. The evaluation set for comparison is the same as that used in PromoterInspector and DPF and is currently a standard for evaluating the performance of promoter recognition system. This set consists of six GenBank genomic sequences with a total length of 1.38 Mb and 35 known TSSs (see Table 3 in [1]). We adopt the same evaluating criterion used by PromoterInspector [1]: A predicted region is counted as correct if a TSS is located within the region or if a region boundary is within 200 bp 5’ of such a TSS. The main results and comparisons are presented in Table 1. In these experiments, PromoterInspector is used with default settings and our system is used by setting tpe=0.06 and tpi=0.75, DPF is used by setting se=0.45. The setting se=0.45 is found to give a balance sensitivity and specificity result. We observed that when the se of DPF is set too high, the number of false positives will increase much more rapidly than the number of true positives. For the same reason, we set t=0.995 for Eponine and p=0.98 for FisrtEF. By comparing the results of DPF with PromoterInspector, we can see that

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AC002397

L44140

D87675

AF017257

AF146793

AC002368

Method PromoterInspector DPF (se=0.45) Eponine (t = 0.995) FirstEF (p = 0.98) PPFB (tpe=0.06, tpi=0.75) PromoterInspector DPF (se=0.45) Eponine (t = 0.995) FirstEF (p = 0.98) PPFB (tpe=0.06, tpi=0.75) PromoterInspector DPF (se=0.45) Eponine (t = 0.995) FirstEF (p = 0.98) PPFB (tpe=0.06, tpi=0.75) PromoterInspector DPF (se=0.45) Eponine (t = 0.995) FirstEF (p = 0.98) PPFB (tpe=0.06, tpi=0.75) PromoterInspector DPF (se=0.45) Eponine(t = 0.995) FirstEF (p = 0.98) PPFB (tpe=0.06, tpi=0.75) PromoterInspector DPF (se=0.45) Eponine (t = 0.995) FirstEF (p = 0.98) PPFB (tpe=0.06, tpi=0.75)

TPa 5 6 8 7 9 6 6 6 6 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

FPb 1 4 1 3 0 14 14 12 11 10 2 3 1 0 1 0 0 3 0 0 2 4 3 3 3 1 3 0 1 0

% TP total predictions 83.3 60 88.8 70 100 30 30 33.3 35.2 44.4 33.3 25 50 100 50 100 100 25 100 100 33.3 20 25 25 25 50 25 100 50 100

% Coveragec 29.4 35.2 47 41.1 52.9 54.5 54.5 54.5 54.5 72.7 100 100 100 100 100 100 100 100 100 100 25 25 25 25 25 100 100 100 100 100

TPa: true positive; FPb: false positive; %Coveragec: the percentage of true promoters in a sequence. Table 2. Results and comparisons of five prediction systems on human Chromosome 22 Method PromoterInspector DPF(se = 0.37) Eponine(t=0.9975) FirstEF ( p=0.98 ) PPFB (tpe=0.1, tpi=0.9) a

TP 239 241 247 242 262

FP 274 482 248 270 246

Se(%)a 60.8 61.3 62.8 61.5 66.6

Sp(%)b 46.6 33.3 49.9 47.2 51.5

Sensitivity : Se=TP/(TP+FN); bSpecificity: Sp=TP/(TP+FP); FN: false negative. TP+FN = 393

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although DPF can predict more promoters it also results in more false positives. Comparing the predicting results of our system with DPF, Eponine, FirstEF and PromoterInspector shows that our method has good performance in terms of both sensitivity and specificity, especially for the sequence AC002397 and L144140. We also evaluate the performance of our system on Release 3.1 of human chromosome 22 with length 35Mb and 393 known genes annotated by the Chromosome 22 Gene Annotation Group at the Sanger Institute (http://www.sanger.ac.uk/ HGP/Chr22). In this experiment, PromoterInspector is used with default settings and our system is used by setting tpe=0.1 and tpi=0.9, DPF is used by setting se=0.37 for giving a comparable sensitivity and specificity result. For the same reason, we set t=0.9975 for Eponine and p=0.98 for FisrtEF. We adopt the same evaluating criterion used by Scherf with PromoterInspector: all the predictions located in the range -2000~+500 around the 5’ extremity of a known gene are considered as a true positive promoter region (TP) and other predictions outside this range are considered as false positives (FP). The recognition results and comparisons are summarized in Table 2. the results showed that our system has better performance compared with other four systems. Although our method otained better results than other models, improvements is not much. This is because our method is based on probability and not applicable to small probable sequences, so our future work will concentrate on conserdering and joinning other feature to predict promoter.

4 Conclusions Eukaryotic promoter predicting is one of the most elusive problems in DNA sequence analysis. Although a number of algorithms have been proposed, most of them suffer from low sensitivity or too many false positives. In this paper, we developed a new algorithm called PPFB for promoter prediction base on the hypothesis: promoter is determined by certain motifs and different promoters consist of different motifs. A new feature selection strategy is based on divergence and a new promoter prediction system that makes use of each feature as classifier and feature boosting is developed. Experimental results show that the performance of our method is better than PromoterInspector, Dragon Promoter Finder, Eponine, and FirstEF. In the future, we will use various different features to construct different weak classifiers and integrate them into one to further improve the prediction accuracy.

Acknowledgments This work is supported by a grant from National Natural Science Foundation of China (project 60772028).

References 1. Scherf, M., Klingenhoff, A., Werner, T.: Highlyspecific localization of promoter regions in large genomic sequences by Promoter Inspector: a novel context analysis approach. J. Mol. Biol. 297, 599–606 (2000) 2. Bajic, V.B., Seah, S.H., Chong, A., Krishnan, S.P.T., Koh, J.L.Y., Brusic, V.: Computer model for recognition of functional transcription start sites in polymerase II promoters of vertebrates. Journal of Molecular Graphics & Modeling 21, 323–332 (2003)

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3. Davuluri, R.V., Grosse, I., Zhang, M.Q.: Computational identification of promoters and first exons in the human genome. Nat. Genet. 29, 412–417 (2001) 4. Down, T.A., Hubbard, T.J.: Computational detection and location of transcription start sites in mammalian genomic DNA. Genome Res. 12, 458–461 (2002) 5. Wu, S., Xie, X., Liew, A.W., Hong, Y.: Eukaryotic promoter prediction based on relative entropy and positional information. Physical Review E 75, 041908-1–041908-7 (2007) 6. Prestridge, D.S., Burks, C.: The density of transcriptional elements in promoter and nonpromoter sequences. Hum. Mol. Genet. 2, 1449–1453 (1993) 7. Hutchinson, G.B.: The prediction of vertebrate promoter regions using differential hexamer frequency analysis. Comput. Appl. Biosci. 12, 391–398 (1996) 8. Cross, S.H., Clark, V.H., Bird, A.P.: Isolation of CpG islands from large genomic closnes. Nucleic Acids Res. 27, 2099–2107 (1999) 9. Ioshikhes, I.P., Zhang, M.Q.: Large-scale human promoter mapping using CpG islands. Nat. Genet. 26, 61–63 (2000) 10. Ponger, L., Mouchiroud, D.: CpGProD: identifying CpG islands associated with transcription start sites in large genomic mammalian sequences. Bioinformatics 18, 631–633 (2002) 11. Schapire, R., Singer, Y.: Improved boosting algorithms using confidence-rated predictions. In: Proc. Of Annual Conf. On Computational Learning Theory, pp. 80–91 (1998) 12. Sergios, T., Konstantinos, K.: Pattern Recognition, 2nd edn. Academic Press, San Diego (2003)

Searching for Interacting Features for Spam Filtering Chuanliang Chen1, Yunchao Gong2, Rongfang Bie1,*, and Xiaozhi Gao3 1

Department of Computer Science, Beijing Normal University, Beijing 100875, China 2 Software Institute, Nanjing University, Nanjing, China 3 Department of Electrical Engineering, Helsinki University of Technology, Otakaari 5 A, 02150 Espoo, Finland [email protected], [email protected], [email protected]

Abstract. In this paper, we introduce a novel feature selection method— INTERACT to select relevant words of emails for spam email filtering, i.e. classifying an email as spam or legitimate. Four traditional feature selection methods in text categorization domain, Information Gain, Gain Ratio, Chi Squared, and ReliefF, are also used for performance comparison. Three classifiers, Support Vector Machine (SVM), Naïve Bayes and a novel classifier— Locally Weighted learning with Naïve Bayes (LWNB) are discussed in this paper. Four popular datasets are employed as the benchmark corpora in our experiments to examine the capabilities of these five feature selection methods and the three classifiers. In our simulations, we discover that the LWNB improves the Naïve Bayes and gain higher prediction results by learning local models, and its performance is sometimes better than that of the SVM. Our study also shows the INTERACT can result in better performances of classifiers than the other four traditional methods for the spam email filtering. Keywords: Interacting features, Feature selection, Naïve bayes, Spam filtering.

1 Introduction The increasing popularity of electronic mails has intrigued direct marketers to flood the mailboxes of millions of users with unsolicited messages. These messages are usually referred to as spam or, more formally, Unsolicited Bulk E-mail (UBE), and may advertise anything, from vacations to get-rich schemes [1]. The negative effect of spam has influenced people’s daily lives: filling mailboxes, engulfing important personal mails, wasting network bandwidth, consuming users' time and energy to solve it, not to mention all the other problems associated with it (crashed mail-servers, pornography advertisements sent to children, etc.). A study in 1997 indicated that the spam messages constituted approximately 10% of the incoming messages to a corporate network [4]. CAUBE.AU reports that their statistics show the volume of spam is increasing at an alarming rate, and some people claim they are even abandoning their email accounts because of spam [3]. This situation seems to be worsening with time, and without appropriate counter-measures, spam messages could eventually undermine the usability of e-mails. These serious threats from spam make the spam filtering, whose task is *

Corresponding author.

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to rule out unsolicited emails automatically from the email stream, more important and in need of solving. In recent years, many studies address the issue of spam filtering based on machine learning, because the attempts to introduce legal measures against spam mailing have limited effect. Several supervised learning algorithms have been successfully applied to spam filtering: Naïve Bayes [5,6,7,8], Support Vector Machine [9,10], Memory Based Learning methods [11,12], and Decision Tree [13]. Among these classification methods, the Naïve Bayes is particularly attractive for spam filtering, as its performance is surprisingly good [12]. The Naïve Bayes classifier has been the filtering engine of many commercial anti-spam software. Therefore, in this paper, we aim at improving the prediction ability of the Naïve Bayes by introducing locally learned model. In order to train or test classifiers, it is necessary to go through large corpus with spam and legitimate emails. E-mails of corpuses have to be preprocessed to extract their words (features) belonging to the message subjects, the bodies and/or the attachments. As the number of features in a corpus can end up being very high, it is usual to choose those features that better represent each message before carrying out the filter training to prevent the classifiers from over-fitting [14]. The effectiveness of the classifiers relies on the appropriate choice of these features, the preprocessing steps of the e-mail features extraction, and the selection of the most representative features are crucial for the performance of the filters [15]. In this paper, a novel feature selection method—INTERACT and a novel classifier—LWNB are introduced to deal with spam filtering. The remainder of this paper is organized as follows. Section 2 demonstrates the INTERACT algorithm for the spam filtering. We explain the principles of the e-mail representation and preprocessing in Section 3. Classifiers used in this paper are presented in Section 4. We report the performances of the four feature selection methods and three classifiers using F measure and accuracy in Section 5. Section 6 concludes our study with a few remarks and conclusions.

2 INTERACT Algorithm Interacting features challenge the current feature selection methods for classification. A feature by itself may have little correlation with the target concept. However, when it is combined with some other features, they can be strongly correlated with the target concept [2]. Many traditional feature selection methods usually unintentionally remove these features, and thus result in the poor classification performances. The INTERACT algorithm can efficiently handle the feature interaction with much lower time cost than the traditional methods. A brief description of the INTERACT algorithm is presented below, and more details can be found in [2]. The INTERACT algorithm searches for the interacting features by solving two key problems: how to update c-contribution effectively, and how to deal with the feature order problem? C-contribution of a feature is an indicator about how significantly the elimination of that feature will affect consistency. Especially, the C-contribution of an irrelevant feature is zero.

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To solve the first problem, the INTERACT algorithm calculates the C-contribution efficiently with a hashing mechanism [2]: each instance is inserted into the hash table, and its values of those features in Slist are used as the hash keys, where Slist is the set of the ranked features not yet eliminated (Slist is initialized with the full set of features). Instances with the same hash keys will be inserted into the same entry in the hash table and cover the old information of the labels. For the second problem, we assume that the set of features can be divided into subset S1 including relevant features, and subset S2 containing irrelevant ones. The INTERACT algorithm intends to remove the features in S2 first, and preserve features in S1, which more probably remain in the final set of selected features. The INTERACT algorithm achieves this target by applying a heuristic to rank the individual features using symmetrical uncertainty (SU) in an descending order so that the (heuristically) most relevant feature is positioned at the beginning of the list. SU has been described in the information theory books and numerical recipes. It is often used as a fast correlation measure to evaluate the relevance of individual features [12,17]. The INTERACT is a filtering algorithm that employs backward elimination to remove the features with no or low C-contribution. Given a full set with N features and a class attribute C, the INTERACT finds a feature subset Sbest for the class concept [2]. The algorithm consists of two major parts: firstly, the features are ranked in the descending order based on their Symmetrical Uncertainty values; secondly, the features are evaluated one by one starting from the end of the ranked feature list. The process is shown as follows. Algorithm 1. INTERACT Algorithm Input:

Output: Process:

F is the full features set with N features{F1,F2, …, FN}; C is the class label; δ is a predefined threshold. Sbest is subset of selected features. Sbest = ∅ for i = 1 to N then calculate SUFi,c for Fi append Fi to Sbest end sort Sbest in descending order according to SUi,c F ← last element of Sbest repeat if F ≠ NULL then p ← c-contribution of F if p ≤ δ then remove F from Sbest end end until F = NULL return Sbest

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3 Preprocessing of Corpus and Message Representation 3.1 Feature Selection Methods for Comparison Other four feature selection methods are used in this paper to test the capability of the INTERACT algorithm. They are Chi Squared (i.e. χ 2 ) Statistic, Information Gain, Gain Ratio, and ReliefF. Their definitions are given as follows. In the following formulas, m is the number of classes (in spam filtering domain, m is 2), and Ci denotes the ith class. V represents the number of partitions a feature can split the training set into. Let N is the total number of samples, and N Ci is that of class

i. In the vth partition, N C( vi ) denotes the number of samples belonging to class i. Chi Squared: The Chi Squared Statistic is calculated by comparing the obtained frequency with the priori frequency of the same class. The definition is: m

i C )2 ( NC( vi ) − N i (v)

V

χ 2 = ∑∑ i =1 v =1

i C( v ) N i

(1)

.

i C( v ) = ( N ( v ) / N ) N denotes the prior frequency. where N i Ci Information Gain: Information Gain is based on the feature’s impact on the decreasing entropy, and is defined as follows: m

InfoGain = [∑ −( i =1

N Ci N

) log(

N Ci N

V

)] − [∑ ( v =1

N C( v ) N C( v ) N (v) m )∑ −( ( vi ) ) log( ( vi ) )] . N i =1 N N

(2)

Gain Ratio: Gain Ratio is firstly used in C4.5, which is defined as (3): m

GainRatio = InfoGain / [∑ −( i =1

N Ci N

) log(

N Ci N

)] .

(3)

ReliefF: The key idea of Relief is to estimate the features according to how well their values distinguish among the instances that are near to each other. The ReliefF is an extension of the Relief, improving the Relief algorithm by estimating the probabilities more reliably and extending to deal with the incomplete and multiclass data sets. More details can be found in [17]. 3.2 Corpus Preprocessing and Message Representation

Each e-mail in the corpora is represented as a set of words. After analyzing all the emails of a corpus, a dictionary with N words/features is formed. Every e-mail is represented as a feature vector including N elements, and the ith word of the vector is a binary variable representing whether this word is in this e-mail. During preprocessing, we perform the word stemming, stop-word removable and Document Frequency Threshold (DFT), in order to reduce the dimension of feature space. The HTML tags

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of the e-mails are also removed during preprocessing. Finally, we extract the first 5,000 tokens of the dictionary according to their mutual information to form the corpora used in this paper.

4 Classifiers for Spam Filtering In this paper, we use three classifiers to test the capabilities of the aforementioned feature selection methods. The three classifiers are Support Vector Machine (SVM), Naïve Bayes, and Locally Weighted learning with Naïve Bayes (LWNB) that is an improvement of Naïve Bayes firstly introduced into spam filtering domain by us. We here only briefly introduce the LWNB, and more details can be found in [1]. In the LWNB, the Naïve Bayes is learned locally in the same way as the linear regression is used in locally weighted linear regression. A local Naïve Bayes model is fit to a subset of the data that is in the neighborhood of the instance, whose class value is to be predicted [1]. The training samples in this neighborhood are weighted, and further ones are assigned with less weight. The classification is then obtained from these Naïve Bayes models. The subset of the data used to train each locally weighted Naïve Bayes model is determined by a nearest neighbors algorithm. In the LWNB, the first k nearest neighbors are selected to form this subset, where k is a user-specified parameter. How to determine the weight of each instance of the subset? As in [1], we use a linear weighting function in our experiments, which is defined as: f linear = 1 − di / d k ,

(4)

where di is the Euclidean distance to the ith nearest neighbor xi. Obviously, by using flinear, the weight decreases linearly with the distance. Empirical study shows the LWNB is not particularly sensitive to the choice of k as long as k is not too small [1]. Too small k may cause the local Naïve Bayes model to fit the noise in the data. The Naïve Bayes calculates the posterior probability of class ci for a test instance with m attribute values a1, a2, …, am as follows:

p(cl | a1 , a2 ,..., am ) =

p (cl ) ∏ mj =1 p (a j | cl )

∑

C

[ p(ci ) ∏ mj =1 p(a j | ci )] i =1

,

(5)

where C is the total number of classes. In the LWNB, the individual probabilities on the right-hand side of (5) are estimated based on the weighted data. The prior probability for class cl becomes: 1 + ∑ i = 0 I (ci = cl ) wi n

p(cl ) =

C + ∑ i = 0 wi n

,

(6)

where ci is the class value of the ith training instance, and the indicator function I(x=y) is 1 iff x = y. The attribute of data is assumed nominal, and as for the numeric attributes, they are discretized. The conditional probability of aj is given by:

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1 + ∑ i = 0 I (a j = aij ) I (ci = cl ) wi n

p(a j | cl ) =

n j + ∑ i = 0 I (a j = aij ) wi n

,

(7)

nj is the number of values of attribute j, and aij is the value of attribute j of ith instance.

5 Experiments and Analysis 5.1 Corpus in Simulations

The experiments are based on four popular benchmark corpora, PU1, PU2, PUA, and Ling Spam, which are all available on [16]. In all PU corpora and Ling Spam corpus, attachments, html tags, and header fields other than the subjects are removed, leaving only subject lines and mail body texts. In order to address privacy, each token of a corpus is encoded to a unique integer. The details about each corpus are given below. PU1 Corpus: The PU1 corpus consists of 1,099 messages, which has 481 spam messages and 618 legitimated ones. The spam rate is 43.77%. PU2 Corpus: The PU2 corpus contains less messages than PU1, which has 721 messages. Among them, there are 579 messages labeled legitimate and 142 spam. PUA Corpus: The PUA corpus has 1,142 messages, half of which, i.e., 571 messages, are marked as spam and the other half legitimate. Ling Spam Corpus: The Ling spam corpus includes 2,412 legitimate messages from a linguistic mailing list and 481 spam ones collected by the author. The spam rate is 16.63%. Different from PU corpora, the messages of Ling spam corpus come from different sources: the legitimate messages are collected from a spam-free, topicspecific mailing list and the spam ones from a personal mailbox. Therefore, the distribution of mails is less similar from the normal user’s mail stream, which makes the messages of Ling spam corpus easily separated. 5.2 Performance Measures

We use two popular evaluation metrics of the text categorization domain to measure the performance of the classifiers: accuracy and F measure. Accuracy: Accuracy is the percentage of the correct predictions in the total predictions. It is defined as follows:

Accuracy =

Pc × 100% . Pt

(8)

where Pc is the number of the correct predictions, and Pt is the number of the total predictions. The higher of the accuracy, the better.

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F measure: The definition of F measure is as follows:

F=

2R × P , R+P

(9)

where R represents Recall, which is the percentage of the messages for a given category that are classified correctly; P is the Precision, the percentage of the predicted messages for a given class that are classified correctly. F measure ranges from 0 to 1, and the higher, the better. 5.3 Results and Analysis

The following classification performance is measured through a 10-fold crossvalidation. We select all of the interacting features, i.e., features with non-negative Ccontribution. Table 1 summarizes the results of dimension reduction after the INTERACT selects the features. Table 1. Summary of results of INTERACT selected features on the four benchmark corpora

Num. of features with non-negative c-contribution

PU1 43

PU2 43

PUA 42

Ling Spam 64

From Table 1, we can find that the dimensions of data have been reduced sharply after removing irrelevant features by the INTERACT. Therefore, we just run the classifiers on these data rather than reducing them further by adjusting the parameter δ. From Table 1, we also can conclude that there are many irrelevant words/features existing in corpus for the spam filtering, and more than 99% of the features are removed by the INTERACT. The following histograms show the performances of the three classifiers, SVM (using linear kernel), Naïve Bayes, and LWNB, on the four corpora. As for other four feature selection methods for comparison, we select the first M features according to the features’ scores, where M is the number of the interacting features found by the INTERACT algorithm. From Fig. 1 and Fig. 2, we discover that the INTERACT algorithm can improve the performances of all the three classifiers. Their performances on the reduced corpus are equal to or better than those on the full corpus, evaluated by the accuracy and F measure. For example, the performances of the SVM on PU1 and PU2 corpora reduced by the INTERACT is equal to those on the full corpora, and its performance on PUA corpus reduced by the INTERACT is better than that on the full corpus. However, the performance of the SVM on Ling Spam corpus reduced by the INTERACT is slightly worse than that on the full corpus. The feature selection capability of the INTERACT is obviously better than the other popular feature selection methods. The competitive performances of the classifiers on the data handled by the INTERACT show that only a few relevant words can still distinguish between the spam and legitimate emails. This is true in practice, for example, it is well known that the words “buy, purchase, jobs, …” usually appear in the spam e-mails, and they thus are useful email category distinguishers.

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Fig. 1. Performances of aforementioned three classifiers and four feature selection methods on PU1, PU2, PUA, and Ling Spam benchmark corpora with accuracy evaluation measure

Fig. 2. Performances of aforementioned classifiers and four feature selection methods on PU1, PU2, PUA and Ling Spam benchmark corpora with F measure evaluation measure

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The performance of the LWNB is also promising. On Ling Spam corpus, its performance is even better than that of the SVM, which is a well-known powerful classifier. On PU1 and Ling Spam corpora, the LWNB successfully improves the performance of the Naïve Bayes by using locally weighted model. However, its performance is worse than that of the Naïve Bayes on PU2 and PUA corpora. The reason may be that the task of the spam filtering suits the hypothesis of the class conditional independence of the Naïve Bayes, that is, given the class label of the others, the frequencies of the words in one email are conditionally independent of one another. Based on a careful observation, we have another question “why the LWNB performs poorly on full corpus”? The reason is: there are many irrelevant features existing on full corpus, which can be also concluded from the feature selection results by performing the INTERACT. When determining the neighbors, all the features take part in calculating distance, and too many irrelevant features conceal the truly useful effects of the relevant features, and therefore result in that the LWNB finds the wrong or irrelevant neighbors to generate locally weighted Naïve Bayes models. However, the LWNB is still a promising classifier for the spam filtering, when combined with some excellent feature selection methods, such as the INTERACT.

6 Conclusions In this paper, we present our work on the spam filtering. Firstly, we introduce the INTERACT algorithm to select interacting words/features for the spam filtering. Other four traditional feature selection methods are also performed in the experiments for performance comparison. Secondly, we introduce a novel classifier LWNB to improve the performance of the Naïve Bayes, a most popular classifier in the spam filtering area, to deal with the spam filtering. Totally, three classifiers, SVM, Naïve Bayes and LWNB, are run on four corpora preprocessed by the five feature selection methods and corresponding full corpora in our simulations. Two popular evaluation metrics, accuracy and F measure, are used to measure the performances of these three classifiers. Our empirical study shows that the INTERACT feature selection can improve all of the three classifiers’ performances, and its feature selection ability is better than that of the four traditional feature selection methods. We briefly analyze the reason why the INTERACT and other four methods can work together to perform well. We also find out that the LWNB can improve the performance of the Naïve Bayes, which is sometimes superior to the SVM. Acknowledgements. The research work presented in this paper was supported by grants from the National Natural Science Foundation of China (Project No. 10601064). Xiaozhi Gao's research work was funded by the Academy of Finland under Grant 214144.

References 1. Frank, E., Hall, M., Pfahringer, B.: Locally Weighted Naive Bayes. In: Proc. of the Conference on Uncertainty in Artificial Intelligence, pp. 249–256 (2003) 2. Zhao, Z., Liu, H.: Searching for Interacting Features. In: Proc. of International Joint Conference on Artificial Intelligence (IJCAI), Hyderabad, India, pp. 1156–1161 (2007)

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3. CAUBE.AU (2006), http://www.caube.org.au/spamstats.html 4. Cranor, L.F., LaMacchia, B.A.: Spam! In: Communications of ACM, pp. 74–83. ACM Press, New York (1998) 5. Sahami, M., Dumais, S., Heckerman, D., Horvitz, E.: A Bayesian Approach to Filtering Junk E-mail. AAAI Technical Report WS-98-05, AAAI 1998 Workshop on Learning for Text Categorization (1998) 6. Schneider, K.M.: A Comparison of Event Models for Naïve Bayes Anti-Spam E-Mail Filtering. In: Proc. of the 10th Conference of the European Chapter of the Association for Computational Linguistics, Budapest, Hungary, pp. 307–314 (2003) 7. Androutsopoulos, I., Paliouras, G., Karkaletsis, V., Sakkis, G., Spyropoulos, C.D., Stamatopoulos, P.: Learning to Filter Spam E-mail: A Comparison of a Naïve Bayesian and a Memory-based Approach. In: Proc. of the Workshop on Machine Learning and Textual Information Access, pp. 1–13 (2000) 8. Zhang, L., Zhu, J., Yao, T.: An Evaluation of Statistical Spam Filtering Techniques. ACM Trans. Asian Lang. Inf. Process 3, 243–269 (2004) 9. Drucker, H., Wu, D., Vapnik, V.N.: Support Vector Machines for Spam Categorization. IEEE Trans. on Neural Networks 10, 1048–1054 (1999) 10. Kolcz, A., Alspector, J.: SVM-based Filtering of E-mail Spam with Content-specific Misclassification Costs. In: Proc. of the TextDM 2001 Workshop on Text Mining - held at the 2001 IEEE International Conference on Data Mining (2001) 11. Sakkis, G., Androutsopoulos, I., Paliouras, G., Stamatopoulos, P.: A Memory-Based Approach to Anti-Spam Filtering for Mailing Lists. Information Retrieval 6, 49–73 (2003) 12. Yu, L., Liu, H.: Feature Selection for High-dimensional Data: A Fast Correlation-based Filter Solution. In: Proc. of the 20th International Conference on Machine Learning, Washington DC, pp. 856–863 (2003) 13. Carreras, X., Marquez, L.: Boosting Trees for Anti-spam Email Filtering. In: Proc. International Conference on Recent Advances in Natural Language Processing (RANLP 2001), Tzigov Chark, Bulgaria, pp. 58–64 (2001) 14. Méndez, J.R., Iglesias, E.L., Fdez-Riverola, F., Díaz, F., Corchado, J.M.: Analyzing the Impact of Corpus Preprocessing on Anti-Spam Filtering Software. Research on Computing Science 17, 129–138 (2005) 15. Méndez, J.R., Fdez-Riverola, F., Díaz, F., Iglesias, E.L., Corchado, J.M.: A Comparative Performance Study of Feature Selection Methods for the Anti-spam Filtering Domain. In: Perner, P. (ed.) ICDM 2006. LNCS (LNAI), vol. 4065, pp. 106–120. Springer, Heidelberg (2006) 16. Email Benchmark Corpus (2006), http://www.aueb.gr/users/ion/publications.html 17. Kononenko, I.: Estimating Attributes: Analysis and Extensions of Relief. In: Proc. of European Conference on Machine Learning, pp. 171–182. Springer, Heidelberg (1994)

Structural Support Vector Machine Hui Xue1 , Songcan Chen1, , and Qiang Yang2 1

2

Department of Computer Science and Engineering, Nanjing University of Aeronautics and Astronautics, 210016, Nanjing, P.R. China Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Hong Kong {xuehui,s.chen}@nuaa.edu.cn, [email protected] http://parnec.nuaa.edu.cn

Abstract. Support Vector Machine (SVM) is one of the most popular classifiers in pattern recognition, which aims to find a hyperplane that can separate two classes of samples with the maximal margin. As a result, traditional SVM usually more focuses on the scatter between classes, but neglects the different data distributions within classes which are also vital for an optimal classifier in different real-world problems. Recently, using as much structure information hidden in a given dataset as possible to help improve the generalization ability of a classifier has yielded a class of effective large margin classifiers, typically as Structured Large Margin Machine (SLMM). SLMM is generally derived by optimizing a corresponding objective function using SOCP, and thus in contrast to SVM developed from optimizing a QP problem, it, though more effective in classification performance, has the following shortcomings: 1) large time complexity; 2) lack of sparsity of solution, and 3) poor scalability to the size of the dataset. In this paper, still following the above line of the research, we develop a novel algorithm, termed as Structural Support Vector Machine (SSVM), by directly embedding the structural information into the SVM objective function rather than using as the constraints into SLMM, in this way, we achieve: 1) to overcome the above three shortcomings; 2) empirically better than or comparable generalization to SLMM, and 3) theoretically and empirically better generalization than SVM. Keywords: Support vector machine, Structural information, Rademacher complexity, Pattern recognition.

1

Introduction

In the past decade, large margin machines have become a hot issue of research in machine learning. Support Vector Machine (SVM)[1], as the most famous one among them, is derived from statistical learning theory[2] and achieves a great success in pattern recognition. 

Corresponding author: Tel: +86-25-84896481 Ext.12106; Fax: +86-25-84498069. This work was supported respectively by NSFC (60773061) and Jiangsu NSF (BK2008xxx).

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Given a training set {xi , yi }ni=1 ∈ Rm × {±1}, the basic objective of SVM is to learn a classifier f = wT x + b which can maximize the margin between classes: 1 min  w 2 w, b 2 s.t. yi (wT xi + b) ≥ 1, i = 1, · · · , n

(1)

If we focus on the constraints in (1), we can immediately capture the following insight about SVM which is easily generalized to the soft margin version: Theorem 1. SVM constrains the scatter between classes as wT Sb w ≥ 4, where Sb = (μ1 − μ2 )(μ1 − μ2 )T , μi is the mean of class i(i = 1, 2). Proof. Without loss of generalization, we assume that the class one has the class label yi = 1, and the other class has yj = −1. Then we reformulate the constraints as wT xi + b ≥ 1, where xi belongs to class one, and wT xj + b ≤ −1, where xj belongs to class two. Let the numbers nof1 theTsamples in theT two classes (w xi + b) = (w μ1 + b) ≥ 1 are respectively n1 and n2 . Then we have n11 i=1 n2 1 T T and − n2 j=1 (w xj + b) = −(w μ2 + b) ≥ 1. Adding the two inequalities, we obtain wT (μ1 − μ2 ) ≥ 2. Squaring the inequality, we further have wT (μ1 −  μ2 )(μ1 − μ2 )T w ≥ 4, i.e. wT Sb w ≥ 4. Consequently, following the above theorem, it is obvious that SVM actually gives a natural lower bound to the scatter between classes, just according with its original motivation that pays more attention to the maximization of margin. However, it discards the prior data distribution information within classes which is also vital for classification. In fact, corresponding to different real-world problems, different classes may have different underlying data structures. It requires that the classifier should adjust the discriminant boundaries to fit the structures which are vital for classification, especially for the generalization capacity of the classifier. However, the traditional SVM does not differentiate the structures, and the derived decision hyperplane lies unbiasedly right in the middle of the support vectors[3, 4], which may lead to a nonoptimal classifier in the real-world problems. Recently, some new large margin machines have been presented to give more concerns to the structural information than SVM. They provide a novel view to design a classifier, that the classifier should be sensitive to the structure of the data distribution, and assume that the data contains clusters. Minimax Probability Machine (MPM)[5] and Maxi-Min Margin Machine (M4 )[3] stress the global structure of the two classes and apply two ellipsoids, i.e. two clusters, to characterize the classes distributions respectively. By using the Mahalanobis distance which combines the mean and covariance of the ellipsoids, they integrate the global structural information into the large margin machines. However, only emphasis on the global structure of the classes is too coarse. In many realworld problems, samples within classes more likely have different distributions. Therefore, Structured Large Margin Machine (SLMM)[4] is proposed to firstly

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apply some clustering methods to capture the underlying structures in each class. As a result, SLMM uses several ellipsoids whose number is equal to the number of the clusters to enclose the training data, rather than only two ellipsoids in M4 respectively corresponding to each class. The optimization problem in soft margin SLMM can be formulated as (2)[4], which introduces the covariance matrices in each cluster into the constraints: |P |+|N |

max ρ − C



ξl

l=1

 | Pi | ρ wT ΣPi w − ξl , xl ∈ Pi , M axP | Nj |  T ρ w ΣNj w − ξl , xl ∈ Nj , − (wT xl + b) ≥ M axN

s.t. (wT xl + b) ≥

wT r = 1, ξl ≥ 0

(2)

where ξl is the penalty for violating the constraints. C is a regularization parameter that makes a trade-off between the margin and the penalties incurred. Pi denotes the ith cluster in class one, i = 1, · · · , CP , and Nj denotes the j th cluster in class two, j = 1, · · · , CN . CP and CN are the numbers of the clusters in the two classes respectively. r is a constant vector to limit the scale of the weight w. By the simple algebraic deduction, MPM, M4 even SVM can all be viewed as the special cases of SLMM. And SLMM also achieves better classification performance among these popular large margin machines experimentally. However, SLMM has much larger time complexity than SVM. Its optimization problems should be solved by SOCP, which handles relatively difficultly in real applications. And the corresponding solution loses the sparsity as in SVM derived from optimizing a QP problem. Consequently, it has poor scalability to the size of the dataset and can not easily be generalized to large-scale or multi-class problems. Furthermore, in the kernel version, SLMM should kernelize the covariance matrix in each cluster within the constraints respectively, which undoubtedly increases extra computational complexity. In this paper, we present a novel classification algorithm that provides a general way to incorporate the structural information into the learning framework of the traditional SVM. We call our method SSVM, which stands for Structural Support Vector Machine. Inspired by the SLMM, SSVM also firstly exploits the intrinsic structures of samples within classes by some unsupervised clustering methods, but then directly introduces the data distributions of the clusters in different classes into the traditional optimization function of SVM rather than in the constraints. The contributions of SSVM can be described as follows:  SSVM naturally integrates the prior structural information within classes

into SVM, without destroying the classical framework of SVM. And the corresponding optimization problem can be solved by the QP just similarly to SVM. Consequently, SSVM can overcome the above shortcomings of SLMM.

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 SSVM empirically has comparable or better generalization to SLMM, since

it considers the separability between classes and the compactness within classes simultaneously. Though SLMM can capture the structural information within classes by some clustering algorithms, it also more emphasizes the separability between classes due to the characteristics of the traditional large margin machines, which more likely does not sufficiently apply the prior information to some extent.  SSVM can be theoretically proved that it has the lower Rademacher complexity than SVM, in the sense that it has better generalization capacity, rather than only validating generalization performance empirically in SLMM. This further justifies that the introduction of the data distribution within classes into the classifier design is essential for better recognition. The rest of the paper is organized as follows. Section 2 presents the proposed Structural Support Vector Machine, and also discusses the kernelization of SSVM. In Section 3, the theoretical analysis of the generalization capacity is deduced. Section 4 gives the experimental results. Some conclusions are drawn in Section 5.

2

Structural Support Vector Machine (SSVM)

Following the line of the research in the SLMM, SSVM also has two steps: clustering and learning. It firstly adopts clustering techniques to capture the data distribution within classes, and then minimizes the compactness in each cluster, which leads to further maximizing the margin in the sense of incorporating the data structures simultaneously. Many clustering methods, such as K-means, nearest neighbor clustering and fuzzy clustering, can be applied in the first clustering step. After the clustering, the structural information is introduced into the objective function by the covariance matrices of the clusters. So the clusters should be compact and spherical for the computation. Following SLMM, here we use the Ward’s linkage clustering in SSVM, which is one of the hierarchical clustering techniques. During the clustering, the Ward’s linkage between clusters to be merged increases as the number of clusters decreases[4]. We can draw a curve to represent this process. Through finding the knee point, i.e. the point of maximum curvature in the curve, the number of clusters can be determined automatically. Furthermore, the Ward’s linkage clustering is also applicable in the kernel space. After clustering, we obtain the c1 and c2 clusters respectively in the two classes. We denote the clusters in the classes as P1 , · · · , Pc1 and N1 , · · · , Nc2 . From Theorem 1, we have proved that SVM gives a natural lower bound to the separability between classes by the constraints. So here we pay more attention to the compactness within classes, that is, the clusters which cover the different structural information in different classes. We aim to maximize the margin and simultaneously minimize the compactness. Accordingly, the SSVM model in the soft margin version can be formulated as:  1 λ  w  2 + wT Σ w + C ξi min 2 w, b 2 i=1 n

Structural Support Vector Machine

s.t. yi (wT xi + b) ≥ 1 − ξi , ξi ≥ 0, i = 1, · · · , n

505

(3)

where Σ = ΣP1 + · · · + ΣPc1 + ΣN1 + · · · + ΣNc2 , ΣPi and ΣNj are the covariance matrices corresponding to the ith and j th clusters in the two classes, i = 1, · · · , c1 , j = 1, · · · , c2 . λ is the parameter that regulates the relative importance of the structural information within the clusters, λ ≥ 0. Compared to SVM, SSVM inherits the advantages of SLMM that incorporates the data distribution information in a local way, that considers the covariance matrices of the clusters in each class which contain the trend of data occurrence in statistics[4]. However, different from SLMM, SSVM directly introduces the prior information into the objective function rather than the constraints. Therefore, SSVM can follow the same techniques as SVM to solve the optimization problem, which mitigates the large computational complexity in SLMM. And the algorithm can efficiently converge to the global optimum which also holds the sparsity and has better scalability to the size of the datasets. Moreover, through minimizing the compactness of the clusters, SSVM more likely further maximizes the margin between classes, which may lead to comparable or better classification and generalization performance than SLMM. We will address these in more details in the following sections. By incorporating the constraints into the objective function, we can rewrite (3) as a primal Lagrangian. Then, we transform the primal into the dual problem following the same steps as SVM: max α

n  i=1

1  αi αj yi yj [xTi (I + λΣ)−1 xj ] 2 i=1 j=1 n

αi −

n

s.t. 0 ≤ αi ≤ C, i = 1, · · · , n n 

αi yi = 0

(4)

i=1

Eq. (4) is a typical convex optimization problem. Using the same QP techniques as SVM, we can obtain the solution . Then the derived classifier function can be formulated as follows, which is used to predict the class labels for future unseen samples x: n  f (x) = sgn[ αi yi xTi (I + λΣ)−1 x + b]

(5)

i=1

It is noteworthy that SSVM boils down to the same solution framework of SVM except adding a regularization parameter λ. When λ = 0, SSVM will degenerate to the traditional SVM. Thus SVM actually can be viewed as a special version case of SSVM. We can also apply the kernel trick in SSVM in order to further improve the classification performance in complex pattern recognition problems. Furthermore, compared to SLMM which has to kernelize each cluster covariance matrix respectively, SSVM can perform complex kernelization through kernelizing the

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covariance matrix sum of all the cluster covariance matrices which makes it simpler and more effective. Assume that the nonlinear mapping function is Φ : Rm → H, where H is a Hilbert space which has high dimension. Then the optimization function of SSVM in the kernel space can be described as: max α

n 

1  αi αj yi yj [Φ(xi )T (I + λΣ Φ )−1 Φ(xj )] 2 i=1 j=1 n

αi −

i=1

n

s.t. 0 ≤ αi ≤ C, i = 1, · · · , n n 

αi yi = 0

(6)

i=1

Due to the high dimension (even infinite), Φ usually can not be explicitly formulated. A solution to this problem is to express all computations in terms of dot products, called as the kernel trick[1]. The kernel function k : Rm ×Rm → R, k(xi , xj ) = Φ(xi )T Φ(xj ) derives the corresponding kernel matrix K ∈ Rn×n , Kij = k(xi , xj ), which is so-called Gram matrix. Consequently, we aim to transform (6) into the form of dot products for adopting the kernel trick. For each covariance matrix in the kernel space, we have  1 Φ T [Φ(xj ) − μΦ ΣiΦ = i ][Φ(xj ) − μi ] | CiΦ | Φ Φ(xj )∈Ci =

→ →T 1 Φ ΦT Φ ΦT T T − T 1 Φ | 1 |C Φ | Ti i i i |C i i | CiΦ |

(7)

where CiΦ denotes the clusters without differentiating the different classes, i ∈ [1, c1 + c2 ]. And TΦ i is a subset of the sample matrix, which is combined with the →

samples belonging to the cluster i in the kernel space. 1 |CiΦ | denotes a | CiΦ |dimensional vector with all the components equal to 1/ | CiΦ |. Then we obtain ΣΦ =

c 1 +c2 i=1

ΣiΦ =

c 1 +c2 i=1

→ →T T 1 Φ ΦT Φ ΦT Φ| 1 T T − T 1  PΦ Ψ PΦ |C |CiΦ | Ti i i i Φ i | Ci |

Φ where PΦ = [TΦ 1 , · · · , Tc1 +c2 ], ⎛ → →T 1 ⎜ |C1Φ | I|C1Φ | − 1 |C1Φ | 1 |C1Φ | ⎜ .. Ψ =⎜ . ⎜ ⎝

⎞

1 I Φ − |CcΦ +c | |Cc1 +c2 | 1

2

→

⎟ ⎟ ⎟ ⎟ ⎠

→T

1 |CcΦ +c | 1 |CcΦ +c 1

and I|CiΦ | is a | CiΦ | × | CiΦ | identity matrix, i ∈ [1, c1 + c2 ].

2

1

2

|

(8)

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(A + UBV)−1 = A−1 − A−1 UB(B + BVA−1 UB)−1 BVA−1

(9)

By the Woodbury’s formula

So T

T

(I + λΣ Φ )−1 = (I + λPΦ Ψ PΦ )−1 = I − λPΦ Ψ (Ψ + λΨ PΦ PΦ Ψ )−1 Ψ PΦ

T

(10)

By substituting (10) into the optimization function (6), we have the kernel form of the dual problem (6) as follows: max α

n  i=1

1  ˜ T Ψ (Ψ + λΨ KΨ ˆ )−1 Ψ K ˜ j] αi αj yi yj [Kij − λK i 2 i=1 j=1 n

αi −

n

s.t. 0 ≤ αi ≤ C, i = 1, · · · , n n 

αi yi = 0

(11)

i=1

˜ K ˜ ij = ˜ i represents the ith column in the kernel Gram matrix K, where K Ct t , x ), x is the sample that is realigned corresponding to the sequence k(xC j i i ˆ is the kernel Gram matrix, K ˆ ij = of the clusters, t = 1, · · · , c1 + c2 . And K Ct Ct k(xi , xj ).

3

Rademacher Complexity

In this section, we will discuss the generalization capacity of SSVM in theory. Different from SLMM which only validates its better generalization performance than SVM empirically by experiments, we will indeed prove that the introduction of the structural information within classes can improve the generalization bound compared to SVM. Here we adopt the Rademacher complexity measure[6] and show the new error bound is tighter. In the traditional kernel machines, we are accustomed to using VC-dimension [2] to estimate the generalization error bound of a classifier. However, the bound involves a fixed complexity penalty which does not depend on the training data, thus can not be universally effective[6]. Recently, Rademacher complexity, as an alternative notion, is presented to evaluate the complexity of a classifier instead of the classical VC-dimension[7]. And for the kernel machines, we can obtain an upper bound to the Rademacher complexity: Theorem 2 [6]. If k : X × X → R is a kernel, and S = {x1 , · · · , xn } is a sample of points from X, then the empirical Rademacher complexity of the classifier FB satisfies  n  2B 2B 

ˆ Rn (FB ) ≤ k(xi , xj ) = tr(K) (12) n n i=1 where B is the bound of the weights w in the classifier.

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Following Theorem 2, we then give the complexity analysis of SSVM compared to SVM. Theorem 3 (Complexity Analysis). The upper bound of the empirical Radˆ SV M (f ) ˆ SSV M (f ) in SSVM is at most the upper bound of R emacher complexity R in SVM, that is, tr(KSSV M ) ≤ tr(KSV M ). Due to limited space, here we omit the proof. The theorem states that there is an advantage to considering the separability between classes and the compactness within classes simultaneously, i.e. the structural information within the clusters, to further reduce the Rademacher complexity of the classifiers being considered. Intuitively, the minimization of the compactness in the clusters more likely leads to the larger margin compared to SVM, which means better generalization performance in practice. Theorem 3 just provides us a theoretical interpretation for the intuition.

4

Experiments

To evaluate the proposed Structural Support Vector Machine (SSVM) algorithm, we investigate its classification accuracies and computational efficiencies in several real-world UCI datasets. Since Structured Large Margin Machine (SLMM)[4] has been shown to be more effective than many relatively modern learning machines, such as Minimax Probability Machine (MPM)[5], MaxiMin Margin Machine (M4 )[3] and Radial Basis Function Networks (RBFN) in terms of classification accuracies, in this experiment we just compare SSVM with SLMM and SVM. For each dataset, we divide the samples into two nonoverlapping training and testing sets, and each set contains almost half of samples in each class respectively. This process is repeated ten times to generate ten independent runs for each dataset and then the average results are reported. Due to the relatively better performance of the kernel version, here we uniformly compare the algorithms in the kernel and soft margin cases. The width parameter σ in the Gaussian kernel, and the regularization parameters C and λ are selected from the set {2−10 , 2−9 , · · · , 29 , 210 } by cross-validation. We apply Sequential Minimal Optimization (SMO) algorithm to solve the QP problems in SSVM and SVM, and SeDuMi program to solve the SOCP problem in SLMM. The experimental results are listed in Table 1. In each block in the table, the first row is the training accuracy and variance. The second row denotes the testing accuracy and variance. And the third one is the average training time in the ten runs after the selection of the parameters. We can make several interesting observations from these results:  SSVM is consistently superior to SVM in the overall datasets both in the

training and testing accuracies, owing to the proper consideration of data distribution information. Furthermore, SSVM also outperforms SLMM in almost all the datasets except in Pima, because that SSVM simultaneously captures the separability between classes and the compactness within classes

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Table 1. The training and testing accuracies (%), variances and average training time (sec.) compared between SSVM and SLMM, SVM in the UCI datasets SSVM SLMM SVM 96.25 ± 0.01 95.31∗ ±0.01 95.63∗ ±0.01 Automobile 91.14 ± 0.00 88.63∗ ±0.03 88.48∗ ±0.01 0.44 3.20 0.36 77.36 ± 0.10 76.03∗ ±0.15 75.68∗ ±0.08 Bupa 76.18 ± 0.04 73.52∗ ±0.12 73.06∗ ±0.06 1.23 18.77 0.89 84.10 ± 0.01 82.59∗ ±0.01 79.87∗ ±0.00 Hepatitis 83.25 ± 0.00 79.82∗ ±0.03 79.61∗ ±0.01 0.58 3.75 0.42 98.46 ± 0.00 96.97∗ ±0.03 96.80∗ ±0.02 Ionosphere 97.52 ± 0.01 95.63∗ ±0.05 95.11∗ ±0.02 1.17 5.71 0.79 79.65 ±0.02 80.63 ± 0.05 76.04∗ ±0.01 Pima 78.63 ±0.01 79.46 ± 0.02 77.08∗ ±0.02 12.53 72.14 7.67 95.58 ± 0.02 95.27 ±0.01 86.54∗ ±0.15 Sonar 87.60 ± 0.07 86.21∗ ±0.11 85.00∗ ±0.13 0.61 3.34 0.50 98.81 ± 0.02 95.61∗ ±0.10 98.47 ±0.02 Water 98.69 ± 0.01 95.49∗ ±0.12 90.51∗ ±0.09 0.39 1.56 0.29 95.96 ± 0.00 94.89∗ ±0.05 92.54∗ ±0.01 Wdbc 95.72 ± 0.00 94.57∗ ±0.03 94.25∗ ±0.01 3.58 43.65 2.77   ∗ Denotes that the difference between SSVM and the other two methods is significant at 5% significance level, i.e., t-value > 1.7341.

rather than only emphasizing the separability in SLMM which may miss some useful classification information. And the gap of the classification accuracies between the two algorithms in Pima is less than one percent.  The training and testing accuracies of SSVM are basically comparable in the datasets, which further provides us an experimental validation for better generalization capacity than SVM, according with the theoretical analysis in Theorem 3. And the variances show the good stability of the SSVM algorithm.  We also report the average training time of the three algorithms. SSVM is slower than SVM due to the clustering pre-processing. However, it is much quicker than SLMM, which adopts the SOCP as the optimizor rather than the QP in the SSVM. Consequently, in view of the efficiency as well as classification performance, SSVM is more likely the best option among the three algorithms.  In order to find out whether SSVM is significantly better than SLMM and SVM, we perform the t -test on the classification results of the ten runs to calculate the statistical significance of SSVM. The null hypothesis H0

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demonstrates that there is no significant difference between the mean number of patterns correctly classified by SSVM and the other two methods. If the hypothesis H0 of each dataset is rejected at the 5% significance level, i.e., the t -test value is more than 1.7341, the corresponding results in Table 1 will be denoted  ∗ . Consequently, as shown in Table 1, it can be clearly found that SSVM possesses significantly superior classification performance compared with the other two methods in almost all datasets, especially according to the testing accuracies. And in Pima, there seems to be no significant difference between SSVM and SLMM, i.e. t -value < 1.7341. This just accords with our conclusions.

5

Conclusion

In this paper, we propose a novel large margin machine called as Structural Support Vector Machine (SSVM). Following the research of SLMM, SSVM also firstly captures the data distribution information in the classes by some clustering strategies. Due to the insights about the constraints in the traditional SVM, we further introduce the compactness within classes according to the structural information into the learning framework of SVM. The new optimization problem can be solved following the same QP as SVM, rather than the SOCP in the recent related algorithms such as MPM, M4 and SLMM. Consequently, SSVM not only has much lower time complexity but also holds the sparsity of the solution. Furthermore, we validate that SSVM has better generalization capacity than SVM both in theory and practice. And it also has better than or comparable classification performance to these related algorithms. Throughout the paper, we discuss SSVM in the binary classification problems. However, SSVM can be easily generalized to the multi-class problems by using the vector labeled outputs techniques, and to large-scale problems through combining with the techniques of minimum enclosing ball[8]. These issues will be our future research.

References 1. Cristianini, N., Shawe-Taylor, J.: An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods. Cambridge University Press, Cambridge (2000) 2. Vapnik, V.: Statistical Learning Theory. Wiley, Chichester (1998) 3. Huang, K., Yang, H., King, I., Lyu, M.R.: Learning Large Margin Classifiers Locally and Globally. In: ICML (2004) 4. Yeung, D.S., Wang, D., Ng, W.W.Y., Tsang, E.C.C., Zhao, X.: Structured Large Margin Machines: Sensitive to Data Distributions. Machine Learning 68, 171–200 (2007) 5. Lanckriet, G.R.G., Ghaoui, L.E., Bhattacharyya, C., Jordan, M.I.: A Robust Minimax Approach to Classfication. JMLR 3, 555–582 (2002)

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6. Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press, Cambridge (2004) 7. Bartlett, P.L., Mendelson, S.: Rademacher and Gaussian Complexities: Risk Bounds and Structural Results. JMLR 3, 463–482 (2002) 8. Tsang, I.W., Kocsor, A., Kwok, J.T.: Simpler Core Vector Machines with Enclosing Balls. In: ICML (2007)

The Turning Points on MLP’s Error Surface Hung-Han Chen 8787 Southside Boulevard, Suite #503 Jacksonville, Florida 32256, USA [email protected]

Abstract. This paper presents a different view on the issue of local minima and introduces a new search method for Backpropagation learning algorithm of Multi-Layer Perceptrons (MLP). As in conventional point of view, Backpropagation may be trapped at local minima instead of finding the global minimum. This concept often leads to less confidence that people may have on neural networks. However, one could argue that most of local minima may be caused by the limitation of search methods. Therefore a new search method to address this situation is proposed in this paper. This new method, “retreat and turn”, has been applied to several different types of data alone or combined with other techniques. The encouraging results are included in this paper. Keywords: MLP neural networks, Backpropagation, Error Surface, Escaping Local Minima, Retreat and Turn.

1 Introduction Neural network has been one of the important methods for problem solving based upon the concept of artificial intelligence. The easy-to-use supervised learning algorithm, Backpropagation, has made Multi-Layer Perceptrons (MLP) neural networks popular for solving pattern recognition problems. However, almost since the beginning, there have been some critics for MLP neural networks regarding different aspects from many intelligent researchers. One of the major critics is that Backpropagation may be trapped at local minima instead of finding the global minimum. “It is both well known and obvious that hill climbing does not always work. The simplest way to fail is to get stuck on a local minimum [1].” When people treat Backpropagation learning algorithm as a variation of hill climbing techniques, often they believe that Backpropagation may be trapped at local minima. However, it is not clear if a situation like that is caused by the limitation of search choices or the gradients really descending nowhere on the error surface. No matter what causes local minima, generating random learning factors and scanning through neighborhood are normally used to escape local minima. However, random learning factor is not efficient, scanning is time consuming, and both do not guarantee to escape local minima within a limited time frame. Therefore, it would be better to save these options as the last resort and to find a more efficient method of escaping local minima instead. This paper presents an innovative search method, “retreat and turn”, to help Backpropagation escape from most of local minima. The turning mechanism from this F. Sun et al. (Eds.): ISNN 2008, Part I, LNCS 5263, pp. 512–520, 2008. © Springer-Verlag Berlin Heidelberg 2008

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proposed method, incorporating with neuron’s firing status, can help Backpropagation to find a meaningful and efficient path on error surface to descend. If Backpropagation with this search method can escape from almost all local minima, eventually it may reach the global minimum without any help from complicate mathematical calculation. Section 2 in this paper discusses the algorithm of Backpropagation and its error surface. Section 3 studies local minima and gradients in details. The “retreat and turn” search method is presented in section 4. Some examples using this innovative search method are included in section 5. Final conclusions are discussed in section 6.

2 Backpropagation and Error Surface The MLP neural networks using Backpropagation learning algorithm constitute of many options of composing structures. In this paper, we are only discussing the form with one nonlinear hidden layer and one linear output layer, as shown in Fig. 1 (a). The error term E is measured in squared form and the transfer function for hidden neurons is the sigmoid function. Backpropagation, in fact, is a gradient descent method used with MLP neural networks. Equation (1) indicates that the weights are updated through gradients on error surface.

w( j , i ) ← w( j , i ) + η ⋅

−∂E ⋅ ∂w( j , i )

(1)

By using chain rule to propagate the error term E from output layer back to hidden layer, the gradients can be generalized with a delta function as in equation (2). The delta function is then defined in equation (3). The delta function can be calculated as in equation (4) for output layer and equation (5) for hidden layer accordingly.

∂E = − δ ( j ) ⋅ O (i ) ⋅ ∂w ( j, i )

δ ( j) =

(2)

−∂E ⋅ ∂net ( j )

(3)

δ (n ) = f ′(net (n )) ⋅ (T (n ) − O(n )) ⋅ δ ( j ) = f ′(net ( j )) ⋅

(4)

∑ δ (n)w(n, j ) = O( j )(1 − O( j )) ⋅ ∑ δ (n)w(n, j ) ⋅ n

(5)

n

The MLP neural networks with Backpropagation learning algorithm may have been claimed with some drawbacks, especially for the chances of being trapped at a local minimum; however, they do, in principal, offer all the potential of universal computing devices. They were intuitively appealing to many researchers because of their intrinsic nonlinearity, computational simplicity and resemblance to the behavior of neurons [1]. Therefore, if the issue of local minima can be resolved, we can see the

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Layer

E +

Input

nonlinear ϕ

ϕ

ϕ

ϕ

Hidden local minimum

global minimum

linear

∑

∑

Output

-

+

Wj

Fig. 1. (a) The structure of a MLP neural network and (b) The error surface for W j

unlimited potential MLP neural networks may have for the future advancement on machine learning and intelligence. For Backpropagation learning algorithm, the purpose of changing weights is to reduce error. How error changes with the direction of gradient as we change a specific weight can be draw on one-dimension error surface as shown in Fig. 1 (b). Therefore the number of hidden neurons and output weights will decide the dimensions of the overall error surface. Since the rest of the weights are also constantly changing with the direction of gradients, the error landscape will change even if this specific weight may stay the same. There are many researches on the topic of MLP error surface. Frasconi et al. [2] lists studies on the surface attached to the cost. Blum [3] identifies local minima for multi-layered networks on the problem of XOR. Hush et al. [4] give interesting qualitative indications on the shape of error surface by investigating on small examples. The error surface is comprised of numerous flat and steep regions where the gradients vary by several orders of magnitude. The flat plateaus could extend to infinity in all directions and the steep troughs may extremely flat in the direction of search. Kordos et al. [5] identify some important error surface properties on the survey of factors influencing MLP error surface. They conclude that error surface depends on network structure, training data, transfer and error functions, but not on training methods. One of the properties is that error surface of networks with hidden layers has a starfish-like structure with many ravines. Another is that global minima are in infinity with MSE error functions, and local minima lie in ravines that asymptotically reach higher error values.

3 Local Minima and Gradients As we know that only limited times an algorithm can search on the error surface to descend, it is very possible that being trapped at a local minimum is simply because the search algorithm hasn’t found the right direction and distance to descend on the

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error surface. In such a case, obviously this is a local minimum only because of the search algorithm but not the topography on error surface. This misunderstanding can be confirmed by the proof and disproof of the local minima for XOR problem using a simple multilayer Perceptrons network. As mentioned earlier, Blum [3] has proven there is a line of local minima on the error surface. However, other researchers have also proven either the points on Blum line are saddle points [6] or there is no local minimum on the XOR error surface [7]. According to them, Blum’s proof is based on incorrect assumptions, and naïve visualization of slices through error surface may fail to reveal the true nature of the error surface. Since Backpropagation algorithm learns through the gradients on error surface, we can examine this issue from the viewpoint of gradients. Lemma 1. Any minimum on the error surface for a gradient descent method, local or −∂E global, should be η⋅ = 0 in vector space. (If using the same learning factor ∂Wi i

∑

on every weight, then

η

can be omitted.) Therefore no further descent can be made ∂E by updating weights through gradients. In such case, either all individual are ∂Wi ∂E are not all zeroes but somehow the directional sum of ∂Wi the gradients becomes zero.

zero or those individual

There are some properties of gradients for MLP neural networks worth noting here. Property 1. Zero gradients for a linear output neuron simply imply that sum of squared errors associated to that neuron is zero in batch mode training. Therefore, zero gradients for all output weights could possibly mean the ultimate minimum E =0. ∂E is large only when output squared errors are Property 2. For hidden neurons, ∂Wi large or the derivatives of the transfer function, f ′(net ( j )) = O ( j )(1 − O( j )) , are large. Since the output errors affect each hidden neurons almost equally, if the effect from output weights is not significantly different, then the biggest gradient mostly occurs when a hidden neuron is the least certain regarding its firing status. In other words, when net function of a hidden neuron mostly falls at the middle of sigmoid function curve, its derivative is often larger and this hidden neuron is less certain to fire or not.

4 Turning Points on Error Surface The remedy to overcome the limitation of Backpropagation search methods, which cause the situation of being trapped at local minima, could lie in the two properties of gradients described in previous section. Obviously, the situation of all zero gradients

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is the final target we are pursuing, E = 0 . Therefore, there is no need and nowhere to descend on the error surface. ∂E What if individual are not all zeroes but somehow the directional sum of the ∂Wi gradients becomes zero? This paper presents a new search method, “retreat and turn”, to find an efficient path on error surface to descend. “Retreat” is a normal reaction on hill climbing technologies when error increases at the current iteration. A common practice on retreat is to retrieve the best weights and reduce the learning factor. However, this line search along the sum of gradient directions sometimes is not enough to find the path for descending on error surface before the learning factor becomes too small and a bigger number needs to be randomly generated. As in ancient Chinese idiom, “If we can’t move the mountain, we can at least make the road turn.” This paper introduces a turning mechanism for MLP search algorithm. Assuming the extreme case that directional sum of all gradients is zero, if we take one ∂E out of the equation, the directional sum of gradients becomes non-zero non-zero ∂Wi again. Then the learning algorithm can find a path to descend on the error surface if the learning step is small enough. ∂E to be removed will create the best effect. The question here would be which ∂Wi In this proposed search method, we choose not to remove any gradient from the output weights since they are directly made by the output errors. Therefore, the better ∂E from the hidden layer. choices will be one or some ∂Wi

From previous sections, it is known that the delta function, δ ( j ) , of each hidden neuron is also a key element to those gradients associated to that neuron. Since the largest δ ( j ) often comes from the least certain of hidden neuron, the turning mecha-

nism presented in this paper is to restrict the hidden neuron with the largest δ ( j ) from updating its weights when MLP encounters an error increase. Fig. 2 is the flowchart describing the process of “retreat and turn” search algorithm. The reason to restrict the hidden neuron with the largest δ ( j ) from updating its weights can be analyzed from the following aspects. The first being the retreat process, the largest δ ( j ) makes the greatest change on weights, from equation (2), therefore it could be the biggest contributor for the error increase. The second being the firing status of hidden neurons, with the least certain neuron, its contribution on reducing the error can go either very good or very bad. And it is certainly very bad when error increases. When the largest δ ( j ) was removed from the equation, the directional sum of gradients is then turning on the error surface. The concept of this search method especially echoes the discovery of ravines and starfish-like error surface from [4] and [5]. Like water always flows to lower ground through “troughs” or “ravines”, the error can descend on the surface by turning away from the walls (when encountering an error increase).

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backpropagation

No error ?

Yes

increase η to double

decrease η to half

add next large

δ

Yes

to pool

If η too small?

No randomly generating a larger η

reset the pool

δ

remove largest δ from pool

Yes

no more δ In the pool?

No

Updating weights according to δ pool

Fig. 2. The process of “Retreat and Turn” search method for hidden layer

5 Some Examples This new search method, “retreat and turn”, has been applied to two different types of data. One is a rare event from healthcare data to predict 3-month inpatient risk for 2.4 million insured members [8]; the other is even distribution data from USGS Land Use/Cover Categories [9]. The prevalence of 3-month inpatient risk in the population is only 1.3%. Medical and pharmacy claim history of one year is summarized and grouped into 53 features by ICD-9 diagnostic codes, CPT-4 procedure codes and NDC pharmacy codes. This featured data is first “divided” into several subgroups by Self-Organized Map (SOM), then “conquered” by MLP neural networks using this “retreat and turn” search algorithm. This combination of two technologies, Chen’s model [8], has been proved to outperform leading commercial risk score software in healthcare industry. Table 1 shows the comparison of the validation. The notation of “5k model” is when 5,000 members are targeted to be the outputs from the model of inpatient risk.

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The data of USGS Land Use/Cover Categories, named comf18, are generated segmented images of four classes. Each segmented region is separately histogram equalized to 20 levels. Then the joint probability density of pairs of pixels separated by a given distance and a given direction is estimated. For each separation, the cooccurrences for the four directions are folded together to form a triangular matrix. From each of the resulting three matrices, six features are computed: angular second moment, contrast, entropy, correlation, and the sums of the main diagonal and the first off diagonal. This results in 18 features for each classification window [9]. Four simulations have been designed to investigate the “retreat and turn” search method with comf18. Two of them use traditional Backpropagation (BP) with adaptive learning factor. Whenever the learning factor becomes too small, a bigger learning

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factor will be randomly generated. They are constructed with 20 and 100 hidden neurons accordingly. Then the “retreat and turn” search method (Chen) is added to conduct two new simulations. Fig. 3 (a) and (b) show the Mean Square Error (MSE) and Accuracy plots from those 4 simulations. The results of this proposed method are comparable to the advanced technique from [9] if running time is not a concern. The smoothness of descending path on the error surface can be measured by the times that learning factors are randomly generated to escape local minima. There are 49 and 37 random learning factors for the first 10,000 iterations from the simulations of BP-18-20-4 and BP-18-100-4 respectively. When the “retreat and turn” search method is added, the random numbers are reduced to 0 and 0 for the first 10,000 iterations. When simulations with the proposed search method are extended to 100,000 iterations, only 9 and 0 random learning factors are found.

6 Conclusions This paper presents an innovative search method, “retreat and turn”, for MLP’s Backpropagation learning algorithm. In order to escape from local minima, or equivalent to find a descending path on error surface within limited times of search, this proposed method incorporates the firing status of each hidden neurons to make a meaningful and efficient turn whenever it encounters an error increase. This proposed method has been tested with different types of data for up to 100,000 iterations without being stuck in a local minimum. In the meantime, this method updates the learning factor in its normal way often for tens of thousands iterations without the need to generate a random one. This means the path for descending on error surface is almost always smooth. The logarithmic scale plots in Fig. 4 have demonstrated the ability of the proposed search method to constantly descend on MLP error surface and maybe reaching the global minimum when the learning curve eventually hits the flat line. Judging by the MSE curves, we also have confidence that

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this method can bring us to a fairly good solution for any error surface within a reasonable amount of time. Undoubtedly, many advanced techniques with complicate mathematical calculation can perform faster or better on solving certain problems. However, the comparable simulation results from this proposed method are quite encouraging since the obstacle of local minima is mostly removed and computational simplicity is still remained. With this proposed search method, Backpropagation MLP can as well be the universal computing device one day if we can run millions of neurons at once when computational speed is greatly improved in the future. One remaining issue for future study could be the speed of learning when errors decrease to a certain level and cause small gradients.

References 1. Minsky, M., Papert, S.: Epilog: the new connectionism. In: Perceptrons, 3rd edn., pp. 247– 280. MIT Press, Cambridge (1988) 2. Frasconi, P., Gori, M., Tesi, A.: Success and Failures of Backpropagation: A Theoretical Investigation. In: Omidvar, O. (ed.) Progress in Neural Networks, vol. 5. Ablex Publishing, Greenwich (1993) 3. Blum, E.K.: Approximation of Boolean Functions by Sigmoidal Networks: Part I: XOR and Other Two-Variable Functions. Neural Computation 1, 532–540 (1989) 4. Hush, D.R., Horne, B., Salas, J.M.: Error Surfaces for Multilayer Perceptrons. IEEE Transactions on Systems, Man, and Cybernetics 22, 1152–1161 (1992) 5. Kordos, M., Duch, W.: On Some Factors Influencing MLP Error Surface. In: 7th International Conference of Artificial Intelligence and Soft Computing, pp. 217–222 (2004) 6. Hamey, L.G.: The Structure of Neural Network Error Surface. In: 6th Australian Conference on Neural Networks, pp. 197–200 (1995) 7. Sprinkhuizen-Kuyper, I.G., Boers, E.J.: A Comment on Paper of Blum: Blum’s “local minima” are Saddle Points. Technical Report 94-34, Department of Computer Science, Leiden University (1994) 8. Chen, H.H., Manry, M.T.: Improving Healthcare Predictive Modeling using NeuroSequences. In: 16th Federal Forecasters Conference (in press, 2008) 9. Abdurrab, A.A., Manry, M.T., Li, J., Malalur, S.S., Gore, R.G.: A Piecewise Linear Network Classifier. In: 20th International Joint Conference on Neural Networks, pp. 1750–1755 (2007)

Parallel Fuzzy Reasoning Models with Ensemble Learning Hiromi Miyajima, Noritaka Shigei, Shinya Fukumoto, and Toshiaki Miike Kagoshima University, 1-21-40 Korimoto, Kagoshima, 890-0065, Japan {miya,shigei}@eee.kagoshima-u.ac.jp