Instantaneous and lagged measurements of linear and nonlinear dependence between groups of multivariate time series: frequency decomposition

Abstract

Measures of linear dependence (coherence) and nonlinear dependence (phase synchronization) between any number of multivariate time series are defined. The measures are expressed as the sum of lagged dependence and instantaneous dependence. The measures are non-negative, and take the value zero only when there is independence of the pertinent type. These measures are defined in the frequency domain and are applicable to stationary and non-stationary time series. These new results extend and refine significantly those presented in a previous technical report (Pascual-Marqui 2007, arXiv:0706.1776 [stat.ME], http://arxiv.org/abs/0706.1776), and have been largely motivated by the seminal paper on linear feedback by Geweke (1982 JASA 77:304-313). One important field of application is neurophysiology, where the time series consist of electric neuronal activity at several brain locations. Coherence and phase synchronization are interpreted as "connectivity" between locations. However, any measure of dependence is highly contaminated with an instantaneous, non-physiological contribution due to volume conduction and low spatial resolution. The new techniques remove this confounding factor considerably. Moreover, the measures of dependence can be applied to any number of brain areas jointly, i.e. distributed cortical networks, whose activity can be estimated with eLORETA (Pascual-Marqui 2007, arXiv:0710.3341 [math-ph]).