Essential Wavelets for Statistical Applications and Data Analysis - Ogden, Todd

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Résumé :
I once heard the book by Meyer (1993) described as a vulgarization of wavelets. While this is true in one sense of the word, that of making a sub? ject popular (Meyer's book is one of the early works written with the non? specialist in mind), the implication seems to be that such an attempt some? how cheapens or coarsens the subject. I have to disagree that popularity goes hand-in-hand with debasement. is certainly a beautiful theory underlying wavelet analysis, there is While there plenty of beauty left over for the applications of wavelet methods. This book is also written for the non-specialist, and therefore its main thrust is toward wavelet applications. Enough theory is given to help the reader gain a basic understanding of how wavelets work in practice, but much of the theory can be presented using only a basic level of mathematics. Only one theorem is for? mally stated in this book, with only one proof. And these are only included to introduce some key concepts in a natural way.

Sommaire:
1 Wavelets: A Brief Introduction.- 1.1 The Discrete Fourier Transform.- 1.2 The Haar System.- Multiresolution Analysis.- The Wavelet Representation.- Goals of Multiresolution Analysis.- 1.3 Smoother Wavelet Bases.- 2 Basic Smoothing Techniques.- 2.1 Density Estimation.- Histograms.- Kernel Estimation.- Orthogonal Series Estimation.- 2.2 Estimation of a Regression Function.- Kernel Regression.- Orthogonal Series Estimation.- 2.3 Kernel Representation of Orthogonal Series Estimators.- 3 Elementary Statistical Applications.- 3.1 Density Estimation.- Haar-Based Histograms.- Estimation with Smoother Wavelets.- 3.2 Nonparametric Regression.- 4 Wavelet Features and Examples.- 4.1 Wavelet Decomposition and Reconstruction.- Two-Scale Relationships.- The Decomposition Algorithm.- The Reconstruction Algorithm.- 4.2 The Filter Representation.- 4.3 Time-Frequency Localization.- The Continuous Fourier Transform.- The Windowed Fourier Transform.- The Continuous Wavelet Transform.- 4.4 Examples of Wavelets and Their Constructions.- Orthogonal Wavelets.- Biorthogonal Wavelets.- Semiorthogonal Wavelets.- 5 Wavelet-based Diagnostics.- 5.1 Multiresolution Plots.- 5.2 Time-Scale Plots.- 5.3 Plotting Wavelet Coefficients.- 5.4 Other Plots for Data Analysis.- 6 Some Practical Issues.- 6.1 The Discrete Fourier Transform of Data.- The Fourier Transform of Sampled Signals.- The Fast Fourier Transform.- 6.2 The Wavelet Transform of Data.- 6.3 Wavelets on an Interval.- Periodic Boundary Handling.- Symmetric and Antisymmetric Boundary Handling.- Meyer Boundary Wavelets.- Orthogonal Wavelets on the Interval.- 6.4 When the Sample Size is Not a Power of Two.- 7 Other Applications.- 7.1 Selective Wavelet Reconstruction.- Wavelet Thresholding.- Spatial Adaptivity.- Global Thresholding.- Estimation of the Noise Level.- 7.2 More Density Estimation.- 7.3 Spectral Density Estimation.- 7.4 Detections of Jumps and Cusps.- 8 Data Adaptive Wavelet Thresholding.- 8.1 SURE Thresholding.- 8.2 Threshold Selection by Hypothesis Testing.- Recursive Testing.- Minimizing False Discovery.- 8.3 Cross-Validation Methods.- 8.4 Bayesian Methods.- 9 Generalizations and Extensions.- 9.1 Two-Dimensional Wavelets.- 9.2 Wavelet Packets.- Wavelet Packet Functions.- The Best Basis Algorithm.- 9.3 Translation Invariant Wavelet Smoothing.- References.- Glossary of Notation.- Glossary of Terms.