Statistical Modeling by Wavelets

Accessible to anyone with a background in advanced calculus and algebra, Statistical Modeling by Wavelets promises to become the standard reference for statisticians and engineers seeking a comprehensive introduction to an emerging field.

BRANI VIDAKOVIC, PhD, is Assistant Professor at the Institute of Statistics and Decision Sciences at Duke University.

Preface

xi

Acknowledgments

xiii

Introduction

1

(22)

Wavelet Evolution

1

(5)

Wavelet Revolution

6

(10)

Wavelets and Statistics

16

(3)

An Appetizer: California Earthquakes

19

(4)

Prerequisites

23

(20)

General

23

(1)

Hilbert Spaces

24

(5)

Projection Theorem

25

(1)

Orthonormal Sets

26

(3)

Reproducing Kernel Hilbert Spaces

29

(1)

Fourier Transformation

29

(6)

Basic Properties

30

(2)

Poisson Summation Formula and Sampling Theorem

32

(1)

Fourier Series

33

(1)

Discrete Fourier Transform

34

(1)

Heisenberg's Uncertainty Principle

35

(1)

Some Important Function Spaces

36

(3)

Fundamentals of Signal Processing

39

(1)

Exercises

40

(3)

Wavelets

43

(58)

Continuous Wavelet Transformation

43

(7)

Basic Properties

44

(4)

Wavelets for Continuous Transformations

48

(2)

Discretization of the Continuous Wavelet Transform

50

(1)

Multiresolution Analysis

51

(9)

Derivation of a Wavelet Function

57

(3)

Some Important Wavelet Bases

60

(20)

Haar's Wavelets

60

(3)

Shannon's Wavelets

63

(2)

Meyer's Wavelets

65

(5)

Franklin's Wavelets

70

(3)

Daubechies' Compactly Supported Wavelets

73

(7)

Some Extensions

80

(16)

Regularity of Wavelets

80

(6)

The Least Asymmetric Daubechies' Wavelets: Symmlets

86

(1)

Approximations and Characterizations of Functional Spaces

87

(2)

Daubechies-Lagarias Algorithm

89

(2)

Moment Conditions

91

(2)

Interpolating (Cardinal) Wavelets

93

(1)

Pollen-Type Parameterization of Wavelets

94

(2)

Exercises

96

(5)

Discrete Wavelet Transformations

101

(18)

Introduction

101

(3)

The Cascade Algorithm

104

(5)

The Operator Notation of DWT

109

(8)

Discrete Wavelet Transformations as Linear Transformations

115

(2)

Exercises

117

(2)

Some Generalizations

119

(48)

Coiflets

120

(2)

Construction of Coiflets

121

(1)

Biorthogonal Wavelets

122

(11)

Construction of Biorthogonal Wavelets

125

(3)

B-Spline Wavelets

128

(5)

Wavelet Packets

133

(7)

Basic Properties of Wavelet Packets

135

(3)

Wavelet Packet Tables

138

(2)

Best Basis Selection

140

(5)

Some Cost Measures and the Best Basis Algorithm

141

(4)

∈-Decimated and Stationary Wavelet Transformations

145

(5)

∈-Decimated Wavelet Transformation

145

(2)

Stationary (Non-Decimated) Wavelet Transformation

147

(3)

Periodic Wavelet Transformations

150

(3)

Multivariate Wavelet Transformations

153

(7)

Discussion

160

(3)

Exercises

163

(4)

Wavelet Shrinkage

167

(50)

Shrinkage Method

168

(2)

Linear Wavelet Regression Estimators

170

(5)

Wavelet Kernels

171

(1)

Local Constant Fit Estimators

172

(3)

The Simplest Non-Linear Wavelet Shrinkage: Thresholding

175

(9)

Variable Selection and Thresholding

177

(1)

Oracular Risk for Thresholding Rules

177

(2)

Why the Wavelet Shrinkage Works

179

(5)

Almost Sure Convergence of Wavelet Shrinkage Estimators

184

(1)

General Minimax Paradigm

184

(4)

Translation of Minimaxity Results to the Wavelet Domain

186

(2)

Thresholding Policies and Thresholding Rules

188

(6)

Exact Risk Analysis of Thresholding Rules

188

(1)

Large Sample Properties of f

189

(2)

Some Other Shrinkage Rules

191

(3)

How to Select a Threshold

194

(13)

Mallat's Model and Induced Percentile Thresholding

194

(1)

Universal Threshold

195

(4)

A Threshold Based on Stein's Unbiased Estimator of Risk

199

(1)

Cross-Validation

200

(2)

Thresholding as a Testing Problem

202

(1)

Lorentz Curve Thresholding

203

(3)

Block Thresholding Estimators

206

(1)

Other Methods and References

207

(6)

Exercises

213

(4)

Density Estimation

217

(30)

Orthogonal Series Density Estimators

217

(2)

Wavelet Density Estimation

219

(6)

δ-Sequence Density Estimators

219

(4)

Bias and Variance of Linear Wavelet Density Estimators

223

(1)

Linear Wavelet Density Estimators in a More General Setting

224

(1)

Non-Linear Wavelet Density Estimators

225

(3)

Global Thresholding Estimator

227

(1)

Non-Negative Density Estimators

228

(8)

Estimating the Square Root of a Density

229

(5)

Density Estimation by Non-Negative Wavelets

234

(2)

Other Methods

236

(7)

Multivariate Wavelet Density Estimators

236

(3)

Density Estimation as a Regression Problem

239

(1)

Cross-Validation Estimator

240

(1)

Multiscale Estimator

240

(1)

Estimation of a Derivative of a Density

241

(2)

Exercises

243

(4)

Bayesian Methods in Wavelets

247

(24)

Motivational Examples

247

(3)

Smooth Shrinkage

250

(5)

Bayesian Thresholding

255

(2)

MAP-Principle

257

(2)

Density Estimation Problem

259

(3)

Full Bayesian Model

262

(3)

Discussion and References

265

(2)

Exercises

267

(4)

Wavelets and Random Processes

271

(28)

Stationary Time Series

272

(1)

Wavelets and Stationary Processes

273

(6)

Wavelet Transformations of Stationary Processes

273

(1)

Whitening of Stationary Processes

274

(2)

Karhunen-Loeve-Like Expansions

276

(3)

Estimation of Spectral Densities

279

(7)

Gao's Algorithm

280

(4)

Non-Gaussian Stationary Processes

284

(2)

Wavelet Spectrum

286

(4)

Wavelet Spectrum of a Stationary Time Series

286

(3)

Scalogram and Periodicities

289

(1)

Long-Memory Processes

290

(6)

Wavelets and Fractional Brownian Motion

290

(2)

Estimating Spectral Exponents in Self-Similar Processes

292

(3)

Quantifying the Whitening Property of Wavelet Transformations for fBm Processes

295

(1)

Discussion and References

296

(1)

Exercises

297

(2)

Wavelet-Based Random Variables and Densities

299

(20)

Scaling Function as a Density

299

(1)

Wavelet-Based Random Variables

300

(8)

Random Densities via Wavelets

308

(2)

Tree Algorithm

308

(2)

Properties of Wavelet-Based Random Densities

310

(2)

Random Densities With Constraints

312

(4)

Smoothness Constraints

312

(1)

Constraints on Symmetry

312

(1)

Constraints on Modality

313

(1)

Skewed Random Densities

313

(3)

Exercises

316

(3)

Miscellaneous Statistical Applications

319

(26)

Deconvolution Problems

319

(4)

Wavelet-Vaguelette Decompositions

323

(2)

Pursuit Methods

325

(4)

Moments of Order Statistics

329

(4)

Wavelets and Statistical Turbulence

333

(5)

K41 Theory

335

(1)

Townsend's Decompositions

336

(2)

Software and WWW Resources for Wavelet Analysis

338

(4)

Commercial Wavelet Software

338

(2)

Free Wavelet Software

340

(1)

Some WWW Resources

341

(1)

Exercises

342

(3)

References

345

(26)

Notation Index

371

(2)

Author Index

373

(6)

Subject Index

379

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Statistical Modeling by Wavelets

Summary

Author Biography

Table of Contents

Supplemental Materials