Wavelet Methods in Statistics with R (eBook)

Preface 7
Contents 9
1 Introduction 11
1.1 What Are Wavelets? 11
1.2 Why Use Wavelets? 12
1.3 Why Wavelets in Statistics? 21
1.4 Software and This Book 23
2 Wavelets 25
2.1 Multiscale Transforms 25
2.2 Haar Wavelets (on Functions) 38
2.3 Multiresolution Analysis 47
2.4 Vanishing Moments 50
2.5 WaveThresh Wavelets (and What Some Look Like) 51
2.6 Other Wavelets 55
2.7 The General (Fast) Discrete Wavelet Transform 60
2.8 Boundary Conditions 65
2.9 Non-decimated Wavelets 67
2.10 Multiple Wavelets 76
2.11 Wavelet Packet Transforms 78
2.12 Non-decimated Wavelet Packet Transforms 85
2.13 Multivariate Wavelet Transforms 86
2.14 Other Topics 88
3 Wavelet Shrinkage 93
3.1 Introduction 93
3.2 Wavelet Shrinkage 94
3.3 The Oracle 95
3.4 Test Functions 98
3.5 Universal Thresholding 98
3.6 Primary Resolution 106
3.7 SURE Thresholding 106
3.8 Cross-validation 108
3.9 False Discovery Rate 110
3.10 Bayesian Wavelet Shrinkage 111
3.11 Linear Wavelet Smoothing 119
3.12 Non-Decimated Wavelet Shrinkage 120
3.13 Multiple Wavelet Shrinkage (Multiwavelets) 128
3.14 Complex-valued Wavelet Shrinkage 130
3.15 Block Thresholding 138
3.16 Miscellanea and Discussion 140
4 Related Wavelet Smoothing Techniques 143
4.1 Introduction 143
4.2 Correlated Data 143
4.3 Non-Gaussian Noise 148
4.4 Multidimensional Data 150
4.5 Irregularly Spaced Data 153
4.6 Confidence Bands 160
4.7 Density Estimation 165
4.8 Survival Function Estimation 168
4.9 Inverse Problems 173
5 Multiscale Time Series Analysis 177
5.1 Introduction 177
5.2 Stationary Time Series 179
5.3 Locally Stationary Time Series 184
5.4 Forecasting with Locally Stationary Wavelet Models 202
5.5 Time Series with Wavelet Packets 207
5.6 Related Topics and Discussion 208
6 Multiscale Variance Stabilization 211
6.1 Why the Square Root for Poisson? 212
6.2 The Fisz Transform 213
6.3 Poisson Intensity Function Estimation 216
6.4 The Haar–Fisz Transform for Poisson Data 217
6.5 Data-driven Haar–Fisz 227
6.6 Discussion 237
A R Software for Wavelets and Statistics 239
B Notation and Some Mathematical Concepts 241
B.1 Notation and Concepts 241
C Survival Function Code 245
References 247
Index 263