Wavelet methods

This overview article motivates the use of wavelets in statistics and introduces the basic mathematics behind the construction of wavelets. Topics covered include the continuous and discrete wavelet transforms, multiresolution analysis, and the non-decimated wavelet transform. We describe the basic mechanics of nonparametric function estimation via wavelets, emphasizing the concepts of sparsity and thresholding. A simple proof of the mean-square consistency of the wavelet estimator is also included. The article ends with two special topics: function estimation with unbalanced Haar wavelets and variance stabilization via the Haar-Fisz transformation. WIREs Comp Stat 2010 2 654-667 DOI: 10.1002/wics.124

[1]  Ricardo H. Nochetto,et al.  A posteriori error analysis for a class of integral equations and variational inequalities , 2010, Numerische Mathematik.

[2]  I. Johnstone,et al.  Wavelet Threshold Estimators for Data with Correlated Noise , 1997 .

[3]  A. Haar.  Zur Theorie der orthogonalen Funktionensysteme , 1910 .

[4]  J. Polzehl,et al.  Adaptive weights smoothing with applications to image restoration , 1998 .

[5]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[6]  Edward H. Adelson,et al.  Orthogonal Pyramid Transforms For Image Coding. , 1987, Other Conferences.

[7]  Martina Wilhelm,et al.  Finite element valuation of swing options , 2008 .

[8]  P. Davies,et al.  Local Extremes, Runs, Strings and Multiresolution , 2001 .

[9]  Debashis Mondal,et al.  Wavelet Variance Analysis for Random Fields on a Regular Lattice , 2012, IEEE Transactions on Image Processing.

[10]  Karsten Urban,et al.  Wavelet Methods for Elliptic Partial Differential Equations , 2008 .

[11]  R. Averkamp,et al.  Wavelet thresholding for non-necessarily Gaussian noise: idealism , 2003 .

[12]  Yoshua Bengio,et al.  High quality document image compression with "DjVu" , 1998, J. Electronic Imaging.

[13]  D. Percival,et al.  Wavelet variance analysis for gappy time series , 2010 .

[14]  Jerome M. Shapiro,et al.  Embedded image coding using zerotrees of wavelet coefficients , 1993, IEEE Trans. Signal Process..

[15]  Christoph Schwab,et al.  Linear Complexity Solution of Parabolic Integro-differential Equations , 2006, Numerische Mathematik.

[16]  Anestis Antoniadis,et al.  Wavelet methods in statistics: Some recent developments and their applications , 2007, 0712.0283.

[17]  Adhemar Bultheel,et al.  Stabilised wavelet transforms for non-equispaced data smoothing , 2002, Signal Process..

[18]  I. Johnstone,et al.  Empirical Bayes selection of wavelet thresholds , 2005, math/0508281.

[19]  Andrew T. Walden,et al.  Wavelet scale analysisof bivariate time series ii:statistical properties for linear processes , 2000 .

[20]  Yves Meyer,et al.  Wavelets - tools for science and technology , 1987 .

[21]  Dominik Schötzau,et al.  hp-discontinuous Galerkin time stepping for parabolic problems , 2001 .

[22]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[23]  W. Sweldens.  The Lifting Scheme: A Custom - Design Construction of Biorthogonal Wavelets "Industrial Mathematics , 1996 .

[24]  C. Schwab,et al.  Wavelet Galerkin pricing of American options on Lévy driven assets , 2005 .

[25]  Wolfgang Dahmen,et al.  Compression Techniques for Boundary Integral Equations - Asymptotically Optimal Complexity Estimates , 2006, SIAM J. Numer. Anal..

[26]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[27]  Donald P. Percival,et al.  On estimation of the wavelet variance , 1995 .

[28]  Christoph Schwab,et al.  On Kolmogorov equations for anisotropic multivariate Lévy processes , 2010, Finance Stochastics.

[29]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[30]  G. Nason.  Wavelet Shrinkage using Cross-validation , 1996 .

[31]  A. Tsybakov,et al.  Wavelets, approximation, and statistical applications , 1998 .

[32]  Nouna Kettaneh,et al.  Statistical Modeling by Wavelets , 1999, Technometrics.

[33]  P. Fryzlewicz.  Unbalanced Haar Technique for Nonparametric Function Estimation , 2007 .

[34]  Marina Vannucci,et al.  Wavelet-Based Nonparametric Modeling of Hierarchical Functions in Colon Carcinogenesis , 2003 .

[35]  Dominik Schötzau,et al.  hp-DGFEM for parabolic evolution problems , 1999 .

[36]  Peter Guttorp,et al.  Wavelet analysis of covariance with application to atmospheric time series , 2000 .

[37]  Rafael C. González,et al.  Local Determination of a Moving Contrast Edge , 1985, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[38]  Amir Averbuch,et al.  Fast adaptive wavelet packet image compression , 2000, IEEE Trans. Image Process..

[39]  Christopher M. Brislawn,et al.  FBI compression standard for digitized fingerprint images , 1996, Optics & Photonics.

[40]  Todd R. Ogden,et al.  Wavelet Methods for Time Series Analysis , 2002 .

[41]  Bernard W. Silverman,et al.  The discrete wavelet transform in S , 1994 .

[42]  B. Silverman,et al.  Incorporating Information on Neighboring Coefficients Into Wavelet Estimation , 2001 .

[43]  Peter Guttorp,et al.  Space‐Time Modelling of Trends in Temperature Series , 2011 .

[44]  T. Fearn,et al.  Bayesian Wavelet Regression on Curves With Application to a Spectroscopic Calibration Problem , 2001 .

[45]  I. Daubechies,et al.  Wavelets on the Interval and Fast Wavelet Transforms , 1993 .

[46]  Jeffrey S. Morris,et al.  Journal of the American Statistical Association Using Wavelet-based Functional Mixed Models to Characterize Population Heterogeneity in Accelerometer Profiles Using Wavelet-based Functional Mixed Models to Characterize Population Heterogeneity in Accelerometer Profiles: a Case Study , 2022 .

[47]  Christoph Schwab,et al.  Wavelet Discretizations of Parabolic Integrodifferential Equations , 2003, SIAM J. Numer. Anal..

[48]  C. Torrence,et al.  A Practical Guide to Wavelet Analysis. , 1998 .

[49]  Thomas R. Fischer,et al.  Quadtree classification and TCQ image coding , 2001, IEEE Trans. Circuits Syst. Video Technol..

[50]  Y. Benjamini,et al.  Adaptive thresholding of wavelet coefficients , 1996 .

[51]  Anestis Antoniadis,et al.  Parametric modelling of thresholds across scales in wavelet regression , 2006 .

[52]  Christoph Schwab,et al.  Fast deterministic pricing of options on Lévy driven assets , 2002 .

[53]  Brani Vidakovic,et al.  Wavelet-based 2D Multifractal Spectrum with Applications in Analysis of Digital Mammography Images , 2007 .

[54]  B. Silverman,et al.  The Stationary Wavelet Transform and some Statistical Applications , 1995 .

[55]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[56]  D. Donoho,et al.  Translation-Invariant De-Noising , 1995 .

[57]  McClanahan,et al.  Tropical pacific forcing of decadal SST variability in the western indian ocean over the past two centuries , 2000, Science.

[58]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[59]  Jeffrey S. Morris,et al.  Wavelet‐based functional mixed models , 2006, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[60]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[61]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[62]  R. A. Madden,et al.  Seasonal Variations of the 40-50 Day Oscillation in the Tropics , 1986 .

[63]  C.-C. Jay Kuo,et al.  Full wavelet transform (FWT) approach to image compression , 1993, Electronic Imaging.

[64]  Fionn Murtagh,et al.  Image Processing and Data Analysis - The Multiscale Approach , 1998 .

[65]  M. Srivastava,et al.  On Tests for Detecting Change in Mean , 1975 .

[66]  David S. Taubman,et al.  High performance scalable image compression with EBCOT. , 2000, IEEE transactions on image processing : a publication of the IEEE Signal Processing Society.

[67]  L. Mark Berliner,et al.  Spatiotemporal Hierarchical Bayesian Modeling Tropical Ocean Surface Winds , 2001 .

[68]  A. Walden,et al.  Wavelet Analysis and Synthesis of Stationary Long-Memory Processes , 1996 .

[69]  William A. Pearlman,et al.  A new, fast, and efficient image codec based on set partitioning in hierarchical trees , 1996, IEEE Trans. Circuits Syst. Video Technol..

[70]  Edward H. Adelson,et al.  The Laplacian Pyramid as a Compact Image Code , 1983, IEEE Trans. Commun..

[71]  Christopher K. Wikle,et al.  Multiresolution Wavelet Analyses in Hierarchical Bayesian Turbulence Models , 1999 .

[72]  A. Walden,et al.  Statistical Properties and Uses of the Wavelet Variance Estimator for the Scale Analysis of Time Series , 2000 .

[73]  Hervé Carfantan,et al.  Time-invariant orthonormal wavelet representations , 1996, IEEE Trans. Signal Process..

[74]  A. Walden,et al.  Wavelet scale analysis of bivariate time series i: motivation and estimation , 2000 .

[75]  G. Nason,et al.  A Haar-Fisz Algorithm for Poisson Intensity Estimation , 2004 .

[76]  T. Fearn,et al.  A decision theoretical approach to wavelet regression on curves with a high number of regressors , 2003 .

[77]  Christopher M. Brislawn,et al.  FBI wavelet/scalar quantization standard for gray-scale fingerprint image compression , 1993, Defense, Security, and Sensing.

[78]  Colm Mulcahy,et al.  Plotting and Scheming with Wavelets , 1996 .

[79]  Christoph Winter,et al.  Wavelet Galerkin schemes for option pricing in multidimensional Lévy models , 2009 .

[80]  M. Delong,et al.  Parkinsonian Symptomatology An Anatomical and Physiological Analysisa a , 1988 .

[81]  R. Lundin,et al.  Solar activity explored with new wavelet methods , 2003 .

[82]  Y. Meyer.  Wavelets and Operators , 1993 .

[83]  G. Nason,et al.  Real nonparametric regression using complex wavelets , 2004 .

[84]  Piotr Fryzlewicz,et al.  Data-driven wavelet-Fisz methodology for nonparametric function estimation , 2007, 0711.0883.

[85]  Christoph Schwab,et al.  Anisotropic Stable Levy Copula Processes - Analytical and Numerical Aspects , 2007 .

[86]  Eric Moulines,et al.  A wavelet whittle estimator of the memory parameter of a nonstationary Gaussian time series , 2008 .

[87]  A. Cohen.  Numerical Analysis of Wavelet Methods , 2003 .

[88]  Thomas R. Fischer,et al.  Image subband coding using arithmetic coded trellis coded quantization , 1995, IEEE Trans. Circuits Syst. Video Technol..

[89]  D. Percival,et al.  Analysis of Subtidal Coastal Sea Level Fluctuations Using Wavelets , 1997 .

[90]  P. Carr,et al.  The Variance Gamma Process and Option Pricing , 1998 .

[91]  Guy P. Nason,et al.  Wavelet Methods in Statistics with R , 2008 .

[92]  Rama Cont.  Encyclopedia of quantitative finance , 2010 .

[93]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .