Larger posterior mode wavelet thresholding and applications

This paper explores the thresholding rules induced by a variation of the Bayesian MAP principle. The MAP rules are Bayes actions that maximize the posterior. The proposed rule is thresholding and always picks the mode of the posterior larger in absolute value, thus the name LPM. We demonstrate that the introduced shrinkage performs comparably to several popular shrinkage techniques. The exact risk properties of the thresholding rule are explored, as well. We provide extensive simulational analysis and apply the proposed methodology to real-life experimental data coming from the field of atomic force microscopy.

[1]  Brani Vidakovic,et al.  BAMS Method: Theory and Simulations , 2001 .

[2]  David L. Donoho,et al.  WaveLab and Reproducible Research , 1995 .

[3]  B. Vidakovic.  Nonlinear wavelet shrinkage with Bayes rules and Bayes factors , 1998 .

[4]  Theofanis Sapatinas,et al.  Empirical Bayes approach to wavelet regression using ϵ-contaminated priors , 2004 .

[5]  Y. Benjamini,et al.  Thresholding of Wavelet Coefficients as Multiple Hypotheses Testing Procedure , 1995 .

[6]  Lianfen Qian,et al.  Nonparametric Curve Estimation: Methods, Theory, and Applications , 1999, Technometrics.

[7]  T. Cai.  Adaptive wavelet estimation : A block thresholding and oracle inequality approach , 1999 .

[8]  Anestis Antoniadis,et al.  Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study , 2001 .

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

[10]  H. Chipman,et al.  Adaptive Bayesian Wavelet Shrinkage , 1997 .

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

[12]  Felix Abramovich,et al.  Bayesian Approach to Wavelet Decomposition and Shrinkage , 1999 .

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

[14]  Marina Vannucci,et al.  Wavelet Thresholding with Bayesian False Discovery Rate Control , 2005, Biometrics.

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

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

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

[18]  M. Clyde,et al.  Flexible empirical Bayes estimation for wavelets , 2000 .

[19]  A. Bruce,et al.  WAVESHRINK WITH FIRM SHRINKAGE , 1997 .

[20]  A. Bruce,et al.  Understanding WaveShrink: Variance and bias estimation , 1996 .

[21]  Gerber,et al.  Atomic Force Microscope , 2020, Definitions.

[22]  Edward H. Adelson,et al.  Noise removal via Bayesian wavelet coring , 1996, Proceedings of 3rd IEEE International Conference on Image Processing.

[23]  M. Clyde,et al.  Multiple shrinkage and subset selection in wavelets , 1998 .

[24]  Brani Vidakovic,et al.  Expansion estimation by Bayes rules , 1999 .

[25]  Sailes K. Sengupta,et al.  Bayesian Inference in Wavelet-Based Models , 2002, Technometrics.

[26]  A. Antoniadis,et al.  Wavelets and Statistics , 1995 .

[27]  Brani Vidakovic,et al.  Γ -Minimax Wavelet Shrinkage: A Robust Incorporation of Information about Energy of a Signal in Denoising Applications , 2001 .

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

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

[30]  F. Ruggeri,et al.  Bayesian Modeling in the Wavelet Domain , 2005 .

[31]  B. Silverman,et al.  Wavelet thresholding via a Bayesian approach , 1998 .

[32]  Calyampudi Radhakrishna Rao,et al.  Bayesian thinking : modeling and computation , 2005 .

[33]  Robert D. Nowak,et al.  Wavelet-based image estimation: an empirical Bayes approach using Jeffrey's noninformative prior , 2001, IEEE Trans. Image Process..

[34]  Edward I. George,et al.  Empirical Bayes Estimation in Wavelet Nonparametric Regression , 1999 .