.

0 (hodnocen0 x )

(1) Půjčeno:1x 

BK

Boca Raton : Chapman & Hall/CRC, 2006

xvii, 392 s. : il., grafy ; 25 cm

ISBN 1-58488-474-6 (váz.)

Texts in statistical science

Bibliografie na s. 379-383, rejstřík

000027007

Preface xv // 1 Linear Models // A simple linear model // Simple least squares estimation // 1.1.1 Sampling properties of 3 // 1.1.2 So how old is the universe? // Adding a distributional assumption // Testing hypotheses about ß // Confidence intervals // 1.2 Linear models in general 10 // 1.3 The theory of linear models 12 // 1.3.1 Least squares estimation of /3 12 // 1.3.2 The distribution of ß 13 // 1.3.3 - 8,)/6a, ~  n-P 14 // 1.3.4 F-ratio results 15 // 1.3.5 The influence matrix 16 // 1.3.6 The residuals, e, and fitted values, (l 16 // 1.3.7 Results in terms of X 17 // 1.3.8 The Gauss Markov Theorem: What’s special about least squares? The geometry of linear modelling // 1.4 .1 Least squares // 4.2 Fitting by orthogonal decompositions // 1.4.3 Comparison of nested models 21 // 1.5 Practical linear modelling 22 // 1.5.1 Model fitting and model checking 23 // 1.5.2 Model summary 28 // 1.5.3 Model selection 30 // 1.5.4 Another model selection example 31 // A follow-up 35 // 1.5.5 Confidence intervals 36 // 1.5.6 Prediction 36 // 1.6 Practical modelling with factors 37 // 1.6.1 Identifiability 38 // 1.6.2 Multiple factors 39 // 1.6.3 â€�Interactions’of factors 40 // 1.6.4 Using factor variables in R 41 // 1.7 General linear model specification in R 44 // 1.8 Further linear modelling theory 45 // 1.8.1 Constraints I: General linear constraints 46 // 1.8.2 Constraints II: ’Contrasts’ and factor variables 46 // 1.8.3 Likelihood 48 // 1.8.4 Non-independent data with variable variance 49 // 1.8.5 AIC and Mallows’statistic 51 // 1.8.6 Non-linear least squares 53 // 1.8.7 Further reading 55 // 1.9 Exercises 55 // 2 Generalized Linear Models 59 // 2.1 The theory of GLMs 60 // 2.1.1 The exponential family of distributions 62 // 2.1.2 Fitting generalized linear models 63 // 2.1.3 The IRLS objective is a quadratic approximation to the log-likelihood 66 //

2.1.4 AICforGLMs 68 // 2.1.5 Large sample distribution of ß 69 // 2.1.6 Comparing models by hypothesis testing 69 // Deviance 70 // Model comparison with unknown d 71 // 2.1.7 d and Pearson’s statistic 71 // 2.1.8 Canonical link functions 72 // 2.1.9 Residuals 73 // Pearson residuals 73 // Deviance residuals 73 // 2.1.10 Quasi-likelihood 74 // 2.2 Geometry of GLMs 76 // 2.2.1 The geometry of IRLS 77 // 2.2.2 Geometry and IRLS convergence 78 // 2.3 GLMs with R 81 // 2.3.1 Binomial models and heart disease 81 // 2.3.2 A Poisson regression epidemic model 87 // 2.3.3 Log-linear models for categorical data 93 // 2.3.4 Sole eggs in the Bristol channel 97 // 2.4 Likelihood 102 // 2.4.1 Invariance 102 // 2.4.2 Properties of the expected log-likelihood 103 // 2.4.3 Consistency 106 // 2.4.4 Large sample distribution of 0 107 // 2.4.5 The generalized likelihood ratio test (GLRT) 108 // 2.4.6 Derivation of 2A ~ x? under Ho 109 // 2.4.7 AIC in general 111 // 2.4.8 Quasi-likelihood results 113 // 2.5 Exercises 115 // 3. Introduction ... GAMs 121 // 3.1 Introduction 121 // 3.2 Univariate smooth functions 122 // 3.2.1 Representing a smooth function: Regression splines 122 // A very simple example: A polynomial basis 122 // Another example: A cubic spline basis 124 // Using the cubic spline basis 126 // 3.2.2 Controlling the degree of smoothing with penalized regression splines 128 // 3.2.3 Choosing the smoothing parameter, A: Cross validation 130 // 3.3 Additive models 133 // 3.3.1 Penalized regression spline representation of an additive // model 134 // 3.3.2 Fitting additive models by penalized least squares 135 // 3.4 Generalized additive models 137 // 3.5 Summary 139 // 3.6 Exercises 140 // 4 Some GAM Theory 145 // 4.1 Smoothing bases 146 // 4.1.1 Why splines? 146 // Natural cubic splines are smoothest interpolators 146 // Cubic smoothing splines 148 //

4.1.2 Cubic regression splines 149 // 4.1.3 A cyclic cubic regression spline 151 // 4.1.4 P-splines 152 // 4.1.5 Thin plate regression splines 154 // Thin plate splines 154 // Thin plate regression splines 157 // Properties of thin plate regression splines 158 // Knot-based approximation ’ 160 // 4.1.6 Shrinkage smoothers 160 // 4.1.7 Choosing the basis dimension 161 // 4.1.8 Tensor product smooths 162 // Tensor product bases 162 // Tensor product penalties 165 // 4.2 Setting up GAMs as penalized GLMs 167 // 4.2.1 Variable coefficient models 168 // 4.3 Justifying P-IRLS 169 // 4.4 Degrees of freedom and residual variance estimation 170 // 4.4.1 Residual variance or scale parameter estimation 171 // 4.5 Smoothing parameter selection criteria 172 // 4.5.1 Known scale parameter: UBRE 172 // 4.5.2 Unknown scale parameter: Cross validation 173 // Problems with ordinary cross validation 174 // 4.5.3 Generalized cross validation 175 // 4.5.4 GCV/UBRE/AIC in the generalized case 177 // Approaches to GAM GCV/UBRE minimization 179 // 4.6 Numerical GCV/UBRE: Performance iteration 181 // 4.6.1 Minimizing the GCV or UBRE score 181 // Stable and efficient evaluation of the scores and derivatives 183 // The weighted constrained case 185 // 4.7 Numerical GCV/UBRE optimization by outer iteration 186 // 4.7.1 Differentiating the GCV/UBRE function 187 // 4.8 Distributional results 189 // 4.8.1 Bayesian model, and posterior distribution of the parameters, // for an additive model 190 // 4.8.2 Structure of the prior 191 // 4.8.3 Posterior distribution for a GAM 192 // 4.8.4 Bayesian confidence intervals for non-linear functions of // parameters 194 // 4.8.5 P-values 194 // 4.9 Confidence interval performance 196 // 4.9.1 Single smooths 196 // 4.9.2 GAMs and their components 200 // 4.9.3 Unconditional Bayesian confidence intervals 202 // 4.10 Further GAM theory 204 //

4.10.1 Comparing GAMs by hypothesis testing 204 // 4.10.2 ANOVA decompositions and nesting 206 // 4.10.3 The geometry of penalized regression 208 // 4.10.4 The "natural" parameterization of a penalized smoother 210 // 4.11 Other approaches to GAMs 212 // 4.11.1 Backfitting GAMs 213 // 4.11.2 Generalized smoothing splines 215 // 4.12 Exercises 217 // 5 GAMs in Practice: mgcv 221 // 5.1 Cherry trees again 221 // 5.1.1 Finer control of gam 223 // 5.1.2 Smooths of several variables 225 // 5.1.3 Parametric model terms 228 // 5.2 Brain imaging example 230 // 5.2.1 Preliminary modelling 232 // 5.2.2 Would an additive structure be better? 236 // 5.2.3 Isotropic or tensor product smooths? 237 // 5.2.4 Detecting symmetry (with by variables) 239 // 5.2.5 Comparing two surfaces 241 // 5.2.6 Prediction with predict. gam 243 // Prediction with Ipmatrix 245 // 5.2.7 Variances of non-linear functions of the fitted model 246 // 5.3 Air pollution in Chicago example 247 // 5.4 Mackerel egg survey example 254 // 5.4.1 Model development 254 // 5.4.2 Model predictions 260 // 5.5 Portuguese larks example 262 // 5.6 Other packages 265 // 5.6.1 Package gam 265 // 5.6.2 Package gss 267 // 5.7 Exercises 270 // 6 Mixed Models and GAMMs 277 // 6.1 Mixed models for balanced data 277 // 6.1.1 A motivating example 277 // The wrong approach: A fixed effects linear model 278 // The right approach: A mixed effects model 280 // 6.1.2 General principles 281 // 6.1.3 A single random factor 282 // 6.1.4 A model with two factors 286 // 6.1.5 Discussion 290 // 6.2 Linear mixed models in general 291 //

6.2.1 Estimation of linear mixed models 292 // 6.2.2 Directly maximizing a mixed model likelihood in R 293 // 6.2.3 Inference with linear mixed models 295 // Fixed effects 295 // Inference about the random effects 296 // 6.2.4 Predicting the random effects 297 // 6.2.5 REML * 298 // The explicit form of the REML criterion 299 // 6.2.6 A link with penalized regression 300 // 6.2.7 The EM algorithm 302 // 6.3 Linear mixed models in R 303 // 6.3.1 Tree growth: An example using Ime 304 // 6.3.2 Several levels of nesting 309 // 6.4 Generalized linear mixed models 310 // 6.5 GLMMswithR 312 // xiv // 6.6 Generalized additive mixed models // 6.6.1 Smooths as mixed model components // 6.6.2 Inference with GAMMs // 6.7 GAMMs with R // 6.7.1 A GAMM for sole eggs // 6.7.2 The temperature in Cairo // 6.8 Exercises // Some Matrix Algebra // A.I Basic computational efficiency // A.2 Covariance matrices // A.3 Differentiating a matrix inverse // A.4 Kronecker product // A.5 Orthogonal matrices and Householder matrices // A.6 QR decomposition // A.7 Choleski decomposition // A.8 Eigen-decomposition // A.9 Singular value decomposition // A. 10 Pivoting // A. 11 Lanczos iteration // B Solutions to Exercises // B.l Chapter 1 � // B.2 Chapter 2 // B.3 Chapter 3 // B.4 Chapter 4 // B.5 Chapters // B.6 Chapter 6 // Bibliography // Index