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Covariance structure of wavelet coefficients: theory and models in a Bayesian perspective
Author(s) -
Vannucci M.,
Corradi F.
Publication year - 1999
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/1467-9868.00214
Subject(s) - wavelet , markov chain monte carlo , covariance , mathematics , hyperparameter , covariance function , prior probability , algorithm , cascade algorithm , bayesian probability , bayesian inference , computer science , artificial intelligence , discrete wavelet transform , wavelet transform , statistics
We present theoretical results on the random wavelet coefficients covariance structure. We use simple properties of the coefficients to derive a recursive way to compute the within‐ and across‐scale covariances. We point out a useful link between the algorithm proposed and the two‐dimensional discrete wavelet transform. We then focus on Bayesian wavelet shrinkage for estimating a function from noisy data. A prior distribution is imposed on the coefficients of the unknown function. We show how our findings on the covariance structure make it possible to specify priors that take into account the full correlation between coefficients through a parsimonious number of hyperparameters. We use Markov chain Monte Carlo methods to estimate the parameters and illustrate our method on bench‐mark simulated signals.