Adaptive Bayesian wavelet shrinkage

Hugh A. Chipman, Eric D. Kolaczyk, Robert McCulloch

Research output: Contribution to journalArticle

380 Citations (Scopus)

Abstract

When fitting wavelet based models, shrinkage of the empirical wavelet coefficients is an effective tool for denoising the data. This article outlines a Bayesian approach to shrinkage, obtained by placing priors on the wavelet coefficients. The prior for each coefficient consists of a mixture of two normal distributions with different standard deviations. The simple and intuitive form of prior allows us to propose automatic choices of prior parameters. These parameters are chosen adaptively according to the resolution level of the coefficients, typically shrinking high resolution (frequency) coefficients more heavily. Assuming a good estimate of the background noise level, we obtain closed form expressions for the posterior means and variances of the unknown wavelet coefficients. The latter may be used to assess uncertainty in the reconstructior. Several examples are used to illustrate the method, and comparisons are made with other shrinkage methods.

Original languageEnglish (US)
Pages (from-to)1413-1421
Number of pages9
JournalJournal of the American Statistical Association
Volume92
Issue number440
StatePublished - Dec 1997
Externally publishedYes

Fingerprint

Wavelet Shrinkage
Wavelet Coefficients
Shrinkage
Coefficient
Posterior Mean
Shrinking
Denoising
Bayesian Approach
Standard deviation
Gaussian distribution
Intuitive
Closed-form
Wavelets
High Resolution
Uncertainty
Unknown
Estimate
Coefficients
Model

Keywords

  • Bayesian estimation
  • Mixture models
  • Uncertainty bands

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability

Cite this

Adaptive Bayesian wavelet shrinkage. / Chipman, Hugh A.; Kolaczyk, Eric D.; McCulloch, Robert.

In: Journal of the American Statistical Association, Vol. 92, No. 440, 12.1997, p. 1413-1421.

Research output: Contribution to journalArticle

Chipman, HA, Kolaczyk, ED & McCulloch, R 1997, 'Adaptive Bayesian wavelet shrinkage', Journal of the American Statistical Association, vol. 92, no. 440, pp. 1413-1421.
Chipman, Hugh A. ; Kolaczyk, Eric D. ; McCulloch, Robert. / Adaptive Bayesian wavelet shrinkage. In: Journal of the American Statistical Association. 1997 ; Vol. 92, No. 440. pp. 1413-1421.
@article{7e28da4bff494e0aa06efe6b799c02a1,
title = "Adaptive Bayesian wavelet shrinkage",
abstract = "When fitting wavelet based models, shrinkage of the empirical wavelet coefficients is an effective tool for denoising the data. This article outlines a Bayesian approach to shrinkage, obtained by placing priors on the wavelet coefficients. The prior for each coefficient consists of a mixture of two normal distributions with different standard deviations. The simple and intuitive form of prior allows us to propose automatic choices of prior parameters. These parameters are chosen adaptively according to the resolution level of the coefficients, typically shrinking high resolution (frequency) coefficients more heavily. Assuming a good estimate of the background noise level, we obtain closed form expressions for the posterior means and variances of the unknown wavelet coefficients. The latter may be used to assess uncertainty in the reconstructior. Several examples are used to illustrate the method, and comparisons are made with other shrinkage methods.",
keywords = "Bayesian estimation, Mixture models, Uncertainty bands",
author = "Chipman, {Hugh A.} and Kolaczyk, {Eric D.} and Robert McCulloch",
year = "1997",
month = "12",
language = "English (US)",
volume = "92",
pages = "1413--1421",
journal = "Journal of the American Statistical Association",
issn = "0162-1459",
publisher = "Taylor and Francis Ltd.",
number = "440",

}

TY - JOUR

T1 - Adaptive Bayesian wavelet shrinkage

AU - Chipman, Hugh A.

AU - Kolaczyk, Eric D.

AU - McCulloch, Robert

PY - 1997/12

Y1 - 1997/12

N2 - When fitting wavelet based models, shrinkage of the empirical wavelet coefficients is an effective tool for denoising the data. This article outlines a Bayesian approach to shrinkage, obtained by placing priors on the wavelet coefficients. The prior for each coefficient consists of a mixture of two normal distributions with different standard deviations. The simple and intuitive form of prior allows us to propose automatic choices of prior parameters. These parameters are chosen adaptively according to the resolution level of the coefficients, typically shrinking high resolution (frequency) coefficients more heavily. Assuming a good estimate of the background noise level, we obtain closed form expressions for the posterior means and variances of the unknown wavelet coefficients. The latter may be used to assess uncertainty in the reconstructior. Several examples are used to illustrate the method, and comparisons are made with other shrinkage methods.

AB - When fitting wavelet based models, shrinkage of the empirical wavelet coefficients is an effective tool for denoising the data. This article outlines a Bayesian approach to shrinkage, obtained by placing priors on the wavelet coefficients. The prior for each coefficient consists of a mixture of two normal distributions with different standard deviations. The simple and intuitive form of prior allows us to propose automatic choices of prior parameters. These parameters are chosen adaptively according to the resolution level of the coefficients, typically shrinking high resolution (frequency) coefficients more heavily. Assuming a good estimate of the background noise level, we obtain closed form expressions for the posterior means and variances of the unknown wavelet coefficients. The latter may be used to assess uncertainty in the reconstructior. Several examples are used to illustrate the method, and comparisons are made with other shrinkage methods.

KW - Bayesian estimation

KW - Mixture models

KW - Uncertainty bands

UR - http://www.scopus.com/inward/record.url?scp=0031315608&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031315608&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0031315608

VL - 92

SP - 1413

EP - 1421

JO - Journal of the American Statistical Association

JF - Journal of the American Statistical Association

SN - 0162-1459

IS - 440

ER -