Time series irreversibility: a visibility graph approach
Abstract
We propose a method to measure realvalued time series irreversibility which combines two different tools: the horizontal visibility algorithm and the KullbackLeibler divergence. This method maps a time series to a directed network according to a geometric criterion. The degree of irreversibility of the series is then estimated by the KullbackLeibler divergence (i.e. the distinguishability) between the in and out degree distributions of the associated graph. The method is computationally efficient, does not require any ad hoc symbolization process, and naturally takes into account multiple scales. We find that the method correctly distinguishes between reversible and irreversible stationary time series, including analytical and numerical studies of its performance for: (i) reversible stochastic processes (uncorrelated and Gaussian linearly correlated), (ii) irreversible stochastic processes (a discrete flashing ratchet in an asymmetric potential), (iii) reversible (conservative) and irreversible (dissipative) chaotic maps, and (iv) dissipative chaotic maps in the presence of noise. Two alternative graph functionals, the degree and the degreedegree distributions, can be used as the KullbackLeibler divergence argument. The former is simpler and more intuitive and can be used as a benchmark, but in the case of an irreversible process with null net current, the degreedegree distribution has to be considered to identifiy the irreversible nature of the series.
pacs:
05.45.Tp, 05.45.a, 89.75.HcI Introduction
A stationary process is said to be statistically time reversible (hereafter time reversible) if for every , the series and have the same joint probability distributions weiss . Roughly, this means that a reversible time series and its time reversed are, statistically speaking, equally probable. Reversible processes include the family of Gaussian linear processes (as well as Fouriertransform surrogates and nonlinear static transformations of them), and are associated with processes at thermal equilibrium in statistical physics. Conversely, time series irreversibility is indicative of the presence of nonlinearities in the underlying dynamics, including nonGaussian stochastic processes and dissipative chaos, and are associated with systems driven outofequilibrium in the realm of thermodynamics Parrondo0 ; Parrondo1 . Time series irreversibility is an important topic in basic and applied science. From a physical perspective, and based on the relation between statistical reversibility and physical dissipation Parrondo0 ; Parrondo1 , recent work uses the concept of time series irreversibility to derive information about the entropy production of the physical mechanism generating the series, even if one ignores any detail of such mechanism Parrondo2 ; Parrondo3 . In a more applied context, it has been suggested that irreversibility in complex physiological series decreases with aging or pathology, being maximal in young and healthy subjects heartbeatPRL ; costa1 ; multiscale , rendering this feature important for noninvasive diagnosis. As complex signals pervade natural and social sciences, the topic of time series reversibility is indeed relevant for scientists aiming to understand and model the dynamics behind complex signals.
The definition of time series reversibility is formal and therefore there is not an a priori optimal algorithm to quantify it in practice. Recently, several methods to measure time irreversibility have been proposed kennel2 ; kennel ; diks ; gaspard ; costa1 ; cammarota ; heartbeatPRL ; andrieux ; wang . The majority of them perform a time series symbolization, typically making an empirical partition of the data range kennel2 (note that such a transformation does not alter the reversible character of the output series kennel ) and subsequently analyze the symbolized series, through statistical comparison of symbol strings occurrence in the forward and backwards series or using compression algorithms kennel ; Parrondo3 ; cover . The first step requires an extra amount of ad hoc information (such as range partitioning or size of the symbol alphabet) and therefore the output of these methods eventually depend on these extra parameters. A second issue is that since typical symbolization is local, the presence of multiple scales (a signature of complex signals) could be swept away by this coarsegraining: in this sense multiscale algorithms have been proposed recently costa1 ; multiscale .
Motivated by these facts, here we explore the usefulness of the horizontal visibility algorithm in such context. This is a time series analysis method which was proposed recently HVA . It makes use of graph theoretical concepts, and it is based on the mapping of a time series to a graph and the subsequent analysis of the associated graph properties pnas ; HVA ; epl ; toral . Here we propose a time directed version of the horizontal visibility algorithm, and we show that it is a simple and well defined tool for measuring time series irreversibility. More precisely, we show that the KullbackLeibler divergence cover between the out and in degree distributions, , is a simple measure of the irreversibility of realvalued stationary stochastic series. Analytical and numerical results support our claims, and the presentation is as follows: The method is introduced in section II. In section III we analyze reversible time series generated from linear stochastic processes, which yield . As a further validation, in section IV we report the results obtained for irreversible series. We first analyze a thermodynamic system (a discrete flashing ratchet) which shows time irreversibility when driven out of equilibrium. Its amount of irreversibility can be increased continuously tuning the value of a parameter of the system, and we find that the method can, not only distinguish, but also quantify the degree of irreversibility. We also study the effect of applying a stalling force in the opposite direction of the net current of particles in the ratchet. In this case the benchmark measure fails predicting reversibility whereas a generalized measure based on degreedegree distributions goes beyond the phenomenon associated to physical currents and still detects irreversibility. We extend this analysis to chaotic signals, where our method distinguishes between dissipative and conservative chaos, and we analyze chaotic signals polluted with noise. Finally, a discussion is presented in section V.
Ii The method
ii.1 The horizontal visibility graph
The family of visibility algorithms is a collection of methods that map series to networks according to specific geometric criteria pnas ; HVA . The general purpose of such methods is to accurately map the information stored in a time series into an alternative mathematical structure, so that the powerful tools of graph theory may eventually be employed to characterize time series from a different viewpoint, bridging the gap between nonlinear time series analysis, dynamical systems, and graph theory epl ; plos ; elsner ; turbulence ; finance .
We focus here on a specific subclass called horizontal visibility algorithm, firstly proposed in HVA , and defined as follows: Let be a realvalued time series of data. The algorithm assigns each datum of the series to a node in the horizontal visibility graph (HVg). Then, two nodes and in the graph are connected if one can draw a horizontal line in the time series joining and that does not intersect any intermediate data height (see figure 1). Hence, and are two connected nodes if the following geometrical criterion is fulfilled within the time series:
(1) 
Some results regarding the characterization of stochastic and chaotic series through this method have been put forward recently HVA ; toral , and the first steps for a mathematically sound characterization of horizontal visibility graphs have been established simone . Interestingly, a very recent work suggests that the method can be used in practice to characterize not only time series but generic nonlinear discrete dynamical systems, sharing similarities with the theory of symbolic dynamics plos .
ii.2 Directed HVg
So far in the literature the family of visibility graphs are undirected, as visibility did not have a predefined temporal arrow. However, as already suggested in the seminal paper pnas , such a directionality can be made explicit by making use of directed networks or digraphs redes . We address such directed version, defining a Directed Horizontal Visibility graph (DHVg) as a HVg, where the degree of the node is now splitted in an ingoing degree , and an outgoing degree, such that . The ingoing degree is defined as the number of links of node with other past nodes associated with data in the series (that is, nodes with ). Conversely, the outgoing degree , is defined as the number of links with future nodes.
For a graphical illustration of the method, see figure 1. The degree distribution of a graph describes the probability of an arbitrary node to have degree (i.e. links) redes . We define the in and out (or ingoing and outgoing) degree distributions of a DHVg as the probability distributions of and of the graph which we call and , respectively.
ii.3 Quantifying irreversibility: DHVg and KullbackLeibler divergence
The main conjecture of this work is that the information stored in the and distributions take into account the amount of time irreversibility of the associated series. More precisely, we claim that this can be measured, in a first approximation, as the distance (in a distributional sense) between the and degree distributions ( and ). If needed, higher order measures can be used, such as the corresponding distance between the and degreedegree distributions ( and ). These are defined as the and joint degree distributions of a node and its first neighbors redes , describing the probability of an arbitrary node whose neighbor has degree to have degree .
We make use of the KullbackLeibler divergence cover as the distance between the in and out degree distributions. Relative entropy or KullbackLeibler divergence (KLD) is introduced in information theory as a measure of distinguishability between two probability distributions. Given a random variable and two probability distributions and , KLD between and is defined as follows:
(2) 
which vanishes if and only if both probability distributions are equal and it is bigger than zero otherwise. Unlike other measures used to estimate time irreversibility kennel2 ; kennel ; cammarota ; heartbeatPRL , the KLD is statistically significant, as it is proved by the ChernoffStein lemma: The probability of incorrectly guessing (via hypothesis testing) that a sequence of data is distributed as when the true distribution is tends to when . The KLD is then related to the probability to fail when doing an hypothesis test, or equivalently, it is a measure of distinguishability: the more distinguishable are and with respect to each other, the larger is .
In statistical mechanics, the KLD can be used to measure the time irreversibility of data produced by nonequilibrium processes but also to estimate the average entropy production of the physical process that generated the data Parrondo0 ; Parrondo2 . Irreversibility can be assessed by the KLD between probability distributions associated to observables in the process and in its time reversal. These measure gives lower bounds to the entropy production, whose accuracy increases as the observables contain a more detailed description of the system. The measure that we present in this work has this limitation: it takes the information from the degree, which is a partial description of the process. Consequently, our technique does not give a tight bound to the entropy production.
Nevertheless, as we will show in several examples, the information of the outgoing degree distribution is sufficient to distinguish between reversible and irreversible stochastic stationary series which are realvalued but discrete in time . We compare the outgoing degree distribution in the actual (forward) series with the corresponding probability in the timereversed (or backward) time series, which is equal to the probability distribution of the ingoing degree in the actual process . The KLD between these two distributions is
(3) 
This measure vanishes if and only if the outgoing and ingoing degree probability distributions of a time series are identical, , and it is positive otherwise. We will apply it to several examples as a measure of irreversibility.
Notice that previous methods to estimate time series irreversibility generally proceed by first making a (somewhat ad hoc) local symbolization of the series, coarsegraining each of the series data into a symbol (typically, an integer) from an ordered set. Then, they subsequently perform a statistical analysis of word occurrences (where a word of length is simply a concatenation of symbols) from the forward and backwards symbolized series andrieux ; wang . Time series irreversibility is therefore linked to the difference between the word statistics of the forward and backwards symbolized series. The method presented here can also be considered as a symbolization if we restrict ourselves to the information stored in the series and . However, at odds with other methods, here the symbolization process (i) lacks ad hoc parameters (such as number of symbols in the set or partition definition), and (ii) it takes into account global information: each coarsegraining is performed using information from the whole series, according to the mapping criterion (1). Hence, this symbolization naturally takes into account multiple scales, which is desirable if we want to tackle complex signals costa1 ; multiscale .
Iii Reversibility
iii.1 Uncorrelated stochastic series
For illustrative purposes, in figure 2 we have plotted the in and out degree distributions of the visibility graph associated to an uncorrelated random series of size : the distributions cannot be distinguished and KLD vanishes (the numerical value of KLD is shown in table 1) which is indicative of a reversible series. In what follows we provide an exact derivation of the associated outgoing and ingoing degree distributions associated to this specific process, showing that they are indeed identical in the limit of infinite size series.
Theorem 1. Let be a biinfinite sequence of independent and identically distributed random variables extracted from a continuous probability density . Then, both the in and out degree distributions of its associated directed horizontal visibility graph are
(4) 
Proof (outdistribution). Let be an arbitrary datum of the aforementioned series. The probability that the horizontal visibility of is interrupted by a datum on its right is independent of ,
where .
The probability of the datum being capable of exactly seeing data may be expressed as
(5) 
where is the probability of seeing at least data. may be recurrently calculated via
(6) 
from which, with , the following expression is obtained
(7) 
which together with equation (5) concludes the proof. An analogous derivation holds for the in case.
Note that this result is independent of the underlying probability density : it holds not only for Gaussian or uniformly distributed random series, but for any series of independent and identically distributed (i.i.d.) random variables extracted from a continuous distribution . A trivial corollary of this theorem is that the KLD between the in and out degree distributions associated to a random uncorrelated process tends asymptotically to zero with the series size, which correctly suggests that the series is time reversible.
iii.2 Correlated stochastic series
In the last section we considered uncorrelated stochastic series which are our first example of a reversible series with . As a further validation, here we focus on linearly correlated stochastic processes as additional examples of reversible dynamics weiss . We use the minimal substraction procedure toral to generate such correlated series. This method is a modification of the standard Fourier filtering method, which consists in filtering a series of uncorrelated random numbers in Fourier space. We study time series whose correlation is exponentially decaying (akin to an OrnsteinUhlenbeck process) and power law decaying . In table 1 we show that the KLD of these series are all very close to zero, and its deviation from zero is originated by finite size effects, as it is shown in figure 3.
Iv Irreversibility
iv.1 Discrete flashing ratchet
We now study a thermodynamic system which can be smoothly driven out of equilibrium by modifying the value of a physical parameter. We make use of the time series generated by a discrete flashing ratchet model introduced in Parrondo2 . The ratchet consists of a particle moving in a one dimensional lattice. The particle is at temperature and moves in a periodic asymmetric potential of height , which is switched on and off at a rate (see Figure 4 for details). The switching rate is independent of the position of the particle, breaking detailed balance Parrondo2 ; Parrondo3 . Hence, switching the potential drives the system out of equilibrium resulting in a directed motion or net current of particles. When using full information of the process, trajectories of the system are described by two variables: the position of the particle and the state of the potential, . The time series are constructed from and variables as follows: if and if .
The dynamics of the system is described by a sixstate Markov chain with transition probabilities , where is the transition rate from to and the sum runs over the accessible states from (see figure 4). All transition rates satisfy the detailed balance condition with respect to the thermal bath at temperature , except the switches between ON and OFF. When the potential is on, and . When it is off, and . On the other hand, switches are implemented with rates that do not depend on the position of the particle and therefore do not satisfy detail balance condition Parrondo3 : , for .
In Figure 5 we depict the values of and as a function of , for state time series of data. Note that for detailed balance condition is satisfied, the system is in equilibrium and trajectories are statistically reversible. In this case both KLD using degree distributions and degreedegree distributions vanish. On the other hand, if is increased, the system is driven out of equilibrium, what introduces a net statistical irreversibility which increases with Parrondo2 . The amount of irreversibility estimated with KLD increases with for both measures, therefore the results produced by the method are qualitatively correct. Interestingly enough, the tendency holds even for high values of the potential, where the statistics are poor and the KLD of sequences of symbols usually fail when estimating irreversibility Parrondo2 . However the values of the KLD that we find are far below the KLD per step between the forward and backward trajectories, which is equal to the dissipation as reported in Parrondo2 . The degree distributions capture the irreversibility of the original series but it is difficult to establish a quantitative relationship between (3) and the KLD between trajectories.
On the other hand, the measure based on the degreedegree distribution takes into account more information of the visibility graph structure than the KLD using degree distributions, providing a closer bound to the physical dissipation as it is expected by the chain rule cover , . The improvement is significant in some situations. Consider for instance the flashing ratchet with a force opposite to the net current on the system Parrondo2 . The current vanishes for a given value of the force usually termed as stalling force. When the force reaches this value, the system is still out of equilibrium () and it is therefore time irreversible, but no current of particles is observed if we describe the dynamics of the ratchet with partial information given by the position .
In Fig. 6 we show how tends to zero when the force approaches to the stalling value. Therefore, our measure of irreversibility (3) fails in this case, as do other KLD estimators based on local flows or currents Parrondo2 . However, captures the irreversibility of the time series, and yields a positive value at the stalling force.
iv.2 Chaotic series
We have applied our method to several chaotic series and found that it is able to distinguish between dissipative and conservative chaotic systems. Dissipative chaotic systems are those that do not preserve the volume of the phase space, and they produce irreversible time series. This is the case of chaotic maps in which entropy production via instabilities in the forward time direction is quantitatively different to the amount of past information lost. In other words, those whose positive Lyapunov exponents, which characterize chaos in the forward process, differ in magnitude with negative ones, which characterize chaos in the backward process kennel . In this section we analyze several chaotic maps and estimate the degree of reversibility of their associated time series using our measure, showing that for dissipative chaotic series it is positive while it vanishes for an example of conservative chaos.
iv.2.1 The Logistic map at is irreversible: analytical derivations
For illustrative purposes, in figure 7 we have plotted the in and out degree distributions of the DHVg associated to a paradigmatic dissipative chaotic system: the Logistic map at . There is a clear distinction between both distributions, as it is quantified by the KLD, which in this case is positive both for degree and degreedegree cases (see table 1). Furthermore, in figure 8 we make a finite size analysis in this particular case, showing that our measure quickly converges to an asymptotic value which clearly deviates from zero, at odds with reversible processes.
Recall that in section III we proved analytically that for a random uncorrelated process , since . Proving a similar result for a generic irreversible process is a major challenge, since finding out exact results for the entire degree distributions is in general difficult toral . However, note that the KLD between two distributions is zero if and only if the distributions are the same in the entire support. Therefore, if we want to prove that this measure is strictly positive, it is sufficient to find that for some value of the degree . Here we take advantage of this fact to provide a rather general recipe to prove that a chaotic system is irreversible.
Consider a time series with a joint probability distribution and support , and denote three (ordered) generic data of the series. By construction,
(8) 
The probability that () is actually the probability that the series increases (decreases) in one step. This probability is independent of time, because we consider stationary series. If the chaotic map is of the form , it is Markovian, and the preceding equations simplify:
(9) 
For chaotic dynamical systems whose trajectories are in the attractor, there exists an invariant probability measure that characterizes the longterm fraction of time spent by the system in the various regions of the attractor. In the case of the Logistic map
(10) 
with parameter , the attractor is the whole interval and the probability measure corresponds to
(11) 
Now, for a deterministic system, the transition probability is simply
(12) 
where is the Dirac delta distribution. Equations (9) for the Logistic map with and reads
(13) 
Notice that, using the properties of the Dirac delta distribution, is equal to one iff , what happens iff , and it is zero otherwise. Therefore the only effect of this integral is to restrict the integration range of to be . Equation (13) reduces to
(14) 
and . We conclude that for the Logistic map and hence the KLD measure based on degree distributions is positive. Recall that is the probability that the series exhibits a positive jump () once in the attractor. These positive jumps must be smaller in size than the negative jumps because, once in the attractor, is constant. The irreversibility captured by the difference between and is then the asymmetry of the probability distribution of the slope of the original time series. The KLD of the degree distributions given by (3) clearly goes beyond this simple signature of irreversibility and can capture more complex and longrange traits.
Series description  

Reversible Stochastic Processes  
uncorrelated  
OrnsteinUhlenbeck ()  
Longrange correlated  
stationary process ()  
Dissipative Chaos  
Logistic map ()  0.377  2.978 
map ()  0.455  3.005 
map ()  0.522  3.518 
Henon map ()  0.178  1.707 
Lozi map  0.114  1.265 
Kaplan Yorke map  0.164  0.390 
Conservative Chaos  
Arnold Cat map 
iv.2.2 Other chaotic maps
For completeness, we consider other examples of dissipative chaotic systems analyzed in sprott2 :

The map: , which reduces to the Logistic and tent maps in their fully chaotic region for and respectively. We analyze this map for .

The 2D Hénon map: , , in the fully chaotic region (, ).

The Lozi map: a piecewiselinear variant of the Hénon map given by in the chaotic regime ( and ).

The KaplanYorke map: .
We generate stationary time series with these maps and take data once the system is in the corresponding attractor. In table 1 we show the value of the KLD between the in and out degree and degreedegree distributions for these series. In every case, we find an asymptotic positive value, in agreement with the conjecture that dissipative chaos is indeed time irreversible.
Finally, we also consider the Arnold cat map: . At odds with previous dissipative maps, this is an example of a conservative (measurepreserving) chaotic system with integer KaplanYorke dimension sprott2 . The map has two Lyapunov exponents which coincide in magnitude and . This implies that the amount of information created in the forward process () is equal to the amount of information created in the backwards process (), therefore the process is time reversible. In figure 9 we show that for a time series of this map asymptotically tends to zero with series size, and the same happens with the degreedegree distributions (see table 1). This correctly suggests that albeit chaotic, the map is statistically time reversible.
iv.3 Irreversible chaotic series polluted with noise
Standard time series analysis methods evidence problems when noise is present in chaotic series. Even a small amount of noise can destroy the fractal structure of a chaotic attractor and mislead the calculation of chaos indicators such as the correlation dimension or the Lyapunov exponents noise . In order to check if our method is robust, we add an amount of white noise (measurement noise) to a signal extracted from a fully chaotic Logistic map (). In figure 10 we plot of its associated visibility graph as a function of the noise amplitude (the value corresponding to a pure random signal is also plotted for comparison). The KLD of the signal polluted with noise is significantly greater than zero, as it exceeds the one associated to the noise in four orders of magnitude, even when the noise reaches the of the signal amplitude. Therefore our method correctly predicts that the signal is irreversible even when adding noise.
V Discussion
In this paper we have introduced a new method to measure time irreversibility of real valued stationary stochastic time series. The algorithm proceeds by mapping the series into an alternative representation, the directed horizontal visibility graph. We have shown that the KullbackLeibler divergence (KLD) between the and degree distributions calculated on this graph is a measure of the irreversibility of the series.
We have shown that the difference between the and distributions at measures the asymmetry in the PDF of the slope of the series. The degree, however, contains information of longrange correlations and structures in the series. In particular, the degreedegree distribution can detect the irreversibility in a series with a symmetric slope, as we have shown for the flashing ratchet at stall force.
Our technique discriminates between conservative and dissipative chaotic maps. The method has been validated by studying both reversible (uncorrelated and linearly correlated stochastic processes as well as conservative chaotic maps) and irreversible (outofequilibrium physical processes and dissipative chaotic maps) series.
We have also shown that the method is robust against noise, in the sense that irreversible signals are well characterized even when these signals are polluted with a significant amount of (reversible) noise. It is also worth emphasizing that it lacks a symbolization process, and hence it can be applied directly to any kind of realvalued time series. This makes our technique of potential interest for several communities. This includes for instance biological sciences, where there is not such a simple tool to discriminate between time series generated by active (irreversible) and passive (reversible) processes.
Acknowledgments We acknowledge financial support from grants MODELICO, Comunidad de Madrid; FIS200913690 (LL, AN and BL) and MOSIACO (ER and JMRP), Ministerio de Educación.
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