NOISE-ASSISTED MULTIVARIATE VARIATIONAL MODE DECOMPOSITION
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NOISE-ASSISTED MULTIVARIATE VARIATIONAL MODE DECOMPOSITION
Charilaos A. Zisou? Georgios K. Apostolidis? Leontios J. Hadjileontiadis?†
?
School of Electrical & Computer Eng., Aristotle University of Thessaloniki, GR-54124, Greece
†
Department of Electrical & Computer Eng., Khalifa University, PO Box 127788, UAE
ABSTRACT amplitudes, i.e. narrow-band signals. The very first method
proposed was the empirical mode decomposition (EMD) [2]
The variational mode decomposition (VMD) is a widely which analyzes an input signal into a set of narrow-band
applied optimization-based method, which analyzes non- components via an iterative sifting process which is driven by
stationary signals concurrently. Correspondingly, its recently the signal’s extrema and does not require any a priori defined
proposed multivariate extension, i.e., MVMD, has shown basis. More recently, a different data-driven approach that has
great potentials in analyzing multichannel signals. However, drawn significant attention is the variational mode decompo-
the requirement of presetting the number of extracted com- sition (VMD) which formulates an optimization problem and
ponents K diminishes the analytic property of both VMD minimizes the collective bandwidth (CBW) of the signal’s
and MVMD methods. This work combines MVMD with the components subject to the full reconstruction constraint. Sev-
noise injection paradigm to propose an efficient alternative eral data-driven signal decomposition methods can be found
for both VMD and MVMD, i.e., the noise-assisted MVMD in the bibliography, many of which have been proposed as
(NA-MVMD), that aims at relaxing the requirement of pre- extensions of the original EMD algorithm [3, 4, 5, 6] as well
setting K, as well as improving the quality of the resulting as alternative approaches, such as the iterative filtering [7],
decomposition. The noise is injected by adding noise vari- the empirical wavelet transform [8], the synchrosqueezed
ables/channels to the initial signal to excite the filter bank transform [9] and the swarm decomposition [10]. Typically,
property of VMD/MVMD on white Gaussian noise. More- the aforementioned methods are designed to analyze univari-
over, an alternative approach of updating center frequen- ate signals. Nevertheless, multivariate non-stationary signal
cies is proposed, which uses the centroid of the generalized analysis using data-driven methods attracts great interest from
cross–spectrum instead of a simple average of the individ- the scientific areas related to multichannel/multisensor signal
ual spectral centroids, showing faster convergence. The processing. Although a multivariate signal can be processed
NA–MVMD is applied to both univariate and multivariate channel-wise, the alignment of frequencies across channels,
synthetic signals, showing improved analytical ability, noise i.e. the mode-alignment, is not guaranteed. Thus, multivari-
intolerance, and less sensitivity in selecting the K parameter. ate extensions of the univariate data-driven decomposition
Index Terms— multivariate analysis, variational mode methods have been proposed, e.g. the multivariate EMD
decomposition, multichannel signals, non-stationary signals (MEMD) [11, 12] and the multivariate VMD (MVDM) [13],
among others [14, 15, 16].
1. INTRODUCTION This work focuses on improving the VMD and MVMD
Over the last two decades, data-driven signal decomposition methods. The main concept of MVMD, similarly to VMD,
methods have proven to be useful tools in non-stationary is to formulate an optimization problem to extract a collec-
signal analysis and have been widely used in a plethora of tion of components that exhibit minimum CBW subject to
problems. The objective of these methods is to output a set full reconstruction of the signal of every channel of the input
of components whose Hilbert transform accurately represents multivariate data. At every optimization step an ensemble of
the instantaneous dynamics, i.e. instantaneous amplitude and components for each channel k is resulted via applying band-
frequency, of an input signal. It is well known that the Hilbert pass filtering around floating dominant frequencies ωk , which
transform can disentangle the instantaneous amplitude and are updated online during the process. The floating dominant
frequency, only in the case of non-overlapping frequency frequency for each component is calculated as the average of
spectra (Bedrosian theorem) [1]. Thus all existing data-driven the spectral centroids across all channels.
methods aim at extracting components with slow-varying The main issue of both VMD and MVMD methods is that
they are highly parameter-sensitive. Specifically, the decom-
Thanks to Projects: Care4MyHeart with ADEK Award Number
AARE18-135 and PROTEIN Grant no.817732 within the H2020 Research
position result is heavily influenced by the number of com-
and Innovation Program for funding this work. (C. Zisou and G. Apostolidis ponents K that must be defined beforehand and arbitrary pre-
contributed equally to this work.). Code: https://github.com/chariszisou setting of K may cause mode splitting. Thus, scholars haveapproached this problem with a focus on estimating the op- Algorithm 1 MVMD
timal parameter K [17, 18, 19, 20]. The aim of this work is procedure MVMD(x(t), K, α, τ , )
not to propose another parameter K estimation approach, but (1) (1)
Initialize {ûk,c }, {ωk }, λ̂(1) , n ← 0
to make the original methods independent from K, under cer- repeat
tain circumstances. An additional objective is to improve the n←n+1
original methods’ decomposition capability, i.e. to enhance for k = 1 : K do
the resulting component orthogonality. To meet these objec- for c = 1 : C do Update component ûk,c :
tives, we exploit the filter bank property of both VMD and (n+1)
ûk,c (ω) ←
MVMD on white Gaussian noise (WGN) [13, 21]. (n)
The new method, the noise-assisted MVMD (NA–MVMD) P (n+1) P (n) λ̂c (ω)
x̂c (ω) − ik ûi,c (ω) +
combines the MVMD with the noise injection paradigm pro- 2
(n)
posed in [6]. In short, NA–MVMD creates a noise injected 1 + 2α(ω − ωk )2
multivariate signal by adding new channels that consist of end for
independent realizations of WGN to the input data channels end for
and then applies the standard MVMD to the new multivari- for k = 1 : K do Update center frequency ωk :
(n+1)
ate signal. Finally, in the update of the floating dominant (n+1) 1 PR∞ |ûk,c (ω)|2 dω
frequencies, the average spectral centroid across all chan- ωk ← ω P ∞ (n+1) dω
C c 0
R
|û (ω)|2 k,c
nels is replaced by the spectral centroid of the generalized c
0
cross-spectrum [22]. To prove the potential of NA–MVMD, end for
experiments are conducted using both univariate and multi- for c = 1 : C do Update λ̂c :
variate synthetic signals. (n+1) (n) P (n+1)
λ̂c (ω) ← λ̂c (ω) + τ (x̂c (ω) − ûk,c (ω))
k
end for
2. MULTIVARIATE VMD P P kûk,c
(n+1) (n)
− ûk,c k22
until convergence: k c (n)3. NOISE-ASSISTED MVMD number of extracted components as well as investigating the
orthogonality condition.
The NA–MVMD method is a refinement of MVMD. The new The synthetic signals generated for this experiment were
elements that NA–MVMD includes are the adoption of the realizations of a stochastic class of non-stationary multi-
noise injection paradigm proposed in [6] and the substitution component signals. This class of signals was defined as
of the center frequency update calculation. The NA–MVMD XM
approach can be applied to both univariate and multivariate s(n) = sm (n) + η(n), n ∈ Z, (10)
m=1
signals. The noise injection aims at exciting the filter bank
character of the original MVMD method to provide a distinct sm (n) = wLm (n − dm ) · Am · cos(ωm n), n ∈ Z, (11)
processing behavior regardless of the input signal characteris- where M is the number of generated components, η(n) is the
tics. The noise injection is performed by adding channels that added WGN, wLm [n − dm ] is a Hanning window of time lo-
consist of independent realizations of WGN, cation dm and length Lm , Am is the amplitude and ωm is the
frequency of mth component. For this experiment M , dm and
x0 (t) = [x(t), η1 (t), · · · , ηN (t)], (6)
Lm were fixed, i.e. M = 3, dm = {250, 125, 375}, Lm =
where x0 (t) is the new multivariate signal, x(t) is the input {500, 125, 100}, whereas, Am and ωm were randomly drawn
univariate or multivariate signal and ηi (t), ∀i = 1, · · · , N , from uniform distributions, i.e. P DF (Am ) = U (0.5, 1.5)
are WGN signals. Then, x0 (t) replaces x(t) in Algorithm and P DF (ωm ) = U (0, π), ∀m.
1. Moreover, the center frequencies update part of MVMD In order to compare both methods in a generalized man-
changes from the average of spectral centroids across all ner, multiple simulations were executed for a sufficient
channels to the centroid of the generalized cross-spectrum amount of iterations, i.e. 500, for three different signal-to-
(GCS) [22]. For a set of N signals {zi (t)}N noise ratios (SNR). Every iteration consisted of the following
i=1 , the GCS is
steps: 1) a multi-component signal (10) was generated, 2)
γ(ω) = (1/(N − 1))(Λmax (Σz (ω)) − 1), (7) the signal was decomposed for different values of K, i.e.
K = {3 : 10} using both VMD and NA–MVMD, 3) the
Sz1 z1 (ω) . . . Sz1 zN (ω) decompositions were characterized as successful or not, and,
Σz (ω) =
.. .. ..
(8) finally, 4) their orthogonality index (OI) was calculated. Suc-
. . .
SzN z1 (ω) . . . SzN zN (ω) cess was defined as a binary variable (1/0). A decomposition
was classified as successful, if all input components (11)
where Σz (ω) contains all the pairwise complex cross– appeared exactly once in the set of the output components.
spectra, Szi,zj (ω) ∀i, j = 1, · · · , N , and Λmax (Σx (ω)) The absence or the splitting of a component was considered
is the largest eigenvalue of the matrix Σz (ω). It is evident incorrect. This definition is very strict, e.g. if 2 out of 3
that for the case of N = 2 the GCS is reduced to the standard components are correctly extracted, the success is set to 0. At
cross–spectrum. Now, let γk (ω) be the GCS of the set of each scenario, after all iterations, the success rate SR was cal-
k th -level components {ûk,c (ω)}C
c=1 at an arbitrary iteration culated as the number of successful decompositions divided
of Algorithm 1 the update of center frequency ωk becomes by the total number of repetitions. Furthermore, the OI was
Z ∞ calculated based on the definition in [2], where for an input
γk (ω)
ωk = ω R∞ dω. (9) signal s(n) and output components um (n), ∀m = 1, · · · , M ,
0 0
γk (ω)dω X XM XM
OI = ( ui (n) · uj (n)/s2 (n)) (12)
The reason for this replacement is that the new approach is n i=1 j=1
a more elegant representation of the components’ collective Except for K, the rest of Algorithm 1 parameters are set as
spectrum and it is a direct extension of the VMD method. α = 1000, τ = 10−4 and = 10−8 . Following (6), one
assisting noise channel is added. Fig. 1 illustrates the analysis
4. EXPERIMENTAL RESULTS of an excerpt signal from the simulations, revealing the main
shortcoming of VMD, i.e. the mode splitting.
To evaluate its performance, NA-MVMD was applied, along Fig. 2 depicts the success of both the proposed and the
with VMD and MVMD, to non-stationary univariate and mul- original method on decomposing the generated synthetic sig-
tivariate synthetic signals. nals versus the parameter K. For high values of SNR, VMD
shows a spike around K = 3, highlighting the necessity of
4.1. Dependence of the parameter K, influence of noise
choosing the correct number of K a priori, since VMD fails
and orthogonality property
at other K. In contrary, the NA–MVMD method successfully
The performance of the NA–MVMD method was tested on analyzes signals independently of K making the a priory se-
synthetic univariate multi-component signals and compared lection of K less restrictive. Moreover, NA–MVMD behavior
with the well-known VMD method. Particular emphasis is remains unchanged for different values of SNR showing good
given on testing the influence of the parameter K, i.e. the noise intolerance.2 SR of NA-MVMD vs. parameter K
1
s(n)
0 SNR=20dB
SNR=15dB
-2 0.8
0 100 200 300 400 500 SNR=10dB
n (samples)
NA-MVMD VMD
0.6
SR
1 1
0.4
u1
0 0
-1 -1
1 1 0.2
u2
0 0
-1 -1
0
1 1 2 3 4 5 6 7 8 9 10
u3
0 0 number of components (K)
-1 -1
1 1 (a)
u4
0 0
-1 -1 SR of VMD vs. parameter K
0 100 200 300 400 500 0 100 200 300 400 500 1
n (samples) n (samples) SNR=20dB
SNR=15dB
0.8 SNR=10dB
Fig. 1. An excerpt signal from the experiment with SN R =
0.6
20dB, ωm = {0.2π, 0.6π, 0.7π} and Am = {0.9, 1.2, 1.4}
SR
and its extracted components applying the NA–MVMD and 0.4
the VMD methods. This example shows the mode splitting 0.2
that occurs in VMD (see components u1 and u2 ) and, the re-
0
spective rectification using NA–MVMD. 2 3 4 5 6 7 8 9 10
number of components (K)
Table 1 depicts the OI versus SNR for both NA–MVMD (b)
and VMD methods for those K that each method has the best Fig. 2. The SR of NA–MVMD and VMD with respect to K.
SR. It is evident that NA–MVMD performs better in terms of
the orthogonality property.
Table 2. Median and standard deviation of the total number
Table 1. The OI×10−2 (mean and standard deviation) of NA- of steps until convergence for the GCS-based (proposed) and
MVMD and VMD for different SNR, using the best perform- the average-based (original) center frequency update.
ing K, i.e K=8 for the NA-MVMD and K=3 for the VMD. SNR (dB) 20 10 5 0
SNR (dB) 20 15 10 GCS 16 ± 6 54 ± 48 63 ± 61 79 ± 79
NA–MVMD 0.06 ± 0.03 0.07 ± 0.03 0.1 ± 0.03 Average 21 ± 2 73 ± 45 80 ± 60 114 ± 106
VMD 0.17 ± 0.13 0.13 ± 0.08 0.17 ± 0.08
4.2. Convergence Analysis 5. CONCLUDING REMARKS
In this experiment, a comparative evaluation regarding the
center frequency update (see Algorithm 1) was conducted. In this work, we presented the NA–MVMD method which
The comparison involved two approaches, the averaging- augments the recently proposed MVMD method incorporat-
based and the GCS-based, i.e. the original and the proposed ing the noise injection paradigm, i.e. adding new channels
one, respectively. For this purpose a synthetic trivariate signal with WGN. The NA–MVMD fully exploits the filter bank
was generated, x(n) = [x1 (n), x2 (n), x3 (n)] + η(n) with, property of MVMD on WGN. The new method allows the
processing of both univariate and multivariate signals mak-
x1 (n) = 0.5 sin(ω1 n) + 0.2 sin(ω2 n) + 0.1 sin(ω3 n), ing it a generalization of both VMD and MVMD. To evaluate
x2 (n) = 0.45 cos(ω1 n) + 0.3 sin(ω3 n) + 0.2 sin(ω4 n), its performance, NA–MVMD was applied to synthetic multi-
component signals showing better analytical ability, noise in-
x3 (n) = 0.4 sin(ω1 n) + 0.25 sin(ω2 n),
tolerance and less sensitivity in selecting the K parameter.
where ω1 = 0.1π, ω2 = 0.2π, ω3 = 0.3π, ω4 = 0.8π, Moreover, an alternative center frequency update, i.e. a core
η(n) is WGN and n ∈ Z. MVMD was applied using both calculation in MVMD algorithm, based on the GCS was pro-
approaches, for different values of SNR. At each SNR level, posed allowing faster convergence.
the process was repeated for 100 iterations and the number Nonetheless, the NA-MVMD should be subjected to
of steps until convergence was monitored. The results are more investigation regarding 1) the better understanding of
presented in Table 2 showing that the proposed approach pro- the noise injection parameters, i.e. number of noisy channels
vides faster convergence especially in high noise levels. This and the noise amplitude, 2) the application to real-life univari-
is explained by the fact that the averaging, as a smoothing op- ate and/or multivariate signals and 3) the combination of the
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