CURVE FITTIN·G FOR PROGRAMMABLE CALCULATORS
by
William M. Kalb
CURVE FITTING
FOR PROGRAMMABLE CALCULATORS
by Wi 11 iam M. Kolb
FIRST EDITION
copyright © 1982 by William M. Rolb, 34 Laughton Street, Marlboro, Maryland 20772, U.S.A.
Upper
All rights reserved. Reproduction or use, without express permission, of editorial or pictorial content, in any manner, is prohibitied. programs may be stored and retrieved electronically for personal use, and may be used in published material if their source is acknowledged. No liability is assumed with respect to the use of programs or information contained herein. While every effort has been made to assure the accuracy of material presented, the author assumes no responsibility for errors or omissions. Neither is any liability assumed for damages reSUlting from the use of the material contained herein. This manuscript is set in Courier 10. author.
ISBN
The cover deSign is by the
8-943494-00-1
IMTEC P.O. Box 1402 Bowie, Maryland 28716 U.S.A.
To Hiroko
III
[V
PREFACE
This book provides all of the essential information needed to fit data to the most common curves. It avoids the usual mathematics and presents instead, straightforward solutions that can be used with most calculators. A basic introduction is included for the novice user of statistical models. The more intrepid explorer may find the sections on derivations, transformations, decomposition and substitution valuable for developing custom curve fitting routines. Users of the Hewlett-Packard HP-4lC/V programmable pocket computer can make immediate use of many models presented with the program and bar code included. The figures used to illustrate the text were drawn on an HP-85 desktop computer with a modified version of the computer's Standard Pac. The cover design was also done on an HP-85 with a program developed by the author. It is a pleasure to acknowledge the contributions of Dr. Harold Balaban in reviewing the manuscript and for his advice over the years on the use of statistical models. I am also greatful to Dr. Robert Groves for the loan of his computer to prepare the manuscript. Robert, Jeanne and Michael deserve very special thanks for their unfailing confidence. I am most indebted, however, to my wife Hiroko whose patience and love is most nearly exponential. April 1982
William M. Kolb
v
VI
TABLE OF CONTENTS Part 1. 2. 3. 4. 5. 6. 7. 8.
I Regression .. ...................... ................ .......... .............. Transformations •••••••••••••••••••••••••••••••••••• Multiple Linear Regression ••••••••••••••••••••••••• Decomposition and Substitution ••••••••••••••••••••• Scaling .................... _................................................................. Goodness of Fit •••••••••••••••••••••••••••••••••••• Significance ••••••••••••••••••••••••••••••••••••••• Getting Started ••••••••••••••.••••••.•••.••••••••••
3 4 5 6 7 7 8 9
Part II Straight Line •••••••••••••••••••••••••••••••••••••• Straight Line Through the Origin ••••••••••••••••••• Straight Line Through a Given Point •••••••••••••••• Alternative Straight Line
Through a Given Point
13 15 17
••••••••••••••••••••••••••••
19
Isotonic Linear Regression ••••••••••••••••••••••••• Reciprocal of a Straight Line •••••••••••••••••• Reciprocal of Straight Line Through a Given Point ...................................................... Combined Linear and Reciprocal ••••••••••••••••••••• Hyperbola ...... .................. ........................ ...... .............................. Hyperbola Through a Given Point •••••••••••••••••••• Reciprocal of a Hyperbola •••••••••••••••••••••••••• Second Order Hyperbola ••••••••••••••••••••••••••••• Parabola •••••••••••••••••••••••••••••••••••••• Parabola Through the Origin ••••••••••••••••••• Parabola Through a Given Point •.••••••••••••••••••• Power .............................. ..............................................................
21 23 25 27 30 32 34 36 49 42 44 46
Modified Power Root • • • • • • • . • .
se
•••.•••••••••••.•••••••••..••••..• . ••••••••••••••••. ••••••• •••••
48
Super Geometric •••••••••••••••.•••••••.••••.•••. Modified Geometric •••••••••••••••••••••• •••••••• Exponential •••••••••••••••••••••••••••••••••••••••• Modified Exponential ••••••••••••••••••••••••••••••• Logarithmic •••••••••••••••••••••••••••••••••••••••• Reciprocal of Logarithmic •••••••••••••••.•••••••••• Hoerl Function ••••••••••••••••••••••••••••••••••••• Modified Hoerl Function •••••••••••••••••••••••••••• Normal Distribution •••••••••••••••••••••••••••••••• Log-Normal Distribution •••••••••••••••••••••••••••• Beta Distribution ••••••••••••••••••••••••••••••••••
62 64 67 7a 73 76
Gamma Distribution
79
•••••••••••••••••••••.•••••••••.•
VII
52 54 56 58 6~
Cauchy Distribution •••••••••••••••••••••••••••••••• Multiple Linear Regression (Two Independent Variables) ••••••••••••••••••••••••••• Multiple Linear Regression (Three Independent Variables) ••••••••••••••••••••••••••• Generalized 2nd Order Polynomial ••••••••••••••••• Generalized 3rd Order Polynomial ••••••••••••••••• Ci rcle .............•........... . ............... . Correction Exponential ••••••••••••••••••••••••••••• LogistiC Curve
82 85 87 9il 92
95 97 •••••••••••.•.•..•••.•.•••.•••••••••• 100
Part III A. B.
Abbreviations and Symbols •••••••••••••••••••••••••• Register Assignments •.•••••••.••.•..••••••.••••
C.
Derivation of a Regression Curve ••••••••••••••••••• Multiple Curve Fitting Program Listing •••••••• Multiple Curve Fitting Program Bar Code •••••••
o.
E. F. G.
References
Index
•••••••••••••••••••• •
. .. . .. . . . . •. . . . .•••. .
VIII
• •••••••••••••••
. .•......••..••.
Al Bl Cl Dl
E1 FI Gl
CURVE FITTING FOR
PROGRAMMABLE CALCULATORS
1
"If you know it to one significant place. then you know it!"
G. S. Shostak
2
INTRODUCTION This is a collection of curve-fitting formulas intended to help anyone who must occasionally perform data analyses. It can be used to find a specific model for your data or as guide in choosing among several possible models. While the book is not exhaustive, it does present a comprehensive collection of the most useful one- and two-variable models. The equations for these models are designed primarily with the programmable scientific calculator in mind. If your calculator has fewer than l~~ registers available, however, you may need to change register assignments before using the formulas. Register numbers are used consistently throughout so that you can easily go from one model to another. Graphs of the various equations are provided to help select an appropriate model and sample problems are included to assist in debugging programs. The book is divided into three major sections. This first section is a general discussion of curve fitting intended as a primer for the beginner. The second contains various statistical models and the calculations necessary for estimating the coefficients. The final section is a series of appendixes that will help you program these models and develop new ones. The remainder of this introduction provides some insight into the techniques used to formulate models and the basics of regression analySis. The uninitiated user of statistical models should review this introduction carefully before drawing conclusions about the significance of any model. 1.
REGRESSION
If you were asked to predict what interest rates would be a year from now, how would you go about it? You could merely hazard a guess or you could employ a mathematical model of some sort to produce a forecast. A simple model would be a plot of interest rates against time for some recent period and a projection of these rates into the future. In this case, there is a dependent variable (interest) and one independent variable (time). A more sophisticated model might include several independent variables such as the national debt, consumer loans, and factory inventories, all of which you may have reason to believe contribute to interest rates.
3
One technique used to formulate a statistical relationship among these variables is regression (the term regression is used for historical reasons). For our simple model, historical data consisting of interest rates and dates would be regressed to supply the "best fitting" line through all of the points. The most common meaning of the term Wbest fit- is the line that minimizes the square of the vertical distances from each data point to the line of regression. The term "least squares W is used to describe this fitting method. For other regression lines -best fit· could mean the line that minimized the square of the horizontal distances, or the line that minimized the perpendicular distances to the data points. The least-squares procedure, however, is not without merit since it minimizes the variance of the error. The most common form of regression is a straight-line fit. In this case, we are trying to find the coefficients WA W and UB" that result in a least squares deviation from the straight line: y A + Bx. Actually, the values obtained for -An and W8 W are only estimates and are sometimes written AI and B' to remind us of this. When we calculate y from a value of x using these coefficients, it too may be written yl. Note that the coefficients derived from this regression are specifically designed to predict y from a knowledge of XI the values obtained for A and B are generally not the best ones to use for estimating x from y. The more data points we have in our sample, the closer our estimate of A and B will be to the expected values. If the data sample is fairly small, as it usually is, we should not be tempted to use more than two or three significant places for A and B. Even if we determined A and B with great accuracy, we should not conclude that our data are actually related by the regression y A + BX. The values obtained for A and B merely reflect our assumption, for the time being, that we can predict y from x using a straight line approximation.
=
2.
TRANSFORMATIONS
Often we can plot our data and see immediately that the assumption of a straight-line fit is not correct. Our intuition or experience may suggest a totally diffe~ent relationship between y and x such as (y-A)*(x+B)=l or y=k*c. Rather than invent a new regression for each case, it is common practice to transform the expression into one that has thi properties of a straight line. Consider the expression y=k*c , for example. By taking the logarithm of both sides, we obtain In y = In k + x*ln c. Since the logarithm of a constant is also a constant, we can re-
4
write the last expression as In y = A + Bx where A~ln k and B-ln c. Thus we have a dependent variable on the left that is linearly related to an independent variable on the right. The transform applied in this case is the logarithm, and instead of ~sing x and y in the regression formula, we input x and In y. The resulting coefficients must then be transformed back to obtain appropriate values for k and c. It should be noted in this example that the transformation produced a least squares regression of In y, rather than y. The transform you choose in other situations may be complex or simple, but the objective is the same--to create a linear relationship between the variables.
3.
MULTIPLE LINEAR REGRESSION
Many phenomena cannot be expressed in terms of a simple linear model. What happens, for example, if there are more than two variables? We need a new regression that does essentially the same thing, i.e., minimizes the sum of squared deviations. When the form of this regression is linear, it is called multiple linear regression. A common example of such an expression is t = A + Bx + Cy + Dz. There are few restrictions on what we use for y or z; x*x and x*x*x could be substituted for y and z in this example. This provides an easy way to fit data to a cubic equation. Where the input of our regression formula calls for x, we input x. Where it calls for y and z, we input x*x and x*x*x, respectively. When the coefficients are computed, we will obtain a best-fit curve for the expression y
Many complex equations can be fitted by transforming the data to an expression that has the form of a multiple linear regression. This process can even be used to develop a new class of models commonly referred to as probability distribution functions. An example of such a distribution is the familiar bellshaped curve, or normal distribution. Unfortunately, not all equations can be transformed by this technique and iterative methods must sometimes be employed to fit data to a curve. The modelS selected for this volume, however, do not require iterative solutions. sometimes an alternative to iterative methods is to divide the curve into two or more pieces, each of Which can be approximated by a separate regression equation. In order to join these curves together, we would like to specify a particular point through which one of the curves must pass. Several regressions included in this book provide a least squares fit through any
5
given point. Others can be developed as desired. The general technique consists of moving the origin (e,~) to the given point (h,k) • The Generalized 2nd Order or Generalized 3rd Order regression formula can then be used to force the curve through this new origin. You will have to solve for a constant term in the general equation by substituting the value of h for x and the value of k for y.
4.
DECOMPOSITION AND SUBSTITUTION
It is possible to express a generalized equation in several different forms by decomposing its coefficients or by substituting other variables. This should be tried before proceeding to develop a new regression formula. The following examples illustrate a few of the variations possible with models presented in this book. General Equation
y
=a
x
y
=a
b
y = a b
Alternate Form
b
y = l/{r xt)
X
y = r e sx
2 X cx
y = r e{X
a b X xc
y
y = a b X xc
y
y
y
=
=a
r
b
X
s
e
r(x/s)
y = r
bx
S)2/t
tx t
e xis
1x
y = s + r*sin -1
y = a + bx y
a
-
X
In y
x
r + sx
The first five examples illustrate decomposition (or modification) of the coefficients. In the first case, the coefficients are related by r=l/a and t--b. In the second case, r=a and s=ln b. The last three examples involve substituting different (or transformed) variables in the equation, e.g. the square-root of x can be substituted for x. These techniques can be applied to-
6
gether to greatly expand the range of equations that can be with just a few basic regression formulas.
5.
fit
SCALING
Most programmable calculators have a ten-digit register capacity. This limitation presents a problem when less significant digits are lost during calculation. The problem is particularly acute in dealing with large numbers. There are two techniques that can sometimes be used to retain greater accuracy in such cases. The first consists of subtracting a common value from each y value before entering the data. After the coefficients have been determined, this value is simply added to the equation for y. Alternatively, y can be divided by a constant amount and the final equation adjusted by using this constant as a multiplier. The same also holds true for values of x. If the values for x are large, either they can be divided by a constant or a constant amount can be subtracted from each. These techniques may be applied simultaneously to reduce both y and x. You must be sure to make appropriate modifications to the final regression equation, however, after the coefficients have been calculated in this manner.
6.
GOODNESS OF FIT
For peace of mind, we would like some assurance that the straight line resulting from our regression analysis is a reasonable approximation or "good fit.· There are a number of ways to evaluate how good the fit is, each with its own advantages and limitations. .Perhaps the most commonly used measure of goodness of fit is the coefficient of determination, RR. (The square-root of RR is called the correlation coefficient.) A useful property of the coefficient of determination is that it can be applied to any linear regression in order to select the best-fitting curve. This is true even when a transform is used to linearize the data. Another property of RR is that it ranges from ~ to 1; it is 1 when all of the data points fall exactly on a straight line, and it is 0 for values of x and y chosen at random. RR has a direct interpretation as well: it is the proportion of the total variation in y explained by the regression line. Thus an RR of 0.80 means that 80% of the observed variation in y can be attributed to variation in x; 20% of the variation in y is unexplained by the regression line. If the data are very "noisy· (i.e., contain significant random errors), an RR of 0.80 may rep-
7
resent a fairly good fit. It is possible to have a coefficient of determination near 1, however, and not have a good fit if the data have very little noise. Even with a high degree of correlation, we cannot infer that the data actually fit a particular curve without assuming something about the distribution of y values. Moreover, a high RR does not prove causality and does not guarantee that new data will necessarily fit the curve. The relationship between weight and age in children, for example, could not be used to accurately predict adult weight. Furthermore, losing weight is not likely to make you younger no matter how good the fit. Whenever RR is used to compare one model with another, it should be corrected to eliminate any bias due to the size of the data sample and the number of coefficients estimated. Appropriate corrections are included for most models in the book.
7.
SIGNIFICANCE
We should always select a model for our data on the basis of either theoretical or empirical knowledge. Whenever pOSSible, we should even design the data-collection procedures so that goodness of fit, lack of fit, and measurement errors can all be tested. Unfortunately, it is not always possible to control what data are collected or to understand the phenomena involved. In such cases, the model must always be suspect until tests of significance are applied to each of the variables. For regressions involving several terms, it is possible to determine the correlation between each pair of variables involved. If two of the variables are highly correlated, we Should consider redoing the regression and omitting one of them since it contributes little toward reducing the variance in y. If a particular variable exhibits little or no correlation to y, we should also consider omitting it from the expression and redoing the regression. It is usually the case that higher order curves with more variables will produce a better coefficient of determination. It is therefore prudent to determine if the improvement in the correlation coefficient is truly significant before opting to use a higher-order curve. There are a number of methods for testing the correlation coefficient, but one of the most commonly used tests of significance is the F-test. Such tests are beyond the scope of this book, however, and the interested reader should consult one of the more advanced texts listed under References.
8
8.
GETTING STARTED
We shall conclude with an example that illustrates how to fit data to a curve using the information in this book. Suppose the heights of a small group of adults were measured as follows: 53, 54,55,56,56,57,57,57,58,58,58,59,59, 61'l, 69, 62, and the frequency of each height summarized in a table. Frequency (y)
Height (x) 53 54 55 56 57 58 59 6" 62
in. in. in. in. in. in. in. in. in.
6.25% 6.25' 6.25% 12.50\ 18.75% 18.75% 12.50% 12.59% 6.25%
We would like to find a curve that fits these data and use it to estimate what percentage of the adult population have a height of 65 inches. The first step Is always to plot the data. Since height is the independent (or given) variable, it is plotted along the x axis, while frequency (or the unknown) is plotted along the y axis. The plot will look something like this:
HO % 3 18
2
12.
r---r--
6.2
e LIM: OJ I/')
We have the option at this point of scaling the data, if it is desirable. Since all of the x values are clustered between 53 and 62, we could simply subtract 50 from each, for example, and
9
remember to add 50 to the final regression equation. These values are not quite large enough to be troublesome, however, so we shall not bother to scale them. The next step is to examine the curve and delete any points appear to be extremes, or outliers. This should be done ~ith discretion so that the results are not biased by preconceptions of what the curve should look like. We should also consider how the curve might behave on either side of the data sample we have plotted; does it reach some limit or does it swing upward again at some point? Knowing or assuming these details will make the selection of a model easier. that
Now we look through the graphs for various equations and pick out ones that are similar to the plotted data. If we already have some idea of the type of curve that will fit the data, the selection will be easier. When it is difficult to find a curve that looks similar to our plotted data, we might replot the data with x values on the Y axis and y values plotted on the X axis to find a better match. In this particular case, it seems likely that the data belong to a normal distribution and replotting isn't necessary. Turning to the normal distribution, the first thing we find is a list of sU~Ration terms that are required. The first sum (RI6) is simply the total of all the x values. The second sum (R17) involves squaring each x value before adding them together. The third Sum requires taking the natural logarithm of each y value and then adding the logarithms together. When each of these summations has been calculated, the results should look like this: Rl6
~
Xi
R17
~
Xi
R21
2
5l4.~89
2
29424.990
n
9.989
RJe
~
In Yi
29.779
R3l
~
In Yi2
49.755
3 R49 - ! Xi
1688344.998
~
x4 i
97194852.geS
R46
!
xi*ln Yi
1189.588
R54
~
x 2i *In y.1
68268.885
R43
~
10
If suggested register assignments are used, the formulas in the text can be applied directly to calculate all of the terms required. In this particular example, the terms become: 620.000 32B6.447 71160.009 39.564 R09 8171892.""" -160.109 R11 5.625 R12 = -0.949 R13 -
Res Re6 = Re7 '" ReB
The coefficients of the normal-distribution calculated from the last three terms.
a b
equation
are
15.929 57.895 -29.586
c =
substituting these coefficients into the general equation gives us the best-fitting normal distribution for our data:
The coefficient of determination is found to be 0.748, meaning that approximately 75% of the observed variance in y is explained by x. For the limited data sample used, this probably represents a fairly good fit. If we were to try another curve with the hope of getting a better fit, it would be necessary to calculate the corrected coefficient of determination for comparison. The curve with the larger corrected value of RR would generally be considered the better-fitting curve. The regression curve we finally settle on should be plotted with our data to check the calculations and goodness of fit. We can value of y. can estimate is 65 inches expect about
now uSe our regression coefficients to estimate any By substituting 65 for x in the above expression, we the expected proportion of the adult population that in height. Based on the data sample used, we would 1.3% of the population to be 65 inches tall.
11
12
STRAIGHT LINE
Y ". a + bx
General Equation:
This is perhaps the most common equation used to fit data. It enjoys widespread use in general forecasting, biology, economics and engineering. It can be used any time y is proportional to x. The following formulas will estimate the coefficients of a linear equation that best fits the data when three or more points are given. X and y may be positive, negative, or equal to zero.
...
Rl6
I
RI?
~ x 2i
A29
RIB
1: Y i
R21 - n
xi
R19 -
w
2
Yi
I Xi *y i
where x and yare the values associated with each data point and n is the total number of points. Res -
RI?*R21 -
(RI6)2
RII •
{R17*R18 - RI6*R28)/R8S
Rl2 -
(R2~*R21
13
- RI6*R18)/R9S
The coefficients of the best-fit straight line are: a ;; Rll b = R12
The goodness of fit (coefficient of determination) is calculated from the following expression: Rll*R18 + R12*R29 - (R18)2/R2l RR
When RR is to be used for compa'rison wi th other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R2l - 1) RRcorrected -
1-
(R2l -
2)
Example:
x ..
10
20
39
49
50
Y =
28
32
46
59
72
R16 150.99 R17 = 5500.89 R18 .2'37.09 R19 12589.9O R20 8260.09
R21 R"5 Rll R12 RR
14
::I
=
5.99 5090.99 12.90 1.15 9.976
STRAIGHT LINE THROUGH THE ORIGIN
General Equation:
y = bx
There are many situations where the relationship between x and y is such that y must be zero when x is zero. If blood pressure is zero, for example, then the volume of blood flow is zero. voltage and current in simple circuits exhibit a similar relationship. This equation can be used to fit a straight line to these kinds of data. It is used any time y is directly proportional to x. The following formulas will estimate the coefficient of a linear equation that best fits the data when two or more points are given. X and y may be positive, negative, or equal to zero.
R19
where x and yare the values associated with each data point.
15
R12 - R29/R17
The coefficient of the best-fit straight line is: b :: R12
Example: 11
17
23
29
y ::
15
23
31
39
R17
17S9.1i19 iSS.IiIS 3236.00 2499.90 1.35
RiS R19 R20 R12
::I
=
16
STRAIGHT LINE THROUGH A GIVEN POINT
'i .. a + bx
General Equation:
This is a variation of the linear equation. It is used to fit data to a straight line which passes through the point h,k. It can be used whenever the value of one pOint is known or assumed to be correct, e.g., surveying from a known tie point or benchmark. The following formulas will estimate the coefficients of a linear equation that best fits the data when three or more points are given. X and y may be positive, negative, or equal to zero.
R16 = Rl7 Rl8
=
2 Yi
I: Xi
Rl9 -
I:
1: x2
R29 -
I Xi *y i
i
1: Y i
where x and yare the values associated with each data point.
17
2 h*k*R16 - h *R18 - k*R17 + h*R2B Rll 2 2*h*R16 - R17 - h *R21
The coefficients of the best-fit
st~aight
line are:
a = Rll b = (k - Rll)/h
Example:
=
lee
y =
140
x
300 230
310
500 400
480
Find the best fitting straight line that passes
th~ough
(30e, 3le) •
RI6 150B.00 Rl7 550000.00 RI8 = 1560.00 R19 = 559000.ge
R20 .. 553000.00 Ril 55.00 b = 9.85
18
the point
ALTERNATIVE STRAIGHT LINE THROUGH A GIVEN POINT
General Equation:
y -
a + bx
These equations will find the best fitting straight line that passes through the point h,k. It produces the same coefficients as the previous method but the technique is somewhat different. It may be used to join two curves at a common point, e.g. a straight line and an arc. The following formulas will estimate the coefficients of a linear equation that best fits the data when three or more points are given. Note that h is subtracted from each x value and k is subtracted from each y value as the sums are calculated. X and y may be positive, negative, or equal to zero.
where x and yare the values associated with each data point and hand k are the coordinates of the given point.
19
RI2 .. R2B/RI7 Rll
~
k - h*R12
The coefficients of the best-fit straight line are: a = Rll b
R12
Example:
x
=
y ,.
140
230
480
Find the best fitting straight line that passes through the point (3B0,3HJ).
RI7 HJB'H'B. "IiJ RI8 10.99 RIg = 723913. BB
R20 Rll R12
8SIHJ0, "13 55.00 0.85
ISOTONIC LINEAR REGRESSION
General Equation:
'1 = a
+ bx
This is a variation of the linear equation based on minimizing the sum of squared deviations as measured perpendicular to the regression line. It corresponds most nearly to a free-hand line drawn through the points. Isotonic regression can be used when there ~re errors in both x andy, e.g., surveying through a number of points that lie on a straight line. The following formulas will estimate the coefficients of a linear equation that best fits these data when three or more points are given. X and '1 may be positive, negative, or equal to zero.
R16 -
t Xi
R19
R17
2 1: Xi
R20
;;
= 1:
y~1
1: xi*Yi
R21 '" n
RIB = 1: Yi
where x andy are the values associated with each data point and n is the total number of points.
21
(R17 - R19)*R2l + (RIB)2 -
{R16}2
R05 2*(R2g*R21 - R16*R18}
~ RI:l5 2
Rll
-Re5 ±
R12 -
(RIB - Rll*R16}/R21
+ 1
There are two possIble lines which satIsfy the regression equation; one is perpendicular to the other. The correct solution for most applications is given by the value of Rll that minimizes the expression: I (R12*R16 - R20) *Rll - R12*RIB I
The coefficients of the best-fit straight line are: a .. R12 Rll
b
Example:
x
=
11313
21313
31313
41313
533
y =
1413
230
3113
41313
483 R2l ". ReS Rll R12 -
R16 15"3 R17 = 55313139 RIB 1560 R19 5591HH' R20 = 553030
5 0.163 -1.176 or 13.B513
664.B8 or 56.96
Since the expression I (R12*R16 - R2g) *Rll - R12*RIBI is minimum (4B6,35l versus 1,559,8513) when Rll equals g.B5, the coefficients of the best-fit line are therefore: a = 57.1313 and b = g.B5.
22
RECIPROCAL OF STRAIGHT LINE 1
y
General Equation:
~
a + bx This equation is the reciprocal of a straight line. It is used when x is inversely proportional to y, e.g., exposure time versus brightness in photography. The following formulas will estimate the coefficients of a reciprocal equation that best fits the data when three or more points are given. Y must not be equal to zero (any small number may be substituted for y when it is) •
2
Rl6
t xi
R25 '"
Rl7
E 2 xi
R34 = 1: xi /y i
E l/Yi
R21 .. n
R24 = 1: l/y i
where x and yare the values associated with each data point and n is the total number of points. Ras RII •
R11*R2l -
(R16)2
(RI7*R24 - RI6*R34)/RaS
23
R12 :
(R21*R34 - R16*R24)/R05
The coefficients of the best-fit reciprocal curve are: a .:: Rll b = R12
The goodness of fit (coefficient of determination) is calculated from the following expression: Rll*R24 + R12*R34 -
(R24)2/R21
RR = R25 -
(R24)2/R21
when RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R2l - 1) 1 -
RRcorrected
(R21 -
2)
Example: x
= 5.1
y
RI6 H17 R21 R24 R25
'" .::
= .. =
2."
3.9
4.0
S.9
3.1
2.2
1.7
1.4
15.000 55.000 5. oI:!!" 2.276 1.205
R34 '" 8.129 R05 50.""9 Rll = 0.965 R12 .. 9.139 RR 1.000
24
RECIPROCAL OF STRAIGHT LINE THROUGH A GIVEN POINT I
General Equation:
y a + bx
This equation is the reciprocal of a straight line forced to pass through the point h,k. It can be used to join two curves through a common point. The following formulas will estimate the coefficients of the reciprocal curve when three or more points are given. Subtract h from each x value and l/k from l/y when calculating the sums. Y must not be equal to zero (any small number ~ay be substituted for y when it 1s).
Rl6
R25
t (xCh)
Rl7 = 1: (xC h ) 2
R24 -
1:
R34
(l/Yi-l/k)
where x and yare the values associated with each data point, h and k are the coordinates of the given point, and n is the total number of points.
25
R12
R34/R17
Rll =
(11k) - h*R12
The coefficients of the best-fit reciprocal curve are:
a
= Rll
b
R12
Example: x '" y
5.1
2.~
3.9
4.0
5."
3.1
2.2
1.7
1.4
Find the best fitting reciprocal curve that passes through po i n t ( 3 , 2) .
RIG = R17 R24 R25
a.""" 1"."0" -0.224
R34
=
Rll .. R12
=
0.180
26
1.302 ".109 ".130
the
COMBINED L[NEAR AND RECIPROCAL
General Equation:
'I
= a + bx + clx
This equation combines a straight line with a reciprocal curve. The following formulas will estimate the coefficients of such an equation when four or more points are given. X must not be equal to zero (any small number may be substi tuted for x when it is).
R16
=
:!:
xl
R22
1:
llx i
R17
=
1:
xi
:2
R23
!
l/x~1
R35 '"
l: 'I ilxl
RIB R19
l: 'I i :::
2
R2l - n
~ 'I i
where x and 'I are the values associated with each data point and n is the total number of points.
27
R~5
..
R96
R21*R35 - R18*R22
R97 ::: R98 R~9
R17*R21 - (RI6)2
:::
(R21)2 - Rl6*R22 R29*R21 - R16*R18 R21*R23 -
(R22)2
The following terms must now be calculated in order to tain the coefficients of the equation:
ob-
Rl3 = (R95*R96 - R97*Re8)/(RS5*R99 - R97 2 ) Rl2 = (R9S - R97*RI3)/R9S RII
(R18 - R12*R16 - RI3*R22)/R21
The coefficients of the best-fit curve are:
a = Rll b
Rl2
c
= R13
The goodness of fit (coefficient of determination) is calculated from the following expression:
RR
=
RII*RI8 + R12*R2e + R13*R35 - (RI8)2/R21 RI9 - (RI8)2/ R21
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
28
(1 - RR)*(R21 - 1) 1 -
RRcorrected
(R21 - 3)
Example: x = ':I
=
R16 R17 RIB R19 R29 R21 R22 R23 R35
'" =
= -
• = =
5
H.I
15
29
25
21
12
15
21
28
ReS Re6 R87 R98Re9 Rll R12 R13 RR
75.99" 1375.990 97.090 2835.999 1579.899 5.U8 9.457 9.9585 8.571:1
29
= ...
..
= ..
1259.980 -1. 447 -9.259 575.888 9.884 -23.628 1.781 178.559 1. ''''9
HYPERBOLA
y = a + b/x
General Equation:
The following formulas can be used to estimate the coefficients of a hyperbolic equation that best fits the data when three or more points are given. X must not be equal to zero (any small number may be substitued for x when it is).
a)0
b<0
~)0
b)e
R23
1:
l/X~
Yi
R35
1;
Yi/xi
l/x i
R21
RIS =
~
Yi
R19
'"
1:
R22
=
1;
2
=
n
where x and yare the values associated with each data point and n is the total number of points.
R0S =
R21*R23 -
(R22)2
Rll = (RIS*R23 - R22*R35)/R05 R12
(R21*R35 - RIS*R22}/R05
30
The coefficients of the best-fit hyperbola are: a =- Rll R12
b
The goodness of fit (coefficient of determination) is calculated from the following expression: Rll*R18 + R12*R35 - (R18)2/R2l RR • RI9 - (RI8)2/R21 When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R2l - 1) 1 -
RRcorrected
(R2I - 2)
Example:
x "" y
=
5.1
2.0
3."
4."
5.9
3.1
2.2
1.7
1.4
RI8 13.5"9 R19 - 43.319 R2l 5.9"9 2.289 R22 = R23 = 1. 46"
R35 RIil5 Rll R12 RR
31
= ~
...
=
8.1)88 2.194 ".613
4.579 9.992
HYPERBOLA THROUGH A GIVEN POINT
y = a + b/x
General Equation:
The following formulas can be used to estimate the coefficients of a hyperbolic equation that passes through a known point h,k when three or more points are given. Subtract k from each y value and l/h from l/x when calculating the sums. X must not be equal to zero (any small number may be substituted for x when it is) •
\
\~ h k
a
RIg
=:
R22 -
~
(l/xi -l/h)
where x and yare the values associated with each data point and hand k are the coordinates of the given point.
32
R12
R35/R23
Rll
k - R12/h
The coefficients of the best-fit hyperbola are: a .. Rll Rl2
b
Example: x -
1.9
2.£1
3.9
4.9
5.0
y -
5.1
3.1
2.2
1.7
1.4
Find the best fitting hyperbola through the point (3,2). RIB • RIg R22 = R23 ~
R35 ..
3.599 11.319 9.617
Rll R12
9.497
33
~
2.355 9.429
4.739
RECIPROCAL OF A HYPERBOLA x
General Equation:
y ax + b
This equation is the reciprocal of a hyperbola. The following formulas will estimate coefficients of the equation that best fits the data when three or more points are given. Neither x nor y can be equal to zero (any small number may be substituted when they are) •
R22 -
I:
l/x i
R2S
t l/y~
R23
!:
l/x~
R26
~l/(Xi*Yi)
R24
~
l/y i
R21
1
=n
where x and yare the values associated with each data point and n is the total number of points. Res
R21*R23
(R22) 2
Rll = (R23*R24
R22*R26) /R"S
R12 = (R21 *R26
R22*R24) /R"S
34
The coefficients of the best-fit curve are:
a ;: Rll b
R12
IS
The goodness of fit (coefficient of determination) Is calculated from the following expression: Rl1*R24 + R12*R26 - (R24)2/ R21 RR • R25 -
(R24)2/R21
When RR is to be used foc comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R21 - 1) RR corrected -
1-
(R21 -
2)
Example:
x y
=
R21 R22 R23 R24 R25
2.8
• =
.. •
2.0
3.0
4.0
5.0
3.2
4.6
5.9
7.2
R26 Re5 • Rll = R12 RR =
5.00"
2.283 1.464 1.195 0.328
35
9.656 2.194 9.120 0.262 0.830
SECOND ORDER HYPERBOLA
y = a + b/x + c/x
General Equation:
2
This is a second degree polynomial where l/x has been substituted for x. It is similar to the hyperbola but generally exhibits steeper descent along the y-axis and levels off faster on the x-axis. The following formulas will estimate the coefficients of such an equation when four or more points are given. X must not be equal to zero (any small number may be substituted when it is).
a>0 b (0 c <0
a>0 b>0 .::>0
R18 .. 1: Y i
R38
::
2
1: Y i / (x ) i
R19
1:
y~1
R4l
1:
l/x:
R22
~
l/x i
R44 •
1:
l/x
R23
1:
l/xf
R21 ,. n
1
where x and yare the values associated with each data point and n is the total number of points.
36
(R22) 2
Res
R21*R23 -
R96
R21*R38 - R18*R23
R07 - R21*R41 - R22*R23 Res .. R21*R35 - RIS*R22 Re9
R21*R44 -
(R23)2
The following terms must now be calculated in order to obtain the coefficients of the equation: Rl3 -
2 (R9S*R96 - R97*R9S)/(R9S*R99 - R97 )
Rl2 = (ReS - R97*RI3)/R95 R1I
(RIB - R13*R23 - R12*R22)/R21
The coefficients of the best-fit hyperbola are: a
=:
Rll
b
z
Rl2
c
=
Rl3
The goodness of fit (coefficient of determination) is calculated from the following expression:
Rll*R18 + R12*R35 + R13*R38 -
(Rl8)2/R2l
RR
Rl9 -
(RI8)2/R21
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
37
(1 - RR)*(R21 - 1) 1 -
RRcorrected
(R21 - 3)
Example: x ..
1.0
2.0
3.0
4.9
5.9
y
2.8
3.2
4.6
5.9
7.2
2.104 R"5 = R96 -10.848 R07 .. 2.586 RIIl8 == -9.873 R09 3.260 Rll :z 11.153 R12 -24.238 R13 15.904 RR = 0.986
RIB = 23.790 R19 125.899 R21 5.000 R22 2.283 R23 1. 464 R35 .. 8.848 R38 = 4.768 R41 .. 1.186 R44 1. 080
38
PARABOLA
Y
General Equation:
::::I
a + bx + cx
2
The parabolic equation belongs to a family of curves known as polynomials. This particular equation is a second degree polynomial with many applications in the physical sciences, e.g., describing the motion of an object under the influence of gravity or acceleration. The following formulas will estimate the coefficients of such an equation when four or more points are given. X and y may be positive, negative, or equal to zero.
Rl6
.
r xi
Rl7
=
t xi
3 R40 = 1: xi
RIa
a
! Yi
R43
2
2
Rl9
!
R20
! Xi·Y i
:E
R36
::
2 ..
xi Yi
T _ X.4 1
R21 : n
Yi
where x and yare the values associated with each data point and n is the total number of points.
39
R9S
a
Rl7*R2l - (Rl6)2
R96 '"
R2l*R36 - R17*RlB
R07
R2l*R40 - R16*Rl7
R08
..
R29*R2l - R16*R18
RIJ9
::I
R21*R43 - (R17)2
The following terms must now be calculated in order to tain the coefficients of the equation:
ob-
Rl2 = (R9S - R07*R13)/R05 Ril
=
(RIB - R13*R17 - Rl2*R16)/R21
The coefficients of the best-fit parabola are: a = Rll
b .. R12 c .. R13
The goodness of fit (coefficient of determination) is calculated from the following expression:
Rll*R18 + R12*R21J + R13*R36 - (RI8)2/ R21 RR ..
R19 - (RIB)2/R2l When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
49
RR
corrected
-
1
-
(1
- RR) * (R21 (R21 - 3)
1)
Example: x ;
1.0
2.0
3.0
4."
5."
Y
2.B
3.2
4.6
5.9
7.2
50.00 R95 R06 :: 353.50" R07 :: 300.""9 57.509 ReB RI/J9 1870.0"0 Rll = 2.14" R12 = 0.421 0.121 R13 RR 0.991
15.""0 R16 • R17 55."0" RI8 23.70" R19 125.899 R29 -= 82.6"" R21 5.1/J"0 R36 = 331.40" R40 - 225. R43 979.00"
"I'"
41
PARABOLA THROUGH THE ORIGIN
y :: ax + bx 2
General Equation:
This equation represents a parabolic curve that is constrained to pass through the origin. It is especially useful for estimating the relationship between elapsed time and counter readings on tape recorders and video recorders. The following formulas will estimate coefficients of the equation that best fits the data when three or more points are given. X and y may be positive, negative, or zero •
• >0 b<0
/
/
/ ..
a>0 b>9
1: xi
2
R4" ,.
1:
3 xi
R28 =
1: xi *y i
R43 ::
l:
xi
R36 .-
l:
Rl7
4
2.
xi Yi
where x and yare the values associated with each data point.
42
R9S
R17*R43 -
(R40)2
R11 •
(R2a*R43 - R36*R413)/R05
R12 -
(R17*R36 - R20*R49)/R95
The coefficients of the best-fit parabola are: a ;;;: Rll
b
R12
Example:
x =
1.0
2.0
3.9
4.0
5.0
y
49
84
113
138
161
R17 R213 R36 R41!1 =
55.99 1913.99 7635.90 225.130
R43 R05 Rll R12
43
979.09
3220.013
=
48.12 -3.26
PARABOLA THROUGH A GIVEN POINT
a + bx + cx 2
y =
General Equation:
This equation represents a parabolic curve that is constrained to pass through the point h,k. The following formulas will estimate coefficients of the equatIon that best fits the data when four or more points are given. Subtract h from each x value and k from each y value when calculating the sums. X and y may be positive, negative, or zero.
Rl7
=
1: (Xi -h)
R2e
-
~
(xi-h)*(Yi- k )
1:
(x -h) 2. (y -k)
R36
i
2
R41J
L
(Xi-h)
3
R43 = E (X - hl 4 i
i
where x and yare the values associated with each data point and hand k are the coordinates of the given point.
44
R05
R17*R43 -
(R40)2
Rll = (R20*R43 - R36*R40)/R05 R12 -
(R17*R36 - R20*R4B)/R05
The coefficients of the best-fit parabola are: a
~
k -
(R11 - h*R12)*h
b = R11 - 2*h*R12
c - R12
Example:
x ,.
y
2.8
2.0
3.0
4.0
5.0
3.2
4.6
5.9
7.2
Find the best fitting parabola through the point (4.5,6.5).
R17 R20 R36 = R40 R43
R05 = Rll R12 a .. b
21. 250 24.700 -70.200 -61. 875 194.313
45
300.625 1.516 0.122 2.139 ".422
POWER
General Equation: This equation is commonly referred to as the learning curve. It describes trends which are geometric in nature and is often applied when y increases at a much faster (geometric) rate than x. The following formulas will estimate the coefficients of a power curve that best fits the data when three or more points are given. X and y must be positive numbers greater than zero.
/
//a>
/
a
b>.
,/
R28
~
In xi
R31
R29
1:
(In xi) 2
R32 '"
R30 ..
l:
In Yi
R21 - n
s
1:(1n Yi)2 Un xi·ln Yi
where x and yare the values associated with each data point and n is the total number of points. R05
R21*R29
Rll
(R29*R30
R12
~
(R28)2 R2S·R32) IRe5
{R21*R32 - R28*R33)/R05
46
The coefficients of the best-fit power curve are:
b
R12
The goodness of fit (coefficient of determination) is calculated from the following expression: Rll*R30 + R12*R32 -
(R30)2/ R2l
RR
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R2l - 1) 1 -
RRcorrected
(R2l -
2)
Example:
x
=
1.0
2.9
3.0
4.0
5.0
y
=
2.8
3.2
4.6
5.9
7.2
. .
R2l R28 R29 R3el = R31 R32
RfIlS = Rll R12 = a RR
5.00~
4.787 6.200 7.468 11.789 8.121
2
47
8.077 0.919 0. 6 f.H!l 2.506
0.917
MODIFIED POWER
General Equation: This equation is a variation of the power curve. It also describes trends which are geometric in nature and is applied when the ratio between successive terms in a series is constant. The follwing formulas will estimate the coefficients of a modified power curve that best fits the data when three or more points are given. Y must be a positive number greater than zero.
R16 =
!
R17 =
! xi
R39
=
xi 2
!
In Yi
R31
!
(In Yi) 2
R46 -
l:
xi * In Yl
R21
=n
where x and yare the values associated with each data point and n is the total number of points. Re5 -
R17*R21 - (R16)2
Rll = (Rl7*R38 - R16*R46)/R95 R12 = (R21*R46 - R16*R39)/R95
4B
The coefficients of the best-fit modified power curve are:
a '"
The goodness of fit (coefficient of determination) lated from the following expression:
is calcu-
RII*R3~ + R12*R46 - (R3~)2/R21 RR ..
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R21 - 1) 1 -
RRcorrected
(R2l - 2)
Example:
x '" y =
RI6 R17 R21 R3((l R31 '" R46
2.8
3.2
3.3
4.0
5.3
4.6
5.9
7.2
ReS Rll = R12 '" a '"
15.a~~
55.~1iI" 5.~1iI~
7.468 11.789 24.904
b
RR =
49
50.1i10~
0.743 0.250 2.103 1. 284 0.984
ROOT
y
General 8quation:
=a
bl / X
This equation is a variation of the modified power curve. It fits the xth root of a constant to the dependent variable, y. The following formulas will estimate the coefficients of this equation when three or more points are given. Y must be a positive number greater than zero and x must not be equal to zero (any small number may be substituted for x when it is).
~>0
b)l
2
R22
l:
l/x 1
R31 -
l:
(In
R23
L
l/X~
R47
1:
(In Y i) /x i
R3ril =
l:
In
R21
yi
y
1)
=n
where x and yare the values associated with each data point and n is the total number of points. Rril5 =
R23*R21 -
(R22)2
Rll = (R23*R30 - R22*R47)/R05 R12 -
(R21*R47 - R22*R30)/Rril5
5ri)
The coefficients of the best-fit curve are:
The goodness of fit (coefficient of determination) lated from the following expression: R11*R3g + R12*R47 -----.----
RR
R3l -
is calcu-
(R39)2/R21
(R39)2/ R21
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R21 - 1) RRcorrected -
1-
(R21 - 2)
Example:
x =
1.0
y =
2.8
R22 ;: R23 "" R21 ,. R30 R3l = R47
3.2
3.0
4.r.l
5.0
4.6
5.9
7.2
2.283
Rr.l5 Rll R12
1. 464
5.00" 7.468 11.789 2.958
a b ;: RR =
Sl
2.104 1. 984
-1. 974 7.271 0.342 0.763
SUPER GEOMETRIC
General Equation: This equation is similar to the power curve but changes much more rapidly. The following formulas will estimate the coefficients of this curve when three or more points are given. X and y must be positive numbers greater than zero.
R39 '"
2;
In Yi
R49
t
(Xi"'ln Xi)2
R31
l:
(In Yi,2
R59
1:
xi * In xi*ln Yi
R48
:!:
xi * In xi
R2l " n
where x and yare the values associated with each data point and n is the total number of points. R9S '"
R21*R49 - (R48,2
Rll
(R30*R49 - R48*RS9)/RBS
R12
(R21*R59 - R30*R48)/R05
52
The coefficients of the best-fit curve are: a = e
Rll
b - R12 The goodness of fit (coefficient of determination) is calculated from the following expression: Rll*R39 + R12*R50 -
(R30)2/ R2l
RR R31 -
(R30)2/R21
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R2l - 1) 1 -
RRcorrected
(R21 - 2)
Example: x '"
1.0
4.0
5.'"
y
2.a 3.'" --------- -- -
2.8
5.9
7.2
::a
..
R21 R30 = R3l = R48 = R49 = RS0 ;:
3.2
4.6
5.000 7.468 11. 789 18.274 108.291 32.37'"
----
R0S = 2"'7.496 1.047 Rll R12 0.122 a = 2.848 RR ,. 9.977
53
MODIFIED GEOMETRIC
General Equation: This equation is another variatIon of the modified power curve. The following formulas will estimate the coefficients of this curve when three or more points are gIven. X and y must be positive numbers greater than zero.
b>e
b<e
R30
E In Yi
R3l =
!:
(In ¥i)
2
R48
R2l
=n
where x and ¥ are the values associated with each data point and n is the total number of poInts. R95 = Rll R12
R21*R53 - (R4B)2 (R30*R53 - R48*R58}/R05
= (R2l*RS8
54
- R3S*R4B)/R95
The coefficients of the best-fit curve are: a '" e
Rll
b • R12
The goodness of fit (coefficient of determination) is calculated from the following expression: Rll*R3a + R12*R58 -
RR
(R30)2/ R21
os
When RR is to be used for comparison with other regression curves, it ~hould be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R21 -
1)
1 -
RRcorrected
(R21 - 2)
Example: x =
1.0
2.0
3.0
4.9
5.0
y =
2.8
3.2
4.6
5.9
7.2
R21 R30 R31 R48 R53 R58
~ ~
5.000 7.468 II. 789 18.274 0.478 2.213
R05
Rll R12
a • b
RR
55
=
=
0.482 1.065 I. 552 2.90" I. 552 0.365
EXPONENTIAL
Ceneral Equation:
y
= a e bx
This equation is used to model many natural phenomenona including: chance or random failures, the time between failures in systems with many independent components, radial errors in centering and bombing, the distribution of lifetimes, and radioactive decay. It is essentially the same as the modified power curve. The following formulas will estimate the coefficients of an exponential curve that best fits the data when three or more points are given. Y must be a positive number greater than zero.
Rl6
=
I:
R17
=
~
R30 =
xi
R31
~' x.2
R46
l;
1
In Yi
-
1:
(In y.) 2
1:
xi *1n Yi
1
R21 "" n
where x and yare the values associated with each data point and n is the total number of points. R05 =
R17*R21 -
(R16)2
RII = (RI7*R30 - RI6*R46)/R05
56
R12
=
(R21*R46 - R16*R3S)/R05
The coefficients of the best-fit exponential curve are: a ..
R12
b
The goodness of fit (coefficient of determination) lated from the following expression: R11*R30 + R46*R12 -
is calcu-
(R30}2/R21
RR = R31 -
(R30)2/ R21
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R2l - 1) 1 -
RRcorrected
(R21 - 2)
Example:
x =
1.0
l.0
3."
4.0
5.0
y
2.8
3.2
4.6
5.9
7.2
Rl6 '" R17 '"' R21 = R30 R31 R46
R05 ~ Rll R12 a = RR
15.000 55.000 5.000 7.468 11. 789 24.994
57
50.000 0.743 0.250 2.103 0.984
EXPONENTIAL
MODI~IED
General Equation: This is a variation of the exponential curve and is ly the same as the root curve. The following formulas mate the coefficients of a modified exponential curve fits the data when three or more points are given. Y positive number greater than zero and x must not be zero.
\
essentialwill estithat best must be a equal to
a>0 b<0
.. >0 b>9
=
l":
(In Yi) 2
~
(In Yi1/xi
R22 ,..
1: l/x
R23
l: l/x.
R47
R30 ,..
r In Yi
R2I ;: n
R3l
i 2 1
where x and yare the values associated with each data point and n is the total number of points. R05'"
R23*R21 -
(R22)2
RII -
(R23*R30 - R22*R47)/R05
R12
(R2l*R47
58
R22*R30l/R05
The coefficients of the best-fit curve are:
a
= e Rll
b
R12
The goodness of fit (coefficient of determination) lated from the following expression: R11*R30 + R12*R47 -
RR
is calcu-
(R39)2/ R21
=:
R31 -
(R30)2/ R21
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R21 -
1)
1 -
RRcorrected
(R21 -
2)
Example:
lC
=
y
=:
R21 '"' R22 = R23 '"' R30 R3l R47 ::
2.B
2.0
3.0
4.0
5.0
3.2
4.6
5.9
7.2
5. 'H) 0 2.283 1. 464 7.468 11 .789 2.958
59
RI!I5 Rll R12 a
::
RR
m
2.U4 1.984 -1.974 7.271 "'.763
LOGARITHMIC
General Equation:
a + b*ln x
y
This equation represents a logarithmic curve. It describes trends where y increases at a much slower rate than x. The following formulas will estimate the coefficients of a logarithmic curve that best fits the data when three or more points are given. X must be a positive number greater than zero.
\
R2B
b)B
=
l:
Rl9
In xi
R29
:!: (In
R18
l:
xi)
2
R5l R2l
Yi
.::
1;
Y~
I Yi *In
=
x·1
n
where x and yare the values associated with each data point and n is the total number of points. R05.::
R21*R29 -
(R28)2
Rll
(R18*R29 - R28*R5l)/R05
R12
(R2l*R5l - R18*R28)/R05
The coefficients of the best-fit logarithmic curve are:
a : Rll b • R12 The goodness of fit (coefficient of determination) is calculated from the following expression: RIl*R18 + R12*R51 -
(R18)2/R21
RR"" Rl9 -
(RI8)2/ R21
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R21 - 1) RRcorrected
a
1-
(R21 - 2)
Example:
y
R21 R18 Rl9 ::: R28 R29 =
I.e
2.0
3.e
4.0
5.0
2.8
3.2
4.6
5.9
7.2
R51 '" R05 Rll R12 .. RR
5.""'0 23.7ge 125.890 4.787 6.200
61
27.939 8.077 2.164 2.699 0.863
RECIPROCAL OF LOGARITHMIC 1
General Equation:
y
z
a + b*ln x This equation is the reciprocal of the logarithmic curve. The following formulas will estimate the coefficients of such a curve when three or more points are given. X must be a positive number greater than zero. Y must not be equal to zero.
b<~
b}0
(In Xi) 2
R29 =
1:
l/Y~
R52
t (In xi)/Y i
In xi
R2l = n
R24 -
t l/Yi
R25
:a
t
R28 -
t
where x and yare the values associated with each data point and n is the total number of points. Re5 =
R21*R29 - (R28)2
Rll = (R24*R29 - R28*R52)/R05 R12
c
(R21*R52 - R24*R28)/R95
62
The coefficients of the best-fit curve are:
a = Rll Rl2
b
The goodness of fit (coefficient of determination) lated from the following expression: R11*R24 + R12*R52 -
is calcu-
{R24)2/ R2l
---------.----------
RR
R25 -
(R24)2/ R21
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
1 -
RRcorrected
(1 - RR)*(R21 - 1) ----.-----(R21 - 2)
Example:
3.e
x ==
y =
R21 R24 R25 R28 R29
2.8
3.2
4.6
5.9
7.2
R52 R135 Rll R12 RR
5.~09
1.195 9.320 4.787 6.2013
63
;; ;;
9.914 8.077 0.376 -13.143 13.950
HOERL FUNCTION
General Equation: This is a generalized form of Hoerl's equation. The following formulas can be used to estimate the coefficients of this equation when four or more points are given. Both x and y must be positive numbers greater than zero (any small number may be substituted for zero).
b
Rl6 = I Xi
R31
2
Rl7 R28
I Xi
-
1:
In Xi
1:
(In Yi) 2
R32
2;
In x.*ln Yi 1
R46
I:
xi*ln Yi
};xi*ln Xl
~
R29 '"
I(ln Xi)2
R48 =
R39
I In y.
R2l
1
=n
where X and yare the values associated with each data point and n is the total number of points.
64
(R16)2
RB5
R17*R21 -
RB6
R21*R32 - R2B*R3B
R37
R21*R4B - R16*R2B
RB8
R21*R46 - R16*R30
RB9 =
R21*R29 -
(R28)2
The following terms must now be calculated in order to tain the coefficients of the equation: R13
(RB5*R~6 - RB7*R0B)/(RB5*R09 - R97 2 )
R12 -
(R3B -
Rll
(R30 - R13*R28 - R12*R16)/R21
ob-
R07*R13)/R05
The coefficients of the best-fit equation are:
a = e
Rll
c = R13 The goodness of fit (coefficient of determination) lated from the following expression: Rll*R30 + R12*R46 + Rl3*R32 -
is calcu-
(R39)2/ R21
RR = R3l -
(R3B)2/R21
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
65
RR
corrected
-
1
-
(I
-
RR)*(R21 (R21
-
-
1)
3)
Example:
IC
:::
Y
R16 R17 R21 R28 R29 R30 R31 R32 R46 R48 RB5
,. :::
= :::
~
1.0
2.0
3.0
4.0
5.0
2.8
3.2
4.6
5.9
7.2
R06 .. R07 R08 Ra9 ::: Rll • R12 R13
15.000 55.090 5.099 4.781 6.29B 7.468 11.789 8.121 24.904 18.274 50.999
a b '"'
RR
66
4.859 19.560 12.504 8.011 0.723 9.288 -9.096 2.060 1.333 9.985
MODIFIED HOERL FUNCTION
General Equation: This is a modified form of Hoarl's equation. The following formulas can be used to estimate the coefficients of this equation when four or more points are given. Both x and y must be positive numbers greater than zero (any small number may be substituted for zero).
~ c<0
~
b>l
c>0
/z _ _ _ _ _ _ _ -. . .:bs. .: >~l .::<13
R22 =
:E
l/xi 2
R23 = !: l/x .
1
R28
~
In xi
R29
~
(In xi)
R30
!: In
2
~
R32 =
rin Xi*lo Yi
R45
2: (In
R47
l:
R2l
Yi
(In y i) 2
R31 =
2
x.) /x.1 1
(In y.) /x. 1
1
n
where x and yare the values associated with each data point and n is the total number of points.
67
R0S
:::I
R21*R23 -
(R22) 2
RIJ6 =
R21*R32 - R28*R30
Rill 7 '"
R21*R45 - R22*R28
R0B -
R21*R47 - R22*R39
R09
R21*R29 -
(R28) 2
The following terms must now be calculated in order to tain the coefficients of the equation: R13
(R05*RIIl6 - R97*R08)/(R05*R09 - R1Il72)
Rl2
{R0B - R07*RI3)/RIIl5
RII
(R311l - R13*R28 - RI2*R22)/R21
ob-
The coefficients of the best-fit equation are: a b
a
e Rll
e
Rl2
c = R13
The goodness of fit (coefficient of determination) is calculated from the following expression: RII*R311l + R12*R47 + R13*R32 -
(R31/l)2/R21
RR
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
68
RRcorrected
;:
1
-
(1 - RR)*(R21 (R21
-
-
1)
3)
Example:
=
1.0
2.0
3.e
4.0
5.e
y =
2.8
3.2
4.6
5.9
7.2
x
R21 R22 R23 R28 R29 R39 R31 R32 R45 R47 R0S
'"
-
= ~
:: =
R96 = R07 Re8 R09 Rll R12 :: R13 a
5.0"" 2.283 1.464 4.787 6.20" 7.468 11. 789 8.121 1.381 2.958 2.104
b
RR -
69
4.859 -4.925 -2.259 8.077 -0.576 1.69" 1.398 ".562 4.954 9.996
NORMAL DISTRIBUTION
General Equation:
y
=a
e( X-b)2/ c
The normal or Gaussian distribution is often used to describe the expected frequency of some characteristic in a large population, e.g., the height of adult males. It applies to many natural and biological phenomena and is particularly appropriate when the data results from many additive factors. The following formulas will estimate the coefficients of a normal distribution when four or more points are given. Y must be a positive number greater than zero.
R16
-
l:
xi
R17
=
l:
x~1
R3~
-
E In Yi
R3l
=
::!:
In Yi
l:
xi3
R49
2
where x and yare the values associated with each data point and n is the total number of points.
70
(Rl6)2
Re5
R17*R21 -
R06
R21*R54 - R17*R39
R07
R21 *R4B - Rl6*Rl7
R0B
R2l*R46 - Rl6*R30
R99 ..
R2l*R43 -
(Rl7)2
The following terms must now be calculated in order to tain the coefficients of the equation:
ob-
2 Rl3 = (R95*R96 - R07*R9S)/(R05*R99 - R97 ) Rl2 Rll
a
(ReS - R07*Rl3)/R95 (R30 - R12*Rl6 - R13*Rl7)/R21
The coefficients of the best-fit normal distribution are:
a '" e b
{Rll -
(Rl2)2/(4*R13)]
-(Rl2)/(2*Rl3)
c -- l/(R13) The goodness of fit (coefficient of determination) is calculated from the following expression:
RR
Rll*R30 + R12*R46 + Rl3*R54 -------------
(R39)2/ R2l
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
71
RRcorrected '"
1
-
(1
-
RR)II(R21 (R21
-
-
1)
3)
Example: x =
1.0
2.0
3.0
4.0
5.0
Y
0.9
3.7
4.7
1.8
0.2
RI6 Rl? R21 R39 R31 R49 R43 R46 R54 ReS Re6
R07 300.I:HH!,1 -18.644 R08 RS9 '" 18?0.S3e -2.746 RIl R12 = 3.236 R13 .. -0.601 a 4.986 b 2.690 c -1. 663 1. 99O RR -
15.1"'0 55.000 5.900 1.729 7.054 225.090 979.090 = 1.458 "" -11.775 59.900 '" -153.965
'" '" -
72
LOG-NORMAL DISTRIBUTION
General Equation:
y = a
The log-normal distribution is often used when the range of data spans several magnitudes. It is frequently applied in economics, biology, and the physical sciences when the data results from a large number of multiplicative factors. The following formulas will estimate the coefficients of a log-normal distribution when four or more points are given. Both x and y must be positive numbers greater than zero •
I
.~
I
',,-
~
I (In x.)3 1
R28 = :!: In xi
R55
1:
R29 = };(lnx i )2
R56
r (In Xi) 4
R39 ..
R57 ..
r (In x i )2"'ln Yi
1:
In y i
R31 ,. ! (In Yi) 2 R32
1:
R21 .. n
In x.*ln Yi 1
where x and yare the values associated with each data point and n is the total number of points.
73
(R28)2
R05 =
R2l*R29 -
R36 -
R21*RS7 - R29*R3"
R37 =
R2l*RS5 - R2S*R29
R0S
R2l*R32 - R2S*R33
R09
R2l*RS6 -
(R29) 2
The following terms must now be calculated in order to tain the coefficients of the equation: Rl3
2 (R0S*R06 - R07*R08)/(R0S*R09 - RB7 )
Rl2
(R38 - R07*R13) /Re5
Rll
(R30 - Rl2*R28 - Rl3*R29)/R2l
The coefficients of the best-fit
log-normal
ob-
distribution
are:
a = e[Rll b
(Rl2)2/(4*Rl3)]
-Rl2/(2*RI3)
c .. 1/ (Rl3) The goodness of fit (coefficient of determination) lated from the following expression: Rll*R30 + Rl2*R32 + R13*R57 -
is calcu-
(R30)2/ R2l
RR
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
74
(1 - RR)*(R21 RRcorrected
1)
1 -
(R21 -
3)
Example: X '"
HHI
3~0
40~
5 'HI
y
339
270
240
220
R07 .. Re8 R99 = Rll Rl2 R13 a "" b
R21 5.00" R28 27.813 R29 .. 156.332 R30 27.976 Rn = 156.634 R32 155.215 R55 .. 887."58 R56 ;z 5076.463 R57 870.288 R05 .. 8.077 R06 -22.034
-
=
c RR
75
87.176 -2.029 942.656 3.268 1.145 -9.129 331.304 4.429 -7.737 ".999
BETA DISTRIBUTION
General Equation: The beta distribution Is sometimes encountered in statistics when the independent variable ranges between ze~o ~nd one. The followinS formulas will estimate the coefficients of a beta distribution when four or more points are given. Y must be a positive number greater than zero. X must be between zero and one.
R28
=
R29 R30
=
~
In xi
R59
I:
In(l-x)i
1:
(In Xi) 2
R69
1:
[In(1-x )]2
R61 '"
l:
(In xi)*ln(l-x i )
E In Yi*ln(l-x i )
r In Yi
R31 =
l:
(In Yi) 2
R62
R32 -
1:
In x i * In Yi
R21 = n
i
where x and yare the values associated with each data point and n is the total number of points.
76
=
R2l*H29 -
(R28)2
R06
R2l*R62 -
R3~*RS9
R07
R2l*R6l - R28*RS9
R0S -
R2l*R32 - R28*R30
R09
R2l*R60
R"S
(RS9) 2
The following terms must now be calculated in order to tain the coefficients of the equation: Rl3
ob-
2 (R05*R06 - R07*R08)/(R0S*R09 - R07 )
Rl2 = (R0S - R07*R13)/R05 Ril = (R30 - Rl2*R28 - Rl3*RS9)/R2l
The coefficients of the best-fit beta distribution are:
b
Rl2
c = R13
The goodness of fit (coefficienL of determination) is calculated from the following expression: Rll*R30 + R12*R32 + Rl3*R62 -
(R3U)2/ R21
RR
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
77
(1 - RR)*(R2I - 1) RRcorrected =
1-
(R2I -
3)
Example:
x 1.0
y -
R21 R28 R29 R30 R31 R32 R59 R60 R6I R62
= .. a
= =
=
9.2
0.3
0.4
0.5
1.6
2.2
2.6
3.1
5.009 -6.725 13.662 3.345 3.036 -3.365 -1.889 0.93e 1.98B -1. 658
Re5 '" R06 R07 = R08 .. R09 Rll RI2 RI3
..
a RR
78
=
8.077 -1. 972 -2.807 5.672 1.079 1.603 9.698 -0.012 4.968 9.999
GAMMA DISTRIBUTION
General Equation: The gamma distribution is commonly used in statistics to describe the Sum of several identical exponential distributions. It Is essentially the same as the Hoerl function. The following formulas can be used to estimate the coefficients of this equation when four or more points are given. Both x and y must be positive numbers greater than zero (any small number may be substituted for zero).
xi
R31
2
Rn
=
R46
~
l:
(In y i) 2
~
In x.*ln Yi 1
R16
l:
R17
r xi
R28
2:
R29
r (In xi) 2
R48
R3" =
r In Yi
R21 = n
In x.1
r xi * In y.1 :!:
x. *In Xi 1
where x and yare the values associated with each data pOint and n is the total number of points.
79
Res =
Rl7*R21 - (Rl6)2
Re6 -
R2l*R32 - R28*R3e
RS7 -
R21*R48 - Rl6*R28
RS8:
R2l*R46 - Rl6*R38
RSg =
R2l*R29 - (R28)2
The following terms must now be calculated in order to tain the coefficients of the equation: Rl3
ob-
(RS5*RS6 - RS7*RS8)/(RSS*RS9 - R97 2 )
Rl2 - (ReB - R07*Rl3)/R85 Rll - (R30 - Rl2*R16 - Rl3*R28)/R2l The coefficients of the best-fit gamma distribution are: a b
= e[Rll
+ Rl3*ln{l/Rl2)]
l/(Rl2)
c - Rl3 The goodness of fit (coefficient of determination) is calculated from the following expression: Rll*R30 + Rl2*R46 + Rl3*R)2 - (R3S)2/R21 RR -
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
80
(1 - RR)*(R21 RRcorrected
=
1-
(R21 -
1)
3)
Example:
y
R16 R17 R21 R28 R29 R30 R31 R32 R48 R46 R05
= = s
= '" =
1.0
2.0
2.8
3.2
4.6
15.000 55.000 5.01"J 4.787 6.200 7.468 11. 789 8.121 18.274 24.994 50.000
4.0
5.0
5.9
7.2
R06 R07 R08 Re9 Rll :0: R12 R13 : ;
a b
RR
81
4.85" 19.560 12.504 8.077 0.723 0.288 -B.096 1. 826 3.475 0.985
CAUCHY DISTRIBUTION 1
General Equation:
y iii
(x + b)2 + c
The Cauchy distribution is sometimes used in statistics to describe distributions that have a mean but no standard deviation. The following formulas will estimate the coefficients of such an equation when four or more points are given. Y must not be equal to zero (any small number may be substituted for y when i t is).
R40 R24 •
t l/Yi
R43
R25 •
I l/y~
R21
1
2
n
where x and yare the values associated with each data point and n is the total number of points.
82
(R16)2
R"5
R17*R21 -
R96 '"
R21*R37 - R17*R24
R97 =
R21*R49 - R16*R17
R9a
=
R21*R34 - R16*R24
R09
R21*R43 -
(RI7)2
The following terms must now be calculated in order to tain the coefficients of the equation:
Rl2
ob-
(R08 - R07*RI3)/R95
RII = (R24 - R12*R16 - R13*RI7}/R21
The coefficients of the best-fit curve are: a
:a
R13
b '" R12/(2*R13) c
= RII
-
2 (RI2 /4*RI3)
The goodness of fit (coefficient of determination) lated from the following expression: RII*R24 + R12*R34 + R13*R37 -
RR
= -----------
is calcu-
(R24)2/ R21
When RR is to be used for comparison with other regression curves. it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
83
(1 - RR)*(R21 - 1) 1 -
RRcorrected
(R21 - 3)
Example:
x ..
1.0
2.9
3.9
4.9
5.9
y ::
2.8
3.2
4.6
5.9
7.2
R16 R17 R21 R24 R25 R34 R37 R40 R43 Re5 R06
=
::
= ::a
= ".
=
15.999 55.099 5.090 1.195 9.320 3.907 9.748 225.900 979.000 59.0ee -17.099
R97 399.00" R08 .. -2.898 R09 1879.099 Rll = 9.451 -0.090 R12 0."05 R13 a 9.0"5 b = -8.388 c 9.972 RR 9.980
= =
84
MULTIPLE LINEAR REGRgSSION TWO INDEPENDENT VARIABLES
z = a + bx + cy
General Equation:
This equation can be used to fit any linear equation involving two independent variables. The following formulas will estimate the coefficients of such an equation when four or more points arB given. X and y may be positive, negative, or equal to zero. Rl6 ,.
=n
~ xi
R21
Rl7 ,. t xi2
R63
Rl8 •
Yi
R64
RI9 = r Yi2
R65
R20 = r xi *y i
R66 '"' ~Y1*zi
L
r zi
..
t
z~1
r xi*zi
where x and yare the values associated with each data point and n is the total number of points.
-
(R16)2
R05 =
R17*R21
Re6 =
R21*R66 - R18*R63
R97 =
R29*R21 - R16*R18
Rea =
R21*R65 - R16*R63
R39 =
R19*R21 - (R18)2
The following terms must now be calculated in order to tain the coefficients of the equation:
R12
(R38
Ril
(R63 - R12*R16 - R13*RI8)/R21
85
ob-
The coefficients of the best-fit curve are: a = Rll b
R12
c ;; R13
The goodness of fit (coefficient of determination) is calculated from the following expression: Rll*R63 + R12*R65 + R13*R66 ---- -------------
RR
R64 -
(R63)2/R21
(R63)2/R21
When RR is to be used for comparison with other regression curves, it should be corrected as fOllows to obtain an unbiased estimate for the coefficient of determination:
RR
1
corrected
-
(1
-
RR)*(R2l (R21
-
-
1)
3)
Example: x =
1.1
2.3
3.2
4.5
5.1
Y
1.7
3.9
5.2
7.1
9.2
z ==
2.8
4.9
3.3
------
4.6
3.8
R16 16.299 R17 ... 63.999 RIB = 26.2B9 RI9 173.989 R20 104.280 R21 5.999 R63 19.499 R64 = 78.349 R65 = 64.790 R66 = U3.47.0
R95 R96 = R07 .. R9B = R.09 Rll
R12
=
R13 ,. RR
=
86
52.569 9.079 96.969 9.229 183.469 2.938 3.364 -1. 728 1.999
MULTIPLE LINEAR REGRESSION THREE INDEPENDENT VARIABLES
General Equation:
t
= a + bx + cy + dz
This regression can be used to fit any linear volving three independent variables. The following estimate the coefficients of such an equation when points are given. X, y, and z may be positive, equal to zero. Rl6
1: xi
R65
Rl7
r x~1
R66
RIB "" r y i
R67
1:
-
2 r Yi
R68
r t~1
R23 =
~
R69 = !Xi*ti
R2l
.::
n
R63
=
~
zi
R64
=
~
2 Zi
RI9
Xi *y i
equation informulas will five or more negative, or
rXi*zi
..
I:Yi*Zi ti
R73 ,. ryi*t i R71 '" r zi*ti
where X, y, z, and t are the values associated with each data point and n is the total number of points. The following terms must now be calculated in order to obtain the coefficients of the equation:
RBI
(R20 2 - R17*RI9)*R63 + (RI7*R66 - R23*R65)*RI8 + (Rl9*R65 - R20*R66)*Rl6
RB2
(R29*R21 - R16*RI8)*R66 + (RI6*RI9 - RIB*R29)*R63 2 + (RlB - RI9*R21)*R65
87
R83
=
(R17*R63 - R16*R65)*R18 + (R21*R65 - R16*R63)*R20 + (R16 2 - R17*R21)*R66 2 (R17*R21 - R16 )*R19 + 2*(R16*R18*R20) 2 - (R17*RlS )
R84
R8S
~ R17*R21 - R16 2
R86
= R21*R69
2 (R21*R20 )
- R16*R67
R87 - R29*R21 - R16*R18 R88 - R21*R65 - R16*R63 R89
=
(R21*R70 - R18*R67)*R85 -
(R86*R87)
(R67*R81 + R69*R82 + R70*R83 + R71*R84) R14
(R63*R81 + R65*R82 + R66*R83 + R64*R84) {(R18*R63 - R21*R66)*R85 + (R87*R88»)*R14 + R89 R13 (R19*R21 - R18 2 )*R85 -
R12
Rll
= =
(R87)2
R86 - R13*R87 - R14*R88 R85 R67 - R12*R16 - R13*R18 - R14*R63 R21
The coefficients of the best-fit
a '"' Rll
b = R12 c - R13 d
=:
R14
88
cu~ve
a~e:
The goodness of fit (coefficient of determination) Is calculated from the following expression:
RI1~R67 + R12*R69 + R13*R70 + R14*R71 - (R67)2/R21 RR = R68 -
(R67)2/R21
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (I - RR)*(R21 - I) I
RRcorrected
-
(R21 - 4)
Example:
x
=
1.0
2.0
3.e
Y
02
2.8
3.2
4.6
4.0
5.~
6.0
5.8
7.2
8.4
1.1
6.8
---
-- - - - - - - 6.4 2.4 8.0 ----
.-~.--.
z =
4.8
2.6
9.4
t
4.0
2.5
8.B
R16 R17 RI8 RI9 = R2" = R21 R63 = R64 R65 '" R66 :z R67 '" R68 R69 R70 R71
-
5.4
RBI R82 ;:(83 RB4 R85 R86 R87 R88 RB9 Rll R12
21.0~O
91.e01:l 32.00e 195.280 132.600 6.0"1.:1 33.6"0 228.880 123.81'}0 186.61:ll') 28. 6~H' 176.300 le3.300 156.041:l 200.020
= -153.768
-0.976 -10.680 38.240 H15.IH'H1 19.201:l 123.601:l == 37.200 -163.920 = 0.941 = 0.998 -1:l.996 R13 IU4 1.01:l8 RR ". 1:l.998
89
= =
-
GENERALIZED 2ND ORDER POLYNOMIAL General Equation: This is a generalized second order polynomial. When r is equal to zero and s equals one, it reduces to a straight line equation. Both rand s must be specified to use this regression. When rand s have values other than zero, the curve will pass through the origin. The following formulas will estimate the coefficients of this equation when three or more points are given. Y may be positive, negative, or equal to zero. If either r or 5 is negative, x must not be equal to zero. If either r or s is non-integer, x must be positive.
~--------------~ 2r Xi
R79
x~s
RIB
~ r+s - Xi
Rl9
R72
1:
R73
I:
R75 R78
:I
1:
1
Yi * Xir
.. ::
I:
y. *x~
I:
Yi
1:
yi
1
1
2
R2l - n
where x and yare the values associated with each data point and n is the total number of points. R12
(R72*R79 - R75*R78)/(R72*R73 - R7S 2 )
90
Rll
(R78 - R12*R75)/R72
The coefficients of the best-fit curve are:
a
Rll
b = R12 The goodness of fit (coeffIcient of determination) lated from the following expression: Rll*R78 + R12*R79 -
is calcu-
(R18)2/ R21
RR
When RR is to be used for comparison with other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination: (1 - RR)*(R21 - 1) 1 -
RRcorrected
(R21 -
2)
Example:
x ,.
1.9
y
2.9
3.0
4.0
5.9
0.9
9.5
0.3
0.2
Given that r = -2 and best fitting curve:
s = -3, find the coefficients of the
r '" -2.e0e s -3.000 R72 1.080 f./73 1. 917 R75 1.037 R78 '" 1. 397 R79 = 1.137
Rl9 RIB R21
2.190 2.900 5.900 Rll 6.191 R12 "" -5.191 RR .. 1.~99
91
GENERALIZED 3RD ORDER POLYNOMIAL General Equation: This is a generalized third order polynomial. When r equals zero, 5 equals one, and t equals 2, it reduces to a classic parabola. The values of c, 5, and t must be specified to use this regression. When r, s, and t have values other than zero, the curve will pass through the origin. The following formulas will estImate the coefficients of this equation when four or more points are given. Y may be positive, negative, or equal to zero. If r, s, or t is negative, x must not be equal to zero. If r, 5, or t is non-integer, x must be positive.
R72
1;
2r xi
R78 .. }; Yi * xir
R73 -
1;
2s xi
R79 -
-~ Yi * xis
R74 =
1;
2t Xi
Rae =
1:
Yi * xit
R75
!
r+s xi
RIa
:!:
yi
R76
E
x~+t
Rl9
~
2 Yi
R17
L xs + t i
R2l
1
n
where x and yare the values associated with each data point and n is the total number of points.
92
R81 =
(R72*R73*R80) + (R7S*R76*R79) + (R7S*R77*R78) - R80*(R75)2 -
R82
(R73*R76*R78) -
(R72*R77*R79)
(R72*R73*R74) + 2*(R7S*R76*R77) - R74*(R75)2 R73*(R76)2 - R72*(R77)2
R13 = (R81/R82) (R75*R76 - R72*R77)*RI3 + (R72*R79)
(R7S*R78)
Rl2 = R72*R73 -
(R7S)2
RII = (R78 - R12*R75 - RI3*R76)/R72
The coefficients of the best-fit curve are:
a - Rll b
Rl2
c ; R13 The goodness of fit (coefficient of determination) is calculated from the following expression: RII*R78 + R12*R79 + RI3*RSe -
(RI8)2/R21
RR '"
\'ihen RR is to be used for comparison wi th other regression curves, it should be corrected as follows to obtain an unbiased estimate for the coefficient of determination:
93
(1 - RR)*(R21 - 1) 1 -
RRcorrected
(R21 -
3)
Example:
x =
2.0
3.0
4.0
5."
-------------------------------y 0.5 1.4 2.8 4.6 6.6 Given the following powers of x, r = 0.2, s = 0.8, and calculate the coefficients of the best-fit curve: r
=
t
=
s ,.
RIa RI9 R21 ,. R72 R73 R74 R75 = R76
0.20" 0.800 200 15.9"0 74.770
R77 R7B R79 R8f/! R81 R82 Rll R12 R13 RR
1.
5. IiH'"
7.516 32.153 95.694 15. "IHJ 24.777
94
=
= ::It
=
-
55.9"" 29.772 47.543 B3.991 4.552 1.914 1.328 -3.209 2.378
1.,,""
t
-
1.2,
CIRCLE
General Equation: This is the general equation for a circle with a radius of r and center at h,ke It is used in surveying to determine the best-fit circle for points which lie on an arc. The circle is best-fit in the sense that the squared deviation of the area is minimized. The following formulas will estimate the coordinates of a circle that best fits the data when four or more points are given. X and y may be positive, negative, or equal to zero.
RI6
.
}; xi
R4fl
~
3 - xi
RI7
- xi
or 2
R36
:!:
Yi *xf
Rl8 =
1;
Yi
R39
~
Xi*Yf
=
1:
Yi
2
R42
}; y~
RI9 R20
R2l
:!: Xi *y i
1
=
n
where x and yare the values associated with each data point and n is the total number of points.
Ras
=
R2~*R21
95
- R16*R18
R06 ,.
(R17 + R19)*R18 -
R97 ..
(R18)2 - R19*R21
RB8 •
(R17 + R19)*R16 -
RB9 =
(R16)2 - R17*R21
R10 •
R"'7*RB9 -
(R36 + R42) *R21
(R39 + R4"')*R21
(R"S)2
The solution is undefined whenever R08 is equal to zero. Otherwise the parameters of the best-fit circle are: h
(R95*R06 + RB7*R0S)/(2*R19)
k ;
(R95*R9S + R06*R09)/(2*Rl"')
After the values of hand k have been determined, the radius of the best-fit circle can be calculated as follows: R17 + R19 - 2*(h*R16 + k*R18) + (h 2 + k 2 )*R21 R21
Example:
x = Y
R16 R17 RIB RIg R20 R21 R36 R39 R49 R42
= •
'"
,.. ,..
=
6
10
13
12
11
8
13
"
56 670 35 345 342 5 3668 3130 8354 3635
14 4
R05 -250 R06 -999 -501i1 R"7 R0S = -589 R09 =: -214 RHl - 4450", h z 6.04 k 4.91 8. iii 1 r = ::0
96
CORRECTION EXPONENTIAL
General Equation:
y = a + b CX
The correction exponential curve is commonly used in economic forecasting. It generally requires iterative techniques to find the coefficients of this curve. If the values of x are evenly spaced, however, the following technique can be applied to estimate the coefficients.
The x,y pairs must first be arranged in ascending order starting with the lowest value of x and ending with the highest value of x. The data must now be divided into three even groups with the same number of points in each group. The first one or two data points can be discarded in order to make the groups come out even. Now replace each x value with the numbers zero through N-l where N is the total number of points left. Sums for each of the groups are calculated as follows: (N-3)/3 RSS
97
(2N-3)/3
L
RI'J6 -
Y1
i"'(N/3)
(N-l) RICl7
z:::
1=(2N/3) The following terms must now be calculated in order to tain the coefficents of the equation: R~6)/(R06
Raa -
(RI'J7 -
Rl3 =
(RI'J8)3/N
Rl2 -
(R~6 - R95)*(RI3 -
ob-
- RI'J5)
1)/(R98 - 1)2
R95 - RI2*[(RI'Ja - ll/(Rl3 - 1)1 Rll =
(N/3)
The coefficients of the best-fit correction exponential are: a
Rll
b = R12 c = Rl3
98
Example:
Year
i
Production
1923 1924 1924 1925 1926
"
33.8 38.9 37.7 42.5 46.3
1 2
3
4
R95
199.2
----
1927 1928 1929 193" 1931
50.6 55.2 58.9 58.0 60.5
5
6 7 8 9
283.2
Rel6
1932 1933 1934 1935 1936
----
62.8 63.5 69.4 63.9 68.2
55 11 12 13 14
R07
318.8
15.""" ".424
N
R08 Rll R12 R13
= =
68.997 -39.917 9.842
99
LOGISTIC CURVE
a General Equation:
y
The logistic curve is commonly used in economic forecasting. It generally requires iterative techniques to find the coefficients of this curve. If the values of x are evenly spaced, however, the following technique can be applied to estimate the coefficients.
a
The x,y pairs must first be arranged in ascending order starting with the lowest value of x and ending with the highest value of x. The data must now be divided into three even groups with the same number of points In each group. The first one or two data points can be discarded in order to make the groups come out even. Now replace each x value with the numbers zero through N-] where N is the total number of points left. Sums for each of the groups are calculated as follows: (N-3)/3
Res i=9
19B
(2N-3)/3 R"6
L
=
l/Yi
i=(N/3)
(N-l)
L
R07..
l/Yi
i=(2N/3)
The following terms must now be calculated in order to tain the coefficents of the equation: HOS
(R,,7 - R06)/(R06 - R"S)
R13 -=
(R08)3/N
R12 =
(R06 - RBS)
- 1)/(R08 - 1)2
R12*[(R0S - 1)/(R13 - 1))
RI'I5 -
Rll
* (R13
::
(N/3 )
The coefficients of the best-fit logistic curve are: a = 1/ (Rll) b
c
R12/Rll
= R13
101
ob-
Example:
Year
i
1956 1957 1958 1959 1960 1961
"
Production 49
1 2 3 4 5
5" 67 88 119 146 13.987
R"5 1962 1963 1964 1965 1966 1967
182 123 273 322 388 475
6
7 8 9
10 11
..
RliJ6
0.925 591
12 13 14 15 16 17
1968 1969 1979 1971
1972 1973 RB7
713
845 983 1143 1256 9.
=
N
R9a Rll R12 R13 a b
= =
,,,n
18.091iJ 0.295 -9.961 E-"S 0.023 = 0.816 -10039.033 -228.532
= ~
192
ABBREVIATIONS AND SYMBOLS
n
the total number of data points
i
an index used to refer to the ith point the x value of the ith point the y value of the ith point the sum of all terms from the first point to the last
e
base of the natural logarithm (2.71828 ••• ) e raised to the x power
In x
natural logarithm of X
RR
coefficient of determination (the square-root of this number is equal to the correlation coefficient) the absolute value of x
Rnn
the contents of calculator Register nn
A-I
REGISTER ASSIGNMENTS
R28
index
Relil
=>
R01
=a
=
1:
xi*Yi
R21 - n
RIil2 - b
R22 -
1:
l/xi
RIil3 = c
R23 •
1:
l/X~
RIil4
R24 - 1: l/Yi
=d
E
2 l/Yi
1:
I/(x.*y.)
ReS • scratch
R25
Re6
= scratch
R26
=
Re7
scratch
R27
=n
1
1
R2B '" 1: In xi
Ra8 .. scratch RS9
scratch
R29 - 1: (In xi)
RHJ
RR
R39 - 1: In Yi
2
Rll - scratch
R31
1:
(In Yi) 2
R12 - scratch
R32 =
~
(In x l )*(ln y 1)
R13 .. scratch
R33
=n
R14
scratch
R34 '"
~
xi/Yi
R15
curve number
R35
l:
Yi/xi
R16 ..
2:
xi
R36
1:
2. xi Yi
.
1:
2 xi
R37
1:
xi/Yi
RI8
1:
Yi
R38 • -~ Yi I x 2I
R19
~ 2 - Yi
R17
R39
B-1
2
}; x.1
*l
1
REGISTER ASSIGNMENTS 3
R40 =
~ Xj
R41
2:
R60 = 1: [In (1 - x i ) ) 2
l/x i
3
3
R61 ,. 1: (In x i )*ln(l
xi)
R62
Xl)
1:
(In Yi)*ln(l
R63
~
21
R44 .. :!: 1/x 1 4
R64 =
E
zi
R45
1:
R65
2:
x1*zi
R46
}; xi" (In Y1 )
R42 = }; Yi R43
}; xi
4
(In xII/xi
::a
::
2
R66 = ~ Yi*zi
R47 = 2: (In Yi)/x i
R67
!
ti
x i *(ln Xi)
R68
};
t~1
R69
}; xi*ti
R48 -
E
2 R49 '"' r (x 1 *ln Xi)
R50 -
}; x." ( In x i )*{ln Yi) 1
R51 = }; Yi*{ln RS2
(In xi)/Y i
R72
~
!
( (l n
R54
-
1:
xt*(ln Yi)
R55
=
I:
(In Xi)
2
(In xi) * ( In Yi)
R58
2:
Yi
R59 .,
~
In (1
x.2r 1
2:
2t Xi
E Xi
R77 =
~
R79
xi)
B-2
r+t
R76
R78
* ( In xII/xI
-
Yi*tl
r+s R75 .. E Xi
R56 .. 1: (In xi) 4 2:
:!:
R74 =
3
RS7
I:
R73 = 2: Xi2s
Xi) /x i 1 2
R53
;:
R71 = 1: z 1* t i
Xi)
1:
R70
= 1:
s+t
Xi
* r Yi Xi
E Y i
.. s Xi
REGISTER ASSIGNMENTS
Rae
=
I
Yi * xit
R99 .. scratch R91
scratch
R82 .. scratch
R92
scratch
R83 .. scratch
R93
scratch
RB4 - scratch
R94
scratch
RBI
scratch
RBS
scratch
R95 .. scratch
R86
scrlltch
R96 - scratch
RB7 = scratch
R97
RBB "" scratch
R98 = scratch
RB9 - scratch
R99 = scratch
B-3
scratch
DERIVATION OF A REGRESSION CURVE
The following example illustrates how a regression curve can be derived for the equation y
ax + bx
2
The estimated values of Y on this line corresponding to Xl' X2 , X3 , •••••• , Xn are
...... ,
aX n +bX n2
The actual values, however, are Yl , Y2 , Y3 , •••••• , Yn ' The least-square regression curve is the one that minimizes the sum of the differences between the estimated Y values and the actual Y values.
s
+ ••••••
The minimum value of S can be determined from the calculus by taking the partial derivatives with respect to a and band equating them to zero.
as
=
ria
as cb
2(aX 1 +
bX~ + 2(aX
n
+ bx2 - Y )X 2
n
C-l
n
n
9
These equQtions can be expressed in summation form as
o
(1 )
(2 )
::
Rewriting (1) and (2) to solve for the coefficients a and b gives us:
(3 )
a
(4)
b
"-
Solve for a and b by first substituting b from equation (4) into equation (3) I and then substituting a from equation (3) into equation (4) • ."
I XY
a
b
=
---
~ X4
l: x2
."
!X4
~ x2
."
2 :EX y
:E x2
."
:EX4
::
C-2
2.
-
x2 y
::: x3 3
3 l: X Y
."
!X
~Xy
."
:EX
:E X 3 y
."
:E X 3
3
!>!l'L T I PLE CURVE !" !TT I NG PROGRk'1
This program fits up to 19 curves to a set of X,Y data points. Data may be added or deleted from the data base at any time. The best-fitting curve can be determined automatically based on the adjusted coefficient of determination, (RR). Values of Y may be computed for any given value of X once a curve has been selected. Any curve may be arbitrarily selected by entering the appropriate curve number. Curve Number
General Eguation
1
Y
a + bX
2
Y
l/(a + bX)
3
Y
4
Y
a + b/X
5
Y
X/ (aX + b)
6
Y
7
Y
a + bX + CX2
8
Y
I/[a(X+b)2 + c]
9
Y
ax
10
y
axbX
11
y
ax b / X
12
Y
13
Y
ab 1 / Xx c
14
Y
ae(b-lnX)2/ c
15
Y
a + b·lnX
16
Y
17
Y
ab
18
Y
ab
19
Y
ae(X-b)£/c
D-l
=a
=
+ bX +c/X
a + b/X + C/X2
b
= abXX c
=
11 (a + b·lnX) X 1/X
MULTIPLE CURVE FITTING PROGRAM
Program Operation The program displays the equation for any selected curve as well as the coefficients and the adjusted coefficient of determination (RR). Errors are easily corrected and accidentally pressing R/S in most cases simply repeats the last function executed. The program requires a QUAD Memory Module when used with the HP-41C. A printer is optional. Limits and Warnings The corrected coefficient of determination is displayed for RR and is used in all comparisons for the best-fitting curve. The 1:- and "DELETE LST X,Y" functions are limited by the internal accuracy Qf the HP-41. The last few digits in certain summations may be in error as a result of using these two functions. Whenever X or Y is zero, it is replaced by 9 E-09. If any value of X is negative, curves 9 through 16 are not used. - If any value of Y is negative, curves 9 through 14 and 17 through 19 are not used. Registers Register assignments are consistent with those in Appendix B except for R39 and R42 which are not computed, and R59 through R69 which are used in certain computations. Flags
FOI is set when finding the best-fit curve. F02 is set i f any value of X is negative. F03 is set i f any value of Y is negative. F21 is used to control the printer when attached.
D-2
MULTIPLE CURVE FITTING PROGRAM
1
2
INPUT
INSTRUCTIONS
STEP
XEO SIZE 070
Set SIZE to 070.
load progr..... "MCF" and "xecuta GTO to
pack
GTO·
. in order
m-v.
3
Initialize program memory and flegs.
4
Key in the X "alue for the first point.
5
Key in the Y R1ue for the first point Ind prell ~+ or RIS. Each entry taJces about 5 seconds. The tollt num· ber of poinu h dispJay.d .n.r ..ell entry.
6
Re.,..t ,tapS 4 and 6 for all points.
71
Correct the last antry by preuing 'b' and then going to steps • end 5.
7b
DISPLAV
FUNCTION
Oal_ any point by entaring X and Y, then pressing ~. key.
the
"
X,ENTER. Y, 1:+
X
ENTER
X
Y
A or RIS
n
b
n· 1
X Y
ENTER
X
•
n·l
Be
Find the best fining curve. The CIIlcubotor displays tha number for nch curve and the corrected ,,11... of R R (coefficient of determination I. The equetion of the batt fitting curva wiN be displeyllCl after IPllroximuelY 90 seconds.
E
i.RR equation
8b
Prell R/s to obtain the coefficientJ of thl bHt fining curve Ind th. corrected coeHiciant of dl1ermlnl1lon (RR).·
R/S RIS RIS RIS
I
9
De1armine the co.fflclantJ for any selactod curve It eny time by entering the curve number II) and pr...• ing'8.' Prom RIS to get each of the CXllfficlants.·
I
B RIS
b c RR equation I b
RIS RIS R/S
c RR
10a
Calculate the "aJus of Y corresponding to any ginn oIue of X by entefing X and pressing ·C.' (This routln. UI81 the ooaffiQena of the last equation dlsplayedJ
X
C
Y
lOb
Calculate additional ....... of Y by entering •• ues of X Ind pr_ing ·c' or RIS.
X
C or R/S
Y
"If the prln.. r il Ittached. the coefficienu ... autornetiallfy printed out.
n-J
MULTIPLE CURVE FITTING PROGRAM
eL·lB:l
.. .....
10'
~
H2 cr .7
...... ,j(~2·
...
~or. ) -K. EM Hi. R. , . c·-
·
,ez .T· ,~J oc, Ie·, .T~ :"
,e6·lffL
10' Rel
8'hl.lilt.
.e<J·L9t.
-roc'
l.,·L,ra )II.
"
g"'
., ,<- xE. .,
" . . 0 B? ,." "0 >-y :.I-e-:-
,~
)«
,~
,.
89
17
'",'>
~
21
Sfl)
rIH.r. C· 1 lit. 11«
.. · '" .. "" ..'" .. ...".. .,.. ., .......••, •• "" •• ~
~~
"X
3•
~T·
Rt"".l
.~
,7
17 2" • T' <~
J2 Jl
~~,
J.
'B
b" OtL
lR
RCI x"
,<-
.T.
lA~Ti"
Xl, ,T' I"';':
H
J? !'IT-
e,
ItCl
51· ,.)(
~T
...., 4(
..
• otl
.~
i>CL
II
, /w
,'K
..
r.1tE't:.
t5' xFU .. 9
~
~7
t·
·.iII LASTX WlL
".
... ·0;"'.
~,
~.
A' .w
" .r • '" " 6" ,. . , •• ., sr· ... +
.... .. .. ~. L~S'T-': :..t~
ST. ltC' oJ ItCL <-~
.H
ST • XT?
71
••
... ~A
lA5T~
,. · 7~
g,
LA!5TX
••,. 0" or· RC'
10
....
~T·
·.,.
~j
46
7<- l.WIII)(
77
'
.~
8. ":'..
... ~4
MeL lCVo')('
... .,.
0'
8?
0>
WlL 8? MCl 81
... ·,T.
~,
~7'
LnlTX 80 • C.L u~
...,.... ~.
~r.
IitCI
ACl X... .:-
~.
·
97
lIt~
U
IT'
9?S
,.... ,,~
sr·
...,., ~2
".
L~~r)(
~£
.,
LAt;lX IIC'
,.n
".
"' u::! 11 j
II·
b
ItCL .. 3
,
...
x_,
0'>
x~o
• TO )( ) Y
x_o-=
II:'.
XEO 8't
lib 117
J:fl1:.G
~Tn
'"
,." ,.. ~
124 J~~
I~G
,27
"e 129
ItCl
·'"'/"
~,
~,
"'"
... .. ..,
~"C,
aI' t:TO
..
"
ACl • 7
z,,8
~~
;,te9
)«
,... 7
)(-
.".,
8" ,
x~Q
"
6'
JT
lM
It
"'
~,
,~
,16 LJ7
~.
"" ,,.,
~1-
"~
I3q ACl
~.~
x<-.7
,,~
><E.O L" t;;TO
.,
zw
>'7' RT" 222.UtL "9
r,:l )( ?:to1 9 [,9 2"l~
2'Z~
~T"
14Z
,10" 14~
'07 ,.ft ,~.
ST-
"X "X AC' ST ReL
'" ,". .... - •• ,,,' ".. ,:,,,
...
,,,,,>
'if -
,
~T-
A'
••
xU
'67
176 ST-
",
.,
".
LW91J<
· , '" ,77 ·
vCl
17~
,
.,
.' ...... ~T-
II.>
,,,
ST
,86
LA~TX
,,a? ,
RC'
,.3 ,.4 ,~~
...
,~
,
,
.. ·
DCl ft. NC' "J ~
STI n$'X
,~~
'" ""
;Sotl
?,
-~
~·D)(·
"",;.c-
z".,
OTN Z63.t.BI A. ?AA xCD ~. 2• 2,.7
..... .••• ,. ItCl
.:'(.?
27"
)(£0 ?C
>71 ,-c
~7.
-
,,'N
••...
.... .to.. r.>
n~
ACL
xco
..
.~
•,..c.,c •• •• ••• -y .. x.-c.x :t7"::t
.........
~~o
II
eTO
203
"b)-
ATH HS·LBl .6 xCO Z6 2.·/ XEO RCl n STn
. ..
.,•• au oe'\. .,
z_ ...... ~
,CL9 33'
r(;":'c I I
~t~
..
JM
RC'
"
3'](·
....
;'41
cro
~:l'i·LDL
.~
'" 17
...", ... • ."
347
...." .........
ux
"
GTO 02 • ..... Ut,,)!!
) eXt, )~ ATM
..."
".7.lPII.
13
".0 KeD
~.
.,"
_0
.,.
. , I RCl " 2 Rl~L
" J Rrt n
x£O
.,~
.'6 .17 .,8 .,?
fC"C
GTn
7~
07
n
>t
,.w,
.. ... ,na ux
.'v
.Z8 .,.,
AL'
~
'20
11
~,
'2
oro
.H
••• .... X.,
()l''tc )
4~.
A'.
-V~.lbTI
-42~LI:n
U? MOV • ?II xc •
.n .,
~
,. ,,"
z~
AGl :>9 STO U 4:', OCl S. .TO 67 '33 ltC' 12 .....l 5S
.,.
... "" 4J5 4.1 ..
14
5'
, ' 7 XEa 2'
}-
:14l1li FR'
.6> OIl
:':4:. eTO 3.4 "'EO 34~
:.:
,.7
:14R ~
~
.,
XED 21
""7 ,-C'''C
XFO ,,? Itr.:L ?IO
,:'C.,r: """ '" nx " 01 .'0
:1':)8 ero :1~' RCl J~2
35' - ...... X'tlbl~" .TN :1~"'.L8I 3~7 Ar...
'0
on
(" 49
•• 8 . . 0
,.~
3"·
<;;TO 62 '0' x£ • 29 '3t.io! A'r:l
:..3
:1'4
l"~
3'. 3.' US 3.~
'J.
0'
~('.,C
U¥ elf) ,s '4& Rf:l 11
44'
"CL
._2 xU
..4
44 :J Ift.:L
)(~O
FC"C
ern
28
'"
"'" ""
CT' STO 8'
~9" GTn .7 -y ..·b"'X ~.,9
~'T.
" S CH1i
0" 441 5T.
.. ,~
~
U-
4':')1
8'0 I I
n "X 'no u
.52 on
.'3 4:'''
Co , .... c.
·V_.F.J
.5~
~""-b)·2
."
"~8 .. 18L
,'\
xEG
?ft
.6' .02
....
X[Q
Itrot l~
)(.
. , 7 "T~
.n
11TH II
3/3 Rr.L .~ 1>TO 61 "Cl
,no '" "" J77
..
.~
3'6
xU 29 3711 "Cl 58
37" ~
30'
II
l'U
fC?C
302 GTV 31.' OCL
.,
2.
,:.'
"
.94 IT> lGS $TO I t
38"
·V_.XT(b
117
trT~
"X)JeA.LDL
38 • ICE" ].e K£O
....-"
.",a,.lBI ~
,.
.. u n
..." ., XEY 25 RCl S7
'72 '6
~7.
XED
,~
JC?C
ero
17 ~ .76 ·Y-I"(." ht.NXI" .77 .T" 478.l HL
'7
,S~
8]
~.a
n
.-.,..,.... ~.,
".0
3'
..........." " ..,. .- .,".. .ez •• 3
"ED 2 ' RCl .6
no
FC"C .1 GIO ACl ~IKI UX STO ReI.. •• 2 rlX 493 ~TO 07 ·V_.h'X49S .TN ...... lOl ,8 ><EO U XFO ?9 .~ ~87
,. .-'"., ....
.:0 J" .or.
,Cl 19' OCl
J.6 x~O 107 JC"C
n. e.o n. aCL
~"M
,~
>~
U
'"
XFY ~. FC?C 01 ero '0 .~" .66 -Y_.·bLH 4 .. :1
W" :1'7'
:'7~·LBL
K
8Z
"0
.7. -oy_.X'tfh
2":\
0-4
..... ,,.(c ..
".
AU
"T~
-;TO . j )
.TM
lO~
2.3 OCl
lI.
."" s,· .,, OT' X s,~
STO 8, IICl
ow~
.. . U
.. cx· ... ) ... 2-
294 Atl 38
~10
• c. .... XT.i-
" '2
3:14
LAf'lX
Z ....
.iI
e~
3~1
J")7
.~
8
GTO J2":. ICl .,1'6- _C,
~H
·
...,,. ;l,
~2]
J"
Ul
~.
.,~
;::76 277
2'l~ •• L8L
2 .. ' 20'
A,
x..o
:S~~
67
xT2
,07
;i • •
II.
.1- 'U
. ..
,9~
J ? .. X[O .-ld .·C"'C 01 • 4.2 ",0 IH
2"
54
e.
~Cl lAS'~
te~
,~,
..
~T··
'G~ .CL
Ifill
,~
,tI
eTn os u J ·v- ... tJo, ... X
,y2 fir.L
'8.
It
.~- .~ LI1.T)f'
I7G
....'A'
"". no..
.. <-
~IO
ot.... :J1l
lAf\Tx RCl. b'
17~
""'~ ~53
".
~.,O
"5
17.
taX)24. ATM 7'4"J.LRI "J xEY 251 X'E& DCl U
....
)~
....., "A
.." • .. .. ,. ,,' ."" ..., '5' .. .,," ,. .,. ,. ". ..... "'" ..
- Y-l'" C....
~,~
,,6 ST·
IT'
.,
Kl
ili'CL
11~
x~o
rr."c::
,,10 1It1:· 20
u.~
'<-" ,.9
.)
x.O ."
or,
~~7
,';1
1;"1
~41
..
...."ft ......
~.,
2"~
.. ."
xca
ttt.:l ,,'9 ltC, AC'
,
?:"'.LOI
LA'TX
51-
Sf DCl
16:"01: ,.4
"
·V . . . . hX·
~TO
t
UI~
~"'" 2"
01
~
... 0 25 317 XF.o.D 2'
~"1;&
~
~4. ~47
LA'STX .CL
.,6
...., .z..... 2
I~a 1~3
?'"
24:'-
,~,
la:( ATH '11~
~.
·y ..... ltJX •
.\2:8 RCI
14:] 7" 1'4:'
"X
eTa
HI>
.rv 21 U' xCO Z.
••
xco AA
311 :J12
~,.
2 •• XFD
t-
(
~il'8 foro 10 229·l.L I t
.,
s:RCG ?7
RCL
3"~ x£o 31M rC"»I".
3"
•
?:t6·L81. ClX
ZJ:. ,L: ">C IoTO
,
x"a R<:' RCL
'114·LB:L
H.
LA!.H<
, ,... >0.
c)(1"?-
.... .. .. . .. · .... .. ,.. ..... ., .., ,.. [)3
z,. ••
Itt A' XC-II-:2'~ SF U
7' •
Z.? ,,"C,
u~
". ,,,. .14
;>3
"EO
..2 ITn
;t7.1
-o,"T-
~Ea
:(.~
3·'-
.,.." .. 2'.
..
l'hl.!BL 07 :1aJ:.' ,.~
08
~.f\.LIA.
,.,~
OC' ACl ~
5T-
An
1!1. 1:.... ,..
3~
:(R
," .. · [J_
~"2
28'
z,~
,. X
'23 GT-
;.;'1111
2'6 If.
110 ~I I " lR!;'X IZI I 7 A-'
~T.
-,..CF'
ero
' 7 A'VII:.W OTM
••
3-:'
£'tX
~.
•• 32
"
11
.y,
."7.
... '"''
0'" ~
,t<
01
MULTIPLE CURVE FITTING PROGRAM
~.J '\a~
)O;C~
rc
"'~
81
'f"
"'3 Cft') -18"
I" I I
~ ...
"0'('
Al
'5117
PI'" I
,?
'5e~
'C' r
t'~'
.,.'" r . . . ..., ~1f1
t",
",,1"1 "*1 ")'. , .... ( 1
-)r>"
":-JI
"lilt"]
I?
-;IJ :o<.CL) ':..14 '51'5
;,(Q'
~., ~,
HCQ
~7'
~lt=. ~I'
RGl
te:.
RC'L
4"
"... Itt ... t.:L ..... : •• " I';·
~'O 8(. ItI,.·L 'ZI RCL of.8
fl." . ""
"IIq
A~(
.•
f:.1
7"~
RCL ••
7"f'> 7.7 7f1"
r1A ANCI.
•
-
fa,ro
V."
€oUt
'HL 65
611
PCI
~
?J
•
AI3 .r.1
"'I
6-14 _C'l .... I!" •
'" ~
.,.
Ctl'? 'STO "9 {'10 IF' 2"1 (.1?
Rl""l
';'7
6~a
:u:.O
~l
~::-I
,,2:2: GTI"l
.,,1
~
oR~~
A.7:1
11
~2"
to'!:
IitCL .'5 626 loKI " "
:H.
rr'Oc ••
"'" ...... 1
I
~I?!-'
)('~
.... 2t
fo'LL
1.J
~_.:l,l
.. T
)(
520
CI'~
A
RCI 4.Z:i xt2 ~TO
62~
... ~i'
..
.. 28
Rr~ ..
"7'~
IWl:L
~Z'9
c:.T.'
fit;'
t..:""
•
5'1.
ST·
X
..,. i I
", ....')
lWeL
"JJ
If(L
5J!
""7. ft> • '..J.:i4
S (l'
fit I
'5:J~
P(I I J ~] ... I"K ~)7 .:;,0 .'1~ '5J9 .y-.£y'l'( .,t~
~41.l.fiL
~
~.Tn
5"':".
lWeL
:t"tt
~\.L
~"7
• Nl L
.. 1
6'5
n-.
~\:L
l.l
RCL
(..1
6'5!5 Rrt 6~6 'fr.1 "~7 •
.,11\ ....HI
.,61
...
no
II
.. ,,~
... rLJ
•• (.~
.,"""1'.. :. .h.... .......... , ,.:........ •... ?I
~~
R:t4
.. I
Ii
~~
7~!
!tCl. ,:t,.
1~;t
STU RTM
~:!
2' 3tt Gl JI f..4
,;ro
' ..... 7'c.'5
.,LL
":'7" "LL "7.~ ~,.
~CL
.
I?
:., ..
I..L)(
"('G ... " fin
~7~
':;10 .. fro,
el \.,
1..7;'
)(, ....
~7"8
tiel
.J.
i!C
66 4:'1
f..1It .1
~f'
\tl ~1
.. F
773
x£u
7704
'Ol'lf
E..C
7,"..
,",r~H
Zl
77'? 778
av,[w • b-
779
788 pfUL
"iT" RUN
,ti' 50(,
"\ .. ~
').,a
o;.TO
(,0""
f..A
(.0,·
~.,
6" 7 I Il~ 1) "IIIW Jo't.L ;;: 1
~c
G I
~? ~
I
",:1 .,.,. ..
~c..l
..,q~\
RCl
'TO
...... ...... "~J
.,,,,. 0:;.,0
htl~
to1t• •
~.1
L
)( . . ·?v
Rl""l
~
...-1
1:-.,'"
.... G
7ti) 7e, 'tt J "" ... 7A:.
8)
'.
.2
AVIEW "I""L eo'
~
)I;({II"
.,;"t..)(
(..9?
I""'j~
01
,.,.,.
I:T(I
Mil
R""" _ "u. 071 _LOL
8.7.,
e ....
wI''''
tl7'J1! R1M IIIIUt_L.8L. 11111 1 .... 1< OO;::-LBt
,.-Il'.'
Re..
C'U ...
OJ}'"
•
-,-
..
"'''6 IIWl.I.
X
eve: OOf· ~A7
•
IIIIU_LDL alit 79<) ·R5l- 7~e aRCL 18 7.,. qVJl:1oI 7.~ Aay 7'"3 "TH t;Ic. .j! 795.LDL C
!'tilt!
.a-tt..
7" ""toll
X~)'I
-
D-5
"1 7.
.2'
VT'\( Rr:1 81
7A? A\oII ....
19. CF" 11 IOJl JiC"L .~ 1'98 (,(\
7'0
1')(
07 .... lRI 77 fJ7" PI"" 1 Itl? j117~. ~c .. y H7ho "'tto: 8l __ Ito a-:L ..
8e~
'OJ"
6"., I
.. 'If:.
ft'lll~
L 6)
"CL "-
""'2 ~'l~
~? I""!'
~'"'~
... ..
,
'jTO "'('.1 Fl"CL
R"ON
"'..J1J'j ~.O
i.e .. (.0'"
I
ARCl
-
~D4:
'NU
17~
.a-CL 0" L ... $IJ(
58"3 :> __ 4
Rio ...
aWl
r"71 7"~
~c,
OJ
I·X RT.l Itl'
I..RIOt
... ~~
ttl
't'X 5lCL
.iJ
ot",
Nl.L
r.."'" .. _41 IW(.L
...... e
IiCL
~.,.
.., ) ....
H"l'"
e~
•
A40!'";"IPI
; I
1
)r.tl
86'
."n
~~.,
RCL
8"'~
I'~
'"\7"_1. "I ~f\tII ::.TO
~7n
0"(. 1l-,7 8:,)~
~
••
I J
"C"L
95 ..... TN 8,"+LI)·\. (,0
"'1
.rw
7'?" ~ro
b"
.. I
.,·JI
1'.-,.,..~ifL
,.".,I-l
':I ....
a ... ?
r."
7'';~
12
~10
el
0<&
-;TI'
II
63
"II
......
7""_' kL HI"L
RCL ... NCL
,.;;rl
e .. 6.LOl. r.. ..
" ..... rl .. ~
IrI .. "J Oc. ..
.,(L
_
~"O
II'CL
corn
7'~.f
1·')0(
05) •
t'btt NIH 761.t.."L I f..:,? tlCl
;:..,~
... TtolI
1·){
ff .. 41 IHHlWt .... :.... F-HTF ..· ..
b6b
~!",
'"
e.;"
"" ..... 07
&"-9 Rfl
e~
)(E:O (,.
.47 RCL
V¥~·LlA.
,,,
0'11
RCL
B2
8:91
)«)y
a')~
l"tx
09:1
XC)'I
01) .. IKL
8l
A~"
vt>c AI)" ,,0)7 IIp('.l A.A .. .a-Tfol ,"",_Llfl
.-"9
..:TN
"~'!":
I,·":
~7""
H I tot
..,~,
1 .'N I."
r\",~
'J~""
?i!.J_Lf'll . , ... )(F'O e~
66 4':t
75-'
I I
181
81
'TO
;-~$
1111
E-t)t; R"CL
.. 1I
~TU
7!\9
l
919 "'" 928
';.1
.n
'''7
':'1
";TI)
.., C
7"~ 7~,r.
Nf,L
~Tro,
l1li ..
eo?
)(1"~
A~.· ~ T M 1I.u,_LBL 6 j
V.J':I 0"18
It~L
R_:I
A4.>.' .....
1~1b
...
~I:J.
'>2
9'~
·,16>
?7 17
?S"_LDL
66 I
';" .. _1.1cI
RTN
,t . .
14
7?
"2 bl
)«( )"(
•
...
81
8l1i1
~"11b
~,,')
'Ie II! rN .ll_LOL
-
ItLL
•• 1 RCL 81 "II_I.II/IL
,Rt':T)(
ttS'~
'(TX
'''14: LH
7"3.' RI 7". RCL
1"
Gc:.J
~7.'"
~77
6"
~4
662
',7.' ,,7'"
;'TO It'M
I"'"
91k:.
q.r
,,~
b.
.,z·)
73"
",Cl
.,.~
.. TN ¥SI.\..tfL 8:1 .... CL
BlJ
74~
.("1.
c.:
.., .. :..
... 1
lit)
2~~
, ..... to
)(
II'I.L
6:1
1S4
I)'~t.
4.'5'" ~CL ~
5'H
PCI
,.::'1 (j.!l2
""3 . ",.
ft7"
sru
';'1
6"'1 'l)ro
"5
8:.7
Rl.L
7 .. "
., .... ,
Es26 ..
7.tot
7,,"_LOt... .:!O 7"18 RCL i!D 7~1 .,.TO t..t
6"b ~.l
OZ4 02!"!o
713
1"0 RTJ-4
RU
822 x"')'f 023
?~
II
......~
Lt.
r ...
7l7.I'tI 2'" 71.8 Irl ~7' 7l? r,.f(, (.I
eTn
a.O-LEtt..
7~t.,RI
r,-:.
81
'It;( 5lLl IU"t
NT~
D • ., AU._I"l .-1 I ftCL I::U ~ • .,1.J: "CL ~14 ~,~ N1N 1I1~_1 Bl _li' )(t:. ... RIA ,,)( . 1 " IITtot 1Il'''.1 tIL. 1I
7"JZ Rel
A.? 6."1
... HI l
~"1
'?J0 IItTH
6'5<e .,r:L
'"1:'''
~70
G"
4:'J
... 2 .rl
:.
557
Pt.:.L
o;TO
(, .. 7 , ('''0 to TO
":.1'1'> ",I;L
-;11=.1
7r':l
..."'...... --
obJ
'U.l
;oi!O! RCl 1:7?;l ,:ITO ".:. "104 kftot '2~-LMI.. ;z .. ,o:t. RCL Ie ,"t? .TO .. J '~tI "'('1. 19
n~
kLL
.,.rl
(.1
z.
•.,lo~.j
[1\ .....
l&
;"~!
11
.1
'5~,!
6f9
7ze RCL
fI" 41 I J
~T(.
,",:;~t
~,~
71".U"I. oe 71~ C)TO 1. 71-' 1tr::1 • • 1"1' o;lO )':) ;"10 RBI 71'.LOL 21
"l
IH;L
)( . . ;:
""
RTll
~TO
I
"\~
e.,
":'ao
IOYlllol
~TO
IH:I
~'2'
?1t"1 ... ..
De., AYIEW Dec., UII7
"I""I
,,:-...;)
:t"~.
WI"
J
c..a.-+-.i.
5'0
.".
:'4l1
IJT
'l(Ft)
.y8-'''' ... 11'(1
r.~ !:t~ '1lw Pl-l I . .. I I x~)" 71.:> )( II\,:>
71"311
91!11
ItiD ~
ft>l flEft ,
MtlI:::
(.'"
'54 ...• :'Tr)
"43 .tll'!
., ....
)Itt:,.
Qoc:.
8""
illd
(.4e
<:0:-·
I b
•.•
~"lA
~f..
L
t .( 1
\1 ....... ~.
•.... 0
.... (~:J~ Itl I
(~l
,~~
".t
CL~
'''2 J.l)( .. 78"
6." • 607
...... 1 7.3
.... ~ "r.1
t\ 1
"'1'"
'S1~.LOL
',,-It
"") ... ~
ftl
73
7ft ,. ...
.1I.1.n .
MULTIPLE CURVE FITTING PROGRAM
};.
~~~~~
IN IT
~s~g~
gggBg agggg I' 4E3Bg BBB~ g~B~
gBBg
E3BBB
0-6
MULTIPLE CURVE FITTING PROGR.\t-,
2 Ie 7;
1IIIflllllllllllllll~IIIIIIIIIII~1111II1I111111111111111111111~IIIIIIIIIIIII
~mllllilllijll/lllllllllmlllllllllllll/llllllllllllllllll"IIIIIIIIIIIIIIIII1IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIImlllllllllili ~rlililjlillililllllllllllllll~IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII111111111111111111 iimill~illi~IIIIII11III1IIIII11II11IIII1I1I11II1II1III1I1111111111111111111111111~llIlllIlllllllImlllllllllllllllmllllllll ROW 5
:~1
0101
ROW \(J 170
1IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIImili mlillllijilliiillllllllllllllllllllllllllllllllllllllllllllllllllllllllllil111111111111111111111111111111111111111111111111111111 ROW 12 IgO' 100)
1I111111111111111111111111111111111111111111111ll1l1lllll1l1lmllllllllllIIIIIIIIIIIIIII~IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII ROW
n
101' 105)
1IIIIIIiillifilliifllllllllllllllllllllllllllllllllllllllllllllllllllllllil1II11111~1I1I111111I1"1II~IIIIIIIIIIIIIIII/IIIIIIIIIII illililliillmillllllllllllllllllllllllllllllllllllllllllllllfllllllllllllli111111111111111111111111111111111111111111111111111
~ijliiljliiillji(III1I11IIIIII1IIIII1III1II1I1IIIII1II11I1II1I~111111I1111111111111111111111111111111111111111111111111111111111 ROW 17 1'35
143)
ROW Ie 11 U
1531
11111111111111111111111111111111111111111111111111111111IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII~IIIIIIIIIIIII
11111111111111111111111111111111111111111111111111111111111111111111111111111111111II~IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII E-l
MULTIPLE CURVE FITTING PROGRAM
ROW,9,';.o
'531
11111111111111111111111111111111111111111111111111111111111111111~1I~11I111111I111"IIIII~~lllllllmlllllllllllllllll1111111111
irlil~lliililllliilllllllllllllllllllllllllllllllllllllllll ~i~liilllllilllillillllllllllllllllllllllllllllllllllllllllllllllll11111II11I1I1I1111I11I11I11nllll~111I1I11II1111II11II11II1111 ~ffiliiljIliilliiilll1i IIIIIIIIIIIIIIIIIIIIUIIII 11111111111111111111111111111111 111I1
1I11II11I111r111ll1
FlOW 23 H!I
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
m~lillll'~lilllliilllllllllllllllllllllllllmlllllllllllllllllllllllllll11111111111111111111111111111111111111111111111111111111 11ililjilllilnlll1iilllllllll~IIIIIII~11111111111111111I1111111111111111111111111111111111111111111111111111111111111111111111I11 ROw 27 Infi' 2331
111111111111111111111111111111111111011111111111111111111111111111111111111111111111111111111111I1II111II11I~mllllllllllllllllll
iililiifilliiflllllllllllllllllillmllllllllllllllllllli11111111~IJIJIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII1111I11 ROW 29 1217. 2'"
1111I11II1I1I1II1I1II11II1I1I1I1II11I1I11I1I~1I1111111111111111111111111111111111111111111~111111111111111111111111111111111111111
iflllrlirlliiiillllllllllllllllllllll~1I111II1I11I1I111I1I11I111II1111I11I1I11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~iil~lt~iIlUIIIIIIIIIIIIIIIIIIIIIIIIIIIIII'"11111111111111111111111111111111111111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ROW 33 '259
2G31
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
~lilliirll~i~11II11I111I11I1111111II1I1I11I11I1I1I11I1I1II1I1I111I11I1I11I11II1I111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~iiliilirnillifliillllllllllllllllllllllllllllllllllllllllllllllllllllllllllillIIIIIIIIIIII~IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII F.OW 36 (273
:2,17",
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
E-2
MULTIPLE CURVE FITTING PROGRAM
ijmliiliiliilliruillllllllllllllllllllmlllllllllllllllllllllWl1Illmllllllllllllllllllllllllllllllllllll"IIIIII~111I11I111 MimlliiiIIII~IIIII"llllllIllIlImllllllllllllllllllll~11I1111II1I11I1111111111111111111111111111[1111111111""111111111 F'OW ag 1286 2911
IIIIII"III"IIII""II11II11II1II1III1""IIIIIIII"III"I"I"m"IIIII~IIIIIIIIII"11IIIIII""IIIIIIIIIlmlllllllllllllll"ll ""IIIIIIIIIIIIIIIIIIII""III~II"IIIII"III"IIIIIIII~II"IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII~IIII~IIIIII11I1IIIIII"III11""11 ~OW
00 i19:/; ZlNtI
~i~l~ilmilllmflllllllllllllllllllll"IIIIIIIII""I11I1~llIlImlmllll1111111111111111111111111111111111111111111111111111 HOW ~~ I~
305)
1111111 •..05.3111
"""IIIIIIIIIIIIIIII""""II"I"IIIII""II11I"IIIIIIIIIIIIIIIIIIIIIIIIII"III~II""1111111111111111111111111111111111111111
li~il~lfijjillil1illll""nlllllllllllllllllllllllllllllllllllllllllllill11111111111111111111111111111111111111111111111111111111 ROW 4~ ~31"
319.
111111111111111111""111111111111111""""111111111111111111111111111"""11111111111111""111111111111111111111111111111111111 AOW CO 1319 J261
IIllIlIIIIIIlIllllIIIIIIIIIIlIllIIllllIIIlIllIlllmllllllllllllllllllllllllllllllllll1IIIIII""II1II1I""III~"'IIJlIIIIIIIIIII 111"IIIIIJl/IIIIIIIIIIIIIIIII""IIII1IIII1II1II1II1III1"III"IIIIIII~lllmll111111111111111111111111111111111111111111111111111 AOW 41 (327 . J36i
flOW ,8 (338· l3IS)
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 .. oo.~
40
~w
SO 'l4l 3·U)
(J.~. J~21
""IIIIIIIIIIIIIIIIII"III"IIIIIIIII""IIIJIIIIIIIIIIIIIIIIIII""~IIIIII1II"'IIIIIIII""""11111111111""111111111111111111
~iUI1iljmillil~i""II1II"lllllIIlmlllll"IIIIIIIIIIIIIIIIIIIIIII~1111II1I1IIIIIIIIIII~IIIIIIIIIIIIIIII"IIII"I"IIIIIIII ROW 52 !3'J4
J601
11111111111111111111111111111111111111111111111111111111111111111'''"1110111111111111111111111111111111111111111111111111111111111
~OOIji lim'lIIiiilllllllllllllllllllllllm"IIIII1III1"IIIIIIIIIIII"~~"'1I111I ~miillflillil~J"IIIIIIIIIUllllrllllllllllllllllllllllllllglllll"llIllI1111111111111111111111111111111111111111111111111111 1111111111111111111111111111111111111""111
E-3
MVLTIPLE CURVE FITTING PROGRAM
ROW 55 1370 371,
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
~i~lijl~~lliiiillllllllllllll ~rlillilillOOillllllllllllllllllllllJIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII~~IIIII~III""HIIIIIIIIIIIIIIJIIIJIIII iililill'rnilli'IIIIIIIIIIIII
i~~I~IIlmf81
~~ijmliillmfllill iirnil'liililillllllllllllll~IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII"11IIIIIIIIII~IIIII~llllIIlmlllmmlll~1I1111111111 I~ot
'14,
ROW 84 ,423
~24!
ROW 62
1I1111~1 ROW 113
111111
"""II11I1IIII11I1JIIIIIIIIIIIIIIIIIIIIII~IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIHIIIIIIIIIWIIIIIIIIIIIUIIIIWIIIIIIIIIIIIIIIIII ROW 55 (424 - 42DI
11111111111111111111111111111
Ilillll~llm1illlllllllllllllllllllllllll~IIIIIIIIIIIIIIIIIII~IIIII~lIIllmlllIIIlIIllmlllllll~III"11II1I11111I111 11~1111111111111111~~IIII"11i
ilI1illmllliii("IIIIIIIIIII~111I11I1111111111111I11I11111111"11~11I1111~IHI~"IIIIIII"mlllll"'IIIII~111I11I1 ROW 158 ,4!\J , .~;
1I11111111111111111111111111111l1mlllllllllllllllllllll'"IIIJIIIUm~lIl11lln~U""llmIlIIIIHIIIIII~I~llIlflllllllll
Illiilf~iillinjlll""IIIIIIII~II"I""II1IJIIIIIIII~llllllllnl1IIIII11Inllllllmllllllnlllll~IIIIIIIIIIIIIIIII"11I1 flOW 7'
(.~O,
402'
IIIIIIIIIIIIIIIIIIIIIIIIIDIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIrllllllm~mlllll"II~IIIIIIIIIIMI~III~I~I~IIJIII~1111I1111
iliiIIOOlliilillllllllllllllllll~mlllll"IIIIII"IIIIIIIII~lmll~ftmm~"""II~IIIHlllllmlll"IIII~IIIIJIIIII E-4
MULTIPLE CURVE FITTING PROGRAM
lililliiiill~liillllmlllllllll""l1ll1lll11lll11ll11ll1ll1l1l11l1flllm""~1I111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~rllliillii~liilii"""IIIIIIIIIIIIIIIIIIIIIIIIIIII"IIIIIIIIII1111111111111111111111111111111111111111111111111111111"'11111111 ROW 76 I 17
~I
liullliiiiIIUiillllllllllllllllllllllllllllllllllllll"~""IIlIInlllllllllmlll"""IIIIIIIIIIIIIIfllllllllllllllllllllllll mDliilmIIIUlII""llllllImllllmlllllllllllllllllll"mlllllmlllllllllllllllII"IIIIIIIIIII'I~II"'I"IIIIIIIIIIIIIIII jiiijli~lliJiiillllllllllllllllllllllllll"IIIIIIIIIIIIIIIIIIIIIIII'"111111II~1"1fI111II1"11 OOiliilliirilliliilllllllllllllllllllllllllllllllllllllllllllllllllmml11111111111111111111111111111111111111111111111111111 ROW 8J
17
5Xi,
liliilli~llfiliilll"IIIIIIIIIIII1IIIII1II111I1I1I1I111I1I1IIIIII"IIJII1111111111111111111111111111111111111111111111111111111 := ROW 85
~381
IIIIIIIIIIIIIIlII/IIIIIIIIIIIII
1111111111111111111111111111111111111111111
lirnlill'millijiillllllllllJlIIIIIIIIIIllIIIIIIIIlIllllIlIlIIlllJlllllllmlllllll1111111111111111111111111111111111111111111
~g]iillmilli'ijlllllllllllllllllml""III1II""IIIIIIIIIII"IIIII~1I1I11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 mfilijllililliiiIlIIllIIllIIDII~mlll"'lIImllllllllllllllll~lmllllllllll~11111111111111111111111111111111111111111111 ROW Ig
!\64t
lil~llIlIlI~iiiIIllIJllllllllllllmmlllllll"'III""IIIIII11111lllll1ml"IIII~II"~II~IIIIIIIIIIIII"IIIIIII"'IIIIIII E-5
l-lULTIPLE CURVE FITTING PROGRAM
liWlillllifill~illlllllllllllllllllllllllllllllll~11 lil~iIMIIUiilllllmlllllllllllllllllllllllllllll~llllIlIlllllIllIll~1IlIlllllllIlIllRllmmllllllllllllllll~1111 liilUllillliliillllllllllll~1111I111I1111111I11I1II11
1111111111
111I111111~~IIIIIIIIII~III~llIIllIllllIllIlm~11I11111111I11I1I11
ROW W7 (630 . 1>121
IIIIIIIII~IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII~IIIIIIIIWIIIIIUIIIIIIIIIIIIIIIIIIIIIIIIIIIIII ROW ~8 (&13
ROW 101
(67~.
5531
8531
11I1II111I11I1II~lIIl1lllllll1ll1mlll
111111111111111111111111
ImiIIO~III~llmfllllllllllllllll~IIIIIIII1IIII1IIIII1II1I11~IIIIIIIMII1""II1IIII11III1II1I1~I~II~III~11111II11111 liiuillljifll~ifllllllllll~llm~lnllllllllllllllllllllllmllllglli11II~lllllIllllllllIlrnlllUlllmlllllll ROW IO~ t7~
71le)
1111111111 FlOW IO~
11111"lllllllllllIIlmIIUIIIIIIIIIII~I~IIIIIIIIIIII~111I11~IIII
liml~ljflilllmflllllllll~IIIIIIIIIII~IIIIIIIIIIIIIIIIIIIII~111111IIIIIIRIIIIUlllllmlllllllllllll~11111111I11I11111 flOW 107 inti' 7:M1
ROW 108 tiM
11
11111111111111111111111111111111111111111111111111111111111111
E-6
MULTIPLE CURVE FITTING PROGRAM
RO'il 101 (142
7411
Ilmllllllllllllllllllllllllllllllllllllllllnllll~IIII"~mMIII~~lmlllllll~~II"lllllmlmllllJllllllmlllll"'1I111
miljljllilIlilliiimlmlllllll~lIIl11mlllllllllllllllllllimmlllllllllmllIIUlIlIIIllUmlDllllnllllllllll1i ROW '"
(156
16JI
FlOW 112 (163
1711
1IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII~~llIlIIllmlll 111111111111111
11111111111
ROW 113 ~'", HI.
11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101 ROW 114
7&41
ROW 115 1715 719)
1IIIfllllllllllllllllll"IIIIIIIIIII~""1I11I1I11I11I1II11I11I111I11I1111I111I1mllllllllllllllllllllllllll"'IIIIIIIIIIIII""1I
illillli~lliil'illlllllllllllll~11111111111I1111I11111111111IIII ROW 1'1
1I111mllllllllllllili
_)
IIIIIIIIIIIIIIIIIIImllllllnl
iillilllijilliiillllllllllllllllllllUllIlIlIllIlIlIlJlIlIlIlIllIIIlIlIlIlIIlJIIlllIlI111111111111111""1111111111111111011 ROW 121
843)
FlOW Il2 (.....
1M)
IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIUOIIIIIIIIIIIIIIII
111111111111111111111111111111111111111111111111
ROW 123 (I&~. 8811
E-7
MULTIPLE CURVE FITTING PROGRAM
PlOW 127 1900: 9111
IIIIIIIIII~IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII~IIIII1IIIIIHI~IIIII~IIIIIIIIIIIIII~Mlmlllmllllllllllll~11111 PlOW 128 1911 : 11271
iliiflllmllli~~IIlIlIlIllIIIIlIIIIHIIIIIIIIIIIIIIII~I~1111111111111~ln~III~WIIII~~UIIII
E-8
REFERENCES
and R.R. COLTON. Statistical Methods, 5th AHKIN, H. New York: Barnes & Noble College Outline series, 197~.
Edition.
BROWNLEE, K.A. Statistical Theory and Methodology in Science and Engineering, 2nd Edition. New York: John Wiley & Sons, Inc., 1965. DANIEL, C. and F.S. WOOD. Fitting Equations to Data. John Wiley & Sons, Inc., 1971.
New York:
DRAPER, N.R. and H. SMITH. Applied Regression Analysis, 2nd Edition. New York: John Wiley & Sons, 1981. DYNACOMP, INC. Regression I. (A software package for home computers). New York: DYNACOMP, Inc. GENERAL ELECTRIC CO. Statistical Analysis System User's Guide, Revision C. Rockville, Maryland: General Electric Company, Information Services Business DiVision, 1975. FERGUSON, G.A. Statistical Analysis In Psychology & 3rd Edition. New York: McGraw-Hill Book Co., 1966. HINES, W.W. and D.C. MONTGOMERY. New York: Roland Press Co., 1972.
Probability
and
Education, Statistics.
KELLY, L.G. Handbook of Numerical Methods and Applications. Reading, Massachusetts: Addison-Wesley Publishing co., 1967. LENTNER, M. Elementary Applied Statistics. New York: Bogden & Quigley, Inc., 1972. MASON, R.D. 3rd Edition.
Tarrytown-on-Hudson,
Statistical Techniques in Business and Economics, Homewood, Illinois: Richard D. Irwin, Inc., 1974.
RESEARCH & EDUCATION ASSOCIATION. The Statistics Problem Solver. New York: Research and Education Association, 1978. SHARP CORPORATION. PC-1211 Sharp Pocket Computer Manual. Osaka, Japan, 1980.
Applications
SPIEG8L, M.R. Schaum's Outline of Theory and Problems of Probability and Statistics. New York: McGraw-Hill Book Co., 1975. SPIEGEL, M.R. Schaum's Outline of Theory and Problems of Statistics. New York: McGraw-Hill Book Co., 1961. STEFFEN, W.W. 11 Curves-Best Fit. Santa Ana, California: Calculator Journal, V8N6, Aug./Dec. 1981. TUKEY, J.W. Exploratory Data Analysis. Addison-Ivesley Publishing Co., 1977.
F-l
PPC
Reading, Massachusetts:
INDEX
beta distribution, 76 Cauchy distribution, 82 circle, 95 coefficient of determination, 7 combined linear & reciprocal, 27 correction exponential, 97 correlation coefficient, 7 cubic equation, 5 decomposition, 6 distributions, 5 beta, 76 Cauchy, 82 gamma, 79 log-normal, 73 normal, 79 exponential, 56 correction, 97 modif i ed, 58 F-test, 8 gamma distribution, 79 generalized polynomial 2nd order, 9" 3rd order, 92 geometric, 46, 52 modified, 54 goodness of fit, 7 Hoerl function, 64, 79 modified, 67 hyperbola, 30 reciprocal, 34 second order, 36 through a point, 32 isotonic linear regression, 21 learning curve, 46 linear regression, 4,13,15,17,19,21
G-l
INDEX
logarithmic, 6~ reciprocal, 62 logistic, 101:1 log-normal distribution, 73 multiple linear regression, 5, 85, 87 normal distribution, 5, HI. 11, 70 parabola, 39 through a point, 44 through the origin, 42 polynomial, 36, 49, 87, 89, 92 power, 46 modified, 48, 50, 56 reciprocal combined linear, 27 hype rbola, 34 logarithmic, 62 straight line, 23, 25 regress ion, 3 root, 50, 58 scaling, 7, 9 second order hyperbola, 37 polynomial, 90 significance, 8 straight line, 13 isotonic, 21 through a point, 17, 19 through the origin, 15 substitution, 6 third order polynomial, 92 transformation, 4
G-2
