Multivariate Variational Mode Decomposition

In this paper, a generic extension of variational mode decomposition (VMD)\nalgorithm for multivariate or multichannel data sets is presented. We first\ndefine a model for multivariate modulated oscillations that is based on the\npresence of a joint or common frequency component among all channels of input\ndata. Using that model for multivariate oscillations, we construct a\nvariational optimization problem that aims to extract an ensemble of\nband-limited modes containing inherent multivariate modulated oscillations\npresent in multivariate input signal. The cost function to be minimized is the\nsum of bandwidths of all signal modes across all input data channels, which is\na generic extension of the cost function used in standard VMD to multivariate\ndata. Minimization of the resulting variational model is achieved through the\nalternate direction method of multipliers (ADMM) approach. That yields an\noptimal set of multivariate modes in terms of narrow bandwidth and\ncorresponding center frequencies that are assumed to be commonly present among\nall channels of a multivariate modulated oscillation. We demonstrate the\neffectiveness of the proposed method through results obtained from extensive\nsimulations involving test (synthetic) and real world multivariate data sets.\nSpecifically, we focus on the ability of the proposed method to yield joint\noscillatory modes in multivariate data which is a prerequisite in many real\nworld applications involving nonstationary multivariate data. We also highlight\nthe utility of the proposed method in two real world applications which include\nthe separation of alpha rhythms in multivariate electroencephalogram (EEG) data\nand the decomposition of bivariate cardiotocographic signals that consist of\nfetal heart rate and maternal uterine contraction (FHR-UC) as its two channels.\n