# Statistical Modeling by Wavelets

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1 Statistical Modeling by Wavelets BRANI VIDAKOVIC Duke University A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto

2 Contents Preface Acknowledgments 1 Introduction 1.1 Wavelet Evolution 1.2 Wavelet Revolution 1.3 Wavelets and Statistics 1.4 An Appetizer: California Earthquakes 2 Prerequisites 2.1 General 2.2 Hilbert Spaces Projection Theorem Orthonormal Sets Reproducing Kernel Hilbert Spaces 2.3 Fourier Transformation Basic Properties Poisson Summation Formula and Sampling Theorem xi xiii V

3 Vi CONTENTS FourierSeries Discrete Fourier Transform Heisenberg 's Uncertainty Principle Some Important Function Spaces Fundamentals of Signal Processing Exercises 40 3 Wavelets Continuous Wavelet Transformation Basic Properties Wavelets for Continuous Transformations Discretization ofthe Continuous Wavelet Transform Multiresolution Analysis Derivation of a Wavelet Function Some Important Wavelet Bases Haar's Wavelets Shannon 's Wavelets Meyer's Wavelets Franklin 's Wavelets Daubechies' Compactly Supported Wavelets Some Extensions Regularity of Wavelets The Least Asymmetrie Daubechies' Wavelets: Symmlets Approximations and Characterizations of Functional Spaces Daubechies-Lagarias Algorithm Moment Conditions Interpolating (Cardinal) Wavelets Pollen-Type Parameterization of'wavelets Exercises 96 4 Discrete Wavelet Transformations Introduction The Cascade Algorithm The Operator Notation of DWT Discrete Wavelet Transformations as Linear Transformations Exercises 117

4 CONTENTS VII 5 Some Generalizations Coiflets Construction of Coiflets Biorthogonal Wavelets Construction of Biorthogonal Wavelets B-Spline Wavelets Wavelet Packets Basic Properties of Wavelet Packets Wavelet Packet Tables Best Basis Selection Some Cost Measures and the Best Basis Algorithm e-decimated and Stationary Wavelet Transformations e-decimated Wavelet Transformation Stationary (Non-Decimated) Wavelet Transformation Periodic Wavelet Transformations Multivariate Wavelet Transformations Discussion Exercises Wavelet Shrinkage Shrinkage Method Linear Wavelet Regression Estimators Wavelet Kernels Local Constant Fit Estimators The Simplest Non-Linear Wavelet Shrinkage: Thresholding Variable Selection and Thresholding Oracular Riskfor Thresholding Rules Why the Wavelet Shrinkage Works Almost Sure Convergence of Wavelet Shrinkage Estimators General Minimax Paradigm Translation of Minimaxity Results to the Wavelet Domain Thresholding Policies and Thresholding Rules Exact Risk Analysis of Thresholding Rules Large Sample Properties of f 189

5 VÜi CONTENTS Some Other Shrinkage Rules How to Select a Threshold Mallat's Model and Induced Percentile Thresholding Universal Threshold A Threshold Based on Stein 's Unbiased Estimator ofrisk Cross-Validation Thresholding as atesting Problem Lorentz Curve Thresholding Block Thresholding Estimators Other Methods and References Exercises Density Estimation Orthogonal Series Density Estimators Wavelet Density Estimation Ö-Sequence Density Estimators Bias and Variance of Linear Wavelet Density Estimators Linear Wavelet Density Estimators in a More General Setting Non-Linear Wavelet Density Estimators Global Thresholding Estimator Non-Negative Density Estimators Estimating the Square Root of a Density Density Estimation by Non-Negative Wavelets Other Methods Multivariate Wavelet Density Estimators Density Estimation as a Regression Problem Cross-Validation Estimator Multiscale Estimator Estimation of a Derivative of a Density Exercises Bayesian Methods in Wavelets Motiv ational Examples Smooth Shrinkage Bayesian Thresholding 255

6 CONTENTS ix MAP-Principle Density Estimation Problem Füll Bayesian Model Discussion and References Exercises Wavelets and Random Processes Stationary Time Series Wavelets and Stationary Processes Wavelet Transformations of Stationary Processes Whitening of Stationary Processes Karhunen-Loeve-Like Expansions 9.3 Estimation of Spectral Densities Gao 's Algorithm Non-Gaussian Stationary Processes 9.4 Wavelet Spectrum Wavelet Spectrum ofa Stationary Time Series Scalogram and Periodicities 9.5 Long-Memory Processes Wavelets and Fractional Brownian Motion Estimating Spectral Exponents in Self-Similar Processes Quantifying the Whitening Property of Wavelet Transformations for fbm Processes Discussion and References Exercises Wavelet-Based Random Variables and Densities Scaling Function as a Density Wavelet-Based Random Variables Random Densities via Wavelets Tree Algorithm Properties of Wavelet-Based Random Densities Random Densities With Constraints Smoothness Constraints Constraints on Symmetry Constraints on Modality Skewed Random Densities 313

7 X CONTENTS 10.6 Exercises Miscellaneous Statistical Applications Deconvolution Problems Wavelet- Vaguelette Decompositions Pursuit Methods Moments of Order Statistics Wavelets and Statistical Turbulence K41 Theory Townsend's Decompositions Software and WWW Resources for Wavelet Analysis Commercial Wavelet Software Free Wavelet Software Some WWW Resources Exercises 342 References 345 Notation Index 371 Author Index 373 Subject Index 379

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