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