IEEE SIGNAL PROCESSING LETTERS, VOL. 19, NO. 2, FEBRUARY 2012 91

Multivariate Multiscale Entropy AnalysisMosabber Uddin Ahmed and Danilo P. Mandic

AbstractMultivariate physical and biological recordings arecommon and their simultaneous analysis is a prerequisite for theunderstanding of the complexity of underlying signal generatingmechanisms. Traditional entropy measures are maximized forrandom processes and fail to quantify inherent long-range de-pendencies in real world data, a key feature of complex systems.The recently introduced multiscale entropy (MSE) is a univariatemethod capable of detecting intrinsic correlations and has beenused to measure complexity of single channel physiological sig-nals. To generalize this method for multichannel data, we firstintroduce multivariate sample entropy (MSampEn) and evaluateit over multiple time scales to perform the multivariate multiscaleentropy (MMSE) analysis. This makes it possible to assess struc-tural complexity of multivariate physical or physiological systems,together with more degrees of freedom and enhanced rigor in theanalysis. Simulations on both multivariate synthetic data and realworld postural sway analysis support the approach.

Index TermsMultivariate embedding, long term correlation,multivariate multiscale entropy (MMSE), multivariate sample en-tropy (MSampEn), multivariate system complexity.

I. INTRODUCTION

R EAL world phenomena are characterized by long termdependencies in their dynamical behaviors, and a numberof criteria have been developed to assess structural complexityof the underlying signal generating mechanisms. Such criteriaare typically based on local predictability, irregularity, self-sim-ilarity, and synchrony [1]. Time delay embedded reconstructionis particularly popular, as it allows for the estimation of invariantquantities (in terms of smooth transformations in state space) ofthe original system, such as attractor dimensions, Lyapunov ex-ponents and various entropy measures [1].Traditional entropy measures, such as Shannon entropy,

KolmogorovSinai (KS) entropy, approximate entropy (ApEn)[2], and sample entropy (SampEn) [3], are maximized forcompletely random processes, and are used to quantify theregularity of univariate time series on a single scale, by e.g.,evaluating repetitive patterns [4]. As a result, using entropy tomeasure complexity of physiological data, which exhibit highdegree of structural richness, yields lower entropy than fortheir randomized surrogates, formed by shuffling the originaldata samples and thus destroying any structure present. This is

Manuscript received November 03, 2011; revised December 10, 2011; ac-cepted December 13, 2011. Date of publication December 20, 2011; date ofcurrent version January 12, 2012. The associate editor coordinating the reviewof this manuscript and approving it for publication was Prof. Z. Jane Wang.The authors are with the Department of Electrical and Electronic En-

gineering, Imperial College London, London SW7 2AZ, U.K. (e-mail:mosabber.ahmed@imperial.ac.uk; d.mandic@imperial.ac.uk).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/LSP.2011.2180713

counterintuitive for a measure of complexity, and the greater en-tropy of the uncorrelated surrogate series also highlights a lackof a straightforward correspondence between regularity andcomplexity. Neither completely predictable (e.g., periodic) norcompletely unpredictable (e.g., uncorrelated random) signalsare truly complex, since at a global level none is structurallyrich. Instead, time series observed from dynamical physical andphysiological systems generally exhibit long-range correlationsat multiple spatial and temporal scales.The multiscale entropy (MSE) method proposed by Costa et

al. [4] explicitly quantifies the amount of structure (correlation)in real world time series, that is, the underlying system com-plexity. The method evaluates sample entropy of coarse grained(averaged over increasing segment lengths) univariate time se-ries; the underlying idea is that coarse graining defines temporalscales, hence a system without structure would exhibit a rapiddecrease in entropy with an increase in time scale. The existingMSE algorithm has been proven to be able to distinguish be-tween physiological time series with different degrees of com-plexity and its extensions have included more rigorous defini-tions of time scales [5]. Applications include the analysis ofpathologic heartbeat conditions like erratic cardiac arrhythmiaand congestive heart failure [4], electroencephalogram (EEG)changes in patients with Alzheimers disease [6], postural swaydynamics analysis [7], and complex dynamics of human redblood cells [8]. All the above results strongly support the generalloss of complexity behavior when a living system undergoeschange from its normal unconstrained state (healthy) to thatunder stress, e.g., due to ageing and disease [9].Recent developments in sensor technology have enabled rou-

tine recording of multivariate time series from both physicaland biological systems, yet when assessing their complexity thestandard MSE analysis can only consider every data channelseparately. This is only appropriate if the components of a mul-tivariate signal are statistically independent or at the very leastuncorrelated (which is usually not the case). For instance, theembedding theorem [10] establishes that the dynamics of a mul-tivariate system can be reconstructed from sufficiently long timedelay embedded vectors of a single time series (seen as a one-dimensional projection of the multivariate system trajectory),however, in practice this is not reliable for systems exhibitingmore than two degrees of freedom. Indeed, based on measure-ments of the -coordinate of the Lorenz system we cannot re-construct its dynamics, as embedding based on the -coordinatedoes not resolve the - symmetry [11].In this work, we illustrate substantial advantages in simul-

taneously analyzing dynamical complexity of several variablesobserved from the same system, especially if there is a large de-gree of uncertainty or coupling underlying the measured systemdynamics. To this end, we first introduce multivariate sample

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92 IEEE SIGNAL PROCESSING LETTERS, VOL. 19, NO. 2, FEBRUARY 2012

Fig. 1. Multivariate sample entropy as a function of data length N, forand in each data channel. Shown are the mean values for 30 simulatedtrivariate time series containing white and noise.

entropy, and evaluate its evolution at different temporal scales.The so introduced multivariate multiscale entropy (MMSE) isshown to operate on any number of data channels, and to pro-vide a robust relative complexity measure for multivariate data.The approach is supported by simulations on both synthetic andreal world multivariate processes.

II. MULTIVARIATE MULTISCALE ENTROPY

To introduce the multivariate multiscale entropy (MMSE)analysis, we propose the following steps.1) Define temporal scales by averaging a -variate time se-ries , over non-overlapping timesegments of increasing length (coarse graining), whereis the number of samples in every channel. This way, forscale , a coarse grainedmultivariate time series is obtainedas , where and thechannel index .

2) Evaluate multivariate sample entropy, , foreach coarse-grained multivariate , and plotas a function of the scale factor .

However, methods to calculate multivariate entropy are stilllacking. To this end, we next propose a multivariate sample en-tropy measure, and use it to introduce the MMSE method.

A. The MSampEn Method

To calculate , recall from multivariate embeddingtheory [11], that for a -variate time series ,

, observed through measurement functions ,the multivariate embedded reconstruction is based on a com-posite delay vector

(1)

where is the embedding vector,the time lag vector, and the composite delay

vector , where .For a -variate time series , , the

MSampEn method is introduced in Algorithm 1, and representsa natural extension of standard univariate sample entropy [3].

Algorithm 1 The Multivariate Sample Entropy

1: Form composite delay vectors , whereand .

2: Define the distance between any two vectors andas the maximum norm, that is,

.

3: For a given composite delay vector and a threshold ,count the number of instances for which, , then calculate the frequency of occurrence,

, and define .

4: Extend the dimensionality of the multivariate delayvector in (1) from to . This can be performedin different ways, as from a space with the embeddingvector the system canevolve to any space for which the embedding vector is

. Thus, a totalof vectors in are obtained, where

denotes any embedded vector upon increasing theembedding dimension from to for a specificvariable .

5: For a given , calculate the number ofvectors , such that ,where , then calculate the frequency ofoccurrence, , and define

.

6: Finally, for a tolerance level , estimate as

B. Effect of Data Length on MSampEnIt has been suggested in [2] that data samples

are sufficient to robustly estimate univariate approximate en-tropy or sample entropy. To assess the sensitivity of the pro-posed multivariate sample entropy to the data length parameter,we evaluated multivariate sample entropy of a trivariate whiteas well as noise as a function of sample size , where foreach channel the embedding dimension was and thethreshold . Fig. 1 shows that for both the white andtrivariate noise, MSampEn estimates were consistent for datalength , illustrating suitability of MSampEn for theanalysis of real world data. This way, the standard deviation ofmultivariate sample entropy estimates (error bars) related to thelength of the simulated series is likely to be smaller than thedeviations related to experimental errors as well as inter- andintra-subject variability for most practical applications.Physically, for the standard univariate sample entropy, the

increase in sample entropy values with an increase in the em-bedding dimension is due to progressively fewer delay vec-tors to compare as increases. On the contrary, for MSampEnthe increase in does not reduce the number of the availabledelay vectors, as the composite multivariate embedded vectorsare constructed in parallel. For instance, for the case in Fig. 1and , we are not taking all the six points from the sameunivariate series, instead we are taking two points (as )from each of the three channels of the trivariate series in hand.

AHMED AND MANDIC: MULTIVARIATE MULTISCALE ENTROPY ANALYSIS 93

Besides, in MSampEn calculation we do not give preference toany particular over when increasing the embedding di-mension from to . Instead, we create embedded vec-tors in all the subspaces and compare them within and acrossthese subspaces to find the -neighbors. This also effectively in-creases the total number of available delay vectors times, andmakes the proposed MSampEn robust to the variation in boththe parameter and data length .

C. Selection of Parameter ValuesThere are several parameters that need to be chosen in

calculation, each introducing their own constraints.For instance, for multimodal data coming from heterogeneousdata channels, the individual channels are likely to exhibitdifferent embedding parameters and . There are severalmethods [12] for determining the optimal embedding param-eters and simultaneously, for a single channel. Futuredevelopments should test for all the and parameters fromthe different data channels simultaneously, but these are not yetavailable. Also, standard sample entropy introduces a practicalconstraint that , which is inherited in MSampEn. Thisdid not influence the results in this work and is sufficient for themajority of real world applications. In MSampEn calculation,the threshold parameter is set to some percentage of thestandard deviation; for MSampEn, we used its multivariategeneralizationthe total variation, , where is thecovariance matrix. To maintain the same total variation forall the multivariate series, the individual data channels werenormalized to unit variance so that the total variation equalsthe number of channels/variables. This way, we take as apercentage (say 15%) of , which is similar to taking

in each channel.

D. Multivariate Complexity AnalysisThe multivariate MSE (MMSE) method assesses relative

complexity of normalized multichannel temporal data byplotting multivariate sample entropy as a function of thescale factor. A multivariate time series is considered morestructurally complex than another if for the majority of timescales its multivariate entropy values are higher than those ofthe other time series. A monotonic decrease in multivariateentropy values with the scale factor reveals that the signal inhand only contains information at the smallest scale, and is thusnot structurally complex. For instance, the standard univariateMSE analysis in [4] showed that for random white noise (un-correlated) the sample entropy values monotonically decreasewith scale, whereas for noise (long-range correlated) thesample entropy remains constant over multiple scales. This hasa physical justification, as by design noise is structurallymore complex than uncorrelated random noise.To illustrate the corresponding behavior for the proposed

multivariate approach, we considered a 4-variate time series,where originally all the data channels were fed with mutuallyindependent white noise. The number of noise channels wasthen gradually decreased (from 4 to 0) with a simultaneousincrease in the number of data channels that represent indepen-dent noise (from 0 to 4). Fig. 2 shows the MMSE curves forall the possible cases: notice that, as desired, when the number

Fig. 2. MMSE analysis for 4-channel data containing independent white andnoise. Each data channel had 10, 000 data points, and the plots represent

an average of 20 independent realizations and error bars the standard deviation(SD).

Fig. 3. Multivariate multiscale entropy (MMSE) analysis for bivariate whiteand noise. Each data channel had 10 000 data points, and the plots representan average of 20 independent realizations and error bars the standard deviation(SD).

of channels representing noise increased, theat higher scales also increased, and when all the four datachannels contained noise, the complexity at larger scaleswas the highest (cf. smallest at scale 1).Cross-channel correlations are a key aspect in multivariate

analysis and in the next set of simulations, we analyzed theability of MMSE to capture cross-channel properties. Fig. 3shows that, as desired, the proposed multivariate extension ofMSE caters for both within- and cross-channel correlations: thecomplexity of the correlated bivariate noise at large scaleswas the highest, followed by the uncorrelated noise, andcorrelated and uncorrelated white noise.

III. EXPERIMENTAL RESULTSMultivariate complexity analysis of real world postural

sway dynamics (time series of the center of pressure (COP)displacement) is next performed for young and elderly subjectsduring quiet standing, and has been compared with the existingunivariate complexity analysis [7]. COP displacement wasrecorded simultaneously for the mediolateral (side-to-side) andanteroposterior (front-to-back) direction [13] for 15 healthyyoung and 12 healthy elderly volunteers at 60 Hz.Since the postural sway time series exhibits high frequency

fluctuations superimposed on low frequency trends, the datawas first detrended using the multivariate empirical modedecomposition technique [14]. The values of the parametersused to calculate MSampEn were , , and

94 IEEE SIGNAL PROCESSING LETTERS, VOL. 19, NO. 2, FEBRUARY 2012

Fig. 4. Multiscale entropy analysis of COP time series for young (redsolidline) and elderly (greendotted line) subjects. Top: Univariate MSE for themediolateral component. Middle: Univariate MSE for the anteroposterior com-ponent. Bottom: Bivariate MMSE analysis. The plots represent mean values ofSampEn/MSampEn for all subjects across all the trials and error bars representstandard error.

(standard deviation of the normalized time series)for each data channel; these parameters were chosen on thebasis of previous studies indicating good statistical repro-ducibility for the univariate SampEn [3], [7]. Since the originaltime series had data points, the highest achievablescale factor had 300 data points; this was sufficient foraccurate analysis, as shown in Fig. 1.The top and middle panel in Fig. 4 show the results obtained

by the univariate MSE performed for the mediolateral (ML) andanteroposterior (AP) component of the COP time series, respec-tively. The bottom panel shows that when both the ML and APcomponent were considered within the multivariate approach,the proposed MMSE was able to discriminate between youngand elderly subjects more efficiently, as indicated by the betterseparation of the MMSE curves and the complexity index (areaunder the MSE curve) being higher for healthy young subjectsthan for their elderly counterparts1. Table I shows the statisticalsignificance of the results obtained by the one-tailed t-test withunequal variances. It is evident that the difference in complexitybetween the young and elderly subjects was statistically signif-icant in the AP direction for a 5% significance level and notsignificant in the ML direction. If, for rigor, we take a 1% sig-nificance level, then none of the univariate analyses gave statis-tically significant results. On the contrary, in the bivariate casewe observed significant differences, even for the significancelevel of . This illustrates significant advan-tages of using MMSE when assessing relative complexity ofreal world multivariate data, and supports the more general con-cept of multiscale complexity loss with ageing and disease orwhen a system is under constraints, as those factors reduce theadaptive capacity of biological organization at all levels [9]. TheMatlab code for the MMSE analysis can be downloaded from[15].

IV. CONCLUSIONThis work has generalized the recently introduced univariate

multiscale entropy (MSE) method to the multivariate case, to1Due to ageing and the associated constraints, the complexity of postural

sway for the elderly is lower than for the young.

TABLE ICOMPLEXITY INDICES AND P VALUES OBTAINED USING ONE-TAILED

STUDENTS T-TEST WITH UNEQUAL VARIANCE

provide complexity analysis of real world biological and phys-ical systems which are typically of multivariate, correlated andnoisy natures. The proposed multivariate multiscale entropy(MMSE) method has been shown to be naturally suited to re-veal long range within- and cross-channel correlations presentin multichannel data. The MMSE method has been validatedon both illustrative benchmark signals and on a simultaneousanalysis of ML and AP component of postural sway dynamicsfrom the young and elderly subjects.

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[15] [Online]. Available: http://www.commsp.ee.ic.ac.uk/~mandic/re-search/Complexity_Stuff.htm