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Nonlinear coherence in multivariate research: invariants and the reconstruction of attractors. (English) Zbl 1201.37106
Summary: 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 are reviewed. Next, updates and multivariate generalizations that use singular-value decomposition are reviewed. Finally, Stewart’s multivariate generalization of the method of false nearest neighbors 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.
37M10Time series analysis (dynamical systems)
37N25Dynamical systems in biology
62M10Time series, auto-correlation, regression, etc. (statistics)
62P10Applications of statistics to biology and medical sciences