The method of MultiScale Entropy (MSE) is an invaluable tool to quantify and compare the complexity of physiological time series at different time scales. Although MSE traditionally employs sample entropy to measure the unpredictability of each coarse-grained series, the same framework can be applied to other metrics. Here we investigate the use of a rank-based entropy measure within the MSE framework. Like in the traditional method, the series are studied in an embedding space of dimension m. The novel entropy assesses the unpredictability of the series quantifying the "amount of shuffling" that the ranks of the mutual distances between pairs of m-long vectors undergo when considering the next observation. The algorithm was tested on recordings from the Fantasia database in a time-varying fashion using non-overlapping 300-samples windows. The method was able to find statistically significant differences between young and healthy elderly subjects at 7 scales/time-windows after accounting for multiple comparisons using the Holm-Bonferroni correction. These promising results suggest the possibility of using this measure to perform a time-varying assessment of complexity with increased accuracy and temporal resolution

Citi, L

Guffanti, G

Mainardi, L

University of Essex Research Repository

Rank-based Multi-Scale Entropy Analysis of Heart Rate VariabilityLuca Citi1, Giulia Guffanti1,2, Luca Mainardi21 School of Computer Science and Electronic Engineering, University of Essex, Colchester, UK2 Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, ItalyAbstractThe method of MultiScale Entropy (MSE) is an invalu-able tool to quantify and compare the complexity of phys-iological time series at different time scales. AlthoughMSE traditionally employs sample entropy to measure theunpredictability of each coarse-grained series, the sameframework can be applied to other metrics.Here we investigate the use of a rank-based entropymeasure within the MSE framework. Like in the traditionalmethod, the series are studied in an embedding space of di-mension m. The novel entropy assesses the unpredictabil-ity of the series quantifying the “amount of shuffling” thatthe ranks of the mutual distances between pairs of m-longvectors undergo when considering the next observation.The algorithm was tested on recordings from the Fan-tasia database in a time-varying fashion using non-overlapping 300-samples windows. The method was ableto find statistically significant differences between youngand healthy elderly subjects at 7 scales/time-windows af-ter accounting for multiple comparisons using the Holm-Bonferroni correction.These promising results suggest the possibility of usingthis measure to perform a time-varying assessment of com-plexity with increased accuracy and temporal resolution.1. IntroductionEntropy is a measure of a system’s uncertainty and candifferentiate deterministic and stochastic systems. It is awidely used method to assess the complexity of physi-ological time-series with important applications in manyfields like cardiology, neurophysiology and human postu-ral sway. The sample entropy (SE) — an important toolto quantify the level of disorder of a time series [1] — hasachieved significant results detecting sepsis from neonatalHR data [2] and diagnosing atrial fibrillation [3]. It can beestimated from a finite-length time series {xi}Ni=1 as:SE = − ln(AB)(1)where B is the total number of vectors vmi =[xi, xi+1, ..., xi+m−1] of m consecutive points that liewithin a distance ρ, while A is the total number of seg-ments ofm+1 points that are still within distance ρ. In thiscontext, the distance between two vectors is the maximumabsolute difference between their corresponding compo-nents, i.e. the distance defined by the infinity norm. Thefraction AB in (1) is an unbiased estimator of the condi-tional probability:P(∥∥vm+1i − vm+1j ∥∥∞ < ρ ∣∣∣ ∥∥vmi − vmj ∥∥∞ < ρ)≡ P(|xi+m − xj+m| < ρ∣∣∣ ∥∥vmi − vmj ∥∥∞ < ρ) (2)where ρ = rσ and σ is the standard deviation of the series.The normalized tolerance r is usually chosen in an intervalrmin < r < rmax. Most authors recommend rmin = 0.1and rmax = 0.25 for the entropy analysis of heart-rate vari-ability [4].The multi-scale entropy (MSE) is a more recent methodproposed in [5][6]. It is based on sample entropy and com-pares the complexity of physiological time series at dif-ferent time scales. The MSE method constructs a coarse-grained variant yτj of the original time series and computesits entropy measure. The coarse-grained series is obtaineddividing the one-dimensional discrete time series into non-overlapping windows of length equal to a scale factor τ ,and averaging the data points within each window:yτj =1ττ∑i=1x(j−1)τ+i, 1 ≤ j ≤ N/τ. (3)The entropy measure is finally calculated for each coarse-grained time series, obtaining the function SE(τ).In this paper, we introduce a novel entropy metric basedon the ranking of mutual distances between pairs of m-long vectors.2. Methods2.1. Rank-based entropyA family of alternative entropy metrics can be definedbased on the “amount of shuffling” that the ranks of themutual distances between pairs of m-long vectors undergowhen considering the next observation.In this paper, we describe one of these metrics which canbe operationally defined in terms of the following steps.We start by computing, for 1 ≤ i < j ≤ N −m, thevectors of mutual distances:dk(i,j) =∥∥vmi − vmj ∥∥∞ (4)d′k(i,j) = |xi+m − xj+m| (5)where k = k(i, j) is the index assigned to each (i, j) pair,with 1 ≤ k ≤ K = (N −m−1) (N −m)/2. We thenconsider the vector dk and find the permutation pi(k) suchthat the vector sk = dpi(k) is sorted in ascending order.Intuitively, if xi+m represents the next observation and vmithe recent “history” (or state) of a deterministic system,1pairs (vmi , vmj ) of close vectors should project into pairs ofclose observations (xi+m, xj+m). As a consequence, thevector s′k = d′pi(k) should be almost sorted too. Conversely,if the system is completely random, the distance betweenpairs of state vectors should bear no information about thecloseness of the following observations, and the vector s′kshould be, in general, unsorted.We quantify this idea into a concrete measure using thenotion of inversion count. Given a vector w, its inversioncount is the number of pairs of indices i and j such thati < j and wi > wj . It represents a measure of the vec-tor’s disorder and it coincides with the number of actions(swaps) that a bubble sort algorithm needs to take to sortthe vector. In assessing the degree of disorder of s′k, weare specifically interested in observation points that resultfrom pairs of close state vectors. Therefore, we determinethe largest index kρ satisfying skρ < ρ and then com-pute the number I of inversions pairs (k1, k2) such thatk1 < kρ, k1 < k2, and s′k1 > s′k2. Finally, we define thenew entropy asRE = − ln(1− I(2K − kρ − 1) kρ/2). (6)The role of the parameter r used to determine ρ = rσ isdifferent in RE than in SE. In a sense, rather than defininga hard threshold for the distance between state vectors, rdetermines the maximum rank of the set of distances thatwill contribute to the final entropy measure. For this rea-son, its choice is less critical than in the case of SE and wesimply set r = rmax = 0.25.The new entropy can also be used within a multi-scaleframework simply by constructing coarse-grained versionsof the original time series as in (3) and computing the en-tropy measure corresponding to each scale factor. In thefollowing we will show the application of this rank-basedMSE to short and noisy artificial data sets as well as toreal-world heart beat series.1Assuming m is sufficiently long to capture the system’s dynamics.2.2. Simulated signalsAn effective entropy metric should be able to measurethe degree of predictability of a time series, and in par-ticular to determine whether it was generated by a deter-ministic or a stochastic system. Ideally, it should be ableto perform this assessment without resorting to the use ofother properties of the signal such as the probability distri-bution of its values or the frequency spectrum. Secondly,a robust metric should yield useful results even when thesystem’s observations are corrupted by a small amount ofadditive noise.To test our algorithm we compared the entropy of agiven time series xi, generated from a deterministic sys-tem in chaotic regime, with that of its surrogate generatedusing the iAAFT algorithm [7], which has approximatelythe same histogram and the same frequency spectrum am-plitude but new uniformly random phases. Clearly, we ex-pect the surrogate signal — which is stochastic — to havea higher entropy than the original deterministic signal.Because the entropy of a system is estimated from finite-length time series, the resulting measure is affected bystatistical fluctuations. We generated M = 200 time se-ries xmi of length N = 400 with different initial condi-tions, and we computed the corresponding entropies RmE .For each series, we also computed the surrogate series{x˜mi } = iAAFT({xmi }) and its entropy R˜mE . Ideally, ourmetric should report a difference between the entropy ofthe surrogates and that of the original data which is sig-nificantly larger than the intrinsic fluctuations due to thefinite-length estimation process. To quantify this desirableproperty, we assessed the discrimination accuracy of ourentropy metric as:a =C[R˜mE]− C[RmE]D[R˜mE]+D[RmE] (7)where C[·] is a measure of central tendency while D[·] isa measure of dispersion. If C[·] and D[·] are, respectively,estimators of mean and standard deviation, the accuracya is the reciprocal of the Davies-Bouldin index, a widelyused metric for evaluating clustering algorithms. In ourcase, we used the median as measure of central tendencyand 1.4826MAD[·] (median absolute deviation) as mea-sure of dispersion.To further validate the metric, we tested its robustness toadditive observation noise. Each series, xmi , was corruptedwith white Gaussian noise, nmi , and finally the entropy of{xmi + nmi } and that of iAAFT({xmi + nmi }) were mea-sured. We repeated the procedure for different values ofthe signal-to-noise ratio (SNR).The first type of signals that we used to test the novelprocedure was generated using the logistic map:xn+1 = µxn(1− xn) (8)where x0 is a real number in the interval [0, 1] and theparameter µ is the rate of growth. We adopted the valueµ = 3.9 for which the map exhibits chaotic behaviour.The second test was conducted on signals generated us-ing the Lorenz flow:dxdt= σ(y−x); dydt= x(ρ− z)−y; dzdt= xy−βz (9)with values σ = 10, β = 83 and ρ = 28 for which the dy-namics evolve around a chaotic attractor. The time serieswas obtained as xi = x(iδ) where the time lag δ = 0.6was chosen using the autocorrelation function.2.3. Real signalsWe also tested our algorithm on real data from the “Fan-tasia” database [8] which is publicly available from Phy-sionet [9]. The series of NN intervals were further pre-processed using an artifact removal filter based on the“filt” function of the “hrv-toolkit” available on Physionet.To break the artificial ties created by the 250Hz sam-pling of the ECG, a random noise uniformly distributed in[−2ms, 2ms] was added to the time series. Then, each sub-ject’s NN series was divided in 15 non-overlapping timewindows of 300 samples each and the rank-based MSEwas independently computed for each time window.3. Results3.1. Simulation resultsFor the logistic map, the rank-based entropy gives avalue 0.130 ± 0.004 (C[·] ± D[·]) while for the surrogatedata it goes up to 0.546 ± 0.014. This corresponds to adiscrimination accuracy a ' 23, which is approximatelytwice that obtained with the sample entropy a ' 12. Thisimprovement stands even in the presence of noise, as re-ported in Fig. 1-left. Remarkably, the entropies computedfrom the different realizations of the logistic map are wellseparated from those of their surrogates even with a SNRas low as 4.4 dB (i.e. when the standard deviation of thenoise is 60% that of the signal).For the series generated from the Lorenz flow, the rank-based entropy is 0.430±0.026 while for the surrogate datait increases to 0.687±0.022. This leads to a discriminationaccuracy a ' 5.5 which is slightly lower than that of thesample entropy a ' 6.5. When the series are contaminatedwith noise, RE seems to perform slightly better than SE asreported in Fig. 1-right.Inf20.014.010.58.06.04.43.11.90.90.005101520SNR [dB]aSEREInf20.014.010.58.06.04.43.11.90.90.00246SNR [dB]aSEREFigure 1: Discrimination accuracy index (see (7)) assess-ing the ability of the two metrics (SE and RE) to detect theincreased entropy of the surrogate data compared to theoriginal series (logistic map on the left, Lorenz flow on theright), as a function of the SNR.0.10.20.30.40.50.60.72 4 6 8τRE(τ)logisticsurrogate0.40.60.81.01.21.41.61.82.02 4 6 8τSE(τ)logisticsurrogateFigure 2: MSE computed using the conventional SE (left)and using RE (right) for the logistic map and for the sur-rogate data. The central thick line represents the medianwhile the thin lines report the first and third quartile of thevalues obtained from the 200 realizations.Finally, the full MSE plots computed using RE and us-ing SE are reported in Fig. 2 and Fig. 3 for the logistic mapand for the Lorenz flow, respectively.3.2. Real data resultsThe entropy values obtained with RE at scales 1 to 4 foryoung and elderly subjects are reported in Fig. 4. We ob-serve that the entropy of the elderly is generally lower thanthat of young subjects, especially for scales 2 and higher,as previously observed in [6]. At the single scale/timelevel, the difference is statistically significant (p < 0.05,using a Mann-Whitney U test) for 2 time windows at scale2 4 6 81.01.21.41.61.82.02.22.42.6SE(τ)τLorenzsurrogate2 4τ6 80.400.450.500.550.600.650.700.75RE(τ)LorenzsurrogateFigure 3: Same as Fig. 2, for the Lorenz flow.0.30.40.50.30.40.50.30.40.55-10 15-20 25-30 35-40 45-50 55-60 65-700.30.40.5window [minutes]REyoung elderly 1234Figure 4: Entropy RE computed at scales 1 (bottom)to 4 (top) on fifteen 300-samples windows. Each plotshows the Median±1.4826MAD /√n for young (dottedline) and elderly subjects (dashed line). The circles denotescales/times for which the difference between young andelderly subjects is statistically significant (p < 0.05). Thestars mark those that remain statistically significant afteraccounting for multiple comparisons.1, for 11 at scales 2 and 3, and for 5 at scale 4. If one isinterested in assessing a statistical difference between thetwo age populations at all scales/times, it is worth notingthat a total of 7 differences remain statistically significanteven after accounting for multiple comparisons using theHolm-Bonferroni correction.Using the conventional MSE, instead, only for 14time/scales out of 60 the differences between the two pop-ulations are statistically significant. None of them remainssignificant after accounting for multiple comparisons.4. ConclusionsWe presented a novel rank-based entropy measurewhich can be used within a multi-scale framework. Testedon simulated time series with known properties, the novelmetric showed a higher robustness to noise when distin-guishing the output of a logistic map from its surrogate.On real data, the rank-based MSE outperformed the con-ventional one in finding statistically significant differencesbetween young and healthy elderly subjects using 300-samples-long RR series. These encouraging results sug-gest the possibility of using this measure to perform a time-varying assessment of complexity with increased accuracyand temporal resolution.References[1] Richman JS, Moorman JR. Physiological time-series anal-ysis using approximate entropy and sample entropy. Am JPhysiol Heart C 2000;278(6):H2039–H2049.[2] Lake DE, Richman JS, Griffin MP, Moorman JR. Sample en-tropy analysis of neonatal heart rate variability. Am J PhysiolReg I 2002;283(3):R789–R797.[3] Lake DE, Moorman JR. Accurate estimation of entropy invery short physiological time series: the problem of atrialfibrillation detection in implanted ventricular devices. Am JPhysiol Heart C 2011;300(1):H319–H325.[4] Castiglioni P, Di Rienzo M. How the threshold r influencesapproximate entropy analysis of heart-rate variability. InComputers in Cardiology. 2008; 561–564.[5] Costa M, Goldberger AL, Peng CK. Multiscale entropy anal-ysis of complex physiologic time series. Phys Rev Lett 2002;89(6):068102.[6] Costa M, Goldberger AL, Peng CK. Multiscale entropy anal-ysis of biological signals. Phys Rev E Feb 2005;71(2 Pt1):021906.[7] Schreiber T, Schmitz A. Improved surrogate data for nonlin-earity tests. Phys Rev Lett Jul 1996;77:635–638.[8] Iyengar N, Peng C, Morin R, Goldberger A, Lipsitz L. Age-related alterations in the fractal scaling of cardiac interbeatinterval dynamics. Am J Physiol Reg I 1996;271(4):R1078–R1084.[9] Goldberger AL, Amaral LA, Glass L, et al. Physiobank,physiotoolkit, and physionet: components of a new researchresource for complex physiologic signals. Circulation 2000;101(23):E215–E220.Address for correspondence:Luca CitiSchool of Computer Science and Electronic EngineeringUniversity of EssexColchester, CO4 3SQ, UKE-mail address: lciti@ieee.org

Rank-based multi-scale entropy analysis of heart rate variability

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