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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 electrocardiogram 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 electrocardiogram data.
Highlights
For a real-life time series it is sometimes difficult to determine whether the underlying process is really stationary using only observations from a section of the process
This article assumes that a time series can be modelled by a locally stationary wavelet (LSW) process and considers the estimation of the associated evolutionary wavelet spectrum (EWS)
Definition 1 (The Locally Stationary Wavelet Process) A LSW process is a sequence of doubly indexed stochastic processes, {Xt, T}t = 0, . . ., T−1, where T = 2J for some J 2 N
Summary
For a real-life time series it is sometimes difficult to determine whether the underlying process is really stationary using only observations from a section of the process. One class of nonstationary models, which we consider here, are the locally stationary processes with slowly evolving second-order structure. This article assumes that a time series can be modelled by a LSW process and considers the estimation of the associated evolutionary wavelet spectrum (EWS).
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