Response Modeling Methodology
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Response Modeling Methodology
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Series on Quality, Reliability and Engineering Statistics
Response Modeling Methodology Empirical Modeling for Engineering and Science
Haim Shore Ben-Gurian University of the Negev, Israel
We World Scientific NEW JERSEY · LONDON · SINGAPORE · BEIJING · SHANGHAI · HONG KONG · TAIPEI · CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Series on Quality, Reliability and Engineering Statistics — Vol. 8 RESPONSE MODELING METHODOLOGY (RMM) Empirical Modeling for Engineering and Science Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-256-102-1
Printed in Singapore.
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RESPONSE MODELING METHODOLOGY
normal distribution, M1(Y) and M2(Y) are partial upper moments of Y, namely,
M k (Y ) =
∫
∞
Med
y k f ( y)dy,
(5.13)
and Med is the median of Y. Note, that given a sample of n observations, an estimate for M1(Y) is the sum of all observations larger than the median, divided by n, and an estimate for M2(Y) is the sum of all squared observations (for observations larger than the median) divided again by n (not n/2!). For example, for the data: y={7,2,15,3}, an estimate of M2(Y) is M2(Y)= (72+152)/4= 68.5. Equation (5.11), with parameters given by (5.12), preserves the first two moments of Y (partial and complete). It has been shown to provide good representation to a wide variety of distributions, and it complies with the requirement that the MSEs of high degree moments, like skewness and kurtosis, are small when sample data are used to estimate its parameters (find details in Shore, 2000a, and also refer to Requirement 6 in Chapter 6). The piece-wise linear transformations, given in (5.8)-(5.12), are addressed in detail, with some numerical examples, in Chapter 21. A second parameter-rich family of distributions is (refer to Shore 2002a) A1 [P/(1-P)]B1 , P < 1/2, y= B2 A 2 {[P/(1-P)] - 1} + A1 , P ≥ 1/2.
(5.14)
This family of distributions looks similar to the generalized Lambda distribution (5.7). Fitting procedures were developed for (5.14) that again require estimates of only first and second degree moments, partial and complete (find details in Shore, 1998a,b, and also relate to Chapter 23). Another type of efforts to derive general families of distributions aimed at assembling under a single “umbrella distribution” several existing distributions. These endeavors have not introduced new
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RESPONSE MODELING METHODOLOGY
that would still deliver a feeling for why moments determine a distribution. A well known statistical function that characterizes a distribution is the moment-generating function (MGF). It is defined by M(t) = E(etY) =
∫
∞
−∞
e ty dF ( y ) ,
(5.19)
where F(y) is the CDF of the r.v., Y. It can be shown mathematically that if the MGFs of two r.v.s, Y1 and Y2, are identical for all values of t in an interval around the point t=0, then the probability density functions of the two r.v.s must be identical. Using a well-known series expansion of ev, (5.20) ev = 1+v+v2/2!+v3/3!+.., we obtain from (5.19) M(t) = E[1+tY+(tY)2/2!+(tY)3/3!+..].
(5.21)
Although the series in (5.20) is infinite, it can be shown under fairly general conditions that the expected value of the sum is equal to the sum of the expected values (a result which is always true for a finite sum of r.v.s). This implies that one can write for (5.21) M(t) = 1+ tE(Y) + t2E(Y2)/2!+ t3E(Y3)/3!+.. . (5.22) Denoting the k-th non-central moment, E(Yk), by µk', we may re-write (5.22) by M(t) = 1+ t µ1' + (t2/2)µ2'+ (t3/6)µ3'+.. .
(5.23)
From (5.23) it can easily be recognized that the n-th derivative of M(t) with respect to t, at the point t=0, is equal to µn'. This is why M(t) is generally related to as "a moment-generating-function". The upshot of all these expressions is that since M(t) uniquely determines the distribution of the r.v., and since M(t) may be represented as a series expansion in terms of its moments, it is reasonable to assume that if the first few moments in this expansion are shared by two r.v.s their characteristic patterns of dispersion, as captured by the respective statistical distributions, will also tend to be similar. Staurt and Ord (1987, Section 3.34) provide a mathematical demonstration of this assertion, and numerous empirical studies repeatedly corroborated this fact.
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(7.6)
RU : = ORJ(<) = (α/λ)[(η+ε1)λ − 1] + µ2 + ε2 = (α/λ)[(η+σε1Ζ1)λ − 1] + µ2 + σε2Ζ2,
(7.7)
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IRU=LQ Ζ2 | ]1 = ρ]1 + (1−ρ2)(1/2) Ζ,
(7.8)
ZKHUH=LVDVWDQGDUGQRUPDOYDULDEOHLQGHSHQGHQWRI=,QWURGXFLQJIRU =IURP LQWR DQG ZHREWDLQUHVSHFWLYHO\ < H[S{(α/λ)[(η+σε1Ζ1)λ − 1] + µ2 + σε2[ρΖ1+(1−ρ2)(1/2) Ζ2]}, (7.9) : ORJ(<) = (α/λ)[(η+σε1Ζ1)λ − 1] + µ2 + σε2[ρΖ1+(1−ρ2)(1/2) Ζ2], (7.10) ZKHUH IRU XQLIRUPLW\ LQ QRWDWLRQ ZH UHSODFHG = ZLWK = QDPHO\ = LV QRZ DVVXPHG WR EH D VWDQGDUG QRUPDO YDULDEOH ZKLFK LV XQFRUUHODWHG LQGHSHQGHQW RI= (TXLYDOHQWO\E\FRQGLWLRQLQJ=RQ= ]ZHREWDLQ : ORJ < җ = (α/λ){[η + σε1[ρΖ2+(1−ρ2)(1/2) Ζ1]]λ − 1} + µ2 + σε2Ζ2. (7.11) (TXDWLRQV DQG GLVSOD\ WKH 500 PRGHO LQ LWV PRVW JHQHUDO IRUP %RWK PRGHOV H[SUHVV WKH UHVSRQVH LQ WHUPV RI WZR XQFRUUHODWHG VWDQGDUG QRUPDO YDULDEOHV DQG WKH WZR PRGHOV DUH HTXLYDOHQWLQWKHVHQVHWKDWRQHPRGHOFDQEHGHULYHGIURPWKHRWKHU 6RPH YDULDWLRQV RI WKH EDVLF PRGHO DV UHSUHVHQWHG E\ RU PD\EHGHYHORSHGE\SXUVXLQJGLIIHUHQWGHILQLWLRQVIRUWKHHUURUV WKLV ZDV EULHIO\ DOOXGHG WR HDUOLHU DQG UHLQWURGXFHG KHUH LQ D PRUH FRQVLVWHQWIDVKLRQ &RQVLGHUWKHHUURUVWUXFWXUHLQ DQG )RUVPDOOHUURUε2 << ZHPD\ZULWH 1 + ε2 ≈ H[S(ε2). )RUVPDOOHUURU ε1 << ηZHPD\ZULWH (η+ε1)λ = ηλ(1+ε1/η)λ ≈ ηλH[S(λε1/η). 7KHVH WZR SRVVLEOH YHUVLRQV IRU HDFK HUURU WHUP JLYH ULVH WR IRXU GLIIHUHQW HUURUVWUXFWXUHV IRU WKH EDVLF PRGHO &RXSOHG ZLWK WKH WZR HTXLYDOHQW YDULDWLRQV ZKHUH WKH FRUUHODWHG HUURUWHUPV KDYH EHHQ UH H[SUHVVHGLQWHUPVRIWZRLQGHSHQGHQWVWDQGDUGQRUPDOYDULDEOHV> DQG @ WKH 500 PRGHO PD\ EH GLVSOD\HG LQ GLIIHUHQW IRUPV 7KHVH DUH LQWURGXFHG EHORZ 2QO\ YDULDWLRQV RI WKH HUURUVWUXFWXUHV DUH
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(7.6D)
0RGHO'LIIHUHQWSUHVHQWDWLRQRIε1 < H[S {(α/λ)[ηλ H[S(λε1 / η) − 1] + µ2 + ε2}. 0RGHO'LIIHUHQWSUHVHQWDWLRQRIERWKε1DQGε2 < H[S {(α/λ)[ηλ H[S (λε1 / η) − 1] + µ2 }(1+ε2).
(7.6E) (7.6F)
)RUHDFKPRGHOZHFDQLQWURGXFH (LWKHUIRUεε2: ε2 = σε2[ρΖ1+(1−ρ2)(1/2) Ζ2],
(7.12)
ZKHUHε1 = σε1=DQG=DQG=DUHXQFRUUHODWHG RUIRUεε1: ε1= σε1[ρΖ2 + (1−ρ2)(1/2) Ζ1],
(7.13)
ZKHUHε2 = σε2=DQG=DQG=DUHXQFRUUHODWHG
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H[S [Uµ2 + Uσε2(1−ρ2)(1/2) ]2] φ(]2)G]2},
(7.14)
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(7.15)
ZKHUH ω ^α, λ ρ, σε σε2` ω ^µ2,σε2,ρ` DQG J DQG J DUH H[SUHVVLRQVWKDWUHODWHWR=DQGWR=UHVSHFWLYHO\ 6LQFH ERWK H[SUHVVLRQV DUH H[SRQHQWLDO IXQFWLRQV RI WKH UHVSHFWLYH QRUPDO YDULDEOHV DQG EHFDXVH RIWKH VWDWLVWLFDO LQGHSHQGHQFH RI = DQG = WKH UWK QRQFHQWUDO PRPHQW RI < PD\ EH GHULYHG DV WKH SURGXFWRI WKH PHDQV RI J DQG RI J ZKHUH IRU WKH PHDQ RI J ZH UHSODFH ^α σε2} E\ ^Uα, Uσε2} DQG VLPLODUO\ IRU WKH PHDQ RI J ZH UHSODFH ^µ σε2`E\^UµUσε2`7KHUWKQRQFHQWUDOPRPHQWRI<LV Ε(
(7.16)
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Ε{[J2(Ζ2;ω2)]U} = H[S[Uµ2 + (1−ρ2)(Uσε2)2/2].
(7.17)
5HIHUIRUH[DPSOHWR6WXDUWDQG2UG 6HFWLRQ ([SDQGLQJ J LQWR D 7D\ORU VHULHV LQ WHUPV RI SRZHUV RI = DURXQG ]HURDQGWDNLQJH[SHFWDWLRQRIWKHILUVWIRXUWHUPVZHREWDLQIRUWKHUWK QRQFHQWUDOPRPHQWRIJ Ε{[J1(Ζ1;θ1)]U} ≅ H[S[(Uα/λ)(ηλ − 1)] [ {1 + (1/2)[(Uα)σε12(λ−1)ηλ − 2 + U2(ασε1ηλ−1 + ρσε2)2]}.
(7.18)
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(7.19))
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(7.20)
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(7.21)
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(7.23)
)RU D ORJQRUPDO YDULDEOH 8 ZLWK SDUDPHWHUV µ1 DQG σ1, WKH PHDQ DQG WKH YDULDQFH DUH UHIHU WR -RKQVRQ .RW] DQG %DODNULVKQDQ &KDSWHURUWR6WXDUWDQG2UG 6HFWLRQ ( 8 H[S(µ1 + σ12/2), 9DU 8 H[S(2µ1 + σ12)[H[S (σ12)−1],
(7.24)
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(7.25)
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136
RESPONSE MODELING METHODOLOGY
individual observations may then be derived numerically either from the quantile function [(8.1) or (8.2)] or from the density function of W. Both methods will now be developed for (8.1). Corresponding expressions may similarly be developed for (8.2). Estimating the Variance of W (Based on Eq. 8.1) Method I Re-write (8.1) as W = Q1(Z1; ω1) + Q2(Z2; ω2),
(8.13)
where Q1(Z1; ω1) = (α/λ)[(η+σε1Z1)λ – 1] + µ2 + σε2ρZ1, Q2(Z2; ω2) = σε2(1-ρ2)(1/2) Z2,
(8.14)
with, respectively, parameters' vectors ω1 = {α, λ, µ2, σε1, σε2ρ}, ω2 = σε2(1-ρ2)(1/2).
(8.15)
Note, that ω1 contains five parameters while ω2 has only one parameter. Since Z1 and Z2 are independent random variables so are Q1 and Q2, and for the r-th non-central moment of W we obtain E(Wr) = µr'(W) =
r
r
j =0
∑ j M
j
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(8.16)
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∫
∞
−∞
{(α/λ)[(η+σε1z)λ – 1] + µ2 + σε2ρz}k φ(z)dz (8.17)
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∫
∞
−∞
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= [σε2(1-ρ2)(1/2)]k µk(Z) where µk(Z) is the k-th standard normal moment.
(8.18)
CHAPTER 8 ESTIMATING THE RELATIONAL MODEL
137
The variance of W for individual observations may be estimated from (8.16)-(8.18), and inserted into the minimization routine (8.10). Method II This method uses an approximate expression for the density function of W, fW, given in (8.20) below. The variance of W may then be estimated from µr'(W) =
∫
∞
−∞
w r f W ( w)dw ,
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(8.19)
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0.792078 0.567448 0.202167
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Canonical Correlation 1 2
0.904635 0.330063
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0.882968 0.164909
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4.5056 0.1223
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154
RESPONSE MODELING METHODOLOGY
Developing fWε1(wε1; θ), in terms of ε1, into a Taylor series expansion around zero, and then applying the expectation operator to the first four terms (namely, including in the approximation terms up to a third degree), we obtain the relatively simple explicit approximate expression for the d.f of W (η=1) ∞
fW(w;θ) = ∫ fWε1(wε1; θ ) fε1(ε1)dε1 ≅ [2π(1−ρ2)σε22]−1/2 exp[-(1/2)Z2] −∞
x {1+(1/2)σ ε12[(1−ρ2)1/2σε2]−2[(α+ρσε2/σε1)2=2-1) + α(λ−1)(1− ρ2)1/2σε2Z]},
(9.10)
where Z = (w-µ2) / [(1-ρ2)σε22]1/2,
(9.11)
and in taking expectation for the Taylor expansion we introduced E(ε1) = 0, E(ε12 ) = σε12, E(ε13 ) = 0. (9.12) From (9.10), all moments of W may be easily calculated by µ'r(W) = E(Wr)=
∫
∞
−∞
w r f W ( w;θ )dw .
(9.13)
Also, from (9.10), an approximate expression for the d.f. of Y may be derived via (9.6). 9.3. Properties of the Error Distribution The properties of the RMM error distribution are explored. In particular, we investigate the shape versatility by studying the behavior of the RMM error distribution in the (√β1,β2) plane, where √β1 and β2-3 are measures of skewness and kurtosis, respectively [the third and fourth standardized cumulants; Refer to (9.26) below for their definitions]. Recall that in Chapter 6 we have related to adequate coverage of the (√β1,β2) plane as one of the most important requirements for the universal applicability of a methodology for empirical relational modeling (therein, Requirement 7). It is therefore natural that studying the coverage associated with the RMM error distribution is of high priority in order to assess its versatility and that of the RMM relational model.
CHAPTER 9 THE RMM ERROR DISTRIBUTION
155
We start by deriving expressions for the moments of either Y or W (the log-transformed response). If the parameters in (9.1) are specified, then the non-central r-th moment of Y is calculated numerically from ∞
µ'r(Y) = E(Yr) = ∫ (y)r fY(y)dy, −∞
(9.14)
where fY(y) is given by (9.6). However, this method requires double numerical integration, which can be a numerically prohibitive task. Alternatively, use can be made of the fact that Z1 and Z2 in (9.2) are independent to derive either exact or approximate explicit expressions for the moments of W and Y. An exact method was detailed in Chapter 8 for the relational model. It is repeated here with η=1. Re-write (9.2) as W = Q1(Z1; ω1) + Q2(Z2; ω2),
(9.15)
where Q1(Z1; ω1) = (α/λ)[(1+σε1Z1)λ – 1] + µ2 + σε2ρZ1, Q2(Z2; ω2)= σε2(1-ρ2)(1/2) Z2,
(9.16)
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ω2 = σε2(1-ρ2)(1/2).
(9.17)
Note, that ω1 contains five parameters while ω2 has only one parameter. Since Z1 and Z2 are independent random variables so are Q1 and Q2, and for the r-th non-central moment of W we obtain E(Wr) = µr'(W) =
r
r
j =0
∑ j M
j
(Q1 ) M r − j (Q2 ) ,
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where Mj(Q1) and Mr-j(Q2) are the non-central j-th and (r-j)-th noncentral moments of Q1 and Q2, respectively: Mk(Q1) =
∫
∞
−∞
{(α/λ)[(1+σε1z)λ – 1] + µ2 + σε2ρz}k φ(z)dz,
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∫
∞
−∞
[σε2(1-ρ2)(1/2) z]k φ(z)dz.
(9.19) (9.20)
From (9.18)-(9.20) all non-central moments of W can be obtained. Note
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CHAPTER 10 FITTING PROCEDURES
169
and Z2, respectively, and ω1= {α, λ, µ2, σε1, σε2ρ}, ω2= {σε2(1-ρ2)(1/2)}, are the parameters that need to be determined (six parameters altogether). Note that QS1 and QS2 differ from Q1 and Q2, respectively, given in (9.16). Since Z1 and Z2 are independent standard normal variables, the r-th moment of WS is Mr(WS) = E[(WS)r] =
r
r
j =0
∑ j M
j
(QS1 ) M r − j (QS 2 ) , (10.10)
where Mj(QS1) and Mr-j(QS2) are the non-central j-th and (r-j)-th moments of QS1 and of QS2, respectively. For the latter we have Mk(QS1) = E[(QS1)k] =
∫
∞
−∞
{{(α/λ)[(η+σε1z)λ – 1] + µ2 + σε2ρz - µ}/σ}k φ(z)dz , (10.11) Mk(QS2) =
∫
∞
−∞
[(σε2/σ)(1-ρ2)(1/2) z]k φ(z)dz
= [(σε2/σ)(1-ρ2)(1/2)]k µk(Z),
(10.12)
where µk(Z) is the k-th moment of the standard normal distribution. For k odd, we have identically µk(Z)=0. For k even, we have for the first few moments µ2(Z) = E(Z2) =1, µ4(Z) = 3, µ6(Z) = 15, µ8(Z) = 105. This implies that to calculate a moment of any desirable degree via (10.10) only numerical integration of (10.11) is needed. To identify the parameter set, {ω1, ω2}, the following routine needs to be applied: Find {ω1, ω2} that minimize the objective function 6
OF = [M1 (WS)]2 +
∑
[Mr (WS)/µr(WS) - 1]2,
(10.13)
2
where µr(WS)= E[(WS)r] is the given exact r-th moment of WS (the standardized W), namely, the r-th moment associated with the distribution that we wish to represent by the RMM model (these moments may be derived from available explicit expressions for the moments, or numerically calculated with the exact density function).
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CHAPTER 10 FITTING PROCEDURES
175
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Figure 10.1. Plots of the density function, exact and fitted (left) and of the error for Gamma (3,2) (based on quantile-matching).
Figure 10.2. Plots of the quantile function (approximate and exact, left) and of the deviation (approximate minus exact, right) for the Gamma (3,2). Horizontal axis is the standard normal quantile (z). Vertical axis (left plot) is the standardized gamma quantile.
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CHAPTER 11 ESTIMATING THE ERROR DISTRIBUTION
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Figure 11.3 displays the relative errors (in %). The mean absolute value of the errors is 0.3888%. Calculating the first four moments of W from the fitted (11.11), we find out that the mean and the variance are well preserved by the fitted equation: Calculated {µ,σ2}: {10.0, 0.007439} Sample {µ,σ2}: {9.970, 0.006929}. The sample skewness and kurtosis are not well preserved. This is to be expected given the small sample size and the well-known high MSE of direct sample estimates of skewness and kurtosis. Consider the KD solution. The authors concluded that the generalized beta distribution (GBD) may model well this dataset (therein, p. 208). The four-parameter d.f and CDF of GBD is given by (therein, p. 119) f(y) = (y-β1)β3(β1+β2y)β4 / [β(β3+1, β4+1)β2(β3+β4+1)], β1 \ β1+β2, F(y) =
∫
y
β1
f (u )du ,
(11.12) (11.13)
where β is the beta function: β(a,b) = Γ(a) Γ(b) / Γ(a+b). For the above data, KD obtain the fitted GLB parameters β1= 18675.3862, β2 = 4880.6411, β3 = -0.4789, β4 = -0.6045. Figure 11.4 displays the relative errors (in %). An error is defined as the difference between the quantile, yi, calculated from solving F(yi)= pi, and the actual i-th order statistic [pi is the CDF value for the i-th order statistic, calculated from (11.9)]. We realize that the fit is about the same as that obtained via (11.11) (refer to Figure 11.3). The mean absolute value of the relative errors is 0.28% (vs. 0.39% for the RMM model). 11.3. Moment-Based Estimation 11.3.1. Introduction Throughout this book reservation has been expressed about estimating a distribution's parameters via four-moment matching. Various sources had been quoted, including our own, where it was asserted that due to the
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CHAPTER 11 ESTIMATING THE ERROR DISTRIBUTION
193
This implies that the approximate expression for E(Y/M) in (11.15) can also be used to approximate E[(Y/M)r], with the appropriate substitution for B and D, namely, E(Yr) ≅ (M)r{1 + (1/2)[rAB2 + r2(AB+D)2] + (3/24){rAB4(1 + 7rA + 6r2A2 + r3A3) + 4r2AB3D(1 + 3rA + r2A2) + 6r3AB2D2(1 + rA) + 4r4ABD3 + r4D4}}.
(11.16)
The reader may easily verify that (11.16) is a highly accurate approximation for the non-central moments of Y. To realize that, calculate for several values of the parameters the approximate moments, given by (11.16), and compare with the exact moments (to derive the latter, multiply Yr, as given by the RHS of (11.14), by the standard normal d.f and integrate). Since (11.16) provides an explicit expression for the r-th non-central moment, no numerical integration is needed during the search routine, and the parameters that deliver moment-matching may be easily identified (either via root-finding or by a minimization procedure). It was earlier explained that the moment-based estimation procedures involve matching of partial moments. Consider approximating the partial moments of Y in (11.14). We may wish to use the same approach that led to the derivation of an approximation to the complete moments (11.16). However, odd-order partial moments of the standard normal variable do not reduce to zero (as complete moments do). Therefore, into the expression derived from the Taylor expansion, we need to introduce for the odd-order power terms of Z partial moments of the standard normal variable. Denoting the r-th upper partial moment of Z by MU(Zr), namely, MU(Zr)=
∫
∞
0
z r φ ( z )dz ,
the first six partial moments are MU(Z) = (2π)-1/2 , MU(Z2) = 1/2 , MU(Z3) = (2/π)1/2 , MU(Z4) = 3/2, MU(Z5) = 4(2/π)1/2 , MU(Z6) = 15/2.
(11.17)
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CHAPTER 11 ESTIMATING THE ERROR DISTRIBUTION
195
From (11.14), we have for the mean of W E(W) = log(M) + (Α)[exp(B2/2) – 1]. Introducing for "A" from (11.18) into (11.14) we obtain W = log(M)+[E(W)-log(M)][exp(BZ) – 1]/[exp(B2/2) – 1]+DZ
(11.18) (11.19)
This expression describes W in terms of log(M), E(W), and two unknown parameters, B and D. Denote by MU(Y) the response upper partial mean, namely, MU(Y) =
∫
∞
M
yf ( y )dy ,
(11.20)
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∫
∞
M
(M)exp{[E(W)-log(M)][exp(Bz)–1]/[exp(B2/2)–1] +Dz}φ(z)dz,
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280
RESPONSE MODELING METHODOLOGY
Figure 17.1. Plot of the Oxygen data (pressure vs. temperature)
Introducing σε1=0 in E[log(P)] (17.3), we obtain, via NL-LS estimation, initial values of α = 603.1, λ = -0.9300, µ2 = -640.9. The estimated value of λ is near the value given by Antoine equation (λ=-1). From (17.1), recalling that E(ε2)= 0, we obtain explicitly for the linear predictor (LP) of observation i ηi + ε1 = {[log(Pi)+640.9](-0.93/603.1)+1}(-1/0.93),
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CHAPTER 17 MODELING A CHEMO-RESPONSE
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Figure 17.4. Experimental data and calculated curve [using (17.12)] for solid Ammonia’s heat capacity
Figure 17.5. Experimental data and calculated curve [using (17.12)] for liquid Butanoic acid’s heat capacity
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318
RESPONSE MODELING METHODOLOGY
19.2. Fitting RMM with Normal or Log-normal Errors In this section the RMM quantile function is fitted to a set of known statistical distributions, pursuing the original formulation of the RMM model, namely, the error terms, ε1 and ε2, are either normal or lognormal. The RMM error quantile-function is fitted using the "QuantileMatching" procedure (refer to Section 10.3 for details). We opt for this approach because when the distribution modeled by RMM is completely specified (no sample estimates are used in the fitting procedure), quantile-matching seems to be most convenient for identifying numerically the RMM parameters. The resulting goodness-of-fit is judged (in this chapter) mainly by how well the first four moments of the fitted RMM error distribution match the exact moments. The moments of the fitted quantile function are numerically evaluated using the simple expression for the r-th non-central moment µr' =
∫
∞
−∞
y ( z ) r φ ( z )dz ,
(19.1)
where y(z) is the fitted quantile function, and φ is the d.f. of the standard normal distribution. From (19.1), the mean, the variance, the skewness measure, Sk, and the kurtosis measure, Ku, are (refer to Stuart and Ord, 1987, Section 3.14) µ = µ1', V(Y) = µ2' - (µ1' )2 , Sk = ¥β1 = E[(Y-µ)3] / σ3 = [ µ3' - 3 µ2' µ1' + 2 (µ1')3] / σ3 , Ku = β2 - 3 = E[(Y-µ)4] / σ4-3 = [µ4' - 4 µ3' µ1' + 6 µ2' (µ1')2 - 3 (µ1')4] / σ4 - 3 .
(19.2)
For the normal distribution: Sk=0, Ku=0. Two versions of the quantile-function are employed in the fitting procedure (refer to Chapter 9): Y = (M) exp{(α/λ)[(1+ε1) λ - 1] + ε2 }, (19.3) Y = (M) exp{(α/λ)[ exp(λε1 ) - 1] + ε2},
(19.4)
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CHAPTER 19 RMM DISTRIBUTIONAL APPROXIMATIONS
329
λ = -0.61228883, S1 = -0.11105481, S2 = 0.44334159, α = -6.37309208.
(19.22)
[the value of α was derived from (19.21) with the above values of S1 and S2; Also the coefficients of z in (19.20), S1 and S2, have different signs, a possibility that is anticipated from (19.7)]. Introducing from (19.22) into (19.20) and thence into (19.17), the error plot in Figure 19.4 is obtained. The maximum absolute error is reduced from 2(10)-6 (Figure 19.3) to 6(10)-7. Although (19.20) has only three parameters that need to be determined from the fitting procedure (α is not), it has accuracy better than the four- parameter approximation (19.18). As a final comment, we note two interesting features of the new approximations for the normal CDF. First, they are non-polynomial. This implies that their high accuracy emanates not from the large number of parameters that characterize polynomial approximations (refer to Johnson et al., 1994), but possibly because the RMM original model is in some way related to the normal distribution. This shows in that the accuracy obtained from
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RESPONSE MODELING METHODOLOGY
A modified approximation for the quatile, u, of U, is [from (21.5)] [(Α − C ) z − hC ], z < 0 u= [(Α + C ) z − hC ], z ≥ 0,
(21.5a)
Let us develop the first three moments of (21.5a). This requires knowledge of the partial moments of Z. Discarding the assumption that Z is normal, however assuming that it is a standardized symmetrically distributed variable, denote by MU(Zr) the r-th upper partial moment of Z, namely, MU(Zr) =
∫
∞
0
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CHAPTER 21 PIECE-WISE LINEAR APPROXIMATIONS
363
is larger than that for z<0, implying that the true relationship is uniformly convex (second derivative of the exact unknown expression for the quantile function is positive). Conversely, when Sk is negative, this implies that the quantile function is concave (second derivative is negative). This observation is probably true also for the quantile function of the RMM error distribution, developed in Chapter 9, and for the inverse normalizing transformations, developed in Chapter 20. Note, however, that this observation is not a rigorous mathematical proof for the relationship between Sk and the convexity (concavity) of the actual quantile relationship between the response, Y, and Z. A mathematical proof for the "inverse" observation, namely, that a convex (concave) transformation of a symmetrically distributed random variable results in positive (negative) skewness may be found in Van Zwet (1964). 21.5. A Fitting Procedure Using First- and Second-Degree Moments To describe the two-moment fitting procedure, re-write (21.4) in terms of the quantile of the un-standardized response, Y: A1 z + B1 , z < 0, y= A z + B , z ≥ 0. 2 2
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∞
M
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RESPONSE MODELING METHODOLOGY
Similarly, from (21.24), the parameters of (21.22) are A1 = 2.023, B1 = 4.976, A2 = 4.864, B2 = 4.756.
(21.28)
Introducing the parameters in (21.27) into (21.5), a piece-wise linear approximation is obtained for the quantile of Y in terms of the standard normal quantile. From this, the r-th non-central moment of Y is µr' = E(Yr) =
∫
0
−4
[σ ( A − C ) z − 0.7978Cσ + µ ]r φ ( z )dz
4
+
∫ [σ ( A + C ) z − 0.7978Cσ + µ ] φ ( z)dz , r
0
(21.29)
where φ(z) is the density function of the standard normal variable, and the limits of integration were set arbitrarily as z=-4 and z=4. Similarly, introducing the parameters in (21.28) into (21.22), we obtain for the r-th non-central moment of Y: µr' = E(Yr) =
∫
0
−4
[ A1 z + B1 ]r φ ( z )dz
4
+
∫ [ A z + B ] φ ( z )dz 0
r
2
2
(21.30)
From (21.29) and (21.30), the first four moments are µ = µ1', σ2 = µ2' − (µ1')2, Sk = [µ3' - 3µ2' µ1' + 2(µ1')3] / σ3, Ku = [µ4' - 4µ3' µ1' + 6µ2' (µ1')2 - 3(µ1')4] / σ4 - 3
(21.31)
Table 21.2 displays the resulting first four moments. In this table, (21.22) [the un-standardized (21.4)] is fitted by the two-moment fitting procedure, with parameters given by (21.24). Also, (21.5) is fitted by the three-moment procedure, with parameters calculated from (21.20) (namely, Sk is approximately preserved). For both models, the first three moments are well preserved. Note, in particular, that for (21.22) only moments of second degree at most took part in the fitting procedure. However, the skewness measure is preserved.
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In this section, we use for inventory analysis a new four-parameter family of distributions (Shore, 1998), which satisfies these requirements. As was done in the previous section, the quantile function and the loss function are first developed. A maximum likelihood estimation procedure is then derived to estimate the parameters of the new distribution. In Section 23.5 the new family of distributions is applied to obtain an optimal solution for a certain inventory problem. Let Y be the response (a r.v.), and assume that Y is bounded by zero from below, and is unbounded, otherwise. Let F(y) be the cumulative distribution function (CDF), and denote by µ and σ the mean and the standard deviation, respectively. Let y be the P-th quantile, namely, F(y)=P. Consider the following inverse distribution function (the quantile function): A [P/(1-P)]B1 , P < 1/2 y(P) = 1 (23.13) B A 2 {[P/(1-P)] 2 - 1} + A1 , P ≥ 1/2, where {Ai, Bi} (i=1,2) are parameters that need to be determined. The motivation for introducing this quantile function, and some of its characteristics, are treated in Shore (1998). The density function (d.f.) of Y is 1-B 1+B [1/(A1 B1 )]P 1 (1-P) 1 , P<1/2, f[y(P)] = (23.14) 1-B 1+B [1/(A1 B1 )]P 2 (1-P) 2 , P ≥ 1/2. Denote by Mi(Y) the i-th upper partial moment of Y, namely, 1
Mi(Y) =
∫
P =1 / 2
[ y ( P)]i dP ,
(23.15)
with y(P) given by (23.13), and dP= f(y)dy. Note that in (23.15) integration is with respect to P (not with respect to y). This practice will often be repeated where P, rather than y, is used as the variable of integration or differentiation. Pursuing the two-moment (partial and complete) matching approach (refer to Chapter 21 for motivation and details), an estimation procedure may be developed to identify the parameters in (23.13) (refer to Shore, 1995b, 1998). However, the procedure requires a numeric search for the
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RESPONSE MODELING METHODOLOGY 1
Li(P) =
∫
P
[AiBi][F/(1-F)]Bi(1/F)dF,
(i=1 for P<1/2, i=2, otherwise).
(23.18)
It may easily be shown that the loss function, associated with (23.13), is L (P) + [L 2 (1/2) - L1 (1/2)], P < 1/2, L Y (P) = 1 L 2 (P), P ≥ 1/2.
(23.19)
This expression will be used in subsequent sections. Suppose now that the moments, needed to determine the parameters of (23.13) via moment matching, are unknown, and have to be estimated from sample observations. It has been previously noted (Shore, 1996, 1998) that estimates of the parameters of (23.13) tend to have large sampling variability due to the fact that B1 and B2 appear as exponents. In particular, B2 tends to be close to zero and is particularly susceptible to sampling deviations. An alternative approach that circumvents this difficulty is needed. To develop the modified procedure, let us impose a requirement for continuity of the d.f. at P=1/2. This implies that [refer to (23.14)]: A1B1= A2B2= K, say, and (23.13) becomes (K/B1 )[P/(1-P)]B1 , P < 1/2, y(P) = B (K/B2 ){[P/(1-P)] 2 - 1} + A1 , P ≥ 1/2.
(23.20)
The parameter "B2" is expected to be close to zero (observe representative values in Shore, 1995b). Since, for non-negative u, Lim λ: [(uλ -1)/λ] = log(u), we may rewrite (23.20) as A [P/(1-P)]B1 , P < 1/2, y(P) = 1 A2 log[P/(1-P)] + B2 , P ≥ 1/2.
(23.21)
Two new parameters are introduced for P≥1/2, to allow for a twomoment fitting. The “penalty" to be paid is the loss of continuity of the quantile function at P=1/2. For a two-moment fitting procedure, using partial and complete
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CHAPTER 23 INVENTORY ANALYSIS
L = [1/(A1B1)] (n/2)
∏
n/2 k =1
407
[ P(k) 1-B1 (1-P(k))1+B1 ]
x [1/(A2B2)] (n/2)
∏
n k = n / 2 +1
[ P(k)1-B2 (1-P(k))1+B2],
(23.25)
where P(k)= F[y(k)], and n is assumed to be an even number. Introducing from (23.13) for P(k) in terms of y(k), (23.25) is expressed in terms of the given observations. The maximum-likelihood estimates for {Ai, Bi} (i=1,2) may now be identified by maximizing log(L) either via differentiation or by a numerical search. A similar procedure can be developed for (23.21). Other fitting and estimation procedures, including procedures for right- and left-censored observations, are developed in Shore (1996). The case of truncated observations may be particularly interesting for models with lost sales. The reader is referred to the above reference for the appropriate fitting procedures. 23.4. First Approach- Newsboy Problem with Order-up-to Policy A general solution for the classical newsboy problem is derived, pursuing the first approach (Section 23.2). We assume that only the first three moments of the demand in the period, D, are known. Let S be the order-up-to level, and IC(S) be the expected inventory cost (ordering, holding and penalty) for the period. It is well known that IC(S) is given by IC(S)= cS + h
∫
S
t =0
(S - t)f D (t)dt + p
∫
∞
t =S
(t - S)f D (t)dt , (23.26)
where fD(d) is the d.f. of D, and {c, h, p} are the unit ordering cost, unit holding cost and unit penalty cost, respectively. It is assumed that penalty cost, p, is applied to excessive demand (that which exceeds available stock and thus create a shortage). It may be easily shown that
∫
S
t =0
(S - t)f D (t )dt = LD(P) + S - E(D),
(23.27)
where LD(P) is the value of the loss function of D at its CDF value FD(S)=P, and E(D) is the expected value of D. Introducing from (23.27)
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CHAPTER 23 INVENTORY ANALYSIS
413
determine the lead-times used to order the components (n decision variable), given relevant costs (that will be detailed shortly). The authors (HS) developed an approximate solution procedure, assuming normality for lead times. An alternative solution procedure, extended for non-normal cases and numerically demonstrated for the normal distribution, is given in Shore (1995a). In the latter reference, the solution procedure is based on the piece-wise linear approximations (Chapter 21). However, as alluded to at the beginning of Section 23.3, these approximations are unbounded in both directions and may deliver unsatisfactory accuracy, for lower-tail quantiles, if the approximated random variable is bounded from below (as lead-time demand is). Additionally, being linear, these approximations fail to deliver satisfactory accuracy if the optimal quantile value resides in the far righttail of the distribution. Here, the family of distributions, introduced in Section 23.3, is employed to derive a general solution for this problem. The accuracy obtained for non-normal populations is demonstrated. Only the first of the two formulations in HS will be addressed, however extension to the other formulation is straightforward. Denote by li the lead-time used to order component i (a decision variable, i=1,2,..,n). Assume that the holding cost per day, for a component that arrives too early, is hi= rpi/365, where r is the annual inventory carrying cost rate, and pi is the cost of component i. It is assumed that assembly cannot be performed early, namely, assembly does not start when the last component has arrived, but according to a pre-planned schedule. However, if at least one component is late, assembly is delayed at a time-independent cost of C. Denote by W(li) the average waiting time for component i: W(li) =
∫
li
y =0
(li - y)dFi(y).
(23.35)
The cost equation is (find further details in HS) Z=
∑
n i =1
hiW(li) + C[1-
Π
n i=1
Pi],
(23.36)
where Pi= Fi(li), namely, Pi is the probability that component i will arrive on time (not be late). It is easy to verify that W(li)= LY(Pi) + li - µ i.
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