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.

We propose a new method for analyzing bivariate nonstationary time series. The proposed method is a statistical procedure that automatically segments the time series into approximately stationary blocks and selects the span to be used to obtain the smoothed estimates of the time-varying spectra and coherence. It is based on the smooth localized complex exponential (SLEX) transform, which forms a library of orthogonal complex-valued transforms that are simultaneously localized in time and frequency. We show that the smoothed SLEX periodograms are consistent estimators, report simulation results, and apply the method to a two-channel electroencephalogram dataset recorded during an epileptic seizure.

We propose to analyze a multivariate non-stationary time series using the SLEX (Smooth Localized Complex EXponentials) library. The SLEX library is a collection of bases; each basis consists of the SLEX waveforms which are orthogonal localized versions of the Fourier complex exponentials. In our procedure, we first build a family of multivariate SLEX models such that every model has a spectral representation in terms of a unique SLEX basis. The SLEX family provides a flexible representation for non-stationary random processes because every SLEX basis is localized in both time and frequency. The next step is to select a model using a penalized log energy criterion which we derive in this paper to be the Kullback-Leibler distance between a model and the empirical time series. In our procedure, we apply SLEX principal components analysis to obtain a decomposition of a possibly highly cross-correlated multivariate data set into non-stationary components with uncorrelated (non-redundant) spectral information. The best model is then selected by computing the log energy criterion based on the SLEX principal components. The proposed SLEX analysis for multivariate non-stationary time series closely parallels traditional Fourier analysis of stationary time series. Hence, our method gives results that are easy to interpret. Moreover, the SLEX method uses computationally efficient algorithms and hence can easily handle massive data sets. We illustrate the SLEX method by its application to a multivariate brain waves data set recorded during an epileptic seizure.

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We develop a test for stationarity of a time series against the alternative of a time-varying covariance structure. Using localized versions of the periodogram, we obtain empirical versions of a reasonable notion of a time-varying spectral density. Coef®cients with respect to a Haar wavelet series expansion of such a time-varying periodogram are an indicator of whether there is some deviation from covariance stationarity. We propose a test based on the limit distribution of these empirical coef®cients.

Many time series in the applied sciences display a time-varying second order structure. In this article, we address the problem of how to forecast these non-stationary time series by means of non-decimated wavelets. Using the class of Locally Stationary Wavelet processes, we introduce a new predictor based on wavelets and derive the prediction equations as a generalisation of the Yule-Walker equations. We propose an automatic computational procedure for choosing the parameters of the forecasting algorithm. Finally, we apply the prediction algorithm to a meteorological time series.

International audienceThere exists a wide literature on modelling strongly dependent time series using a longmemory parameter d, including more recent work on semiparametric wavelet estimation. As a generalization of these latter approaches, in this work we allow the long-memory parameter d to be varying over time. We embed our approach into the framework of locally stationary processes. We show weak consistency and a central limit theorem for our log-regression wavelet estimator of the time-dependent d in a Gaussian context. Both simulations and a real data example complete our work on providing a fairly general approach

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