Nonlinear Robust Regression
With Application using R.
My New R-Package "nlr" and Book entitled "Robust nonlinear regression with
application using R" is released.
- The Book is a comprehensive guide for theories and program guide for "nlr"
package.
- The "nlr" package in R, I created for the book include all computation
methods discussed in our book.
- The Book is available for sale in John Wiley Inc
International production website at:
- the "nlr| package is free available in R-CRAN at the
following Link.
https://cran.r-project.org/package=nlr
Package Documentary:
The package which I called "nlr" as abbreviation of "Non-Linear Robust" is set
of tools for fitting nonlinear regression models using robust methods, in
cllasical model assumtion, autocorrelated errors and heteroscedastic error
cases, and include tools to detect outliers.
Fourteen years of my long life passed to create it as the bellow package
history:
- I started the first programs in S-PLUS at 2005, the
objects, numerical methods beside theories I implemented at my Ph.D study during
2006-2009.
- From 2009 to 2012 I created new methods in
Heteroscedasticity and added to set of SPLUS program, but still it was not
written as a package.
- In my postdoc from 20012 to 2014 I moved it to R and
created first package and documentation files.
- From 2014 to March 2018, I completed, corected
errors, adjusted for the book, and transfered to LINUX for publishing.
- At 14-March 2018 I submitted to CRAN for first time.
- At last Package accepted and published at
21-Mar-2018
The package first functions written at 2005 and to 2018 took 14 years of long
continuous research and hard working. Without it my theories, and the book is
absolutely meaningful. That I can claim is more important than the book. It is
because this area is applied and lack computation tools, which without
computation tools the theories will not have effect.
During all these years many researchers from several countries contacted me and
request the programs, therefore I decided to implement the package and the book.
The package brings the nonlinear regression to a new era, in the sense that it
create theories, computation tools, and output of the package is object formats
that can be used by researchers not only to fit their own numerical examples but
also develop new theories and new computer tools, based on them.
Table of Content
Part One
Theories
1
1
Robust
Statistics
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
4
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
Robust
Approach in Linear Regression
18
1.8
S-Estimator
23
1.9
Least Absolute and Quantile Esimates
25
1.10
Outlier Detection in Linear Regression
26
1.10.1
Studentized and Deletion Studentized Residuals
27
1.10.2
Hadi Potential
27
1.10.3
Elliptic Norm (Cook
Distance)
28
1.10.4
Difference in
Fits
28
1.10.5
Atkinson’s Distance
28
1.10.6
DFBETAS
28
2
Nonlinear Models
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
37
2.4
A Nonlinear Model Example
39
3
Robust
Estimators in Nonlinear Regression
41
3.1
Outliers in Nonlinear Regression
41
3.2
Breakdown Point in Nonlinear Regression
42
3.3
Parameter Estimation
44
3.4
Least Absolute and Quantile Estimates
44
3.5
Quantile Regression
44
3.6
Least Median of
Squares
45
3.7
Least Trimmed Squares
47
3.8
Least Trimmed
Differences
48
3.9
S-Estimator
48
3.10
τ
-Estimator
50
3.11
MM-Estimate
50
3.12
Environmental Data Examples
53
3.13
Nonlinear models
54
3.14
Carbon Dioxide Data
60
3.15
Conclusion
61
4
Heteroscedastic Variance
65
4.1
Definitions and Notations
66
4.2
Weighted Regression for
the Nonparametric
Variance Model
67
4.3
Maximum Likelihood Estimates
68
4.4
Variance Modeling and Estimation
70
4.5
Robust
Multistage Estimate
72
4.6
Least Squares Based
Estimate of Variance Parameters
73
4.7
Robust
Least Squares Based Estimate of
the Structural Variance
Parameter
75
4.8
Weighted M-Estimate
76
4.9
Chicken Growth Data Example
77
4.10
Toxicology Data Example
81
4.11
Evaluation and Comparison of Methods
84
5
Autocorrelated Errors
85
5.1
Introduction
85
5.2
Nonlinear Autocorrelated Model
86
5.3
The Classic Two-stage Estimator
87
5.4
Robust
Two-stage Estimator
88
5.5
Economy Data
89
CONTENTS
5
5.6
ARIMA(1,0,1)(0,0,1)7 Autocorrelation Function
93
6
Outlier Detection in
Nonlinear Regression
103
6.1
Introduction
103
6.2
Estimation Methods
104
6.3
Point Influences
105
6.3.1
Tangential Plan Leverage
106
6.3.2
Jacobian Leverage
107
6.3.3
Generalized and Jacobian Leverage
for M-Estimator
108
|
6.4
|
Outlier
|
Detection Measures
|
111
|
|
|
6.4.1
|
Studentized and
Deletion Studentized Residuals
|
112
|
|
|
6.4.2
|
Hadi’s Potential
|
112
|
|
|
6.4.3
|
Elliptic Norm (Cook Distance)
|
113
|
|
|
6.4.4
|
Difference in Fits
|
113
|
|
|
6.4.5
|
Atkinson’s
Distance
|
113
|
|
|
6.4.6
|
DFBETAS
|
114
|
|
|
6.4.7
|
Measures based on
Jacobian and MM-Estimators
|
114
|
|
|
6.4.8
|
Robust Jacobian Leverage and
Local Influences
|
115
|
|
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6.4.9
|
Overview
|
116
|
6.5
Simulation Study
117
6.6
Numerical Example
118
6.7
Variance Heteroscedasticity
120
6.7.1
Heteroscedastic Variance Studentized Residual
132
6.7.2
Simulation Study, Hetroscedastic Variance
133
6.8
Conclusion
134
Part
Two Computations
137
7
Optimization
139
7.1
Optimization Overview
139
7.2
Iterative Methods
140
7.3
Wolfe Condition
142
7.4
Convergence Criteria
143
7.5
Mixed
Algorithm
143
7.6
Robust
M-Estimator
144
7.7
The Generalized M-Estimator
145
7.8
Some Mathematical Notation
145
7.9
Genetic Algorithm
146
8
nlr
Package
147
8.1
Overview
147
8.2
nl.form
Object
148
8.2.1
selfStart
Initial Values
154
8.3
Model Fit
by
nlr
155
8.3.1
Output Objects,
nl.fitt
159
8.3.2
Output Objects,
nl.fitt.gn
162
8.3.3
Output Objects,
nl.fitt.rob
164
8.3.4
Output Objects,
nl.fitt.rgn
164
8.4
nlr.control
165
8.5
Fault
Object
167
8.6
Ordinary Least Squares
168
8.7
Robust
Estimators
171
8.8
Heteroscedastic Variance Case
174
8.8.1
Chicken
Data Example
175
8.8.2
National Toxicology Study Program Data
179
8.9
Autocorrelated Errors
180
8.10
Outlier Detection
191
8.11
Initial Values and Self-start
198
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9
|
Robust
Nonlinear Regression
in R
|
205
|
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9.1
Lakes
Data Examples
|
205
|
|
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9.2
Simulated Data Examples
|
208
|
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A
|
”nlr” Database
|
215
|
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A.1
Data
Set used in The Book
215
|
|
A.1.1
|
Chicken
Growth Data
|
215
|
|
A.1.2
|
Environmental Data
|
216
|
|
A.1.3
|
Lakes Data
|
217
|
|
A.1.4
|
Economy Data
|
217
|
|
A.1.5
|
National Texicology Program(NTP) Data
|
218
|
|
A.1.6
|
Cow Milk Data
|
219
|
|
A.1.7
|
Simulated Outliers
|
219
|
|
A.1.8
|
Artificially Contaminated Data
|
220
|
|
A.2
|
Nonlinear Regression Models
|
221
|
|
A.3
|
Robust Loss functions Data Bases
|
221
|
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A.4
|
Heterogeneous Variance Models
|
222
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