Linear least squares computations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Dekker
1988
|
| Edition: | 1. print. |
| Series: | Statistics
91 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 293 S. |
| ISBN: | 0824776615 |
Staff View
MARC
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| 100 | 1 | |a Farebrother, Richard William |d 1946- |e Verfasser |0 (DE-588)12497841X |4 aut | |
| 245 | 1 | 0 | |a Linear least squares computations |
| 250 | |a 1. print. | ||
| 264 | 1 | |a New York [u.a.] |b Dekker |c 1988 | |
| 300 | |a XIII, 293 S. | ||
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| adam_text | Contents
PREFACE iii
1 THE GAUSS AND GAUSS JORDAN
METHODS: ASPECTS OF COMPUTER
PROGRAMMING 1
1.1 Introduction 1
1.2 Gauss s Method 3
1.3 Gauss s Method with Row Interchanges 8
1.4 The Gauss Jordan Method 10
1.5 Arithmetical Cost 11
1.6 Efficient Programming 13
1.7 Computer Representation of Numbers 15
1.8 A Measure of Computational Accuracy 16
1.9 Gauss s Method with Integer Coefficients 17
Exercises 18
References 20
2 MATRIX ANALYSIS OF GAUSS S METHOD:
THE CHOLESKY AND DOOLITTLE
DECOMPOSITIONS 23
2.1 Matrix Representation 23
2.2 Matrix Multiplication 26
vii
viii Contents
2.3 Matrix Inversion 27
2.4 Elementary Matrices 30
2.5 Matrix Analysis of Gauss s Method 32
2.6 The Determinant 34
2.7 Doolittle s LqU^, Decomposition 35
2.8 The U 0DU0 Decomposition 37
2.9 Cholesky s U U Decomposition 38
2.10 Horst s Method 40
Exercises 42
References 43
3 THE LINEAR ALGEBRAIC MODEL: THE
METHOD OF AVERAGES AND THE METHOD
OF LEAST SQUARES 45
3.1 The Linear Algebraic Model 45
3.2 The Method of Averages 46
3.3 The Method of Least Squares 49
3.4 Accuracy 50
3.5 Empirical Condition Number 51
3.6 Longley s Test Problem 52
Exercises 53
References 54
4 THE CAUCHY BIENAYME, LAPLACE, AND
SCHMIDT PROCEDURES 59
4.1 The Cauchy Bienayme Procedure I 59
4.2 The Cauchy Bienayme Procedure II 63
4.3 The Laplace Orthogonalization Procedure 66
4.4 The Schmidt Orthogonalization Procedure 68
4.5 Comparison of the Schmidt and Laplace
Procedures 69
4.6 Laplace s Procedure with Column Interchanges 71
4.7 Uniqueness of the UqDU0 Decomposition 72
4.8 Partial Orthogonalization and Scaling 73
Exercises 75
References 76
Contents ix
5 HOUSEHOLDER S PROCEDURE 81
5.1 Householder s Procedure 81
5.2 Householder s Transformation Matrix 84
5.3 Comparison of the Householder and Laplace
Procedures 89
5.4 Further Remarks on Householder s Procedure 90
5.5 Maindonald s Variant of Householder s
Procedure 91
5.6 Householder s Procedure with Column
Interchanges 93
Exercises 94
References 95
6 GIVENS S PROCEDURE 97
6.1 Givens Transformation Matrix 97
6.2 Givens s Procedure 99
6.3 A Revised Version of Givens s Procedure 101
6.4 Partial Orthogonalization 103
6.5 Square Root Free Variants of Givens s
Procedure 106
6.6 Gentleman s Procedure 109
6.7 The Weighted Least Squares Estimator 112
6.8 Deleting Observations 113
6.9 Imposing Constraints 115
6.10 Stirling s Procedure 118
6.11 Generalized Givens Transformations with
Integer Coefficients 120
Exercises 123
References 126
7 UPDATING THE QU DECOMPOSITION 127
7.1 Adding Rows 127
7.2 Deleting Rows 129
7.3 Adding Columns 131
7.4 Deleting Columns 132
7.5 Permuting Columns 133
7.6 All Possible Regressions 135
x Contents
7.7 Adding Dummy Rows and Dummy Columns 137
Exercises 139
References 140
8 PSEUDO RANDOM NUMBERS 141
8.1 Data Precision 141
8.2 Multiplicative Congruential Pseudo Random
Number Generators 142
8.3 Uniformly Distributed Pseudo Random
Numbers 145
8.4 Mean and Variance 146
8.5 Normally Distributed Pseudo Random
Numbers 147
Exercises 148
Project: A Simulation Study 149
References 151
9 THE STANDARD LINEAR MODEL 153
9.1 The Linear Statistical Model 153
9.2 The Expectation and Variance of a Random
Variable 155
9.3 The Standard Linear Model 156
9.4 The Expectation and Variance of an Estimator 158
9.5 The Expectation and Variance of /? 159
9.6 The Least Squares Estimator of a2 161
9.7 The Expected Results of the Simulation Study 162
Project: A Least Squares Computer Program 164
References 165
10 CONDITION NUMBERS 167
10.1 Theoretical Condition Numbers 167
10.2 Empirical Condition Numbers 170
10.3 Perturbations of the Full Data Set 170
Exercises 171
References 172
Contents x;
11 INSTRUMENTAL VARIABLE ESTIMATORS 173
11.1 The Instrumental Variable Estimator I 173
11.2 A Nonsymmetric Variant of Householder s
Procedure 174
11.3 Nonsymmetric Householder Transformation
Matrices 177
11.4 The Expectation and Variance of an
Instrumental Variable Estimator 179
11.5 The Instrumental Variable Estimator II 180
11.6 Computing Instrumental Variable Estimators
by Householder s Procedure 182
Exercises 183
References 184
12 GENERALIZED LEAST SQUARES
ESTIMATION 185
12.1 Grouped Data 185
12.2 Three Linear Models 188
12.3 The Weighted Least Squares and Generalized
Least Squares Estimators 189
12.4 Elementary Computational Procedures 191
12.5 Generalized Least Squares Estimation by
Householder Transformations 192
12.6 A Reexamination of Householder s Procedure 195
12.7 The Reverse Cholesky Decomposition 196
12.8 Generalized Least Squares Estimation by
Givens Transformations I 198
12.9 Generalized Least Squares Estimation by
Givens Transformations II 201
12.10 Updating the Generalized Least Squares
Estimates 202
12.11 Special Methods for the Reverse Cholesky
Decomposition 206
12.12 Generalizations of the Generalized Least
Squares Problem 208
12.13 Estimation in Rank Deficient Models 211
Exercises 214
References 216
xii Contents
13 ITERATIVE SOLUTIONS OF LINEAR AND
NONLINEAR LEAST SQUARES PROBLEMS 217
13.1 Iterative Refinement of the Generalized Least
Squares Estimator 217
13.2 Iterative Refinement of the Ordinary Least
Squares Estimator 219
13.3 Iterative Solution of Nonlinear Least Squares
Problems 223
Exercises 226
References 227
14 CANONICAL EXPRESSIONS FOR THE LEAST
SQUARES ESTIMATORS AND TEST
STATISTICS 229
14.1 The Canonical Form of the Standard Linear
Model 229
14.2 Unbiased Estimators of P and a2 231
14.3 The Minimum Variance Unbiased Linear
Estimator of P 232
14.4 A Statistical Test for Deleted Regressors I 234
14.5 Distributional Assumptions 235
14.6 A Statistical Test for Deleted Regressors II 238
14.7 A Statistical Test for Additional Observations 239
Exercises 240
Project: A Simulation Study of the Deletion
Test 241
References 242
15 TRADITIONAL EXPRESSIONS FOR THE
LEAST SQUARES UPDATING FORMULAS
AND TEST STATISTICS 245
15.1 Adding Regressors I 245
15.2 Adding Regressors II 247
15.3 The Inverse of a Partitioned Matrix 251
15.4 Adding Observations I 252
15.5 Adding Observations II 253
15.6 Deleting Observations 256
Contents xiii
15.7 Imposing Constraints 257
15.8 A Statistical Test for Linear Constraints 259
Exercises 261
References 263
16 LEAST SQUARES ESTIMATION SUBJECT TO
LINEAR EQUALITY CONSTRAINTS 265
16.1 Inconsistent Constraints 265
16.2 The General Form of the Constrained Least
Squares Estimator 267
16.3 A Particular Form of the Constrained Least
Squares Estimator 268
16.4 The Constrained Minimum Variance
Unbiased Linear Estimator of P 269
16.5 Traditional Expressions for the Constrained
Least Squares Estimator 271
16.6 The Distribution of the Test for Linear
Constraints 272
Exercises 273
References 274
SUPPLEMENTARY READING 275
GLOSSARY OF MATRIX ALGEBRA 277
AUTHOR INDEX 283
SUBJECT INDEX 287
|
| adam_txt |
Contents
PREFACE iii
1 THE GAUSS AND GAUSS JORDAN
METHODS: ASPECTS OF COMPUTER
PROGRAMMING 1
1.1 Introduction 1
1.2 Gauss's Method 3
1.3 Gauss's Method with Row Interchanges 8
1.4 The Gauss Jordan Method 10
1.5 Arithmetical Cost 11
1.6 Efficient Programming 13
1.7 Computer Representation of Numbers 15
1.8 A Measure of Computational Accuracy 16
1.9 Gauss's Method with Integer Coefficients 17
Exercises 18
References 20
2 MATRIX ANALYSIS OF GAUSS'S METHOD:
THE CHOLESKY AND DOOLITTLE
DECOMPOSITIONS 23
2.1 Matrix Representation 23
2.2 Matrix Multiplication 26
vii
viii Contents
2.3 Matrix Inversion 27
2.4 Elementary Matrices 30
2.5 Matrix Analysis of Gauss's Method 32
2.6 The Determinant 34
2.7 Doolittle's LqU^, Decomposition 35
2.8 The U'0DU0 Decomposition 37
2.9 Cholesky's U'U Decomposition 38
2.10 Horst's Method 40
Exercises 42
References 43
3 THE LINEAR ALGEBRAIC MODEL: THE
METHOD OF AVERAGES AND THE METHOD
OF LEAST SQUARES 45
3.1 The Linear Algebraic Model 45
3.2 The Method of Averages 46
3.3 The Method of Least Squares 49
3.4 Accuracy 50
3.5 Empirical Condition Number 51
3.6 Longley's Test Problem 52
Exercises 53
References 54
4 THE CAUCHY BIENAYME, LAPLACE, AND
SCHMIDT PROCEDURES 59
4.1 The Cauchy Bienayme Procedure I 59
4.2 The Cauchy Bienayme Procedure II 63
4.3 The Laplace Orthogonalization Procedure 66
4.4 The Schmidt Orthogonalization Procedure 68
4.5 Comparison of the Schmidt and Laplace
Procedures 69
4.6 Laplace's Procedure with Column Interchanges 71
4.7 Uniqueness of the UqDU0 Decomposition 72
4.8 Partial Orthogonalization and Scaling 73
Exercises 75
References 76
Contents ix
5 HOUSEHOLDER'S PROCEDURE 81
5.1 Householder's Procedure 81
5.2 Householder's Transformation Matrix 84
5.3 Comparison of the Householder and Laplace
Procedures 89
5.4 Further Remarks on Householder's Procedure 90
5.5 Maindonald's Variant of Householder's
Procedure 91
5.6 Householder's Procedure with Column
Interchanges 93
Exercises 94
References 95
6 GIVENS'S PROCEDURE 97
6.1 Givens Transformation Matrix 97
6.2 Givens's Procedure 99
6.3 A Revised Version of Givens's Procedure 101
6.4 Partial Orthogonalization 103
6.5 Square Root Free Variants of Givens's
Procedure 106
6.6 Gentleman's Procedure 109
6.7 The Weighted Least Squares Estimator 112
6.8 Deleting Observations 113
6.9 Imposing Constraints 115
6.10 Stirling's Procedure 118
6.11 Generalized Givens Transformations with
Integer Coefficients 120
Exercises 123
References 126
7 UPDATING THE QU DECOMPOSITION 127
7.1 Adding Rows 127
7.2 Deleting Rows 129
7.3 Adding Columns 131
7.4 Deleting Columns 132
7.5 Permuting Columns 133
7.6 All Possible Regressions 135
x Contents
7.7 Adding Dummy Rows and Dummy Columns 137
Exercises 139
References 140
8 PSEUDO RANDOM NUMBERS 141
8.1 Data Precision 141
8.2 Multiplicative Congruential Pseudo Random
Number Generators 142
8.3 Uniformly Distributed Pseudo Random
Numbers 145
8.4 Mean and Variance 146
8.5 Normally Distributed Pseudo Random
Numbers 147
Exercises 148
Project: A Simulation Study 149
References 151
9 THE STANDARD LINEAR MODEL 153
9.1 The Linear Statistical Model 153
9.2 The Expectation and Variance of a Random
Variable 155
9.3 The Standard Linear Model 156
9.4 The Expectation and Variance of an Estimator 158
9.5 The Expectation and Variance of /? 159
9.6 The Least Squares Estimator of a2 161
9.7 The Expected Results of the Simulation Study 162
Project: A Least Squares Computer Program 164
References 165
10 CONDITION NUMBERS 167
10.1 Theoretical Condition Numbers 167
10.2 Empirical Condition Numbers 170
10.3 Perturbations of the Full Data Set 170
Exercises 171
References 172
Contents x;
11 INSTRUMENTAL VARIABLE ESTIMATORS 173
11.1 The Instrumental Variable Estimator I 173
11.2 A Nonsymmetric Variant of Householder's
Procedure 174
11.3 Nonsymmetric Householder Transformation
Matrices 177
11.4 The Expectation and Variance of an
Instrumental Variable Estimator 179
11.5 The Instrumental Variable Estimator II 180
11.6 Computing Instrumental Variable Estimators
by Householder's Procedure 182
Exercises 183
References 184
12 GENERALIZED LEAST SQUARES
ESTIMATION 185
12.1 Grouped Data 185
12.2 Three Linear Models 188
12.3 The Weighted Least Squares and Generalized
Least Squares Estimators 189
12.4 Elementary Computational Procedures 191
12.5 Generalized Least Squares Estimation by
Householder Transformations 192
12.6 A Reexamination of Householder's Procedure 195
12.7 The Reverse Cholesky Decomposition 196
12.8 Generalized Least Squares Estimation by
Givens Transformations I 198
12.9 Generalized Least Squares Estimation by
Givens Transformations II 201
12.10 Updating the Generalized Least Squares
Estimates 202
12.11 Special Methods for the Reverse Cholesky
Decomposition 206
12.12 Generalizations of the Generalized Least
Squares Problem 208
12.13 Estimation in Rank Deficient Models 211
Exercises 214
References 216
xii Contents
13 ITERATIVE SOLUTIONS OF LINEAR AND
NONLINEAR LEAST SQUARES PROBLEMS 217
13.1 Iterative Refinement of the Generalized Least
Squares Estimator 217
13.2 Iterative Refinement of the Ordinary Least
Squares Estimator 219
13.3 Iterative Solution of Nonlinear Least Squares
Problems 223
Exercises 226
References 227
14 CANONICAL EXPRESSIONS FOR THE LEAST
SQUARES ESTIMATORS AND TEST
STATISTICS 229
14.1 The Canonical Form of the Standard Linear
Model 229
14.2 Unbiased Estimators of P and a2 231
14.3 The Minimum Variance Unbiased Linear
Estimator of P 232
14.4 A Statistical Test for Deleted Regressors I 234
14.5 Distributional Assumptions 235
14.6 A Statistical Test for Deleted Regressors II 238
14.7 A Statistical Test for Additional Observations 239
Exercises 240
Project: A Simulation Study of the Deletion
Test 241
References 242
15 TRADITIONAL EXPRESSIONS FOR THE
LEAST SQUARES UPDATING FORMULAS
AND TEST STATISTICS 245
15.1 Adding Regressors I 245
15.2 Adding Regressors II 247
15.3 The Inverse of a Partitioned Matrix 251
15.4 Adding Observations I 252
15.5 Adding Observations II 253
15.6 Deleting Observations 256
Contents xiii
15.7 Imposing Constraints 257
15.8 A Statistical Test for Linear Constraints 259
Exercises 261
References 263
16 LEAST SQUARES ESTIMATION SUBJECT TO
LINEAR EQUALITY CONSTRAINTS 265
16.1 Inconsistent Constraints 265
16.2 The General Form of the Constrained Least
Squares Estimator 267
16.3 A Particular Form of the Constrained Least
Squares Estimator 268
16.4 The Constrained Minimum Variance
Unbiased Linear Estimator of P 269
16.5 Traditional Expressions for the Constrained
Least Squares Estimator 271
16.6 The Distribution of the Test for Linear
Constraints 272
Exercises 273
References 274
SUPPLEMENTARY READING 275
GLOSSARY OF MATRIX ALGEBRA 277
AUTHOR INDEX 283
SUBJECT INDEX 287 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:18:34Z |
| indexdate | 2024-07-09T20:51:31Z |
| institution | BVB |
| isbn | 0824776615 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015371581 |
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| physical | XIII, 293 S. |
| publishDate | 1988 |
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| publisher | Dekker |
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| series | Statistics |
| series2 | Statistics |
| spelling | Farebrother, Richard William 1946- Verfasser (DE-588)12497841X aut Linear least squares computations 1. print. New York [u.a.] Dekker 1988 XIII, 293 S. txt rdacontent n rdamedia nc rdacarrier Statistics 91 Kleinste-kwadratenmethode gtt Matrices gtt Moindres carrés Statistische modellen gtt Least squares Methode der kleinsten Quadrate (DE-588)4038974-1 gnd rswk-swf (DE-588)4143389-0 Aufgabensammlung gnd-content Methode der kleinsten Quadrate (DE-588)4038974-1 s DE-604 Statistics 91 (DE-604)BV000003265 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015371581&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Farebrother, Richard William 1946- Linear least squares computations Statistics Kleinste-kwadratenmethode gtt Matrices gtt Moindres carrés Statistische modellen gtt Least squares Methode der kleinsten Quadrate (DE-588)4038974-1 gnd |
| subject_GND | (DE-588)4038974-1 (DE-588)4143389-0 |
| title | Linear least squares computations |
| title_auth | Linear least squares computations |
| title_exact_search | Linear least squares computations |
| title_exact_search_txtP | Linear least squares computations |
| title_full | Linear least squares computations |
| title_fullStr | Linear least squares computations |
| title_full_unstemmed | Linear least squares computations |
| title_short | Linear least squares computations |
| title_sort | linear least squares computations |
| topic | Kleinste-kwadratenmethode gtt Matrices gtt Moindres carrés Statistische modellen gtt Least squares Methode der kleinsten Quadrate (DE-588)4038974-1 gnd |
| topic_facet | Kleinste-kwadratenmethode Matrices Moindres carrés Statistische modellen Least squares Methode der kleinsten Quadrate Aufgabensammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015371581&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003265 |
| work_keys_str_mv | AT farebrotherrichardwilliam linearleastsquarescomputations |