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Multivariate Variational Mode Decomposition

Naveed ur Rehman, Hania Aftab

arXiv:1907.04509v1eess.SP

TL;DR

The paper addresses the need to decompose nonstationary multichannel data while aligning shared frequency content across channels. It introduces MVMD, a variational extension of VMD based on common multivariate oscillations, and demonstrates mode alignment on synthetic and real-world signals. The method is applied to EEG alpha-rhythm separation and bivariate cardiotocographic signal decomposition.

  • Problem

    Nonstationary multivariate data require methods that align common frequency content across channels, while existing approaches have limitations in robustness, mode separation, or predefined filter boundaries.

  • Method

    MVMD extends VMD to multichannel data by modeling common multivariate oscillations and minimizing collective bandwidth while fully reconstructing all input channels.

  • Results

    MVMD exhibits mode alignment across synthetic, multivariate wGn, EEG, and cardiotocographic signals, demonstrating initial potential for data fusion.

  • Takeaways & Limitations

    The method provides joint oscillatory modes for multivariate data and is demonstrated in biomedical applications including EEG and cardiotocographic signal analysis.

  • Takeaways & Limitations

    MVMD inherits VMD’s limitations, including requiring the number of modes in advance, difficulty separating DC components, and challenges with sudden signal onsets.

Abstract

from arXiv · show

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.

I. INTRODUCTION

Nonstationary multivariate data require data-driven methods that can align shared frequency content across channels. MVMD extends VMD to address this need through a variational multichannel decomposition.

  • Motivation: Linear time-frequency transforms use fixed atoms, which can blur nonstationary signals because of limited joint time-frequency resolution.This motivates data-driven decomposition methods with fewer a priori assumptions.
  • Existing methods: EMD extracts oscillatory modes through recursive sifting but depends heavily on interpolation and stopping criteria and lacks strong theoretical guarantees.These issues limit robustness and mathematical analysis of the decomposition.
  • Multivariate challenges: Multivariate time-frequency methods must align common frequency information across channels and incorporate interchannel correlation.Direct operation in the multidimensional signal space is described as a way to satisfy these requirements.
  • Existing methods: MEMD, multivariate SST, and multivariate EWT have limitations including mode mixing, absent mode separation, and reliance on predefined filterbank boundaries.The stated shortcomings motivate a fully multivariate variational alternative.
  • Proposed method: MVMD extends VMD to any number of channels by seeking common multivariate oscillations with minimum collective bandwidth while reconstructing all input channels.Its model uses a shared frequency component across channels and inherits VMD’s variational framework and ADMM optimization.

II. UNIVARIATE AND MULTIVARIATE MODULATED OSCILLATIONS

The paper represents univariate modulated oscillations with amplitude and phase functions, then uses Hilbert-transform analytic signals to obtain a canonical representation. The original real signal is recovered from the analytic signal’s real part.

  • A. Univariate Oscillations: A univariate AM-FM signal is described by an amplitude function and a time-varying phase function associated with frequency modulation.The amplitude-phase representation is not unique for a real-valued signal.
  • A. Univariate Oscillations: Pairing a signal with its Hilbert transform produces an analytic signal with a canonical amplitude and phase representation.This construction is used to define a unique set of amplitude and phase functions.
  • A. Univariate Oscillations: The analytic signal’s Fourier transform yields a unilateral signal spectrum.The Hilbert-transform construction underlies the frequency-domain representation used in the decomposition framework.
  • A. Univariate Oscillations: The original real-valued signal is recovered by taking the real part of its complex-valued analytic signal.This connects the complex representation back to the observed signal.

B. Multivariate Oscillations

The multivariate model represents each channel as an AM-FM component and applies the Hilbert transform componentwise. MVMD then assumes a common instantaneous frequency component across all channels.

  • B. Multivariate Oscillations: A multivariate signal is represented as a set of C real-valued AM-FM signals, with channel-specific amplitude and phase functions.The amplitude and phase functions correspond to the individual signal components.
  • B. Multivariate Oscillations: The Hilbert transform is applied separately to each channel to form an analytic signal vector.This yields canonical amplitude and phase functions for each component.
  • B. Multivariate Oscillations: The original real-valued multivariate signal is recovered as the real part of the analytic signal vector.The complex vector representation therefore retains a direct reconstruction relation to the observed channels.
  • B. Multivariate Oscillations: The proposed multivariate oscillation model assumes one common instantaneous frequency component among all data channels.This model is used to formulate the multivariate variational decomposition in the following section.

III. VARIATIONAL MODE DECOMPOSITION

VMD decomposes a univariate input into a predefined number of modes by minimizing their total bandwidth while reconstructing the signal. It estimates bandwidth through analytic-signal formation, frequency shifting, and iterative ADMM updates.

  • VMD decomposes an input into a finite, predefined K number of intrinsic principal modes.
  • The variational model minimizes the sum of mode bandwidths while fully reconstructing the input signal or doing so in the least-squares sense.
  • Each mode’s bandwidth is estimated by forming its unilateral analytic signal, shifting it to baseband by ωk, and measuring the squared L2 norm of the resulting gradient.
  • The constrained optimization uses a quadratic reconstruction-fidelity penalty and Lagrange-multiplier terms enforcing the constraints.
  • ADMM solves the augmented-Lagrangian saddle-point problem through iterative sub-optimization problems for individual parameters or functions.
  • The mode and center-frequency updates are performed in the spectral domain, with each ωk updated as the center of gravity of its associated mode power spectrum.

IV. MULTIVARIATE VARIATIONAL MODE DECOMPOSITION

MVMD extends VMD to multichannel data by extracting multivariate modes with minimum collective bandwidth while reconstructing every channel. Its model enforces a common frequency component across channels and uses ADMM for optimization.

  • MVMD extracts a predefined K number of multivariate modulated oscillations from an input containing C data channels.
  • The objective minimizes the sum of extracted-mode bandwidths while requiring their sum to exactly recover the original multichannel signal.
  • MVMD represents each channel-mode component analytically and extends the VMD cost function to multichannel signal matrices.
  • A single ωk is used to harmonically mix every channel in a mode, thereby modeling oscillations with one common frequency component across channels.
  • The multichannel bandwidth is measured after shifting all channel spectra by ωk and taking the resulting matrix’s Frobenius norm.
  • ADMM converts the constrained MVMD optimization into simpler suboptimization problems corresponding to the update equations.

A. Mode update

The MVMD mode-update subproblem follows the form of VMD’s mode update and is solved conveniently in the Fourier domain.

  • A. Mode update: The MVMD mode update is formulated as a subproblem analogous in form to the original VMD mode update.
  • A. Mode update: Solving the mode subproblem in the Fourier domain yields a convenient frequency-domain update relation.

B. Center frequency update

MVMD updates each mode’s center frequency through a Fourier-domain optimization within ADMM. Unlike univariate VMD, the update incorporates power-spectrum contributions from all channels.

  • B. Center frequency update: The center-frequency update subproblem isolates the terms of the Lagrangian that depend on ωk.
  • B. Center frequency update: The center-frequency optimization is transformed into the Fourier domain using the Plancherel theorem.
  • B. Center frequency update: Setting the derivative of the resulting quadratic objective to zero produces the MVMD frequency-update relation.
  • B. Center frequency update: Each MVMD center-frequency update accounts for power-spectrum contributions from all C channels.
  • B. Center frequency update: The mode and center-frequency relations are optimized iteratively in the frequency domain using ADMM as specified by Algorithm 2.

V. EXPERIMENTAL RESULTS

The experiments evaluate MVMD on synthetic and real multivariate signals, emphasizing separation of multivariate oscillations and alignment of common frequency scales across channels.

  • Experiments span real-world bivariate oscillations, synthetic multichannel tone mixtures, white Gaussian noise, and comparisons with VMD and MEMD.The evaluation also includes multivariate EEG and bivariate cardiotocographic data.
  • MVMD is evaluated primarily for aligning common frequency scales across multiple data channels.The paper identifies this as a critical requirement in applications including image fusion, biomedical classification, and denoising.

A. Separation of multivariate modulated oscillations

MVMD separates multivariate modulated oscillations in real bivariate data and aligns shared frequency content across channels more coherently than channelwise VMD.

  • In an oceanographic float position record, MVMD decomposes the bivariate signal into K = 4 modes representing its principal multivariate oscillations.The modes are shown through both channelwise time plots and two-dimensional plots.
  • The first two modes contain rotating multivariate oscillations that may correspond to a presumed coherent vortex.Non-rotating modes lack a joint frequency scale in one channel.

B. Mode-alignment property

MVMD aligns common oscillations across channels while producing a filterbank structure distinct from MEMD and stronger quasi-orthogonality in the reported wGn experiments.

  • Mode-alignment property: Mode-alignment means placing common or joint oscillations with similar frequency content across multiple channels within a single mode.The property supports coherent multivariate time-frequency analysis for fusion, denoising, and classification.
  • MVMD vs MEMD filterbanks for wGn: For the four-channel wGn process, MEMD shows a quasi-dyadic filterbank, whereas MVMD modes have different bandwidths in the log-frequency domain.The comparison averages power spectral density plots over 100 realizations.
  • MVMD vs MEMD filterbanks for wGn: Frequency content is aligned across channels within similar modes for both methods, appearing slightly more prominent in MVMD and yielding better-defined subband filters.
  • Quasi-orthogonality of MVMD modes: MVMD’s K = 8 mode correlation matrix is almost diagonal, indicating strong quasi-orthogonality, while MEMD shows adjacent-mode leakage through a tridiagonal structure.The matrices are averaged over 100 realizations of four-channel wGn.

E. Noise robustness

MVMD retains mode alignment and quasi-orthogonality as noise and the number of input channels increase, while outperforming MEMD in separating noisy multivariate tones.

  • Noise robustness: MVMD more clearly aligns common 2-Hz and 288-Hz modes across noisy channels than MEMD.MEMD exhibits information leakage across channels and spreads the 288-Hz tone across its first three IMFs.
  • Noise robustness: MVMD recovers the common 2-Hz and 288-Hz tones in both channels while retaining the channel-specific 24-Hz tone in channel 2.The tones appear in modes 1 and 3, while the 24-Hz component appears only in channel 2 of mode 2.
  • Noise robustness: MVMD can suppress noisy-input reconstruction through τ = 0, whereas MEMD has no mechanism to prevent noise from entering the decomposition.This control reflects the algorithms’ treatment of the reconstruction constraint in noisy data.
  • Performance evaluation for increasing number of channels: Increasing channels from C = 2 to 16 does not affect MVMD’s filterbank structure, mode alignment, or quasi-orthogonality.The evaluation used 200 realizations of white Gaussian noise for C = 2, 4, 8, and 16.

G. Real World Examples

MVMD provides aligned, narrow-band decompositions in multichannel EEG and cardiotocographic data, supporting alpha-rhythm separation and FHR-UC analysis while retaining VMD’s known limitations.

  • 1) Separation of α rhythms in EEG:: MVMD localizes the EEG α-rhythm to mode u2 across all four channels.The α-rhythm corresponds to 8–12 Hz, and aligned PSD peaks across channels support the mode-alignment observation.
  • 2) Decomposition of fetal heart rate (FHR) and maternal uterine contraction (UC) recordings from cardiotocographic (CTG) signal:: MVMD produces narrow-band FHR-UC modes without the mode-mixing artifacts observed in BEMD.Across the selected modes, MVMD also exhibits mode alignment across both channels.
  • VI. CONCLUSION: MVMD is a multivariate extension of VMD that minimizes collective bandwidth while estimating common center frequencies across channels.Its variational formulation targets common multivariate modulated oscillations and uses the VMD framework for narrow-band modes.
  • VI. CONCLUSION: Mode alignment is demonstrated on synthetic, white-Gaussian-noise, EEG, and cardiotocographic signals, indicating initial potential for data fusion.The paper frames aligned frequency content within a single mode as necessary for multivariate multiscale data-driven approaches.
  • VI. CONCLUSION: MVMD inherits VMD limitations, including requiring a predefined mode count K, difficulty separating a DC component, and challenges with sudden signal onset.The authors report observing these limitations in the proposed multivariate method.
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