SLEX analysis of multivariate non-stationary time series

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 ran- dom 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 de- composition 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 com- ponents. 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