functional data analysis 书籍：Smoothing Splines Methods and Application 超级棒

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Contents
1 Introduction 1
1.1 Parametric and Nonparametric Regression . . . . . . . 1
1.2 Polynomial Splines . . . . . . . . . . . . . . . . . . . . . 4

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1.3 Scope of This Book . . . . . . . . . . . . . . . . . . . . 7
1.4 The assist Package . . . . . . . . . . . . . . . . . . . . 9
2 Smoothing Spline Regression 11
2.1 Reproducing Kernel Hilbert Space . . . . . . . . . . . . 11
2.2 Model Space for Polynomial Splines . . . . . . . . . . . 14
2.3 General Smoothing Spline Regression Models . . . . . . 16
2.4 Penalized Least Squares Estimation . . . . . . . . . . . 17
2.5 The ssr Function . . . . . . . . . . . . . . . . . . . . . 20
2.6 Another Construction for Polynomial Splines . . . . . . 22
2.7 Periodic Splines . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Thin-Plate Splines . . . . . . . . . . . . . . . . . . . . . 26
2.9 Spherical Splines . . . . . . . . . . . . . . . . . . . . . . 29
2.10 Partial Splines . . . . . . . . . . . . . . . . . . . . . . . 30
2.11 L-splines . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 Smoothing Parameter Selection and Inference 53
3.1 Impact of the Smoothing Parameter . . . . . . . . . . . 53
3.2 Trade-Offs . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 Unbiased Risk . . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Cross-Validation and Generalized Cross-Validation . . . 64
3.5 Bayes and Linear Mixed-Effects Models . . . . . . . . . 67
3.6 Generalized Maximum Likelihood . . . . . . . . . . . . 71
3.7 Comparison and Implementation . . . . . . . . . . . . . 72
3.8 Confidence Intervals . . . . . . . . . . . . . . . . . . . . 75
3.8.1 Bayesian Confidence Intervals . . . . . . . . . . . 75
3.8.2 Bootstrap Confidence Intervals . . . . . . . . . . 81
3.9 Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . 84
3.9.1 The Hypothesis . . . . . . . . . . . . . . . . . . . 84
3.9.2 Locally Most Powerful Test . . . . . . . . . . . . 85
3.9.3 Generalized Maximum Likelihood Test . . . . . . 86
3.9.4 Generalized Cross-Validation Test . . . . . . . . 87
3.9.5 Comparison and Implementation . . . . . . . . . 87
4 Smoothing Spline ANOVA 91
4.1 Multiple Regression . . . . . . . . . . . . . . . . . . . . 91
4.2 Tensor Product Reproducing Kernel Hilbert Spaces . . 92
4.3 One-Way SS ANOVA Decomposition . . . . . . . . . . 93
4.4 Two-Way SS ANOVA Decomposition . . . . . . . . . . 98
4.5 General SS ANOVA Decomposition . . . . . . . . . . . 110
4.6 SS ANOVA Models and Estimation . . . . . . . . . . . 111
4.7 Selection of Smoothing Parameters . . . . . . . . . . . 114
4.8 Confidence Intervals . . . . . . . . . . . . . . . . . . . . 116
4.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5 Spline Smoothing with Heteroscedastic and/or Correlated Errors 139
5.1 Problems with Heteroscedasticity and Correlation . . . 139
5.2 Extended SS ANOVA Models . . . . . . . . . . . . . . 142
5.3 Variance and Correlation Structures . . . . . . . . . . . 150
5.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6 Generalized Smoothing Spline ANOVA 163
6.1 Generalized SS ANOVA Models . . . . . . . . . . . . . 163
6.2 Estimation and Inference . . . . . . . . . . . . . . . . . 164
6.2.1 Penalized Likelihood Estimation . . . . . . . . . 164
6.2.2 Selection of Smoothing Parameters . . . . . . . . 167
6.2.3 Algorithm and Implementation . . . . . . . . . . 168
6.2.4 Bayes Model, Direct GML and Approximate
Bayesian Confidence Intervals . . . . . . . . . . . 170
6.3 Wisconsin Epidemiological Study of Diabetic
Retinopathy . . . . . . . . . . . . . . . . . . . . . . . . 172
6.4 Smoothing Spline Estimation of Variance Functions . . 176
6.5 Smoothing Spline Spectral Analysis . . . . . . . . . . . 182
6.5.1 Spectrum Estimation of a Stationary Process . . 182
6.5.2 Time-Varying Spectrum Estimation of a Locally
Stationary Process . . . . . . . . . . . . . . . . . 183
6.5.3 Epileptic EEG . . . . . . . . . . . . . . . . . . . 185
7 Smoothing Spline Nonlinear Regression 195
7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 195
7.2 Nonparametric Nonlinear Regression Models . . . . . . 196
7.3 Estimation with a Single Function . . . . . . . . . . . . 197
7.3.1 Gauss–Newton and Newton–Raphson Methods . 197
7.3.2 Extended Gauss–Newton Method . . . . . . . . . 199
7.3.3 Smoothing Parameter Selection and Inference . . 201
7.4 Estimation with Multiple Functions . . . . . . . . . . . 204
7.5 The nnr Function . . . . . . . . . . . . . . . . . . . . . 205
7.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8 Semiparametric Regression 227
8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 227
8.2 Semiparametric Linear Regression Models . . . . . . . . 228
8.2.1 The Model . . . . . . . . . . . . . . . . . . . . . 228
8.2.2 Estimation and Inference . . . . . . . . . . . . . 229
8.2.3 Vector Spline . . . . . . . . . . . . . . . . . . . . 233
8.3 Semiparametric Nonlinear Regression Models . . . . . . 240
8.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 247
9 Semiparametric Mixed-Effects Models 273
9.1 Linear Mixed-Effects Models . . . . . . . . . . . . . . . 273
9.2 Semiparametric Linear Mixed-Effects Models . . . . . . 274
9.3 Semiparametric Nonlinear Mixed-Effects Models . . . . 283
9.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 288
References 347
Author Index 355
Subject Index 359

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关键词：Application Functional smoothing Analysis function Smoothing Splines Methods FDA

审批中

Statistical analysis often involves building mathematical models that
examine the relationship between dependent and independent variables.
This book is about a general class of powerful and flexible modeling
techniques, namely, spline smoothing.
Research on smoothing spline models has attracted a great deal of
attention in recent years, and the methodology has been widely used in
many areas.

This book provides an introduction to some basic smoothing
spline models, including polynomial, periodic, spherical, thin-plate, L-,
and partial splines, as well as an overview of more advanced models, including
smoothing spline ANOVA, extended and generalized smoothing
spline ANOVA, vector spline, nonparametric nonlinear regression, semiparametric
regression, and semiparametric mixed-effects models. Methods
for model selection and inference are also presented.
The general forms of nonparametric/semiparametric linear/nonlinear
fixed/mixed smoothing spline models in this book provide unified frameworks
for estimation, inference, and software implementation. This book
draws on the theory of reproducing kernel Hilbert space (RKHS) to
present various smoothing spline models in a unified fashion. On the
other hand, the subject of smoothing spline in the context of RKHS
and regularization is often regarded as technical and difficult. One of
my main goals is to make the advanced smoothing spline methodology
based on RKHS more accessible to practitioners and students. With this
in mind, the book focuses on methodology, computation, implementation,
software, and application. It provides a gentle introduction to the
RKHS, keeps theory at the minimum level, and provides details on how
the RKHS can be used to construct spline models.

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