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.

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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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We describe a nonlinear regression problem, where the regression functions have an additive structure and the dependent variable is a one-dimensional time series. Multivariate time series with unknown time delay operators are used as independent variables. By fitting a feedforward neural network with block structure to the data, we estimated the additive regression function and, parallel to this, the time lags. We present the consistency proof of neural network weights estimator and the time lag estimator independently from each other. In the practical part of the article, we present the useful feature of blocked neural networks to estimate the relevance measures of each input variable in a simple way. Furthermore, we propose an approach to solve the well-known variable selection problem for the class of nonlinear multivariate beta-mixing time series models considered here. Finally, we apply the methodology to an artificial example.

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