Statistical analysis of nonstationary time series
Hernando Ombao, Jonathan A. Raz · Deep Blue (University of Michigan) · 1999
In the dissertation, we propose (i) a new method for analyzing a bivariate non-stationary time series; (ii) a new span selector for periodogram smoothing and (iii) a new model of a non-stationary random process. The dissertation was motivated by the need of neurologists to study changes in the electrical activity of the brain during an epileptic seizure. This can be directly approached by estimating the time-varying spectra and coherence of electroencephalograms (EEGs) or brain waves that are recorded during an epileptic seizure. The proposed method is a statistical procedure that automatically segments the time series into approximately stationary blocks and automatically computes the span for obtaining smoothed estimates of the time-varying spectra and coherence. The proposed method is based on the SLEX (S&barbelow;mooth L&barbelow;ocalized Complex EXponential) transform which forms a library of orthonormal transforms. The SLEX transform is based on the SLEX vectors which are localized in time and frequency. Automatic segmentation in the Auto-SLEX method is obtained by applying the Best Basis Algorithm of Coifman and Wickerhauser (1991). Kernel smoothed estimates are obtained by using a span selector that is developed for generalized additive models (Hastie and Tibshirani, 1990). One feature of the SLEX basis vectors is that they can be made arbitrarily close to the Fourier complex exponentials. Thus, the Auto-SLEX method parallels the methods developed for stationary time series and yields results that are easy to interpret. We also propose a new model of a non-stationary random process that has a representation that uses the SLEX basis vectors as stochastic building blocks. The representation generalizes the Cramer spectral representation by specifying a transfer function that changes with time and by using a basis from the SLEX library that is orthonormal and localized in both time and frequency. Under the SLEX model, we defined a spectrum that gives a time-frequency decomposition of power. Thus, the SLEX model is in the spirit of traditional spectral analysis because it gives a time-dependent analogue of the spectrum of a stationary random process.