Automatic Statistical Analysis of Bivariate Nonstationary Time Series

Abstract

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.

Failure data analysis of a large-scale heterogeneous server environment

“…The first important issue of interest concerns the stationarity of these error and failure processes, where a time series is said to be stationary if it is invariant to shifts in time with respect to statistical measures. Motivated by its effective use in a recent Web traffic study [13], we use a spe-cific statistical procedure, called Auto-SLEX [12], to partition the error and failure rate and increment processes into stationary intervals. Auto-SLEX has the advantages that it does not identify changes in the level of a time series and it only identifies changes in the correlation structure, because the partitioning is based on second-order properties of the process and ignores the level information contained in the spectral value at zero frequency; refer to [12,13] for additional technical details.…”

“…Motivated by its effective use in a recent Web traffic study [13], we use a spe-cific statistical procedure, called Auto-SLEX [12], to partition the error and failure rate and increment processes into stationary intervals. Auto-SLEX has the advantages that it does not identify changes in the level of a time series and it only identifies changes in the correlation structure, because the partitioning is based on second-order properties of the process and ignores the level information contained in the spectral value at zero frequency; refer to [12,13] for additional technical details. The corresponding results for error/failure rate and increment processes demonstrate that these processes are clearly nonstationary and that they contain relatively long time intervals which are stationary.…”

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

“…Dahlhaus (2000) introduces a method for modelling multivariate processes based on the locally stationary process model (Dahlhaus, 1997). Other methods for the analysis of non-stationary bivariate time series include the Auto-SLEX method (Ombao et al, 2001). Using this method, the data is automatically segmented into approximately stationary dyadic blocks.…”

Forecasting non-stationary time series by wavelet process modelling

“…Certainly the simplest approach consists in assuming piecewise stationarity, or approximate piecewise stationarity, where the challenge is to find the stretches of homogeneity optimally in a data-driven way (Ombao et al, 2001). The resulting estimate of the time-varying second order structure is, necessarily, rather blocky over time, so some further thoughts on how to cope with these potentially artificially introduced discontinuities are needed.…”