Bayesian wavelet shrinkage with beta priors
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Bayesian wavelet shrinkage with beta priors Alex Rodrigo dos S. Sousa1
· Nancy L. Garcia2 · Brani Vidakovic3
Received: 24 June 2020 / Accepted: 9 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In wavelet shrinkage and thresholding, most of the standard techniques do not consider information that wavelet coefficients might be bounded, although information about bounded energy in signals can be readily available. To address this, we present a Bayesian approach for shrinkage of bounded wavelet coefficients in the context of non-parametric regression. We propose the use of a zero-contaminated beta distribution with a support symmetric around zero as the prior distribution for the location parameter in the wavelet domain in models with additive gaussian errors. The hyperparameters of the proposed model are closely related to the shrinkage level, which facilitates their elicitation and interpretation. For signals with a low signal-to-noise ratio, the associated Bayesian shrinkage rules provide significant improvement in performance in simulation studies when compared with standard techniques. Statistical properties such as bias, variance, classical and Bayesian risks of the associated shrinkage rules are presented and their performance is assessed in simulations studies involving standard test functions. Application to real neurological data set on spike sorting is also presented.
1 Introduction Wavelet-based methods are increasingly applied in many fields, such as mathematics, signal and image processing, geophysics, bioinformatics, and many others. In statistics, applications of wavelets arise mainly in the tasks involving non-parametric regression, density estimation, assessment of scaling, functional data analysis and stochastic processes. These methods basically utilize the possibility of representing functions that belong to certain functional spaces as expansions in a wavelet basis, similar to others expansions such as splines or Fourier, among others. However, the
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Alex Rodrigo dos S. Sousa [email protected]
1
University of São Paulo, São Paulo, Brazil
2
University of Campinas, Campinas, Brazil
3
Texas A&M University, College Station, TX, USA
123
A. R. d. S. Sousa et al.
wavelet expansions have characteristics that make them particularly useful: they are localized in both time and scale/frequency in an adaptive way, their coefficients are typically sparse, the coefficients can be obtained by fast computational algorithms, and the magnitudes of the coefficients can be linked to the smoothness properties of the functions they represent. These properties of wavelet representations enable adaptive time/frequency data analysis, bring computational advantages, and allow for statistical data modeling at varying resolution scales. Wavelet shrinkage methods are used to estimate the underlying signal when its noisy version is observed. The noisy signal is transformed to a wavelet domain, the resulting wavelet coefficients are shrunk, and the inverse transform of the sh
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