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Ali KOMATY, Abdel-Ouahab BOUDRAA, John NOLAN, Delphine DARE - On the behavior of EMD and MEMD in presence of symmetric alphastable noise IEEE Signal Processing Letters -Vol. 22, n°7, p.818-822 - 2015
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On the behavior of EMD and MEMD in presence
of symmetric
α
-stable noise
A. Komaty, A.O. Boudraa, Senior Member IEEE, J.P. Nolan and D. Dare
Abstract—Empirical Mode Decomposition (EMD) and its ex-tended versions such as Multivariate EMD (MEMD) are data-driven techniques that represent nonlinear and non-stationary data as a sum of a f nite zero-mean AM-FM components referred to as Intrinsic Mode Functions (IMFs). The aim of this work is to analyze the behavior of EMD and MEMD in stochastic situations involving non-Gaussian noise, more precisely, we examine the case of Symmetric α-Stable (SαS) noise. We report numerical experiments supporting the claim that both EMD and MEMD act, essentially, as f lter banks on each channel of the input signal in the case of SαS noise. Reported results show that, unlike EMD, MEMD has the ability to align common frequency modes across multiple channels in same index IMFs. Further, simulations show that, contrary to EMD, for MEMD the stability property is well satisf ed for the modes of lower indices and this result is exploited for the estimation of the stability index of the SαS input signal.
Index Terms—EMD, MEMD, f lter banks, symmetric α-stable noise.
I. INTRODUCTION
E
MPIRICAL mode decomposition (EMD) is a fully adap-tive data-driven approach for the decomposition of non-stationary signals [1]. This technique decomposes any signal into a linear combination of a f nite number of basis functions called intrinsic mode functions (IMFs). Being proven eff cient when dealing with deterministic signals of oscillatory nature, EMD also reveals interesting properties when dealing with random signals. Dealing with such signals, their properties, their transformations, and their characterization in time and frequency domains has gained enormous attention in the last decade. Since a random signal is not repeatable in a predictable manner, it may only be described probabilistically or in terms of its average behavior. To be able to devise mathematical tools for this purpose, one needs to assume a statistical model which best describes the data. Evaluation of the performances of such methods depends upon the ability to determine the probability density function (pdf) of a function of the data samples, either analytically or numerically. When this is not possible, one must resort to Monte Carlo computer simula-tions. Among various probability distributions, the Gaussian distribution plays a predominant role in signal processing [2]. Many of the theorems of communications, estimation and detection theory have been formulated based on the Gaussian assumption thanks to the Central Limit Theorem (CLT), which Ali Komaty, Abdel Boudraa and Delphine Dare are with IRENav, Ecole Navale, BCRM Brest, CC 600, 29240 BREST Cedex 9, France and John P. Nolan is with American University, Washington DC, USA. The work of John P. Nolan was supported by an agreement with Cornell University, Operations Research & Information Engineering under contract W911NF-12-1-0385 from the Army Research Development and Engineering Command.holds for a large variety of distributions. Unfortunately, a broad class of phenomena encountered in practice are undeniably non-Gaussian and can be characterized by their impulsive nature [3]. Random f uctuations of gravitational f elds, un-derwater acoustic noise of snapping shrimp, radar clutter, economic market indexes, Internet traff c or man-made noise have been found to belong to this class. Signals of this class are more likely to exhibit sharp spikes or bursts of outlying measurements than one would expect from normally distributed signals [4]-[6]. Impulsive perturbations of these signals are commonly modeled by symmetric α-stable (SαS) distributions. More precisely SαS distribution describes a large class of impulsive random variables with heavy-tailed distri-butions. This family possesses strong theoretical justif cations according to the Generalized CLT (GCLT) which extends the CLT to the case when the summands are heavy-tailed [7]. Up to now the behavior of EMD has been analyzed in presence of fractional Gaussian noise (fGn) [8] and its extended version, Multivariate EMD (MEMD) [9], in white Gaussian noise case [10]. But much less attention has been paid to situations involving processes that generate impulsive signals or noise bursts using such decompositions. Thus, for more real world applications, it is important to investigate how such signal decompositions behave in the presence of SαS noise. Because Gaussian and stable non-Gaussian distributions are invariant under linear operations, they are very important in signal processing. Hence the importance of studying their characteristics when decomposed using EMD and MEMD.
II. BASICS OFEMDANDMEMD
EMD: Standard EMD breaks down any real-valued
sig-nal x(t) into a reduced number of oscillating modes (AM-FM) called Intrinsic Mode Functions (IMFs) and a residual r(t) consisting of all local trends [1]. By construction, each IMF is a zero-mean waveform whose number of zero-crossings (ZCs) differs at most by one from the number of its extrema. More precisely, EMD ends up with an expansion of the form:
x(t) =
M
X
m=1
cm(t) + r(t) (1)
where cm(t)is the mthIMF and M is the number of extracted
modes. The number of extrema of x(t) is decreased when going from one residual to the next.
MEMD: Standard EMD considers only 1D signals and
the local mean is calculated by averaging the upper and lower envelopes obtained by interpolating between the local maxima and minima respectively. MEMD has been developed
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both in the number and in scale properties [17]. The higher the CC, the less signif cant the splitting in separate IMFs. Thus, CC between normalized IMFs (leakage between sub-bands) may cause blurred time-frequency estimates such as IF. Using this quantitative evaluation, it has been shown that EMD and MEMD generate approximately mono-component and locally orthogonal data-driven basis functions in presence of white Gaussian noise (α = 2) [12]. Figure 2 shows CC of IMFs averaged over J=1000 realizations of SαS 2-channel process of length L=1000 samples using EMD (channel-wise) and MEMD. The CC estimates Θ[m, m′] are calculated for
IMFs obtained from MEMD and EMD as follows: Θ[m, m′] := 1 J J X j=1 Θj[m, m′] pΘj[m, m]Θj[m′, m′] Θj[m, m′] := 1 L L X k=1 cjm(k)c j m′(k) (4)
By def nition, we have 0 ≤ Θ[m, m′]≤ 1.
(a) EMD α = 1.2 (b) EMD α = 1.4 (c) EMD α = 1.8
(d) MEMD α = 1.2 (e) MEMD α = 1.4 (f) MEMD α = 1.8
Fig. 2. CCs of IMFs for a bivariate SαS distribution using (a-c) EMD (channel-wise) and (d-f) MEMD.
We report in Fig. 2 alignment results of three typical values of α. Figures 2(d)-2(f) show that, on average, MEMD has almost the same behavior for all α ∈ [1, 2]. Larger values along the diagonal (m = m′) suggest that the IMFs in MEMD are
well aligned. For α = 1.8 and α = 1.4, both decompositions produce correlograms with diagonal-dominant elements while being more pronounced in the case of MEMD. For α = 1, unlike MEMD, EMD does not exhibit a pronounced diagonal dominance, concluding that EMD does not produce same index IMFs with the same scale when α deviates from 2. As shown in Fig. 2(c), for more impulsive cases, signif cant values of CC estimates are observed off-diagonal (m 6= m′)
indicating missaligned IMFs. This suggests that standard EMD is not well suited for decomposing signals of high impulsive nature.
C. Stability test
EMD or MEMD are data-driven projections of a signal on some space, thus it is important to check if the stability property is preserved or not using these decompositions. As with any other family of distributions, it is not possible to prove that a given set is or is not stable, even for normal-ity this is still an active research f eld [11]. A solution to
this problem is to check whether or not data are consistent with stability hypothesis. More precisely, for plausibly stable smoothed density of data (Fig. 3(a),3(c)) the f tted distribution is compared to data using Quantile-Quantile (Q-Q) plot as shown in Figs. 3(b) and 3(d). Q-Q plot is designed to show the closeness of two distributions [11]. If the f tting is consistent, stability parameters (α, β, γ, δ) are estimated. Four methods are used: Maximum Likelihood (ML) [19], Quantile, Empiri-cal Characteristic Function (ECF) and Fractional Lower Order Moments (FLOM). Developing these estimation methods goes beyond the scope of this paper, and the reader is referred to [4],[13]-[15] and [19] for more details. If the estimates (ˆα, ˆβ,γ, ˆˆ δ) differ signif cantly, the data are considered not stably distributed. While non-Gaussian stable distributions are heavy-tailed, most heavy-tailed distributions are not stable. In many cases, it is not appropriate to f t heavy-tailed data with a stable distribution. As shown in Fig. 3, the pdf of the f rst extracted IMF (averaged over 150 realizations) by EMD is bimodal, while the corresponding one of MEMD is unimodal. However, in both cases, an α-stable f tting is used to approximate the f rst mode even though, for the EMD case, this f tting is not accurate (one cannot f t a bimodal distribution using a unimodal stable one). Nevertheless, this test was made to prove that, even if the f rst IMF is not stable, when f tted using a stable distribution, its estimate ˆα is approximately equal to the original index α. The Q-Q plot of Fig. 3 shows that this f tting is more consistent in MEMD than in EMD. However, estimates (ˆα, ˆβ, ˆγ, ˆδ), averaged over 150 realizations, reported in Table 1 for the f rst IMF results in a signif cantly different results using the four methods in the case of EMD, but the parameters retrieved in MEDM are on the average the same and close to the true parameters (α = 1.5, β = 0, γ = 1, δ = 0). Therefore, these results support the claim that, in f rst approximation, stability is more preserved by MEMD than by EMD and more particularly the stability index α. −10 −5 5 10 pdf smoothed data SαSf t −30 −20 −10 10 20 30 −100 −50 50 SαS data
(a) EMD:Density plot of stable distribution vs Data. (b) EMD: Q-Q plot of stable distribution vs Data.
−10 −5 5 10 pdf smoothed data SαSf t −30 −20 −10 10 20 30 −60 −40 −20 20 40 60 SαS data
(c) MEMD:Density plot of stable distribution vs Data. (d) MEMD: Q-Q plot of stable distribution vs Data.
Fig. 3. Density plot and Q-Q plot of the f rst IMF for a SαS i.i.d. signal with α = 1.5 and a data length of 10000 samples.
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