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.
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.
We consider time series which change their structure repeatedly between a finite number of states, and we discuss algorithms to detect the changepoints for two particular situations. In the first ...case, the observed time series is a nonparametric autoregression of order p and the autoregression function changes sometimes. Here, we use a system of neural networks to estimate the autoregression functions and to detect the changepoints. In the second case, the time series is a piecewise linear process with stable innovations, where we assume that the various processes represent different dominating local frequencies, and we use wavelet packet coefficients to detect the changepoints.
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