Essential Wavelets for Statistical Applications and Data Analysis

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.