Bayesian Wavelet Shrinkage of the Haar-Fisz Transformed Wavelet Periodogram

TL;DR

This paper introduces a Bayesian wavelet shrinkage method applied to Haar-Fisz transformed spectra for modeling evolving spectral structures in non-stationary time series, providing stable estimates and credible intervals.

Contribution

It combines Haar-Fisz transformation with Bayesian wavelet shrinkage to improve spectral estimation of locally stationary time series, offering credible intervals for spectral uncertainty.

Findings

Produces stable spectral estimates on simulated and real data

Provides credible intervals that give insights into spectral uncertainty

Demonstrates effectiveness on infant ECG data

Abstract

It is increasingly being realised that many real world time series are not stationary and exhibit evolving second-order autocovariance or spectral structure. This article introduces a Bayesian approach for modelling the evolving wavelet spectrum of a locally stationary wavelet time series. Our new method works by combining the advantages of a Haar-Fisz transformed spectrum with a simple, but powerful, Bayesian wavelet shrinkage method. Our new method produces excellent and stable spectral estimates and this is demonstrated via simulated data and on differenced infant ECG data. A major additional benefit of the Bayesian paradigm is that we obtain rigorous and useful credible intervals of the evolving spectral structure. We show how the Bayesian credible intervals provide extra insight into the infant ECG data.