Robust Nonlinear Regression with Applications using R

The first book to discuss robust aspects of nonlinear regression—with applications using R software

Robust Nonlinear Regression: with Applications using R covers a variety of theories and applications of nonlinear robust regression. It discusses both parts of the classic and robust aspects of nonlinear regression and focuses on outlier effects. It develops new methods in robust nonlinear regression and implements a set of objects and functions in S-language under SPLUS and R software. The software covers a wide range of robust nonlinear fitting and inferences, and is designed to provide facilities for computer users to define their own nonlinear models as an object, and fit models using classic and robust methods as well as detect outliers. The implemented objects and functions can be applied by practitioners as well as researchers. 

The book offers comprehensive coverage of the subject in 9 chapters: Theories of Nonlinear Regression and Inference; Introduction to R; Optimization; Theories of Robust Nonlinear Methods; Robust and Classical Nonlinear Regression with Autocorrelated and Heteroscedastic errors; Outlier Detection; R Packages in Nonlinear Regression; A New R Package in Robust Nonlinear Regression; and Object Sets.

• The first comprehensive coverage of this field covers a variety of both theoretical and applied topics surrounding robust nonlinear regression
• Addresses some commonly mishandled aspects of modeling
• R packages for both classical and robust nonlinear regression are presented in detail in the book and on an accompanying website

Hossein Riazoshams, PhD, is a full-time Faculty member at the Department of Mathematics and Statistics, Lamerd Islamic Azad University of Iran; Stockholm University, Sweden; and University of Putra, Malaysia.

Habshah Midi, PhD, is Professor at the Department of Mathematics, Faculty of Science and Institute for Mathematical Research, University of Putra, Malaysia.

Gebrenegus Ghilagaber, PhD, is Professor and Head at the Department of Statistics, Stockholm University, Sweden.

Preface xi

Acknowledgements xiii

About the Companion Website xv

Part One Theories 1

1 Robust Statistics and its Application in Linear Regression 3

1.1 Robust Aspects of Data 3

1.2 Robust Statistics and the Mechanism for Producing Outliers 4

1.3 Location and Scale Parameters 5

1.3.1 Location Parameter 5

1.3.2 Scale Parameters 9

1.3.3 Location and Dispersion Models 10

1.3.4 Numerical Computation of M-estimates 11

1.4 Redescending M-estimates 13

1.5 Breakdown Point 13

1.6 Linear Regression 16

1.7 The Robust Approach in Linear Regression 19

1.8 S-estimator 23

1.9 Least Absolute and Quantile Esimates 25

1.10 Outlier Detection in Linear Regression 27

1.10.1 Studentized and Deletion Studentized Residuals 27

1.10.2 Hadi Potential 28

1.10.3 Elliptic Norm (Cook Distance) 28

1.10.4 Difference in Fits 29

1.10.5 Atkinson’s Distance 29

1.10.6 DFBETAS 29

2 NonlinearModels: Concepts and Parameter Estimation 31

2.1 Introduction 31

2.2 Basic Concepts 32

2.3 Parameter Estimations 34

2.3.1 Maximum Likelihood Estimators 34

2.3.2 The Ordinary Least Squares Method 36

2.3.3 Generalized Least Squares Estimate 38

2.4 A NonlinearModel Example 39

3 Robust Estimators in Nonlinear Regression 41

3.1 Outliers in Nonlinear Regression 41

3.2 Breakdown Point in Nonlinear Regression 43

3.3 Parameter Estimation 44

3.4 Least Absolute and Quantile Estimates 44

3.5 Quantile Regression 45

3.6 Least Median of Squares 45

3.7 Least Trimmed Squares 47

3.8 Least Trimmed Differences 48

3.9 S-estimator 49

3.10 ��-estimator 50

3.11 MM-estimate 50

3.12 Environmental Data Examples 53

3.13 NonlinearModels 55

3.14 Carbon Dioxide Data 61

3.15 Conclusion 64

4 Heteroscedastic Variance 67

4.1 Definitions and Notations 69

4.2 Weighted Regression for the Nonparametric Variance Model 69

4.3 Maximum Likelihood Estimates 71

4.4 VarianceModeling and Estimation 72

4.5 Robust Multistage Estimate 74

4.6 Least Squares Estimate of Variance Parameters 75

4.7 Robust Least Squares Estimate of the Structural Variance Parameter 78

4.8 Weighted M-estimate 79

4.9 Chicken-growth Data Example 80

4.10 Toxicology Data Example 85

4.11 Evaluation and Comparison of Methods 87

5 Autocorrelated Errors 89

5.1 Introduction 89

5.2 Nonlinear Autocorrelated Model 90

5.3 The Classic Two-stage Estimator 91

5.4 Robust Two-stage Estimator 92

5.5 Economic Data 93

5.6 ARIMA(1,0,1)(0,0,1)7 Autocorrelation Function 103

6 Outlier Detection in Nonlinear Regression 107

6.1 Introduction 107

6.2 Estimation Methods 108

6.3 Point Influences 109

6.3.1 Tangential Plan Leverage 110

6.3.2 Jacobian Leverage 111

6.3.3 Generalized and Jacobian Leverages for M-estimator 112

6.4 Outlier DetectionMeasures 115

6.4.1 Studentized and Deletion Studentized Residuals 116

6.4.2 Hadi’s Potential 117

6.4.3 Elliptic Norm (Cook Distance) 117

6.4.4 Difference in Fits 118

6.4.5 Atkinson’s Distance 118

6.4.6 DFBETAS 118

6.4.7 Measures Based on Jacobian and MM-estimators 119

6.4.8 Robust Jacobian Leverage and Local Influences 119

6.4.9 Overview 121

6.5 Simulation Study 122

6.6 Numerical Example 128

6.7 Variance Heteroscedasticity 134

6.7.1 Heteroscedastic Variance Studentized Residual 136

6.7.2 Simulation Study, Heteroscedastic Variance 140

6.8 Conclusion 141

Part Two Computations 143

7 Optimization 145

7.1 Optimization Overview 145

7.2 Iterative Methods 146

7.3 Wolfe Condition 148

7.4 Convergence Criteria 149

7.5 Mixed Algorithm 150

7.6 Robust M-estimator 150

7.7 The Generalized M-estimator 151

7.8 Some Mathematical Notation 151

7.9 Genetic Algorithm 152

8 nlr Package 153

8.1 Overview 153

8.2 nl.form Object 154

8.2.1 selfStart Initial Values 159

8.3 Model Fit by nlr 161

8.3.1 Output Objects, nl.fitt 164

8.3.2 Output Objects, nl.fitt.gn 167

8.3.3 Output Objects, nl.fitt.rob 169

8.3.4 Output Objects, nl.fitt.rgn 169

8.4 nlr.control 170

8.5 Fault Object 172

8.6 Ordinary Least Squares 172

8.7 Robust Estimators 175

8.8 Heteroscedastic Variance Case 179

8.8.1 Chicken-growth Data Example 179

8.8.2 National Toxicology Study Program Data 183

8.9 Autocorrelated Errors 184

8.10 Outlier Detection 193

8.11 Initial Values and Self-start 201

9 Robust Nonlinear Regression in R 207

9.1 Lakes Data Examples 207

9.2 Simulated Data Examples 211

A nlr Database 215

A.1 Data Set used in the Book 215

A.1.1 Chicken-growth Data 216

A.1.2 Environmental Data 216

A.1.3 Lakes Data 218

A.1.4 Economic Data 221

A.1.5 National Texicology Program(NTP) Data 223

A.1.6 CowMilk Data 223

A.1.7 Simulated Outliers 225

A.1.8 Artificially Contaminated Data 227

A.2 Nonlinear Regression Models 227

A.3 Robust Loss FunctionsData Bases 229

A.4 Heterogeneous Variance Models 229

References 233

Index 239

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