Frederick David Abraham, Blueberry Brain Institute, Waterbury Center, VT

Abstract: First, some linear techniques in multivariate time-series analysis in EEG research are reviewed to highlight the problem of estimating the dimensionality of the state space (embedding dimension), the reconstruction of an attractor, and the evaluation of invariant properties of the attractor. The traditional linear techniques included the usual spectral and cospectral measures of power, phase, and coherence to which stepwise discriminant analysis was applied for canonical representation of the attractor. Then, some traditional nonlinear techniques of attractor reconstruction and dimensional analysis which use the time-lagged univariate approach of Ruelle and Takens (Takens, 1981) are reviewed. Next, updates and multivariate generalizations that use singular-value decomposition (Broomhead King, 1986) are reviewed. Finally, Stewart's (1995, 1996) multivariate generalization of the method of false nearest neighbors (Abarbanel, Brown, Sidorowich, Tsimring, 1993; Kennel, Brown, Abarbanel, 1992) is reviewed. These are particularly relevant for evaluating multivariate coherence in research on the complex cooperative dynamical systems found in neuroscience, psychology, and social science when time series of sufficient length are investigated.

Keywords: dynamical system, nonlinear, time series, false nearest neighbors, Lyapunov exponents, singular-value decomposition, singular-value analysis, attractor reconstruction, attractor invariants, system invariants, chaos