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Tables of Contents for Fitting Equations to Data
Chapter/Section Title
Page #
Page Count
Introduction
1
4
Flow diagram of procedure used
1
1
Role of computer
1
2
Sequence of subjects discussed
3
2
Assumptions and Methods of Fitting Equations
5
14
Assumptions
5
1
Methods of fitting equations
6
1
Least squares
6
3
Linear least-squares estimation
9
1
Nonlinear least-squares estimation
9
10
Linear least-squares estimates--one independent variable
10
1
Assumptions
10
1
Basic idea and derivation
10
2
Confidence regions
12
1
Linear least-squares estimates--general case
13
1
Estimates of coefficients
13
2
Variance and standard errors of coefficients
15
1
Computer computations
16
1
Partitioning sums of squares
16
1
Multiple correlation coefficient squared, Ry2
17
1
F-value
17
1
Bias and random error
17
1
Residual mean square
17
1
Residual root mean square
18
1
One Independent Variable
19
31
Plotting data and selecting form of equation
19
1
Plots of linearizable equations
19
3
Plots of nonlinearizable equations
22
2
Statistical independence and clusters of dependent variable
24
1
Allocation of data points
25
1
Outliers
25
1
Use of computer program
26
1
Study of residuals
27
5
Dealing with error in the independent variable
32
1
Example of fitting a straight line to data-one independent variable
32
18
Cumulative distribution plots of random normal deviates
33
1
Example of fitting a straight line to data--one independent variable
33
17
Two or More Independent Variables
50
10
Inadequacies of x-y plots with two or more independent variables
50
3
Equation forms and transformations
53
2
Variances of estimated coefficients, bt, and fitted values, Yj
55
1
Use of indicator variables for discontinuous or qualitative classifications
56
1
One qualitative factor at two levels
One qualitative two-level factor and one quantitative (continuous) factor
Discrete factors at more than two levels
Discrete factors interacting with continuous factors
Allocation of data in factor space
57
3
Linear dependences among the xi
The outermost points in data space
Nested data
Fitting an Equation in Three Independent Variables
60
23
Introduction
60
1
First trials
60
1
Possible causes of disturbance
61
4
Outlier or logged response or squared independent variable
65
7
Random error estimated from near neighbors
72
1
Independence of observations
73
2
Systematic examination of alternatives discovered sequentially
75
2
Remote points in factor space
77
1
Equation using ``lined-out'' data
77
4
Conclusions on stack loss problem
81
1
General conclusions
82
1
Selection of Independent Variables
83
38
Introduction
83
3
Assumptions
Obvious imperfections
2K possible equations from K candidate variables
On stepwise regression
F-test
All 2K equations and fractions
Total squared error as a criterion for goodness of fit-CP
86
3
Definition
Derivation of the CP statistic
Mallows' graphical method of comparing fitted equations
A four-variable example
89
2
Disposition of data in factor space
91
4
A six-variable example
95
26
Search for all 2K equations
TK, t- directed search
Fractional replicates
Computer printouts of four-variable example
104
1
Computer printouts of six-variable example
104
1
Fractional replication for 2K equations
104
17
Some Consequences of the Disposition of the Data Points
121
113
Introduction
121
1
Description of example with ten independent variables
122
1
First steps
Interior analysis I. ``Component effects'' table
123
1
Interior analysis II. Component and component-plus-residual plots
124
2
Search of all 2K possible subset equations
126
1
tK, t-directed search
126
1
Test for an outlier
127
2
Interior analysis III. Weighted-squared-standardized-distance to look for far-out points of influential variables
129
2
Interior analysis IV Variance ratio to look for far-out points of uninfluential variables
131
2
Interior analysis V Error estimation from near neighbors
133
3
Cross verification
136
1
Other examples of error estimation from near neighbors
137
2
Six-variable example of Chapter 6
Stack loss example of Chapter 5
Additional examples of using component and component-plus-residual plots
139
6
Example of the use of plots in choosing the form of equation
Distribution of observations over each independent variable
Four-variable example-Distribution of independent variables
Eleven-variable example-Inner and outer observations
Other conditions identified by component and component-plus-residual plots
145
1
Summary
146
88
Computer printouts of ten-variable example
150
49
Computer printout of six-variable example
199
1
Computer printouts of stack loss example
200
7
Computer printouts of four-variable octane example
207
11
Computer printouts of eleven-variable example
218
14
Critical values for studentized residual to test for single outlier
232
2
Selection of Variables in Nested Data
234
33
Background of example
234
2
Fitting equation
236
1
Recognition of nested data
236
1
Data identification and entry
236
1
Fitting equation to nested data
236
4
Use of indicator (dummy) variables
Test for significance of added variables; ``individual'' versus ``common'' slopes
Fitting equation among sets of nested data
240
3
Selecting variables based on total error, recognizing bias and random error
243
1
Properties of final equation
244
3
Comparison of equations
247
20
Data preparation and computer printouts of example
248
19
Nonlinear Least Squares, a Complex Example
267
71
Introduction
267
1
Background of example
268
3
Replicates
271
2
Potential equations
273
3
Nonlinear fit-observations of individual cements versus time
276
1
Fit with indicator variables
277
3
Fit with composition
280
13
Possible nesting within cements
293
2
Comparison with previous linear least-square fits
295
1
Fit using composition in mol percents
296
3
Conclusions
299
39
Computer printout of nonlinear example
300
38
Glossary
338
16
Conventions
338
1
Symbols
338
5
Computer terms
343
11
LINWOOD User's Manual
354
66
LINWOOD, a computer linear least-squares curve-fitting program
Abstract
354
1
General information
355
1
Input
356
20
Control card entry
356
10
Standard data card entry
366
1
Problems using transformation option
367
7
Order of cards
374
2
Sample problem
376
5
Example of weighted observations
381
15
Example to measure the precision of calculations
396
10
Example of using plots in choosing the form of equation
406
9
Execution time
415
3
Summary of control cards
418
1
Summary of order of cards
419
1
NONLINWOOD User's Manual
420
29
NONLINWOOD, a computer nonlinear least-squares curve-fitting program
Abstract
420
1
General information
421
1
Input
422
6
Control card
422
2
Format card
424
1
Starting values of coefficients
424
1
Data cards
424
1
Order of cards
425
1
Control and data card entry forms
425
1
Equation subroutine
425
2
Example of transformations
427
1
Example 1 2 coefficients 2 variables
428
14
Example 2 43 coefficients 16 variables
442
2
Example 3 19 coefficients 7 variables
444
2
Summary of control cards
446
1
Example of subroutine model
447
1
Summary of order of cards
447
2
Bibliography
449
4
Index
453