SLEX Analysis of Multivariate Nonstationary Time Series

Abstract

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.

Estimating linear dependence between nonstationary time series using the locally stationary wavelet model

“…Existing methods of assessing multivariate dependence include those of Dahlhaus (2000), who extended univariate locally stationary processes to a multivariate setting, Ombao et al (2005) which is an extension to the SLEX framework and Guo and Dai (2006) who use a Cholesky decomposition approach. The application of our method to multivariate cases is possible using the current methodology, by taking pairwise combinations of multivariate series.…”

Testing parametric assumptions of trends of a nonstationary time series

“…Giurcanu & Spokoiny (2004) treated nonstationarity by assuming that correlation functions could be well approximated by those of stationary processes. Ombao et al (2005) generalized the framework of Dahlhaus (1997) by utilizing the smooth localized complex exponentials. Here we shall follow the framework in Draghicescu et al (2009) and Zhou & Wu (2009) and assume that the error sequence {e i } n i=1 is generated from the model…”

Testing additive assumptions on means of regular monitoring data: a multivariate nonstationary time series approach

“…, n, as a multivariate nonstationary process. The problem of modeling nonstationary processes has been studied in the literature by means of spectral representations (Priestley (1965), Dahlhaus (1996Dahlhaus ( , 1997), pseudo-differential operators (Mallat, Papanicolaou, and Zhang (1998)), discrete non-decimated wavelets (Nason, von Sachs, and Kroisandt (2000)) and smooth localized complex exponentials (Ombao, von Sachs, and Guo (2005)). Other contributions can be found in Cheng and Tong (1998), Giurcanu and Spokoiny (2004), and Draghicescu, Guillas, and Wu (2009), among others.…”