Non-stationary time series (i.e., time series with statistical properties that vary over time) arise in many areas of neuroscience research and clinical practice. For example, the spectral properties of electroencephalograms (EEGs) vary with brain state, and frequently this variation is of central clinical or scientific importance. Existing methods of spectra analysis assume that a time series is a realization of a stationary random process. These methods can be extended to non-stationary processes using windowed Fourier transforms, but the number and size of the windows must be chosen subjectively. We propose to develop improved, automatic statistical methods for analysis of non-stationary multivariate time series. We will evaluate the methods in applications to realistic simulated data and to real multi-channel EEG data required from patients with brain disorders. We will compare results from our automatic methods with the clinical judgments.
Our specific aims are to develop, evaluate, apply, implement, and distribute the following statistical methods: (1) an estimator of the time-varying power spectrum of a univariate random process; (2) estimators of the time-varying spectral density matrix, coherences, and phase spectra of a multivariate random process; (3) time-frequency principal component analysis; (4) time-frequency filters; (5) cycle-spinning to reduce bias due to the dyadic structure of our estimators; and (6) univariate and multivariate processes that are smooth in both time and frequency domains. Our first proposed statistical methods are based on the Smooth Localized complex Exponential (SLEX) transform, which provides a rich selection of orthogonal transforms. The structure of the SLEX transform allows us to use the computationally efficient Best Basis algorithm of Coifman and Wickerhauser to automatically select a particular transform, which represents a segmentation of a non-stationary time series into approximately stationary intervals. Our second proposed approach (Aim 6) takes a new path. Unlike the traditional approaches that focus on modeling the Periodiograms, we propose to model the transfer function directly as a smooth function in both frequency and time using smoothing splines and to use a signal-plus-noise model. By modeling the transfer function directly, we alloy simultaneous smoothing in time and frequency within the Fourier transformation. Unlike the periodiogram, the transfer function preserves the phase information, and therefore the time-varying cross-spectra, coherence, and phase can be directly calculated from the transfer functions.
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