Generalized Additive Models: An Introduction with R

by Wood; Simon N.

Edition: 1st

ISBN13: 9781584884743

ISBN10: 1584884746

Format: Hardcover

Pub. Date: 2006-02-27

Publisher(s): Chapman & Hall/

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Summary

Now in widespread use, generalized additive models (GAMs) have evolved into a standard statistical methodology of considerable flexibility. While there is an outstanding research monograph on GAMs, there has been a long-standing need for an accessible introductory treatment of the subject.

Author Biography

Simon N. Wood is a professor of Statistics at the University of Bath, UK.

Preface	p. xv
Linear Models	p. 1
A simple linear model	p. 2
Simple least squares estimation	p. 3
Sampling properties of [beta]	p. 3
So how old is the universe?	p. 5
Adding a distributional assumption	p. 7
Testing hypotheses about [beta]	p. 7
Confidence intervals	p. 9
Linear models in general	p. 10
The theory of linear models	p. 12
Least squares estimation of [beta]	p. 12
The distribution of [beta]	p. 13
[characters not reproducible]	p. 14
F-ratio results	p. 15
The influence matrix	p. 16
The residuals, [epsilon], and fitted values, [mu]	p. 16
Results in terms of X	p. 17
The Gauss Markov Theorem: What's special about least squares?	p. 17
The geometry of linear modelling	p. 18
Least squares	p. 19
Fitting by orthogonal decompositions	p. 20
Comparison of nested models	p. 21
Practical linear modelling	p. 22
Model fitting and model checking	p. 23
Model summary	p. 28
Model selection	p. 30
Another model selection example	p. 31
A follow-up	p. 35
Confidence intervals	p. 36
Prediction	p. 36
Practical modelling with factors	p. 37
Identifiability	p. 38
Multiple factors	p. 39
'Interactions' of factors	p. 40
Using factor variables in R	p. 41
General linear model specification in R	p. 44
Further linear modelling theory	p. 45
Constraints I: General linear constraints	p. 46
Constraints II: 'Contrasts' and factor variables	p. 46
Likelihood	p. 48
Non-independent data with variable variance	p. 49
AIC and Mallow's statistic	p. 51
Non-linear least squares	p. 53
Further reading	p. 55
Exercises	p. 55
Generalized Linear Models	p. 59
The theory of GLMs	p. 60
The exponential family of distributions	p. 62
Fitting generalized linear models	p. 63
The IRLS objective is a quadratic approximation to the log-likelihood	p. 66
AIC for GLMs	p. 68
Large sample distribution of [beta]	p. 69
Comparing models by hypothesis testing	p. 69
Deviance	p. 70
Model comparison with unknown [phi]	p. 71
[phi] and Pearson's statistic	p. 71
Canonical link functions	p. 72
Residuals	p. 73
Pearson residuals	p. 73
Deviance residuals	p. 73
Quasi-likelihood	p. 74
Geometry of GLMs	p. 76
The geometry of IRLS	p. 77
Geometry and IRLS convergence	p. 78
GLMs with R	p. 81
Binomial models and heart disease	p. 81
A Poisson regression epidemic model	p. 87
Log-linear models for categorical data	p. 93
Sole eggs in the Bristol channel	p. 97
Likelihood	p. 102
Invariance	p. 102
Properties of the expected log-likelihood	p. 103
Consistency	p. 106
Large sample distribution of [theta]	p. 107
The generalized likelihood ratio test (GLRT)	p. 108
Derivation of 2[lambda tilde X superscript 2 subscript r] under H[subscript 0]	p. 109
AIC in general	p. 111
Quasi-likelihood results	p. 113
Exercises	p. 115
Introducing GAMs	p. 121
Introduction	p. 121
Univariate smooth functions	p. 122
Representing a smooth function: Regression splines	p. 122
A very simple example: A polynomial basis	p. 122
Another example: A cubic spline basis	p. 124
Using the cubic spline basis	p. 126
Controlling the degree of smoothing with penalized regression splines	p. 128
Choosing the smoothing parameter, [lambda]: Cross validation	p. 130
Additive models	p. 133
Penalized regression spline representation of an additive model	p. 134
Fitting additive models by penalized least squares	p. 135
Generalized additive models	p. 137
Summary	p. 139
Exercises	p. 140
Some GAM Theory	p. 145
Smoothing bases	p. 146
Why splines?	p. 146
Natural cubic splines are smoothest interpolators	p. 146
Cubic smoothing splines	p. 148
Cubic regression splines	p. 149
A cyclic cubic regression spline	p. 151
P-splines	p. 152
Thin plate regression splines	p. 154
Thin plate splines	p. 154
Thin plate regression splines	p. 157
Properties of thin plate regression splines	p. 158
Knot-based approximation	p. 160
Shrinkage smoothers	p. 160
Choosing the basis dimension	p. 161
Tensor product smooths	p. 162
Tensor product bases	p. 162
Tensor product penalties	p. 165
Setting up GAMs as penalized GLMs	p. 167
Variable coefficient models	p. 168
Justifying P-IRLS	p. 169
Degrees of freedom and residual variance estimation	p. 170
Residual variance or scale parameter estimation	p. 171
Smoothing parameter selection criteria	p. 172
Known scale parameter: UBRE	p. 172
Unknown scale parameter: Cross validation	p. 173
Problems with ordinary cross validation	p. 174
Generalized cross validation	p. 175
GCV/UBRE/AIC in the generalized case	p. 177
Approaches to GAM GCV/UBRE minimization	p. 179
Numerical GCV/UBRE: Performance iteration	p. 181
Minimizing the GCV or UBRE score	p. 181
Stable and efficient evaluation of the scores and derivatives	p. 183
The weighted constrained case	p. 185
Numerical GCV/UBRE optimization by outer iteration	p. 186
Differentiating the GCV/UBRE function	p. 187
Distributional results	p. 189
Bayesian model, and posterior distribution of the parameters, for an additive model	p. 190
Structure of the prior	p. 191
Posterior distribution for a GAM	p. 192
Bayesian confidence intervals for non-linear functions of parameters	p. 194
P-values	p. 194
Confidence interval performance	p. 196
Single smooths	p. 196
GAMs and their components	p. 200
Unconditional Bayesian confidence intervals	p. 202
Further GAM theory	p. 204
Comparing GAMs by hypothesis testing	p. 204
ANOVA decompositions and nesting	p. 206
The geometry of penalized regression	p. 208
The "natural" parameterization of a penalized smoother	p. 210
Other approaches to GAMs	p. 212
Backfitting GAMs	p. 213
Generalized smoothing splines	p. 215
Exercises	p. 217
GAMs in Practice: mgcv	p. 221
Cherry trees again	p. 221
Finer control of gam	p. 223
Smooths of several variables	p. 225
Parametric model terms	p. 228
Brain imaging example	p. 230
Preliminary modelling	p. 232
Would an additive structure be better?	p. 236
Isotropic or tensor product smooths?	p. 237
Detecting symmetry (with by variables)	p. 239
Comparing two surfaces	p. 241
Prediction with predict.gam	p. 243
Prediction with lpmatrix	p. 245
Variances of non-linear functions of the fitted model	p. 246
Air pollution in Chicago example	p. 247
Mackerel egg survey example	p. 254
Model development	p. 254
Model predictions	p. 260
Portuguese larks example	p. 262
Other packages	p. 265
Package gam	p. 265
Package gss	p. 267
Exercises	p. 270
Mixed Models and GAMMs	p. 277
Mixed models for balanced data	p. 277
A motivating example	p. 277
The wrong approach: A fixed effects linear model	p. 278
The right approach: A mixed effects model	p. 280
General principles	p. 281
A single random factor	p. 282
A model with two factors	p. 286
Discussion	p. 290
Linear mixed models in general	p. 291
Estimation of linear mixed models	p. 292
Directly maximizing a mixed model likelihood in R	p. 293
Inference with linear mixed models	p. 295
Fixed effects	p. 295
Inference about the random effects	p. 296
Predicting the random effects	p. 297
REML	p. 298
The explicit form of the REML criterion	p. 299
A link with penalized regression	p. 300
The EM algorithm	p. 302
Linear mixed models in R	p. 303
Tree growth: An example using lme	p. 304
Several levels of nesting	p. 309
Generalized linear mixed models	p. 310
GLMMs with R	p. 312
Generalized additive mixed models	p. 316
Smooths as mixed model components	p. 316
Inference with GAMMs	p. 318
GAMMs with R	p. 319
A GAMM for sole eggs	p. 319
The temperature in Cairo	p. 321
Exercises	p. 325
Some Matrix Algebra	p. 331
Basic computational efficiency	p. 331
Covariance matrices	p. 332
Differentiating a matrix inverse	p. 332
Kronecker product	p. 333
Orthogonal matrices and Householder matrices	p. 333
QR decomposition	p. 334
Choleski decomposition	p. 334
Eigen-decomposition	p. 335
Singular value decomposition	p. 336
Pivoting	p. 337
Lanczos iteration	p. 337
Solutions to Exercises	p. 341
Chapter 1	p. 341
Chapter 2	p. 346
Chapter 3	p. 351
Chapter 4	p. 353
Chapter 5	p. 360
Chapter 6	p. 369
Bibliography	p. 379
Index	p. 385
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