Nonparametric regression and generalized linear models: a roughness penalty approach
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
London u.a.
Chapman & Hall
1994
|
| Ausgabe: | 1. ed. |
| Schriftenreihe: | Monographs on statistics and applied probability
58 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006392140&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | IX, 182 S. Ill., graph. Darst. |
| ISBN: | 0412300400 |
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| 100 | 1 | |a Green, Peter J. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Nonparametric regression and generalized linear models |b a roughness penalty approach |c P. J. Green and B. W. Silverman |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
Preface xi
1 Introduction 1
1.1 Approaches to regression 1
1.1.1 Linear regression 1
1.1.2 Polynomial regression 2
1.2 Roughness penalties 2
1.2.1 The aims of curve fitting 2
1.2.2 Quantifying the roughness of a curve 4
1.2.3 Penalized least squares regression 5
1.3 Extensions of the roughness penalty approach 7
1.4 Computing the estimates 9
1.5 Further reading 9
2 Interpolating and smoothing splines 11
2.1 Cubic splines 11
2.1.1 What is a cubic spline? 11
2.1.2 The value second derivative representation 12
2.2 Interpolating splines 13
2.2.1 Constructing the interpolating natural cubic spline 15
2.2.2 Optimality properties of the natural cubic spline
interpolant 16
2.3 Smoothing splines 17
2.3.1 Restricting the class of functions to be considered 18
2.3.2 Existence and uniqueness of the minimizing
spline curve 18
2.3.3 The Reinsch algorithm 19
2.3.4 Some concluding remarks 21
2.4 Plotting a natural cubic spline 22
vi CONTENTS
2.4.1 Constructing a cubic given values and second
derivatives at the ends of an interval 22
2.4.2 Plotting the entire cubic spline 23
2.5 Some background technical properties 24
2.5.1 The key property for g to be a natural cubic spline 24
2.5.2 Expressions for the roughness penalty 24
2.6 Band matrix manipulations 25
2.6.1 The Cholesky decomposition 26
2.6.2 Solving linear equations 26
3 One dimensional case: further topics 29
3.1 Choosing the smoothing parameter 29
3.2 Cross validation 30
3.2.1 Efficient calculation of the cross validation score 31
3.2.2 Finding the diagonal elements of the hat matrix 33
3.3 Generalized cross validation 35
3.3.1 The basic idea 35
3.3.2 Computational aspects 35
3.3.3 Leverage values 36
3.3.4 Degrees of freedom 37
3.4 Estimating the residual variance 38
3.4.1 Local differencing 38
3.4.2 Residual sum of squares about a fitted curve 39
3.4.3 Some comparisons 40
3.5 Weighted smoothing 40
3.5.1 Basic properties of the weighted formulation 41
3.5.2 The Reinsch algorithm for weighted smoothing 41
3.5.3 Cross validation for weighted smoothing 42
3.5.4 Tied design points 43
3.6 The basis functions approach 44
3.6.1 Details of the calculations 46
3.7 The equivalent kernel 47
3.7.1 Roughness penalty and kernel methods 47
3.7.2 Approximating the weight function 47
3.8 The philosophical basis of roughness penalties 49
3.8.1 Penalized likelihood 50
3.8.2 The bounded roughness approach 51
3.8.3 The Bayesian approach 51
3.8.4 A finite dimensional Bayesian formulation 54
3.8.5 Bayesian inference for functionals of the curve 55
3.9 Nonparametric Bayesian calibration 57
3.9.1 The monotonicity constraint 58
CONTENTS vii
3.9.2 Accounting for the error in the prediction
observation 59
3.9.3 Considerations of efficiency 59
3.9.4 An application in forensic odontology 60
4 Partial splines 63
4.1 Introduction 63
4.2 The semiparametric formulation 64
4.3 Penalized least squares for semiparametric models 65
4.3.1 Incidence matrices 65
4.3.2 Characterizing the minimum 65
4.3.3 Uniqueness of the solution 66
4.3.4 Finding the solution in practice 68
4.3.5 A direct method 69
4.4 Cross validation for partial spline models 70
4.5 A marketing example 71
4.5.1 A partial spline approach 72
4.5.2 Comparison with a parametric model 73
4.6 Application to agricultural field trials 75
4.6.1 A discrete roughness penalty approach 75
4.6.2 Two spring barley trials 77
4.7 The relation between weather and electricity sales 79
4.7.1 The observed data and the model assumed 79
4.7.2 Estimating the temperature response 81
4.8 Additive models 83
4.9 An alternative approach to partial spline fitting 85
4.9.1 Speckman s algorithm 85
4.9.2 Application: the marketing data 86
4.9.3 Comparison with the penalized least squares
method 86
5 Generalized linear models 89
5.1 Introduction 89
5.1.1 Unifying regression models 89
5.1.2 Extending the model 90
5.2 Generalized linear models 91
5.2.1 Exponential families 91
5.2.2 Maximum likelihood estimation 93
5.2.3 Fisher scoring 94
5.2.4 Iteratively reweighted least squares 95
5.2.5 Inference in GLMs 95
5.3 A first look at nonparametric GLMs 98
viii CONTENTS
5.3.1 Relaxing parametric assumptions 98
5.3.2 Penalizing the log likelihood 98
5.3.3 Finding the solution by Fisher scoring 99
5.3.4 Application: estimating actuarial death rates 101
5.4 Semiparametric generalized linear models 104
5.4.1 Maximum penalized likelihood estimation 105
5.4.2 Finding maximium penalized likelihood esti¬
mates by Fisher scoring 105
5.4.3 Cross validation for GLMs 107
5.4.4 Inference in semiparametric GLMs 110
5.5 Application: tumour prevalence data 111
5.6 Generalized additive models 112
6 Extending the model 115
6.1 Introduction 115
6.2 The estimation of branching curves 115
6.2.1 An experiment on sunflowers 116
6.2.2 The estimation method 116
6.2.3 Some results 118
6.3 Correlated responses and non diagonal weights 120
6.4 Nonparametric link functions 121
6.4.1 Application to the tumour prevalence data 124
6.5 Composite likelihood function regression models 125
6.6 A varying coefficient model with censoring 126
6.7 Nonparametric quantile regression 129
6.7.1 The LMS method 129
6.7.2 Estimating the curves 130
6.8 Quasi likelihood 134
7 Thin plate splines 137
7.1 Introduction 137
7.2 Basic definitions and properties 137
7.2.1 Quantifying the smoothness of a surface 138
7.3 Natural cubic splines revisited 139
7.4 Definition of thin plate splines 142
7.5 Interpolation 143
7.5.1 Constructing the interpolant 143
7.5.2 An optimality property 144
7.5.3 An example 144
7.6 Smoothing 147
7.6.1 Constructing the thin plate spline smoother 147
7.6.2 An example 148
CONTENTS ix
7.6.3 Non singularity of the defining linear system 148
7.7 Finite window thin plate splines 150
7.7.1 Formulation of the finite window problem 150
7.7.2 An example of finite window interpolation and
smoothing 151
7.7.3 Some mathematical details 153
7.8 Tensor product splines 155
7.8.1 Constructing tensor products of one dimensional
families 155
7.8.2 A basis function approach to finite window
roughness penalties 156
7.9 Higher order roughness functionals 159
7.9.1 Higher order thin plate splines 160
8 Available software 163
8.1 Routines within the S language 163
8.1.1 The S routine smooth, spline 163
8.1.2 The new S modelling functions 164
8.2 The GCVPACK routines 165
8.2.1 Details of the algorithm 166
References 169
Author index 175
Subject index 177
|
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| id | DE-604.BV009666151 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T09:40:12Z |
| institution | BVB |
| isbn | 0412300400 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006392140 |
| oclc_num | 29635089 |
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| physical | IX, 182 S. Ill., graph. Darst. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Chapman & Hall |
| record_format | marc |
| series | Monographs on statistics and applied probability |
| series2 | Monographs on statistics and applied probability |
| spellingShingle | Green, Peter J. Silverman, Bernard W. Nonparametric regression and generalized linear models a roughness penalty approach Monographs on statistics and applied probability Analyse de régression ram Lineaire modellen gtt Non-parametrische statistiek gtt Regressieanalyse gtt Statistique non-paramétrique ram Regression analysis Nonparametric statistics Analyse de régression Statistique non-paramétrique Nichtparametrisches Verfahren (DE-588)4339273-8 gnd Lineare Regression (DE-588)4167709-2 gnd Lineares Regressionsmodell (DE-588)4127971-2 gnd |
| subject_GND | (DE-588)4339273-8 (DE-588)4167709-2 (DE-588)4127971-2 |
| title | Nonparametric regression and generalized linear models a roughness penalty approach |
| title_auth | Nonparametric regression and generalized linear models a roughness penalty approach |
| title_exact_search | Nonparametric regression and generalized linear models a roughness penalty approach |
| title_full | Nonparametric regression and generalized linear models a roughness penalty approach P. J. Green and B. W. Silverman |
| title_fullStr | Nonparametric regression and generalized linear models a roughness penalty approach P. J. Green and B. W. Silverman |
| title_full_unstemmed | Nonparametric regression and generalized linear models a roughness penalty approach P. J. Green and B. W. Silverman |
| title_short | Nonparametric regression and generalized linear models |
| title_sort | nonparametric regression and generalized linear models a roughness penalty approach |
| title_sub | a roughness penalty approach |
| topic | Analyse de régression ram Lineaire modellen gtt Non-parametrische statistiek gtt Regressieanalyse gtt Statistique non-paramétrique ram Regression analysis Nonparametric statistics Analyse de régression Statistique non-paramétrique Nichtparametrisches Verfahren (DE-588)4339273-8 gnd Lineare Regression (DE-588)4167709-2 gnd Lineares Regressionsmodell (DE-588)4127971-2 gnd |
| topic_facet | Analyse de régression Lineaire modellen Non-parametrische statistiek Regressieanalyse Statistique non-paramétrique Regression analysis Nonparametric statistics Nichtparametrisches Verfahren Lineare Regression Lineares Regressionsmodell |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006392140&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002494005 |
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