Multivariate Variational Mode Decomposition

We present a generic extension of variational mode decomposition (VMD) algorithm to multivariate or multichannel data. The proposed method utilizes a model for multivariate modulated oscillations that is based on the presence of a joint or common frequency component among all channels of input data. We then formulate a variational optimization problem that aims to extract an ensemble of band-limited modes containing inherent multivariate modulated oscillations present in the data. The cost function to be minimized is the sum of bandwidths of all signal modes across all input data channels, which is a generic extension of the cost function used in standard VMD to multivariate data. Minimization of the resulting variational model is achieved through the alternating direction method of multipliers (ADMM) that yields an optimal set of multivariate modes in terms of narrow bandwidth and corresponding center frequencies. The proposed extension is elegant as it does not require any extra user-defined parameters for its operation i.e., it uses the same parameters as standard VMD. We demonstrate the effectiveness of the proposed method through results obtained from extensive simulations involving test (synthetic) and real world multivariate data sets. Specifically, we highlight the utility of the proposed method in two real world applications which include the separation of alpha rhythms in multivariate electroencephalogram (EEG) data and the decomposition of bivariate cardiotocographic signals that consist of fetal heart rate and maternal uterine contraction (FHR-UC) as its two channels.