Multivariate Variational Mode Decomposition (1907.04509v1)

Abstract: In this paper, a generic extension of variational mode decomposition (VMD) algorithm for multivariate or multichannel data sets is presented. We first define 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. Using that model for multivariate oscillations, we construct a variational optimization problem that aims to extract an ensemble of band-limited modes containing inherent multivariate modulated oscillations present in multivariate input signal. 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 alternate direction method of multipliers (ADMM) approach. That yields an optimal set of multivariate modes in terms of narrow bandwidth and corresponding center frequencies that are assumed to be commonly present among all channels of a multivariate modulated oscillation. 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 focus on the ability of the proposed method to yield joint oscillatory modes in multivariate data which is a prerequisite in many real world applications involving nonstationary multivariate data. We also 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.

• The paper introduces a novel multivariate VMD (MVMD) method that extends standard VMD to multichannel signals using a variational optimization framework.
• The method utilizes ADMM to jointly optimize bandwidth across channels, enabling robust mode alignment and effective noise mitigation.
• Empirical results in EEG and cardiotocography illustrate MVMD’s effectiveness for diverse multichannel signal processing applications.

Overview of Multivariate Variational Mode Decomposition

The paper "Multivariate Variational Mode Decomposition" by Naveed ur Rehman and Hania Aftab introduces a significant extension of the variational mode decomposition (VMD) algorithm to handle multivariate or multichannel data. The work builds upon the existing VMD technique, which is primarily designed for univariate or single-channel signal analysis, to address the complexities associated with multivariate data processing.

Key Contributions and Methodology

The core contribution of this research is the proposal of a generic extension of the VMD algorithm, termed multivariate VMD (MVMD), that is tailored for datasets with multiple channels. This enhancement is underpinned by the formulation of a variational optimization problem aimed at extracting an ensemble of band-limited modes intrinsic to the multivariate input signal. The proposed model considers multivariate oscillations predicated on a joint frequency component common to all channels.

The approach optimizes the collective bandwidth of the modes across all data channels, which extends the original cost function used in standard VMD to a multivariate framework. The minimization of this variational model is effectively tackled using the alternate direction method of multipliers (ADMM), leading to a solution that uncovers optimal sets of multivariate modes with narrow bandwidths and corresponding center frequencies.

Numerical and Empirical Results

The proposed MVMD is demonstrated to be effective through comprehensive simulations involving both synthetic and real-world multivariate datasets. These simulations focus on several crucial aspects:

1. Mode Alignment: MVMD exhibits the ability to align similar frequency content across various channels within a single mode, a property known as mode alignment. This is critical for applications where coherent multivariate time-frequency analysis is paramount.
2. Robustness to Noise: The method inherits VMD's robustness to noise, allowing for reliable decomposition even in the presence of noisy multichannel data.
3. Applications: The utility of MVMD is showcased in practical scenarios, such as separating alpha rhythms in multivariate electroencephalogram (EEG) data and decomposing bivariate cardiotocographic signals containing fetal heart rate and maternal uterine contractions.

Implications and Future Directions

The introduction of MVMD marks an important step in multivariate signal processing, offering a robust framework for dealing with the complexities inherent in multichannel data. This work opens up several avenues for future research and application, including:

• Further exploration of MVMD's filterbank structure, which appears distinct from the quasi-dyadic structure observed in multivariate EMD.
• Investigation into the algorithm's performance with varying channel counts and its scalability to higher-dimensional datasets.
• Practical implementations in various engineering and biomedical domains, where the mode-alignment property and robustness to noise could greatly enhance processing and analysis capabilities.

In conclusion, the paper provides a substantial extension to the VMD methodology, equipping it to handle multichannel data effectively. This development is poised to benefit a wide range of applications that require precise and coherent multivariate signal decompositions.