Curve Fitting for Programmable Calculators

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

b0 b0 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 b0 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.

b0

/z _ _ _ _ _ _ _ -. . .:bs. .: >~l .::!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.

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71

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E-6

MULTIPLE CURVE FITTING PROGRAM

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MULTIPLE CURVE FITTING PROGRAM

PlOW 127 1900: 9111

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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