decomposeMVMD: Multivariate Variational Mode Decomposition

In xemd: eXtended Empirical Mode Decomposition

Multivariate Variational Mode Decomposition

Description

Decomposes the channels of a sampled multichannel signal into modes whose channel coordinates share one center frequency.

Usage

decomposeMVMD(signal, samplingInterval, initialCenters, alpha, tau,
              fixedZeroCenter, tolerance, maxIterations = 499)

Arguments

signal	Numeric matrix of finite values, samples by channels, with at least one row and one column, as composed by the caller and without standardization.
samplingInterval	Sampling interval \Delta t in seconds, a finite number greater than zero.
initialCenters	Numeric vector of initial center frequencies in Hz, each in [0, 1/(2\Delta t)]; its length is the number of modes K. No default.
alpha	Penalty \alpha > 0 in s^2, as in decomposeVMD. No default; PySDKit takes it on cycles per sample without the factor 2.
tau	Dual ascent step \tau \ge 0; 0 solves the penalized problem with zero multipliers. No default.
fixedZeroCenter	TRUE keeps the first center at 0, while its channel coordinates are still updated, and requires initialCenters[1] == 0. No default.
tolerance	Threshold \epsilon > 0 of the relative change summed over modes and channels. No default; the 1e-7 of PySDKit belongs to another measure.
maxIterations	Whole number \ge 1 of sweeps; default 499, the limit of 500 states of the authors' MVMD.m 1.0.1.

Details

Each channel is extended and transformed as in decomposeVMD. In each sweep the coordinate c of mode k is

\hat u_{k,c} = \frac{\hat x_c - \sum_{i \ne k} \hat u_{i,c} + \hat\lambda_c / 2}{1 + 2 \alpha (\omega - \omega_k)^2},

with a denominator common to the channels; the new center weights |\omega| by the power summed over the channels, and the multiplier of each channel is updated with step \tau. The iteration stops with "tolerance" when \sum_k \sum_c \|\hat u_{k,c}^{(n+1)} - \hat u_{k,c}^{(n)}\|^2 / \|\hat u_{k,c}^{(n)}\|^2 < \epsilon, which cannot happen before the second sweep, or with "iterationLimit" after maxIterations sweeps. A coordinate without energy in the previous sweep leaves its ratio undefined, so the test fails until maxIterations. The channels keep the order received. The joint bandwidth formulation does not impose the geometrical IMF conditions of EMD. A shared spectral center does not require identical amplitudes or instantaneous frequencies across channels.

Value

A list with

modes	Double array, samples by modes by channels.
fitResidual	Numeric matrix, samples by channels, x_c - \sum_k u_{k,c}.
centers	Numeric vector of the K shared centers in Hz.
iterations	Integer vector repeating, for every mode, the number of sweeps.
converged	Logical vector repeating, for every mode, the result of the joint stopping test.
stopReason	"tolerance" or "iterationLimit".

Conditions

A mode without energy over all its channels leaves its center undefined and stops the function with an error of class xemdZeroEnergy.

References

ur Rehman, N. and Aftab, H. (2019). Multivariate Variational Mode Decomposition. arXiv:1907.04509.

See Also

decomposeVMD, decomposeSVMD, decomposeTSL

Examples

SamplingInterval <- 1 / 1000
Time <- seq_len(1000) * SamplingInterval
Signal <- cbind(cos(2 * pi * 2 * Time) + 0.25 * cos(2 * pi * 24 * Time) +
                    0.0625 * cos(2 * pi * 288 * Time),
                cos(2 * pi * 2 * Time),
                0.25 * cos(2 * pi * 24 * Time))
OUT <- decomposeMVMD(Signal, samplingInterval = SamplingInterval,
                     initialCenters = (0:2) / (6 * SamplingInterval),
                     alpha = 2000 * SamplingInterval^2 / (8 * pi^2),
                     tau = 0.5, fixedZeroCenter = FALSE, tolerance = 1e-6,
                     maxIterations = 98)
dim(OUT$modes)
OUT$centers
OUT$stopReason

xemd documentation built on Nov. 9, 2026, 1:09 a.m.