Detection of time reversibility in time series by ordinal patterns analysis
DDetection of time reversibility in time series by ordinal patterns analysis
J. H. Mart´ınez, a) J. L. Herrera-Diestra, and M. Chavez INSERM-UM1127, Sorbonne Universit´e, Institut du Cerveau et de la Moelle Epini`ere. France ICTP South American Institute for Fundamental Research, IFT-UNESP. Brazil CNRS UMR7225, Hˆopital Piti´e Salpˆetri`ere. France (Dated: 13 September 2018)
Time irreversibility is a common signature of nonlinear processes, and a fundamental property of non-equilibrium systems driven by non-conservative forces. A time series is said to be reversible if its statisticalproperties are invariant regardless of the direction of time. Here we propose the Time Reversibility fromOrdinal Patterns method (TiROP) to assess time-reversibility from an observed finite time series. TiROPcaptures the information of scalar observations in time forward, as well as its time-reversed counterpart bymeans of ordinal patterns. The method compares both underlying information contents by quantifying its(dis)-similarity via Jensen-Shannon divergence. The statistic is contrasted with a population of divergencescoming from a set of surrogates to unveil the temporal nature and its involved time scales. We tested TiROPin different synthetic and real, linear and non linear time series, juxtaposed with results from the classicalRamsey’s time reversibility test. Our results depict a novel, fast-computation, and fully data-driven method-ology to assess time-reversibility at different time scales with no further assumptions over data. This approachadds new insights about the current non-linear analysis techniques, and also could shed light on determiningnew physiological biomarkers of high reliability and computational efficiency.PACS numbers: 05.45.Tp, 05.70.Ln, 89.75.Kd, 87.23.-n, 87.19.le, 89.65.Gh, 87.10.VgKeywords: Time reversibility, Time series, Ordinal patterns analysis, Nonlinearity, Surrogate data
Most time series observed from real systems areinherently nonlinear, thus detecting this propertyis of full interest in natural or social sciences. Onefeature that ensures the nonlinear character of asystem is the time irreversibiliity. A time series issaid to be reversible if its statistical properties areinvariant regardless of the direction of time. Herewe propose the Time Reversibility from OrdinalPatterns (TiROP) method to assess the tempo-ral symmetry of linear and nonlinear time seriesat different scales. Our approach is based on afast-computing symbolic representation of the ob-served data. Here, TiROP is compared with aclassical time-reversibility test in a rich varietyof synthetic and real time series from differentsystems, including ecology, epidemiology, econ-omy and neuroscience. Our results confirm thatTiROP has a remarkable performance at unveil-ing the time scales involved in the temporal irre-versibility of a broad range of processes.
I. INTRODUCTION
A time series is said to be reversible if its statisti-cal properties are invariant regardless of the direction oftime. Time irreversibility is a fundamental property ofnon-equilibrium systems and dynamics resulting from a) Electronic mail: [email protected] non-conservative forces (memory) , therefore, it is ex-pected to be present in the scalar observation of differentbiological and physical systems. Indeed, time irreversibil-ity has been reported in ecological and epidemiologi-cal time series , in tremor time series of patients withParkinson’s disease , in electroencephalographic (EEG)recordings of epileptic patients , or in cardiac inter-beat interval time series extracted from patients andhealthy subjects under different cardiac conditions .Any time series that is a realisation of a stationary,linear Gaussian process is time reversible, because of thesymmetry of their covariance functions . Neverthe-less, a non-Gaussian amplitude distribution could be dueto a static nonlinear transformation of a stationary lin-ear Gaussian process, and by itself is no proof of tempo-ral irreversibility. Furthermore, non-Gaussian processesmodeled as outputs of linear systems are reversible .In contrast, the output of a non-linear system excitedby non-Gaussian noises is time irreversible . Non-linearand non-Gaussian linear models typically have temporaldirectionality as a property of their higher-order depen-dency . The study of time reversibility properties oftime series might therefore provide meaningful insightsinto the underlying nonlinear mechanisms of the observeddata.Classical time reversibility tests require higher-ordermoments of the studied signal X t to be finite . Othertests have been devised by directly comparing the dis-tribution of vectors { X t , X t +1 , · · · , X t + D } and itstime-reversed version { X t + D , X t + D − , · · · , X t } , or fromthe projection of dynamics onto a finite number ofplanes . In the last years, some works have proposedstatistical tests for irreversibility based on the so-calledvisibility graphs , i.e., the mutual visibility relationships a r X i v : . [ phy s i c s . d a t a - a n ] S e p between points in a one-dimensional landscape represent-ing X t . These works show that irreversible dynamicsresults in an asymmetry between the probability distri-butions of graph properties (e.g. links or paths-basedcharacteristics). Recently, this approach has been ex-tended for the study of non-stationary processes .For real-valued time series, some studies have proposedtime-reversibility tests based on different symbolizationprocedures to characterize the dimensional phase spacesof X t and its time-reversed version . These sym-bolic transformations are generally done by defining aquantization procedure to transform the time series intoa discrete sequence of unique patterns or symbols .Some of these reversibility tests use a priori binomialstatistics to assess statistical significance of findings .Nevertheless, such tests assume independence of the ob-served symbols, which is unlikely to occur in real datawith temporal correlations. In case of such serial correla-tions, a rigorous theoretical framework cannot be derivedand Monte Carlo simulations (e.g. parametric or non-parametric re-sampling) must be performed to estimatethe significance level of time reversibility tests .In this work we propose a novel procedure, the TimeReversibility from Ordinal Patterns method (TiROP),that compares the empirical distributions of the forwardand backward statistics of a time series. To estimatethe asymmetry between both probability distributions weuse the ordinal symbolic representations . In contrastwith other approaches based on symbolic analysis, the or-dinal patterns analysis used here is fully data-driven, i.e.,the symbolic transformation does not require any a pri-ori threshold, or any knowledge about the data sequence.We complete our time reversibility test with surrogatedata analysis without making assumptions on the under-lying generating process .The proposed framework is validated on synthetic datasimulated with linear, nonlinear, non-Gaussian stochas-tic and deterministic processes. The method is also illus-trated on a collection of different real time series. Thereliability and performances of our method are also com-pared with those obtained by a classical moment-basedmethod. The remainder of the paper is organized asfollows: Section II describes the proposed framework,as well as the comparative method used to benchmarkour solution. Experimental results and evaluation of themethod in synthetic time series are in Section III; whilethe evaluation of the test on real data is provided in Sec-tion IV. Finally, we conclude the paper with a discussionin Section V. II. METHODSCapturing information dynamics from time series
Symbolisation procedures map a time series X t ontoa discretized symbols sequence by extracting its am-plitudes’ information . Among several symbolisation original time series time-reversed signal X t X t ´ 𝜋 𝛿 (P( 𝜋 ),P( 𝜋 ))´P( 𝜋 ) P( 𝜋 )´ 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 𝜋 (a)(b)(c) FIG. 1. Main steps of the TiROP algorithm for evaluatingthe time-reversibility of a time series X t . (a) (Left) Originaltime series represented in blue. (Right) The time reversed sig-nal X (cid:48) t represented in orange. (b) Patterns π ’s extracted from X t and X (cid:48) t for D = 3. (c) Probability distributions P ( π ) and P (cid:48) ( π ) extracted from X t and X (cid:48) t , respectively. The Jensen-Shannon δ captures the dissimilarity between the informationcontent in both distributions proposals , we considered here the dynamical transfor-mation by Band and Pompe . This method maps atime series X t with t = 1 , . . . , T to a finite number of pat-terns that encode the relative amplitudes observed in the D -dimensional vector X t = { X t , X t + τ , . . . , X t +( D − τ } .The elements of the vector X t are mapped uniquely ontothe permutation π = ( π , π , . . . , π D − ) of (0 , , . . . , D −
1) that fulfills X t + π τ (cid:54) X t + π τ , (cid:54) . . . (cid:54) X t + π D − τ .Each order pattern (permutation) represents thus a sub-set of the whole embedding state space.The set of all possible ordinal patterns derived froma time series is noted as S t , whose cardinality is D ! atmost. The whole sequence of ordinal patterns extractedfrom X t is known as the symbolic representation of thetime series. The information content of X t is captured bythe probability density P ( π ) of finding a particular pat-tern of order D in S t . The higher the order is, the moreinformation is captured from the time series. To sam-ple the empirical distribution of ordinal patterns denselyenough for a reliable estimation of its probability distri-bution we follow the condition T (cid:62) ( D + 1)!The analysis of ordinal representations has some prac-tical advantages : i) it is computationally efficient, ii) it is fully data-driven with no further assumptions aboutthe data range to find appropriate partitions and, iii) asmall D is generally useful in descriptive data analysis .Furthermore, this symbolisation method is known to berelatively robust against noise, and useful for time serieswith weak stationarity . TABLE I. Synthetic models. LGP and AR(2) are two linear reversible processes. The non-linear (non-reversible) AR modelsare driven by a Laplacian and bimodal noise distribution, respectively. Two Self-Exciting Threshold AutoRegressive models,SETAR(2; 2,2) and SETAR(2; 3,2), are non linear models with regime switching behavior. The last two models, R¨ossler andLorenz oscillators, are set under chaotic regime.Model EquationLGP Gaussian noise with distribution N (0 , a x t +2 = 0 . x t +1 + 0 . x t + (cid:15) t N-AR(2) b x t = 0 . x t − − . x t − + 0 . y t − + 0 . x t − + 0 . y t − + 0 . η (cid:48) t y t = sin(4 πt ) + sin(6 πt ) + 0 . η (cid:48)(cid:48) t SETAR(2; 2,2) a x t = (cid:40) .
62 + 1 . x t − − . x t − + 0 . (cid:15) t if x t − ≤ . .
25 + 1 . x t − − . x t − + 0 . (cid:15) t otherwiseSETAR(2; 3,2) a x t = (cid:40) .
733 + 1 . x t − − . x t − + 0 . x t − + 0 . (cid:15) t if x t − ≤ . .
983 + 1 . x t − − . x t − + 0 . (cid:15) t otherwiseR¨ossler ˙ x = − y − z ˙ y = x + 0 . y ˙ z = 0 . z ( x − . x = 10( y − x )˙ x = x (28 − z ) − y ˙ x = xy − . z a (cid:15) t denotes white noise processes. b noises { η (cid:48) t , η (cid:48)(cid:48) t } are iid. See main text for the parameters Assessing time reversibility
A time series X t is said to be time-reversible if thejoint distributions of vectors X t = { X t , X t +1 , · · · , X t + D } and X (cid:48) t = { X t + D , X t + D − , · · · , X t } for D are equal forall t , i.e., the statistical properties of the process arethe same forward and backward in time. All Gaus-sian processes (and all static transformations of a lin-ear Gaussian process) are time-reversible since their jointdistributions are determined by the covariance functionwhich is symmetric . On the contrary, linear pro-cesses driven by non-Gaussian innovations and the non-linear processes with regime-switching structures, such asthe self-exciting threshold autoregressive (SETAR) pro-cess , are generally time irreversible .Time reversibility implies that the differences of the se-ries being tested have symmetric marginal distributions,i.e. if X t is time reversible, the distribution of Y t,τ = X t − X t − τ is symmetric about the origin for every τ .Time reversibility also implies that all the odd momentsof Y t,τ , if exist, are zero . A simple measure for a de-viation from reversibility for a certain time lag τ was in-troduced by Ramsey . Time reversibility is assessed bychecking the difference between the sample bi-covariancesfor zero mean time series γ ( τ ) = (cid:104) X t X t − τ (cid:105) − (cid:104) X t X t + τ (cid:105) .This method is a benchmark test for time-reversibilityand it has been proved to be effective at detecting non-linearity and reversibility in different time series, such ashearth rates, economical data, or even in SETAR mod-els . Nevertheless, moment-based tests for time re-versibility are not really applicable because they require higher-order moments of X t to be finite, which may ruleout many real time series . Furthermore, it is quite pos-sible to encounter a situation in which the individual teststatistics are significant for some lags but insignificant forothers.In this work, we propose the Time Reversibility fromOrdinal Patterns method (TiROP) as a procedure to as-sess for time-irreversibility with no assumptions aboutthe process or the observed signal X t (see the generalscheme in Fig. 1). Ordinal symbolic representations arenot symbols ad hoc, but they encode information aboutthe temporal structure of the underlying data. Instead ofcomparing empirical distributions from X t and its time-reversed version X (cid:48) t , we compare the permutation parti-tion (i.e., the symbolic representation) of the embeddingstate spaces spanned by X t and X (cid:48) t . The idea behindTiROP is to compare the distribution P ( π ) of ordinalpatterns obtained from the original signal, i.e., the distri-bution of the ordinal transformation of vectors X t ; withthe probability P (cid:48) ( π ) resulting from its time reversed ver-sion X (cid:48) t .To quantify the (dis)-similarity between both informa-tion contents, we use the Jensen-Shannon divergence δ ( P ( π ) , P (cid:48) ( π )) = D ( P ( π ) , M ( π )) + D ( P (cid:48) ( π ) , M ( π ))),where M ( π ) = ( P ( π ) + P (cid:48) ( π )) and D ( U, W ) = (cid:80) i U ( i ) log U ( i ) W ( i ) is the divergence from distribution U to W . Time reversibility implies that distributions of vec-tors X t and X (cid:48) t , and therefore the distributions of theirordinal transformations, are the same.To rule out the possibility that large values of δ couldaccount for non-Gaussian distributions, or large autocor- * * * * * * * * * * * * * * * (a) (b) * * * * (c) * * * * (d) * * * * * (e) (f) (g) (h) dimension Ddimension D d i s t a n c e s FIG. 2. TiROP test applied to synthetic models. Yellow dots indicate the original δ values for each D . Dashed red lines arevisual guides but do not represent continuity. Blue dots represent the distributions of { δ s } at different scales. Black asterisksindicate the dimension D for which the value of δ is statistically different from { δ s } . The different models are: (a) LinearGaussian process; (b) linear autoregressive model; (c) non-linear AR model driven by a Laplacian noise; (d) non-linear systemexcited by a noise with a bi-modal distribution; (e) SETAR with two regimes, each one with second order delays; (f) SETARmodel with two regimes, with delays of third and second order; (g) chaotic R¨ossler system; and (h) chaotic Lorenz model. relation values at different time lags in signal X t , the sta-tistical significance of δ values is assessed by a z-test toquantify the statistical deviation from values obtained inan ensemble of surrogate data . An ensemble { X st } of surrogate time series are created directly from the orig-inal dataset through replication of the linear autocor-relation and amplitudes distribution. In this work, weuse the so-called Iterative Amplitude Adjusted FourierTransform (IAAFT) that preserves power spectrumdensity and amplitude distribution of original data, whileall other higher-order statistics are destroyed. For each X st , we repeat the procedure of Fig. 1 to compute aset of { δ s } dissimilarities. If the original dissimilarity isstatistically distant from the distribution of { δ s } we canassume that X t comes from a nonlinear system with atime irreversible dynamics.The reliability and performances of our TiROP methodare also compared with those obtained by the Ramsey’stime-reversibility test γ ( τ ) based on moments and dis-cussed above . All significance tests are set at p < . D or time lags τ ). III. TIME REVERSIBILITY IN SYNTHETIC TIMESERIES
In this section, we evaluate the performance of theTiROP method on synthetic time series, simulated withdifferent classes of models (see Table. I):
Time-reversible linear systems: a linear Gaussianprocess (LGP), and a linear auto-regressive model of sec-ond order driven by a white noise.
Non-reversible coupled non-linear systems:
Twonon-reversible nonlinear AR models (N-AR) driven bynon-Gaussian noises . We first consider a non-linearsystem driven by Laplacian noises drawn from the dis-tribution p ( η ) = b exp (cid:16) −| η − µ | b (cid:17) , with µ = 0 and b = 1. Then, we use the same non-linear model ex-cited by a noise that follows the bi-modal distribution p ( η ) = 0 . N ( η | µ, σ ) + 0 . N ( η | − µ, σ ), with µ = 0 . σ = 1. Non-reversible switched nonlinear systems:
Two processes from the family of Self-Exciting Thresh-old AR (SETAR) models, which are largely used to modelecological systems and are characterized for having jumpsbetween different non-linear regimes, each one with dif- (g) (h) * * * * * * * * * ** * * * * * * * ** * * * * * * * * ** * * * * * * * * * * * ** * delaydelay (a) (b) (c) (d)(e) (f)
FIG. 3. Ramsey’s reversibility test of synthetic models. Blue dots indicate the original γ values for each time-lag τ . Dashedred lines are visual guides but do not represent continuity. Distributions of { γ s } at different scales are represented by the pointsinside the yellow plots. Black asterisks indicate at which time-lag, γ is statistically different from γ s . Same stipulations as inthe caption of Fig. 2. ferent delays . The fifth model is a SETAR withtwo regimes, each one with second order delays. Thesixth one is a SETAR with two regimes with delays ofthird and second order. Chaotic, non-reversible systems:
The last twomodels are the classical R¨osler and Lorenz systems intheir corresponding chaotic regimes . The analyzed timeseries correspond to the evolution of the y and z vari-ables, from the R¨ossler and Lorenz systems, respectively.For each model, the length of each time series is set to T = 10 , after discarding the first 1000 points to avoidpossible transients. Contrary to phase state reconstruc-tion, which requires to select a dimension D and timedelay τ embedding according to some criteria, in ordinaltime-series analysis the criteria are computational costand statistical significance in view of the amount of dataavailable . We therefore do not make any assump-tion regarding the dimension, and use different values of D depending on the data length. Although, larger de-lays can provide additional scale-dependent informationabout the time series under study, we set τ = 1 through-out this work .For the assessment of statistical significance we gener-ate 50 surrogates from each original sequence. For dif-ferent scales ( D = 3 , ..., δ and the set of { δ s } for all surrogate time series considered. We calcu-late a z -statistics for each D as | δ −(cid:104){ δ s }(cid:105) σ ( { δ s } ) | and we checkfor irreversibility by testing the null hypothesis H ofa time-reversible process with significance level α (cid:54) . τ ’s. Fig. 2 (3) shows the results for TiROP (Ramsey)methodology along different scales (delays).As expected, the statistical properties of the LPG pro-cess are the same forward and backward in time and thusthe null hypothesis of reversibility is never rejected byboth tests. Interestingly, whereas the TiROP methodcorrectly diagnoses the AR model as a reversible pro-cess, Ramsey’s statistics yields false positives and falselyrejects H in two non-continuous delays.Whereas non-Gaussian processes modeled as outputsof linear systems are reversible , the output of a non-linear system excited by non-Gaussian noises is time ir-reversible. For the case of non-linear AR (N-AR) processexcited by a Laplacian noise, the null hypothesis of time-reversibility is correctly rejected by our TiROP method,while Ramsey’s test fails to detect time-irreversibilityalong all time-lags. Similar to the previous, the output ofthe N-AR model driven by a bi-modal noise is detectedas irreversible by TiROP for all dimensions D >
1, while
10 y2000 lynx cases days500 USD * * * * ** * * * * (a) (b) dimension D dimension D d i s t a n c e s (c) dimension D FIG. 4. Time-reversibility test on different real data. Insets shows the collection of samples of each process, its temporal andamplitude scales. (a) Time series of lynx returns. (b) Weekly Mexican reported cases of dengue. (c) Daily S&P closing prices.Yellow dots indicate the original δ values for each D . Dashed red lines are visual guides but do not represent continuity. Bluedots represent the distributions of { δ s } at different scales. Black asterisks indicate the dimension D for which the value of δ isstatistically different from { δ s } . Ramsey’s test only detects irreversibility in the first threedelays.For the SETAR and chaotic models, both TiROP andRamsey’s tests correctly reject the time-reversible hy-pothesis, in agreement with previous studies at iden-tifying the intrinsic time irreversibility of such mod-els . To notice, however, that Ramsey’s statis-tics yields a false negative at the first delay for the R¨osslersystem.To further evaluate the performance of the TiROPmethod, we consider short sample sizes. Numerical simu-lations show that our TiROP test can correctly detectedirreversibility in SETAR and chaotic models when thedata length is, at least, ten times the fundamental pe-riod T of SETAR ( T (cid:39) T (cid:39)
52 samples). Forthese sample sizes, the Ramsey’s method increases dra-matically the number of incorrect rejections of true nullhypothesis, as well as the number of false negatives inchaotic systems.
IV. TIME REVERSIBILITY IN REAL DATA
To further demonstrate the potentials of our test, weapply it to real data of different nature: ecology (the timeseries of lynx abundance), epidemiology (dengue preva-lence), economy (the S&P price-index series) and neu-roscience (electroencephalographic data from an epilep-tic patient). As data have different length we applyTiROP in different dimensions, following the condition T (cid:62) ( D + 1)!Inset in Fig. 4-(a) shows the well-known time series x t of fur returns of the Canadian lynx, a valuable collectionrepresenting the regularity and rhythm of lynx popula-tion in Canada. Each amplitude represents the amountof lynx furs that trappers caught and brought into postsin the same hunting season. T = 114 samples were col-lected during 1821-1914 near Mackenzie river region .Notice that this dataset was used to fit the SETAR mod-els’ parameters used in this work . Before applying thetime-reversibility test, we applied the variance stabiliz-ing transformation y t = log ( x t + 1). Despite its shortdata length, our results suggest irreversibility in this timeseries, in full agreement with previous works .Inset in Fig. 4-(b) depicts M = 678 epidemiologi-cal weeks of reported cases of Dengue in Mexico dur-ing the years 2000-2015 . As for the lynx time series,time reversibility was assessed on the transformed data y t = log ( x t + 1). Based on nonlinear prediction tech-niques, different studies have proposed evidence for timereversibility in different ecological and epidemiologicaltime series . For the time series of dengue prevalenceconsidered here, the TiROP method rejects the null hy-pothesis of time reversibility for all scales. This result in-dicates that such dengue’s dynamics cannot be analyzedby conventional linear models.The inset in Figure 4-(c) shows M = 5444 samplesfrom the Standard & Poor’s Index encompassing thedaily historical closing prices from January 1990 to Au-gust 2011 . This is the most representative index ofthe real situation of market in USA based on the cap-italization of 500 large companies with common stocksin NYSE and NASDAQ. Although S&P-500 time se-ries has been suggested to be irreversible and chaotic ,some works have showed that moment-based methodsfail at detecting irreversibility . To account for the non-stationarity of original data, we extracted the log-returns y t = log( x t +1 ) − log( x t ), and then we checked for time-reversibility at different scales up to D = 7. Our methodrejects the hypothesis of a time-reversible process, whichagrees with previous findings suggesting that irreversibil-ity in economical time series is a rule instead of a simpleexception . * * * * (a) (b) dimension D d i s t a n c e s dimension D FIG. 5. Time-reversibility test on EEG data. Insets showten seconds in the same scales (a) before and (b) during theepileptic episode. Same stipulations as in the caption of Fig. 4
As many others time series in biology and medicine,electroencephalographic (EEG) signals display strongnonlinearities during different cognitive or pathologicalstates . Time-reversibility can be a useful property ofinterictal EEG signals, as it can serve as a marker of theepileptogenic zone . Here, we applied our TiROP testto scalp EEG recordings from a pediatric subject withintractable epileptic seizures . Figs. 5(a)-(b) showthe time series corresponding to the interictal and ictal(seizure) periods, respectively. Our results confirm pre-vious findings suggesting that interictal EEG dynamicscan be associate to a reversible linear process, whereastime irreversibility characterizes epileptic seizures . V. CONCLUSIONS
In this work we have addressed the problem of detect-ing, from scalar observations, the time scales involvedin temporal irreversibility. Based on the ordinal pat-terns analysis, the TiROP method compares the infor-mation content of the symbolic representation of X t andthe counterpart of its time-reversed version X (cid:48) t . In con-trast with other approaches based on symbolic analysis,the approach proposed here has the key practical advan-tage that it is fully data-driven and it does not requireany a priori thresholds, or any knowledge about the datasequence for its symbolic representation, which is veryuseful in real-world data analysis.Results confirm that TiROP provides an interestingand promising approach to the analysis of complex timeseries. The applicability and advantages of our methodwas demonstrated by many examples from synthetic andreal, linear and nonlinear models. The method outper-forms a classical moment-based test, which often failsto detect time-irreversibility along different time-lags.Our results confirm temporal irreversibility in economicaltime series, and suggest this property as a common signa-ture in epidemiological data. This would imply that ad-ditional nonlinear analysis techniques should be appliedfor a more complete characterization of such time series.The results indicates that time irreversibility can also beobserved at scalp EEG recordings of epileptic seizures inhumans.To conclude, this study shows that the detection oftemporal irreversibility in time series can be successfullyaddressed using ordinal symbolic representation. Themain advantage of our proposal relies on its simplicity,reliability and computational efficiency thanks to the or-dinal patterns transformation and analysis. The detec-tion of temporal irreversibility in other data (e.g. cardiacor climate time series) might provide meaningful insightsinto the underlying process generating the observed timeseries. This framework could also add new functional-ity to current non-linear analysis techniques, but also itcould open the way to define physiological biomarkers. ACKNOWLEDGMENTS
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