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ON EMPIRICAL MODE DECOMPOSITION

AND ITS ALGORITHMS

Gabriel Rilling, Patrick Flandrin and Paulo Goncalves

Laboratoire de Physique (UMR CNRS 5672), Ecole Normale Superieure de Lyon46, allee dItalie 69364 Lyon Cedex 07, France

{grilling,flandrin}@ens-lyon.frProjet IS2, INRIA Rhone-Alpes, 655, avenue de lEurope 38330 Montbonnot, France

[email protected]

ABSTRACT

Huangs data-driven technique of Empirical ModeDecomposition (EMD) is presented, and issues re-

lated to its effective implementation are discussed.A number of algorithmic variations, including newstopping criteria and an on-line version of the al-gorithm, are proposed. Numerical simulations areused for empirically assessing performance elementsrelated to tone identification and separation. Theobtained results support an interpretation of themethod in terms of adaptive constant-Qfilter banks.

1 INTRODUCTION

A new nonlinear technique, referred to as EmpiricalMode Decomposition(EMD), has recently been pio-

neered by N.E. Huanget al. for adaptively represent-ing nonstationary signals as sums of zero-mean AM-FM components [2]. Although it often proved re-markably effective [1, 2, 5, 6, 8], the technique is facedwith the difficulty of being essentially defined by analgorithm, and therefore of not admitting an analyt-ical formulation which would allow for a theoreticalanalysis and performance evaluation. The purposeof this paper is therefore to contribute experimen-tally to a better understanding of the method andto propose various improvements upon the originalformulation. Some preliminary elements of experi-

mental performance evaluation will also be providedfor giving a flavour of the efficiency of the decompo-sition, as well as of the difficulty of its interpretation.

2 EMD BASICS

The starting point of the Empirical Mode Decompo-sition (EMD) [2] is to consider oscillations in signalsat a very local level. In fact, if we look at the evolu-tion of a signalx(t) between two consecutive extrema(say, two minima occurring at times t

and t+), we

can heuristically define a (local) high-frequency part

{d(t), t

t t+}, or local detail, which corre-sponds to the oscillation terminating at the two min-ima and passing through the maximum which nec-

essarily exists in between them. For the picture tobe complete, one still has to identify the correspond-ing (local) low-frequency part m(t), or local trend,so that we have x(t) =m(t) +d(t) for t

t t+.

Assuming that this is done in some proper way for allthe oscillations composing the entire signal, the pro-cedure can then be applied on the residual consistingof all local trends, and constitutive components of asignal can therefore be iteratively extracted.

Given a signalx(t), the effective algorithm of EMDcan be summarized as follows [2] (see also emd.pptin [9]):

1. identify all extrema ofx(t)2. interpolate between minima (resp. maxima),

ending up with some envelopeemin(t) (resp. emax(t))3. compute the meanm(t) = (emin(t) + emax(t))/24. extract the detail d(t) =x(t) m(t)5. iterate on the residual m(t)In practice, the above procedure has to be refined

by asiftingprocess [2] which amounts to first iterat-ing steps 1 to 4 upon the detail signal d(t), until thislatter can be considered as zero-mean according tosome stopping criterion. Once this is achieved, thedetail is referred to as an Intrinsic Mode Function(IMF), the corresponding residual is computed and

step 5 applies. By construction, the number of ex-trema is decreased when going from one residual tothe next, and the whole decomposition is guaranteedto be completed with a finite number of modes.

Modes and residuals have been heuristically intro-duced on spectral arguments, but this must notbe considered from a too narrow perspective. First,it is worth stressing the fact that, even in the caseof harmonic oscillations, the high vs. low frequencydiscrimination mentioned above applies only locallyand corresponds by no way to a pre-determined sub-


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signal

Empirical Mode Decomposition

imf1

imf2

imf3

imf4

imf5

imf6

imf7

imf8

imf9

res.

time

frequency

signal

time

frequency

mode #1

time

frequency

mode #2

time

frequency

mode #3

Figure 1: EMD of a 3-component signal. The an-alyzed signal (first row of the top diagram) is thesum of 2 sinusoidal FM components and 1 Gaus-sian wavepacket. The decomposition performed byEMD is given in the 8 IMFs plotted below, the lastrow corresponding to the final residue. The time-frequency analysis of the total signal (top left of the4 bottom diagrams) reveals 3 time-frequency signa-tures which overlap in both time and frequency, thusforbidding the components to be separated by any

non-adaptive filtering technique. The time-frequencysignatures of the first 3 IMFs extracted by EMDevidence that these modes efficiently capture the 3-component structure of the analyzed signal. (Alltime-frequency representations are reassigned spec-trograms [3, 9].)

band filtering (as, e.g., in a wavelet transform). Se-lection of modes rather corresponds to an automaticand adaptive (signal-dependent) time-variant filter-

signal

Empirical Mode Decomposition

imf1

imf2

imf3

res.

Figure 2: EMD of a 3-component signal Nonlin-

ear oscillations. The analyzed signal (first row of thediagram) is the sum of 3 components: a sinusod ofsome medium period T superimposed to 2 triangu-lar waveforms with periods smaller and larger thanT. The decomposition performed by EMD is given inthe 3 IMFs plotted below, the last row correspondingto the final residue.

ing. An example in this direction, where a signalcomposed of 3 components which significantly over-lap in time and frequency is successfully decomposed

by the method, is given in Figure 1. This figurehas been obtained by running the Matlab scriptemd_fmsin.m.1

Another example (emd_triang.m) that puts em-phasis on the potentially non-harmonic nature ofEMD is given in Figure 2. In this case, both linearand nonlinear oscillations (associated respectivelywith 1 sinusod and 2 triangular waveforms) are ef-fectively identified and separated, whereas any har-monic analysis (Fourier, wavelets,. . . ) would endup in the same context with a much less compactand physically less meaningful decomposition.

3 ALGORITHMIC VARIATIONS

As it has been defined in Section 2, the EMD algo-rithm depends on a number of options which haveto be controlled by the user and which require someexpertise. Our purpose here is to make more precisethe rationale for such choices, as well as to base uponthis analysis variations on the initial formulation.

1Matlab scripts developed for the algorithms and exam-ples described throughout this paper are available as freewarecodes [9].


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3.1 Sampling, interpolation and border ef-

fects

A basic operation in EMD is the estimation of up-per and lower envelopes as interpolated curves be-tween extrema. The nature of the chosen interpo-lation plays an important role, and our experimentstend to confirm (in agreement with what is recom-mended in [2]) thatcubic splinesare to be preferred.Other types of interpolation (linear or polynomial)tend to increase the required number of sifting itera-tions and to over-decompose signals by spreadingout their components over adjacent modes.

A second point is that, since the algorithm op-erates in practice on discrete-time signals, some spe-cial attention has to be paid to the fact that extremamust be correctly identified, a pre-requisite which re-quires a fair amount ofoversampling(this point will

be investigated further in Section 4 below).

Finally, a third issue that has to be taken intoaccount is related to boundary conditions, so as tominimize error propagations due to finite observa-tion lengths. To this end, we obtain good results bymirrorizing the extrema close to the edges.

3.2 Stopping criteria for sifting

The extraction of a mode is considered as satisfactorywhen the sifting process is terminated. Two condi-tions are to be fulfilled in this respect [2]: the firstone is that the number of extrema and the numberof zero-crossings must differ at most by 1; the secondone is that the mean between the upper and lowerenvelopes must close to zero according to some cri-terion.

The evaluation of how small is the amplitude of themean has to be done in comparison with the ampli-tude of the corresponding mode, but imposing a toolow threshold for terminating the iteration processleads to drawbacks similar to the ones mentioned pre-viously (over-iteration leads to over-decomposition).As an improvement to the criteria that have been

considered so far [2], we propose in emd.m [9] to in-troduce a new criterion based on 2 thresholds1 and2, aimed at guaranteeinggloballysmall fluctuationsin the mean while taking into account locally largeexcursions. This amounts to introduce themode am-plitudea(t) := (emax(t) emin(t))/2 and the evalu-ation function (t) := |m(t)/a(t)| so that sifting isiterated until (t)< 1 for some prescribed fraction(1 ) of the total duration, while (t)< 2 for theremaining fraction. One can typically set 0.05,1 0.05 and2 10 1 (default values in emd.m).

3.3 Local EMD

In the classical EMD implementation, sifting itera-tions apply to the full length signal, and they arepursued as long as there exists a local zone wherethe mean of the envelopes is not considered as suffi-

ciently small. However, as it has been already men-tioned, it turns out that over-iterating on the wholesignal for the sake of a better local approximationhas the drawback of contaminating other parts ofthe signal, in particular in uniformizing the ampli-tude and in over-decomposing it by spreading outits components over adjacent modes. Moreover, thehierarchical and nonlinear nature of the princeps al-gorithm can by no means guarantee that the EMDof concatenated signals would be the concatenationof individual EMDs.

This observation suggests therefore a first varia-

tion upon the initial EMD formulation. This vari-ation, referred to as local EMD (local_emd.m),introduces an intermediate step in the sifting pro-cess: those local zones where the error remains largeare identified and isolated, and extra-iterations areapplied only to them. This is achieved by introduc-ing a weighting function w(t) such that w(t) = 1 onthose connected time supports where(t)> 1, witha soft decay to 0 outside those supports. Step 4 ofthe algorithm described in Section 2 is thus simplyreplaced by d(t) =x(t) w(t) m(t).

3.4 On-line EMD

A second variation is based on the observation thatthe sifting step relies on interpolations between ex-trema, and thus only requires a finite number of them(5 minima and 5 maxima in the case of cubic splines)for being operated at a given point. This suggeststhat the extraction of a mode could therefore be pos-sible blockwise, without the necessary knowledge ofthe whole signal (or previous residual). This remarkpaved the road for our development of an EMD al-gorithm which operateson-lineand can therefore beapplied to data flows (emd_online.m).

A pre-requisite for the blockwise extraction of a

mode is to apply the same number of sifting stepsto all blocks in order to prevent possible discontinu-ities Since this would require the knowledge of thewhole signal, the number of sifting operations is pro-posed to be fixed a priori, and it proved in fact thata few iterations (less than 10, typically 4) are gener-ally sufficient to extract a meaningful IMF. The ef-fective application of the on-line version of the EMDalgorithm that we propose is obtained by means ofa sliding window operating on top of the local algo-rithm described above. The front edge of the window


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progresses when new data become available, whereasthe rear edge progresses by blocks when the stoppingcriterion is met on a block. Based on this principle,an IMF and its corresponding residual can be com-puted sequentially. The whole algorithm can there-

fore be applied to this residual, thus allowing for anextraction of the next mode with some delay.

An example of how the algorithm works on can beappreciated by running ex_online.m, an example inwhich the analyzed signal is the periodization of the3-component signal used in Figure 1. In this case,the final decomposition obtained on-line on 16000data points clearly appears as the periodization ofthe decomposition obtained by decomposing the ele-mentary block of 2000 data points.

Besides the essential usefulness of an on-line al-gorithm for decomposing data flows, one can alsopoint out its advantage over standard (block) al-gorithms in terms of computational burden, whichquickly becomes very heavy when dealing with longdata records.

4 PERFORMANCE ELEMENTS

Since EMD is essentially defined by an algorithm anddoes not admit an analytical definition, its perfor-mance evaluation is difficult and requires extensivesimulation experiments. We will report here on twosituations: yet elementary, they both point on non-trivial features that are to be known (and better un-derstood) prior applying EMD to real data and try-

ing to interpret the decomposition.

4.1 Tones and sampling

When analyzing a pure tone, EMD is expected to bethe identity operator, with only 1 mode (supposed tobe identical to the tone), and no residual. Even whenkeeping apart possible border effects, this happensnot to be true because of the unavoidable influenceof sampling, which may create jittered extrema whendealing with only a few points per period.

Figure 3 (emd_sampling.m) aims at quantifyingthis phenomenon by plotting, as a function of the

tone frequency f, the relative error

e(f) =

n

(xf[n] d1[n])2/n

x2f[n]

1/2, (1)

whered1[n] stands for the 1st EMD mode extractedfrom the tone xf[n] of (normalized) frequency f. Itresults that such a tone estimation is heavily depen-dent on f: while the error reaches minima when thetone period is an evenmultiple of the sampling pe-riod, we globally observe that e(f) C f2.

-8 -7 -6 -5 -4 -3 -2 -1-16

-14

-12

-10

-8

-6

-4

-2

0

log2(frequency)

log2

(error)

Figure 3: EMD of 1 tone Estimation error. Circlescorrespond to eq.(1) evaluated on 256 points tones.The superimposed straight line has a slope of 2 inthis (base 2) log-log plot, thus evidencing that erroris upper bounded by a quadratic increase.

4.2 Tones Separation

In the case of 2 superimposed tones

x[n] = a1cos 2f1n + a2cos 2f2n

with f2 < f1 < 1/2, EMD is expected to extractthem via its first 2 modes, although moderate sam-pling may require more that one mode for extractingf1, thus locating f2 in modes of index larger than2. Errors in the extraction can be quantified via aweighted extension of the criterion (1), with f1 com-pared to mode 1, and f2 to the subsequent modewith smallest error (emd_separation.m).

The result is plotted in Figure 4, evidencing anintricate structure with entire domains where tonesseparation is difficult, in particular when f1 > 1/4.The observed patterns clearly depend on the ampli-tudes ratio := a1/a2 but, in a first approximation(and when both f1 and f2 are sufficently small so

that no sampling issue enters the problem), they allshare a common feature: most errors are containedwithin a triangular domain limited by two lines pass-ing through the origin. In other words, for a givenfrequency f1, there exist for each amplitude ratio a fixed < 1 that defines a confusion frequencybandB(f1) := [f1, f1] such thatf1 and f2 B(f1)cannot be separated. This supports an interpreta-tion of EMD in terms of a constant-Q filter bank, inclose agreement with the findings reported in [1, 4, 7]in stochastic situations involving broadband noise.


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f1> f

2

f2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

f1> f

2

f2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

f1> f

2

f2

0

0.2

0.4

0.6

0.8

1

Figure 4: EMD of 2 tones Estimation error. Aweighted extension of the criterion (1) is evaluated inthe lower triangle of the normalized frequency square[0, 1/2] [0, 1/2], and plotted on a linear gray-levelscale (darker pixels correspond to larger errors). Am-plitude ratios are set to = 4, 1 and 1/4 (from topto bottom).

5 CONCLUSION

EMD is a promising new addition to existing tool-boxes for nonstationary and nonlinear signal process-ing, but it still needs to be better understood. Thispaper discussed algorithmic issues aimed at more ef-fective implementations of the method, and it pro-posed some preliminary performance measures.

The results reported here are believed to providewith new insights on EMD and its use, but they aremerely of an experimental nature and they clearlycall for further studies devoted to more theoreticalapproaches.

References

[1] K.T. Coughlin and K.K. Tung, 11-year solarcycle in the stratosphere extracted by the empir-ical mode decomposition method, Adv. Space

Res., Nov. 2002 (submitted).

[2] N.E. Huang, Z. Shen, S.R. Long, M.L. Wu,H.H. Shih, Q. Zheng, N.C. Yen, C.C. Tung andH.H. Liu, The empirical mode decompositionand Hilbert spectrum for nonlinear and non-stationary time series analysis,Proc. Roy. Soc.London A, Vol. 454, pp. 903995, 1998.

[3] P. Flandrin, Time-Frequency/Time-Scale Anal-ysis, Academic Press, 1999.

[4] P. Flandrin, G. Rilling and P. Goncalves, Em-pirical Mode Decomposition as a filter bank,

IEEE Sig. Proc. Lett., 2003 (in press).

[5] R. Fournier, Analyse stochastique modale dusignal stabilometrique. Application a letude delequilibre chez lHomme, These de Doctorat,Univ. Paris XII Val de Marne, 2002.

[6] E.P. Souza Neto, M.A. Custaud, C.J. Cejka,P. Abry, J. Frutoso, C. Gharib and P. Flan-drin, Assessment of cardiovascular autonomiccontrol by the Empirical Mode Decomposition,4th Int. Workshop on Biosignal Interpretation,Como (I), pp. 123-126, 2002.

[7] Z. Wu and N.E. Huang, A study of the charac-teristics of white noise using the Empirical ModeDecomposition method, Proc. Roy. Soc. Lon-don A, Dec. 2002 (submitted).

[8] Z. Wu, E.K. Schneider, Z.Z. Hu and L. Cao,The impact of global warming on ENSO vari-ability in climate records, COLA Technical Re-port, CTR 110, Oct. 2001.

[9] www.ens-lyon.fr/~flandrin/software.html

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