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Detection of Time Reversibility in Time Series by Ordinal Patterns Analysis J Detection of time reversibility in time series by ordinal patterns analysis J. H. Mart´ınez,1, a) J. L. Herrera-Diestra,2 and M. Chavez3 1)INSERM-UM1127, Sorbonne Universit´e,Institut du Cerveau et de la Moelle Epini`ere. France 2)ICTP South American Institute for Fundamental Research, IFT-UNESP. Brazil 3)CNRS UMR7225, H^opitalPiti´eSalp^etri`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 statistical properties are invariant regardless of the direction of time. Here we propose the Time Reversibility from Ordinal Patterns method (TiROP) to assess time-reversibility from an observed finite time series. TiROP captures the information of scalar observations in time forward, as well as its time-reversed counterpart by means 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 divergences coming from a set of surrogates to unveil the temporal nature and its involved time scales. We tested TiROP in different synthetic and real, linear and non linear time series, juxtaposed with results from the classical Ramsey'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 approach adds new insights about the current non-linear analysis techniques, and also could shed light on determining new 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.Vg Keywords: Time reversibility, Time series, Ordinal patterns analysis, Nonlinearity, Surrogate data Most time series observed from real systems are non-conservative forces (memory)5, therefore, it is ex- inherently nonlinear, thus detecting this property pected to be present in the scalar observation of different is of full interest in natural or social sciences. One biological and physical systems. Indeed, time irreversibil- feature that ensures the nonlinear character of a ity has been reported in ecological and epidemiologi- system is the time irreversibiliity. A time series is cal time series6,7, in tremor time series of patients with said to be reversible if its statistical properties are Parkinson's disease8, in electroencephalographic (EEG) invariant regardless of the direction of time. Here recordings of epileptic patients9{11, or in cardiac inter- we propose the Time Reversibility from Ordinal beat interval time series extracted from patients and Patterns (TiROP) method to assess the tempo- healthy subjects under different cardiac conditions12{15. ral symmetry of linear and nonlinear time series Any time series that is a realisation of a stationary, at different scales. Our approach is based on a linear Gaussian process is time reversible, because of the fast-computing symbolic representation of the ob- symmetry of their covariance functions16{18. Neverthe- served data. Here, TiROP is compared with a less, a non-Gaussian amplitude distribution could be due classical time-reversibility test in a rich variety to a static nonlinear transformation of a stationary lin- of synthetic and real time series from different ear Gaussian process, and by itself is no proof of tempo- systems, including ecology, epidemiology, econ- ral irreversibility. Furthermore, non-Gaussian processes omy and neuroscience. Our results confirm that modeled as outputs of linear systems are reversible19. TiROP has a remarkable performance at unveil- In contrast, the output of a non-linear system excited ing the time scales involved in the temporal irre- by non-Gaussian noises is time irreversible20. Non-linear versibility of a broad range of processes. and non-Gaussian linear models typically have temporal directionality as a property of their higher-order depen- dency18. The study of time reversibility properties of time series might therefore provide meaningful insights I. INTRODUCTION into the underlying nonlinear mechanisms of the observed arXiv:1809.04377v1 [physics.data-an] 12 Sep 2018 data. A time series is said to be reversible if its statisti- Classical time reversibility tests require higher-order 21{23 cal properties are invariant regardless of the direction of moments of the studied signal Xt to be finite . Other time. Time irreversibility is a fundamental property of tests have been devised by directly comparing the dis- 1{3 4,9,24 non-equilibrium systems and dynamics resulting from tribution of vectors fXt;Xt+1; ··· ;Xt+Dg and its time-reversed version fXt+D;Xt+D−1; ··· ;Xtg, or from the projection of dynamics onto a finite number of planes14,25. In the last years, some works have proposed a)Electronic mail: [email protected] statistical tests for irreversibility based on the so-called visibility graphs26, i.e., the mutual visibility relationships 2 between points in a one-dimensional landscape represent- (a) 27{29 ing Xt . These works show that irreversible dynamics results in an asymmetry between the probability distri- ´ butions of graph properties (e.g. links or paths-based Xt Xt characteristics). Recently, this approach has been ex- tended for the study of non-stationary processes30,31. original time series time-reversed signal For real-valued time series, some studies have proposed (b) time-reversibility tests based on different symbolization procedures to characterize the dimensional phase spaces 12,32,33 � of Xt and its time-reversed version . These sym- bolic transformations are generally done by defining a (c) ´ quantization procedure to transform the time series into �(P(�),P(�)) a discrete sequence of unique patterns or symbols13,15,34. P(�) P(´�) Some of these reversibility tests use a priori binomial statistics to assess statistical significance of findings32,33. Nevertheless, such tests assume independence of the ob- served symbols, which is unlikely to occur in real data �1 �2 �3 �4 �5 �6 �1 �2 �3 �4 �5 �6 with temporal correlations. In case of such serial correla- tions, a rigorous theoretical framework cannot be derived FIG. 1. Main steps of the TiROP algorithm for evaluating the time-reversibility of a time series Xt. (a) (Left) Original and Monte Carlo simulations (e.g. parametric or non- time series represented in blue. (Right) The time reversed sig- parametric re-sampling) must be performed to estimate 0 nal Xt represented in orange. (b) Patterns π's extracted from the significance level of time reversibility tests12,13,15. 0 Xt and Xt for D = 3. (c) Probability distributions P (π) and 0 0 In this work we propose a novel procedure, the Time P (π) extracted from Xt and Xt, respectively. The Jensen- Reversibility from Ordinal Patterns method (TiROP), Shannon δ captures the dissimilarity between the information that compares the empirical distributions of the forward content in both distributions and backward statistics of a time series. To estimate the asymmetry between both probability distributions we 40 use the ordinal symbolic representations35,36. In contrast proposals , we considered here the dynamical transfor- 35 with other approaches based on symbolic analysis, the or- mation by Band and Pompe . This method maps a dinal patterns analysis used here is fully data-driven, i.e., time series Xt with t = 1;:::;T to a finite number of pat- the symbolic transformation does not require any a pri- terns that encode the relative amplitudes observed in the ori threshold, or any knowledge about the data sequence. D-dimensional vector Xt = fXt;Xt+τ ;:::;Xt+(D−1)τ g. We complete our time reversibility test with surrogate The elements of the vector Xt are mapped uniquely onto data analysis without making assumptions on the under- the permutation π = (π0; π1; : : : ; πD−1) of (0; 1;:::;D − 37,38 lying generating process . 1) that fulfills Xt+π0τ 6 Xt+π1τ ; 6 ::: 6 Xt+πD−1τ . The proposed framework is validated on synthetic data Each order pattern (permutation) represents thus a sub- simulated with linear, nonlinear, non-Gaussian stochas- set of the whole embedding state space. tic and deterministic processes. The method is also illus- The set of all possible ordinal patterns derived from trated on a collection of different real time series. The a time series is noted as St, whose cardinality is D! at reliability and performances of our method are also com- most. The whole sequence of ordinal patterns extracted pared with those obtained by a classical moment-based from Xt is known as the symbolic representation of the method. The remainder of the paper is organized as time series. The information content of Xt is captured by follows: Section II describes the proposed framework, the probability density P (π) of finding a particular pat- as well as the comparative method used to benchmark tern of order D in St. The higher the order is, the more our solution. Experimental results and evaluation of the information is captured from the time series. To sam- method in synthetic time series are in Section III; while ple the empirical distribution of ordinal patterns densely the evaluation of the test on real data is provided in Sec- enough for a reliable estimation of its probability distri- 41 tion IV. Finally, we conclude the paper with a discussion bution we follow the condition T > (D + 1)! in Section V. The analysis of ordinal representations has some prac- tical advantages36: i) it is computationally efficient, ii) it is fully data-driven with no further assumptions about II. METHODS the data range to find appropriate partitions and, iii) a small D is generally useful in descriptive data analysis35. Capturing information dynamics from time series Furthermore, this symbolisation method is known to be relatively robust against noise, and useful for time series with weak stationarity39,42{46.
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