Wavelet Threshold Estimators for Data with Correlated Noise

Wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level‐dependent soft threshold to the coefficients in the wavelet transform. A variety of threshold choices is proposed, including one based on an unbiased estimate of mean‐squared error. The practical performance of the method is demonstrated on examples, including data from a neurophysiological context. The theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable `bench‐mark', that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity, and more generally is obtained by the use of an `oracle' that provides information that is not actually available in the data. It is shown that the level‐dependent threshold estimator performs well relative to the bench‐mark risk, and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator. The wavelet domain structure of both short‐ and long‐range dependent noise is considered, and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes, adapting automatically to the regularity properties of the underlying model. The proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem.

[1]  Michel Loève,et al.  Probability Theory I , 1977 .

[2]  C. G. Khatri,et al.  On Certain Inequalities for Normal Distributions and their Applications to Simultaneous Confidence Bounds , 1967 .

[3]  Z. Šidák.  ON MULTIVARIATE NORMAL PROBABILITIES OF RECTANGLES: THEIR DEPENDENCE ON CORRELATIONS' , 1968 .

[4]  Murad S. Taqqu,et al.  Weak convergence to fractional brownian motion and to the rosenblatt process , 1975, Advances in Applied Probability.

[5]  M. Taqqu.  Weak convergence to fractional brownian motion and to the rosenblatt process , 1975, Advances in Applied Probability.

[6]  H. Teicher,et al.  Probability theory: Independence, interchangeability, martingales , 1978 .

[7]  C. Stein.  Estimation of the Mean of a Multivariate Normal Distribution , 1981 .

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

[9]  Peter Hall,et al.  Nonparametric regression with long-range dependence , 1990 .

[10]  G. Weiss,et al.  Littlewood-Paley Theory and the Study of Function Spaces , 1991 .

[11]  J. Hart.  Kernel regression estimation with time series errors , 1991 .

[12]  Wavelet coefficients of a gaussian process and applications , 1992 .

[13]  David L. Donoho,et al.  Interpolating Wavelet Transforms , 1992 .

[14]  I. Daubechies,et al.  Multiresolution analysis, wavelets and fast algorithms on an interval , 1993 .

[15]  Patrick Flandrin,et al.  Time-scale analyses and self-similar stochastic processes , 1994 .

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

[17]  D. L. Donoho,et al.  Ideal spacial adaptation via wavelet shrinkage , 1994 .

[18]  David R. Brillinger,et al.  Uses of cumulants in wavelet analysis , 1994, Optics & Photonics.

[19]  Pierre Moulin.  Wavelet thresholding techniques for power spectrum estimation , 1994, IEEE Trans. Signal Process..

[20]  B. Eisenberg,et al.  Biological signals that need detection: currents through single membrane channels , 1994, Proceedings of 16th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[21]  D. Donoho.  Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition , 1995 .

[22]  D. Fredkin,et al.  Persistence: a new statistic for characterizing ion-channel activity. , 1995, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[23]  I. Johnstone,et al.  Wavelet Shrinkage: Asymptopia? , 1995 .

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

[25]  D. Donoho,et al.  Translation-Invariant DeNoising , 1995 .

[26]  Kai Schneider,et al.  Wavelet Smoothing of Evolutionary Spectra by Non-Linear Thresholding , 1996 .

[27]  Michael H. Neumann.  SPECTRAL DENSITY ESTIMATION VIA NONLINEAR WAVELET METHODS FOR STATIONARY NON‐GAUSSIAN TIME SERIES , 1996 .

[28]  Yazhen Wang.  Function estimation via wavelet shrinkage for long-memory data , 1996 .

[29]  David R. Brillinger,et al.  Some uses if cumulants in wavelet analysis , 1996 .

[30]  Rainer von Sachs,et al.  Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra , 1997 .

[31]  R. Dahlhaus.  Fitting time series models to nonstationary processes , 1997 .

[32]  I. Johnstone,et al.  Minimax estimation via wavelet shrinkage , 1998 .

[33]  S. Mallat,et al.  Adaptive covariance estimation of locally stationary processes , 1998 .

[34]  Vo Anh,et al.  Semiparametric regression under long-range dependent errors , 1999 .