This paper de®nes and studies a new class of non-stationary random processes constructed from discrete non-decimated wavelets which generalizes the Crame Âr (Fourier) representation of stationary time series. We de®ne an evolutionary wavelet spectrum (EWS) which quanti®es 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.
This article develops a wavelet decomposition of a stochastic process which parallels a time{localized Cram er (Fourier) spectral representation. We provide a time{scale instead of a time{frequency decomposition and, hence, instead of thinking as scale in terms of \inverse frequency" we start from genuine time{scale building blocks or \atoms". Using this class of locally stationary wavelet (LSW) processes, a doubly{indexed array of processes fX t;T g t=1;:::;T ; T 1, we develop a theory for the estimation of the \evolutionary wavelet spectrum". Our asymptotics are based on rescaling in time{ location which allows us to perform rigorous estimation theory starting from a single stretch of observations of fX t;T g. This evolutionary wavelet spectrum measures the local power in the variance{covariance decomposition of the process fX t;T g at a certain
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