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Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum

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This paper defines and studies a new class of non-stationary random processes constructed from discrete non-decimated wavelets which generalizes the Cramér (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time-localized autocovariance. We illustrate our theory with a pedagogical example based on discrete non-decimated Haar wavelets and also a real medical time series example.
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Keywords: Local stationarity; Non-linear wavelet shrinkage; Non-stationary time series; Wavelet periodogram; Wavelet processes; Wavelet spectrum

Document Type: Research Article

Affiliations: 1: University of Bristol, UK 2: Université Catholique de Louvain, Belgium 3: Universität Kaiserslautern, Germany

Publication date: February 1, 2000

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