Bayesian wavelet de-noising with the caravan prior^*

¹ Biometris, Wageningen University & Research, Postbus 16, 6700 AA Wageningen, The Netherlands.
² Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Van Mourik Broekmanweg 6, 2628 XE Delft, The Netherlands.
³ Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, 412 96 Göteborg, Sweden.
⁴ Korteweg-de Vries Institute for Mathematics, University of Amsterdam, PO Box 94248, 1090 GE Amsterdam, The Netherlands.
⁵ Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, Nijmegen, The Netherlands.

^** Corresponding author: smoritz@chalmers.se

Received: 24 October 2018
Accepted: 8 August 2019

Abstract

According to both domain expert knowledge and empirical evidence, wavelet coefficients of real signals tend to exhibit clustering patterns, in that they contain connected regions of coefficients of similar magnitude (large or small). A wavelet de-noising approach that takes into account such a feature of the signal may in practice outperform other, more vanilla methods, both in terms of the estimation error and visual appearance of the estimates. Motivated by this observation, we present a Bayesian approach to wavelet de-noising, where dependencies between neighbouring wavelet coefficients are a priori modelled via a Markov chain-based prior, that we term the caravan prior. Posterior computations in our method are performed via the Gibbs sampler. Using representative synthetic and real data examples, we conduct a detailed comparison of our approach with a benchmark empirical Bayes de-noising method (due to Johnstone and Silverman). We show that the caravan prior fares well and is therefore a useful addition to the wavelet de-noising toolbox.