Instantaneous oscillatory direction and phase for multivariate timeseries

TL;DR

This paper generalizes the analytic signal approach to multivariate time series, enabling the calculation of instantaneous oscillatory direction and phase, which is useful for phase synchronization analysis.

Contribution

It introduces a novel method for defining instantaneous amplitude and phase in multivariate signals, extending the classical analytic signal concept.

Findings

Provides a new procedure for local oscillatory direction detection

Facilitates phase synchronization analysis in multivariate data

Useful as a preprocessing step for complex data analysis

Abstract

This text describes a generalization of the analytic signal (Gabor, 1946) approach for the definition of instantaneous amplitude and phase to the case of multivariate signals. It was originally written as an appendix for another paper, where the determination of the locally dominant oscillatory direction (the instantaneous amplitude) described here is used as a preprocessing step for another kind of data analysis. The text is reproduced in a 'standalone' form because the procedure might prove useful in other contexts too, especially for the purpose of phase synchronization analysis (Rosenblum et al., 1996) between two (or more) multivariate sets of time series (Pascual-Marqui, 2007).