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Time-reversibility of linear stochastic processes
Abstract
Time-reversibility is defined for a process X(t) as the property that {X(t1), …, X(tn
)} and {X(– t1), …, X(– tn
)} have the same joint probability distribution. It is shown that, for discrete mixed autoregressive moving-average processes, this is a unique property of Gaussian processes.
... From a time series viewpoint, it is important to characterize the properties underlying stochastic processes associated with the time-reversal symmetry and its breakdown [17][18][19] (see [20] for a review). It has been shown that reversibility implies stationarity [18] and that linear Gaussian random processes and static non-linear transformations of such processes are reversible, whereas time irreversibility implies non-linear dynamics, linear or non-linear non-Gaussian processes as possible generative processes [17][18][19]. ...
... From a time series viewpoint, it is important to characterize the properties underlying stochastic processes associated with the time-reversal symmetry and its breakdown [17][18][19] (see [20] for a review). It has been shown that reversibility implies stationarity [18] and that linear Gaussian random processes and static non-linear transformations of such processes are reversible, whereas time irreversibility implies non-linear dynamics, linear or non-linear non-Gaussian processes as possible generative processes [17][18][19]. Indeed, memory breaks time-reversal symmetry, acting as a dissipative force, whereas noise results in a loss of irreversibility [21]. ...
... • Pomeau. One of the first tests proposed to detect irreversibility in a time series (see also [17,61]) was introduced by Yves Pomeau in 1982 [1]. It is based on calculating a time-asymmetric function on the data, defined as follows: ...
Many physical and biological phenomena are characterized by time asymmetry, and are referred to as irreversible. Time-reversal symmetry breaking is in fact the hallmark of systems operating away from equilibrium and reflects the power dissipated by driving the system away from it. Time asymmetry may manifest in a wide range of time scales; quantifying irreversibility in such systems thus requires methods capable of detecting time asymmetry in a multiscale fashion. In this contribution we review the main algorithmic solutions that have been proposed to detect time irreversibility, and evaluate their performance and limitations when used in a multiscale context using several well-known synthetic dynamical systems. While a few of them have a general applicability, most tests yield conflicting results on the same data, stressing that a “one size fits all” solution is still to be achieved. We conclude presenting some guidelines for the interested practitioner, as well as general considerations on the meaning of multiscale time irreversibility.
... Time reversibility describes the property whereby a process is invariant under reverse time scale, whereas amplitude reversibility describes the invariant probabilistic properties of a process with respect to amplitude reversal [1]. The irreversibility metrics, namely time irreversibility (TIR) and amplitude irreversibility (AIR), are fundamental properties of nonequilibrium systems [2,3,4,5], and serve as reliable parameters for the determination of nonlinearity, a necessary condition for chaotic behavior. If the fluctuations in real-world systems are not symmetric, the related time series is intuitively irreversible. ...
... A process is defined as time-reversible if it is invariant under the reversal of time scale [2,9,41] and as amplitude-reversible if it is invariant with respect to amplitude reversal [1]. Let us first introduce their statistical definitions. ...
... Statistically, if a process and its time-reversal form have the same joint probability distributions, it is timereversible; otherwise, it is time-irreversible. Weiss [2] defined a stationary process X(t) as time-reversible if {X(t 1 ), X(t 2 ), . . . , X(t m )} and {X(−t 1 ), X(−t 2 ), . . . ...
Although time irreversibility (TIR) and amplitude irreversibility (AIR) are relevant concepts for nonequilibrium analysis, their association has received little attention. This paper conducts a systematic comparative analysis of the relationship between TIR and AIR based on statistical descriptions and numerical simulations. To simplify the quantification of TIR and AIR, the amplitude permutation and global information of the associated vector are combined to produce a joint probability estimation. Chaotic logistic, Henon, and Lorenz series are generated to evaluate TIR and AIR according to surrogate theory, and the distributions of joint permutations for these model series are measured to compare the degree of TIR and AIR. The joint permutation TIR and AIR are then used to investigate nonequilibrium pathological features in epileptic electroencephalography data. Test results suggest that epileptic brain electrical activities, particular those during the onset of seizure, manifest higher nonequilibrium characteristics. According to the statistical definitions and targeted pairs of joint permutations in the chaotic model data, TIR and AIR are fundamentally different nonequilibrium descriptors from time- and amplitude-reversibility, respectively, and thus require different forms of numerical analysis. At the same time, TIR and AIR both provide measures for fluctuation theorems associated with nonequilibrium processes, and have similar probabilistic differences in the pairs of joint permutations and consistent outcomes when used to analyze both the model series and real-world signals. Overall, comparative analysis of TIR and AIR contributes to our understanding of nonequilibrium features and broadens the scope of quantitative nonequilibrium measures. Additionally, the construction of joint permutations contributes to the development of symbolic time series analysis.
... Time reversibility describes the property whereby a process is invariant under reverse time scale, whereas amplitude reversibility describes the invariant probabilistic properties of a process with respect to amplitude reversal [1]. The irreversibility metrics, namely time irreversibility (TIR) and amplitude irreversibility (AIR), are fundamental properties of nonequilibrium systems [2][3][4][5], and serve as reliable parameters for the determination of nonlinearity, a necessary condition for chaotic behavior. If the fluctuations in real-world systems are not symmetric, the related time series is intuitively irreversible. ...
... A process is defined as time-reversible if it is invariant under the reversal of time scale [2,9,41] and as amplitudereversible if it is invariant with respect to amplitude reversal [1]. Let us first introduce their statistical definitions. ...
... Statistically, if a process and its time-reversal form have the same joint probability distributions, it is time-reversible; otherwise, it is time-irreversible. Weiss [2] defined a stationary process X (t) as time-reversible if {X(t 1 ), X (t 2 ), . . . , X (t m )} and {X(−t 1 ), X (−t 2 ), . . . ...
Although both time irreversibility (TIR) and amplitude irreversibility (AIR) are relevant concepts for nonequilibrium analysis, their association has received little attention. This paper describes a systematic comparative analysis of the relationship between TIR and AIR based on a statistical description and numerical simulations. To simplify the quantification of TIR and AIR, the amplitude permutation and global information of the associated vector are combined to produce a joint probability estimation. Chaotic logistic, Henon, and Lorenz series are generated to evaluate TIR and AIR according to surrogate theory, and the distributions of the joint permutations for these model series are measured to compare the degree of TIR and AIR. The joint permutation TIR and AIR are then used to investigate nonequilibrium pathological features in epileptic electroencephalography data. The results suggest that the epileptic brain electrical activities, particular those during the onset of a seizure, manifest higher nonequilibrium characteristics. According to the statistical definitions and targeted pairs of joint permutations in the chaotic model data, TIR and AIR are fundamentally different nonequilibrium descriptors from time- and amplitude-reversibility, respectively, and thus require different forms of numerical analysis. Both TIR and AIR provide measures for fluctuation theorems associated with nonequilibrium processes, and have very similar probabilistic differences in the pairs of joint permutations and consistent outcomes when used to analyze both the model series and real-world signals. Overall, the comparative analysis of TIR and AIR contributes to our understanding of nonequilibrium features and broadens the scope of quantitative nonequilibrium measures. Additionally, the construction of joint permutations contributes to the development of symbolic time series analysis.
... However such a modelling approach cannot reproduce irreversibility. Weiss [2] showed that if the process x(t) is Gaussian, then it is time reversible. As a result, a directional process cannot be Gaussian. ...
... Therefore, other models should be used to accurately model this behaviour [3]. 2 of 11 In hydrology, it is known that the ascending part of a hydrograph is steeper than the descending one [4]. This pattern clearly reflects time's arrow and can be modelled as a stochastic property. ...
... The process is time reversible or time symmetric if its joint distribution does not change after a reflection of time about the origin [2], i.e., if for any n, t 1 ; t 2 ; . . . ; t n−1 ; t n , F(x 1 , x 1 , . . . ...
We investigate the impact of time’s arrow on the hourly streamflow process. Although time asymmetry, i.e., temporal irreversibility, has been previously implemented in stochastics, it has only recently attracted attention in the hydrological literature. Relevant studies have shown that the time asymmetry of the streamflow process is manifested at scales up to several days and vanishes at larger scales. The latter highlights the need to reproduce it in flood simulations of fine-scale resolution. To this aim, we develop an enhancement of a recently proposed simulation algorithm for irreversible processes, based on an asymmetric moving average (AMA) scheme that allows for the explicit preservation of time asymmetry at two or more time-scales. The method is successfully applied to a large hourly streamflow time series from the United States Geological Survey (USGS) database, with time asymmetry prominent at time scales up to four days.
... The definition in (2) is adapted from Weiss (1975). Assuming that the process x(t) is stationary and that times are equidistant, i.e. t i -t i-1 = D, and shifting the time by τ = 2t 1 + (n -1)D, we get F x 1 ; x 2 ; . . . ...
... differenced process has a negative value (meaning that the next value of the original process would be lower than the current; solution 2). We note that in this example, because the entire setting is nonlinear, with reference to Weiss's (1975) result mentioned in the Introduction, there is no inconsistency with the constraint that the original process be Gaussian in its marginal distribution. As seen in Figure 3, the two synthetic time series so generated are characterized by a few steep increases followed by systematic milder decreases. ...
... It can easily be verified that the SMA model always results in time symmetric processes, irrespective of the specific vector a. This property of symmetric linear filters was first observed by Weiss (1975). However, if we apply the SMA scheme to the differenced process ~ x τ :¼ x τ À x τÀ1 , then we can easily reproduce its skewness (and higher order moments; see Koutsoyiannis 2019, Dimitriadis andKoutsoyiannis 2018) and hence the characteristics of the temporal asymmetry. ...
Time’s arrow has important philosophical, scientific and technical connotations and is closely related to randomness as well as to causality. Stochastics offers a frame to explore, characterize and simulate irreversibility in natural processes. Indicators of irreversibility are different if we study a single process alone, or more processes simultaneously. In the former case, description of irreversibility requires at least third-order properties, while in the latter lagged second-order properties may suffice to reveal causal relations. Several examined data sets indicate that in atmospheric processes irreversibility is negligible at hydrologically relevant time scales, but may exist at the finest scales. However, the irreversibility of streamflow is marked for scales of several days and this highlights the need to reproduce it in flood simulations. For this reason, two methods of generating time series with irreversibility are developed, from which one, based on an asymmetric moving average scheme, proves to be satisfactory.
... Time reversibility describes the invariant statistical properties of processes under the reversal time scale [12]. A time series is said to be irreversible if its probabilistic properties depend on the time direction. ...
... Definition I. In the definition of G. Weiss [12], a stationary process X(t) is time reversible if {X(t 1 ), X(t 2 ), · · · , X(t m )} and {X(−t 1 ), X(−t 2 ), · · · , X(−t m )} have the same joint probability distributions for every t and m. ...
... As for the condition of stationarity imposed to stochastic process in the definitions, we should note that the stationarity is in fact a separate concept to the general definitions of time irreversibility. There exists nonstationary process that is time reversible [13], and there are examples of stationary processes which are not time reversible [12]. The two concepts, stationarity and time irreversibility, therefore, do not imply each others. ...
Time irreversibility (temporal asymmetry) is one of fundamental properties that characterize the nonlinearity of complex dynamical processes, and our brain is a typical complex dynamical system manifested with nonlinearity. Two subtraction-based parameters, Ys and X2, are employed to measure the probabilistic differences of permutations instead of raw vectors for the simplified quantification of time irreversibility, which is validated by chaotic and reversible processes and the surrogate data. We show that it is equivalent to quantify time irreversibility by measuring probabilistic difference between the forward and its backward processes and between the symmetric permutations. And we detect time irreversibility of two groups of epileptic EEGs, from the Nanjing General Hospital (NJGH) and from the public Bonn epileptic database. In our contribution, the manifestation of nonlinearity of whether healthy or diseased brain electrical activities is highlighted, and the highest time irreversibility of epileptic EEG during seizures is demonstrated. NJGH epileptic EEGs during seizure-free intervals of have lower time irreversibility than the control data while those of the Bonn data sets have higher nonlinearity than the healthy brain electrical activities. For the inconsistent results, we conduct multi-scale analysis and elucidate from the circadian rhythms in epileptic nonlinearity, however, more targeted researches are needed to verify our assumptions or to determine if there are other reasons leading to the inconsistency.
... only interested in exploiting the regular Fourier transform, when we are dealing with the spatial domain, or when the underlying data is a time-reversible stochastic process that is invariant under the reversal of the time scale [22]. In such cases, the data can be represented using an undirected cycle graph, see Fig. 1. ...
... Using (22), we can express the observations y 0 = Φ 0 x as ...
In this paper the focus is on subsampling as well as reconstructing the second-order statistics of signals residing on nodes of arbitrary undirected graphs. Second-order stationary graph signals may be obtained by graph filtering zero-mean white noise and they admit a well-defined power spectrum whose shape is determined by the frequency response of the graph filter. Estimating the graph power spectrum forms an important component of stationary graph signal processing and related inference tasks such as Wiener prediction or inpainting on graphs. The central result of this paper is that by sampling a significantly smaller subset of vertices and using simple least squares, we can reconstruct the second-order statistics of the graph signal from the subsampled observations, and more importantly, without any spectral priors. To this end, both a nonparametric approach as well as parametric approaches including moving average and autoregressive models for the graph power spectrum are considered. The results specialize for undirected circulant graphs in that the graph nodes leading to the best compression rates are given by the so-called minimal sparse rulers. A near-optimal greedy algorithm is developed to design the subsampling scheme for the non-parametric and the moving average models, whereas a particular subsampling scheme that allows linear estimation for the autoregressive model is proposed. Numerical experiments on synthetic as well as real datasets related to climatology and processing handwritten digits are provided to demonstrate the developed theory.
... Another relevant property in the context of financial systems is that of time reversibility, i.e. the degree of dynamical invariance under time reversal. Statistical time reversibility refers to the situation where the statistical properties of a certain process are invariant under time reversal [18], this being a more natural concept to explore in noisy series than strict reversibility, and intuitively measures our capacity to detect the correct arrow of time in erratically evolving dynamics. This purely statistical concept has been recently found to have deep links with the physics of information, as physical systems operating away from equilibrium are shown to produce entropy at a rate which is proportional to a suitable measure of time irreversibility of adequate physical observables [20,21]. ...
... , x n } and S − = {x −1 , x −2 , . . . , x −n } (where n denotes time) generated by this process have asymptotically the same joint distribution [18]. In the concrete case where the process is stationary, the time series {x −1 , x −2 , . . . ...
The relation between time series irreversibility and entropy production has been recently investigated in thermodynamic systems operating away from equilibrium. In this work we explore this concept in the context of financial time series. We make use of visibility algorithms to quantify in graph-theoretical terms time irreversibility of 35 financial indices evolving over the period 1998-2012. We show that this metric is complementary to standard measures based on volatility and exploit it to both classify periods of financial stress and to rank companies accordingly. We then validate this approach by finding that a projection in principal components space of financial years based on time irreversibility features clusters together periods of financial stress from stable periods. Relations between irreversibility, efficiency and predictability are briefly discussed.
... In contrast, there is an arrow of time in Economy B (for example, the economy never transitions from state 3 to state 2). This is a feature of our framework that is not shared by stationary Gaussian models, which are time reversible ( Weiss, 1975 ). ...
... As a result, it can be used to address certain issues that are assumed away in Gaussian models. Stationary Gaussian models are timereversible ( Weiss, 1975 ), for example, so the conclusions reached by an econometrician living in a stationary Gaussian world would be the same whether time runs forward or backward. Such models exclude the possibility that, say, interest rates "go up by the stairs and down by the elevator" (or the converse). ...
We study the properties of the yield curve under the assumptions that (i) the fixed-income market is complete and (ii) the state vector that drives interest rates follows a finite discrete-time Markov chain. We focus in particular on the relationship between the behavior of the long end of the yield curve and the recovered time discount factor and marginal utilities of a pseudo-representative agent; and on the relationship between the “trappedness” of an economy and the convergence of yields at the long end.
... Time irreversibility is an important alternative to characterize nonlinear complex system [11]. Time series is directional, or irreversible, if its probabilistic properties depend on the direction of time, otherwise it is said to be reversible [12,13]. Mathematically speaking, if X(t) is time reversible, the joint distributions of {X(t 1 ), · · · , X(t n )} and {X(−t 1 ), · · · , X(−t n )} should be same for every t and n. ...
... A stochastic process is said to be time reversible if its probabilistic properties are invariant with respect to time reversal. In Weiss G. definition, a stationary process X(t) is time reversal if {X(t 1 ), X(t 2 ), · · · , X(t n )} and {X(−t 1 ), X(−t 2 ), · · · , X(−t n )} have same joint probability distributions for every t and n, that the process is invariant under the time-scale reversal [12]. It also works for {X(t 1 ), X(t 2 ), · · · , X(t n )} and {X(t −1+m ), X(t −2+m ), · · · , X(t −n+m )}, for every n and m, and when m is n + 1, the definition implies has same joint probability distributions to its inversion [14,16]. ...
Time irreversibility, a fundamental property of complex system, can be characterized by the directional asymmetry of probabilistic distributions, which is quantified by equalities-involved permutation relative entropy in our nonlinear dynamics analysis of heart rates. Equal values are not rare in discrete heartbeats because of the limits of resolution of signals collection, and more importantly equal states contain underlying important cardiac regulation information which is neglected by the original permutative method and some measurements of time irreversibility. The relative entropy of modified permutation associated with equal states shows advantages to these time irreversibility parameters and has great improvements to that of the original permutation. The temporal irreversibility researches on cardiac activities suggest the highest probabilistic asymmetry of the healthy young heart rates and highlight the facts that heart diseases and aging broke asymmetry, or reduce the nonlinear dynamical complexity, of human heartbeats.
... 2) Time Reversibility: A stochastic process is defined as time-reversible if it is invariant under the reversal of the time scale [45]. An example of time-reversible process is a linear Gaussian random process. ...
... where S is a signal with N samples, and τ = 1 [45], [46]. According to the original study, 99 surrogate data are created by iterative amplitude-adjusted Fourier transform in order to match the same PSD of the original signal [46]. ...
Objective:
Preterm birth is a large-scale clinical problem involving over 10% infants. Diagnostic means for timely risk assessment are lacking and the underlying physiological mechanisms unclear. To improve the evaluation of pregnancy before term, we introduce dedicated entropy measures derived from a single-channel electrohysterogram (EHG).
Methods:
The estimation of Approximate Entropy (ApEn) and Sample Entropy (SampEn) is adjusted to monitor variations in the regularity of single-channel EHG recordings, reflecting myoelectrical changes due to pregnancy progression. In particular, modifications in the tolerance metrics are introduced for improving robustness to EHG amplitude fluctuations. An extensive database of 58 EHG recordings with 4 monopolar channels in women presenting with preterm contractions was manually annotated and used for validation. The methods were tested for their ability to recognize the onset of labor and the risk of preterm birth. Comparison with the best single-channel methods according to the literature was performed.
Results:
The reference methods were outperformed. SampEn and ApEn produced the best prediction of delivery, although only one channel showed a significant difference (p<0.04) between labor and non labor. The modified ApEn produced the best prediction of preterm delivery, showing statistical significance (p<0.02) in 3 channels. These results were also confirmed by the area under the receiver operating characteristic curve and 5-fold cross-validation.
Conclusion:
The use of dedicated entropy estimators improves the diagnostic value of EHG analysis earlier in pregnancy.
Significance:
Our results suggest that changes in the EHG might manifest early in pregnancy, providing relevant prognostic opportunities for pregnancy monitoring by a practical single-channel solution.
... According to these considerations, one possible indicator of the aforementioned transitory periods can be the deviation from a stationary linear-stochastic Gaussian process. One key feature of such processes is (statistical) time-reversibility, i.e. that every multi-point statistics is invariant under a change of the direction of time (Weiss, 1975). In this spirit, time intervals during which time-reversibility is temporarily lost can serve as indicators of dynamical anomalies. ...
Recent work has provided ample evidence that nonlinear methods of time series analysis potentially allow for detecting periods of anomalous dynamics in paleoclimate proxy records that are otherwise hidden to classical statis- tical analysis. Following upon these ideas, in this study we systematically test a set of Late Holocene terrestrial paleoclimate records from Northern Europe for indications of intermittent periods of time-irreversibility during which the data are incompatible with a stationary linear-stochastic process. Our analysis reveals that the onsets of both the Medieval Climate Anomaly and the Little Ice Age, the end of the Roman Warm Period and the Late Antique Little Ice Age have been characterized by such dynamical anomalies. These findings may indicate qualitative changes in the dominant regime of inter-annual climate variability in terms of large-scale atmospheric circula- tion patterns, ocean-atmosphere interactions and external forcings affecting the climate of the North Atlantic region.
... , x n } is called statistically time reversible if the time series S − = {x −1 , x −2 , . . . , x −n } has the same joint distribution as S [8]. Of course by definition, non-stationary series are (infinitely) irreversible: the statistical properties of a non-stationary process vary with time, and therefore S and S − have different statistics that increase over time without bounds. ...
Visibility algorithms are a family of methods to map time series into networks, with the aim of describing the structure of time series and their underlying dynamical properties in graph-theoretical terms. Here we explore some properties of both natural and horizontal visibility graphs associated to several non-stationary processes, and we pay particular attention to their capacity to assess time irreversibility. Non-stationary signals are (infinitely) irreversible by definition (independently of whether the process is Markovian or producing entropy at a positive rate), and thus the link between entropy production and time series irreversibility has only been explored in non-equilibrium stationary states. Here we show that the visibility formalism naturally induces a new working definition of time irreversibility, which allows to quantify several degrees of irreversibility for stationary and non-stationary series, yielding finite values that can be used to efficiently assess the presence of memory and off-equilibrium dynamics in non-stationary processes without needs to differentiate or detrend them. We provide rigorous results complemented by extensive numerical simulations on several classes of stochastic processes.
... This observation has prompted researchers to investigate time-reversibility more closely. Weiss (1975) showed that (i) a stationary Gaussian process is always time-reversible; (ii) for stationary ARMA(p,q) with p ≥ 1, time-reversibility and Gaussianity are equivalent; (iii) the same equivalence holds for MA(q) processes with non-symmetric or non-antisymmetric coefficients (a MA(q) process of the form X t = q i=0 b i ε t−i with i.i.d. ε t 's has symmetric or antisymmetric coefficients if b i = ±b q−i , i = 0, . . . ...
Time-reversibility is a crucial feature of many time series models, while time-irreversibility is the rule rather than the exception in real-life data. Testing the null hypothesis of time-reversibilty, therefore, should be an important step preliminary to the identification and estimation of most traditional time-series models. Existing procedures, however, mostly consist of testing necessary but not sufficient conditions, leading to under-rejection, or sufficient but non-necessary ones, which leads to over-rejection. Moreover, they generally are model-besed. In contrast, the copula spectrum studied by Goto et al. ( 2022, : 3563--3591) allows for a model-free necessary and sufficient time-reversibility condition. A test based on this copula-spectrum-based characterization has been proposed by authors. This paper illustrates the performance of this test, with an illustration in the analysis of climatic data.
... It has been said that a stationary process X (t) is statistically time reversible if for every n the series {x 1 , ..., x n } and {x n , ..., x 1 } have the same joint probability distributions [25,45]. Examples of reversible processes include white noise, linear Gaussian processes, and near-equilibrium thermodynamic systems. ...
El Niño-Southern Oscillation (ENSO) is a complex climate phenomenon that results from ocean–atmosphere interactions and exhibits nonlinearity in both spatial and temporal evolution. The directed horizontal visibility graph (DHVG) and Kullback–Leibler divergence (KLD) efficiently characterize the nonlinearity of complex systems without requiring additional symbolization processes. Our study utilizes this method to quantify sea surface temperature (SST) irreversibility across multiple El Niño events and identified similar fluctuation patterns. Subsequently, we map the irreversibility series into visibility graphs and analyze multiple topological properties to investigate the fluctuation structure of SST irreversibility. The irreversibility fluctuation structure effectively explains the periodic changes of El Niño and its nonlinear evolution over time. Results show that KLD exhibits a sharp increase during the strengthening and ending phases of El Niño events and remains in a lower value after the peak month of Niño events; the visibility graph of SST irreversibility exhibits scale-free and small-world properties, indicating that the KLD series has scale invariance and self-similarity, and its fluctuations are not random but correlated; Hurst exponent analysis revealed long-range anti-persistence and mean reversion characteristics in the KLD series. This study utilizes complex network and information theory to investigate the fluctuation pattern and structure of SST irreversibility during El Niño events, providing novel insights into the evolution of ENSO over time scales.
... From a statistical viewpoint, time-reversal symmetry quantifies the extent to which it is possible to discern a preferred time direction in the realization of some stationary stochastic process (48). Observed phenomena are thought of as realizations of a stochastic dynamical process, and the goal is to try to extract information on the statistical properties of these processes from the time-reversal symmetry and its breakdown (49)(50)(51). For instance, linear Gaussian random processes and static non-linear transformations of such processes are time-reversible, so that time irreversibility implies ruling out Gaussian linear models and their static nonlinear transformations as possible generative models. ...
We study how obsessive-compulsive disorder (OCD) affects the complexity and time-reversal symmetry-breaking (irreversibility) of the brain resting-state activity as measured by magnetoencephalography (MEG). Comparing MEG recordings from OCD patients and age/sex matched control subjects, we find that irreversibility is more concentrated at faster time scales and more uniformly distributed across different channels of the same hemisphere in OCD patients than in control subjects. Furthermore, the interhemispheric asymmetry between homologous areas of OCD patients and controls is also markedly different. Some of these differences were reduced by 1-year of Kundalini Yoga meditation treatment. Taken together, these results suggest that OCD alters the dynamic attractor of the brain's resting state and hint at a possible novel neurophysiological characterization of this psychiatric disorder and how this therapy can possibly modulate brain function.
... From a statistical viewpoint, time-reversal symmetry quantifies the extent to which it is possible to discern a preferred time direction in the realization of some stationary stochastic process (48). Observed phenomena are thought of as realizations of a stochastic dynamical process, and the goal is to try to extract information on the statistical properties of these processes from the time-reversal symmetry and its breakdown (49)(50)(51). For instance, linear Gaussian random processes and static non-linear transformations of such processes are time-reversible, so that time irreversibility implies ruling out Gaussian linear models and their static nonlinear transformations as possible generative models. ...
We study how obsessive-compulsive disorder (OCD) affects the complexity and time-reversal symmetry-breaking (irreversibility) of the brain resting-state activity as measured by magnetoencephalography (MEG). Comparing MEG recordings from OCD patients and age/sex matched control subjects, we find that irreversibility is more concentrated at faster time scales and more uniformly distributed across different channels of the same hemisphere in OCD patients than in control subjects. Furthermore, the interhemispheric asymmetry between homologous areas of OCD patients and controls is also markedly different. Some of these differences were reduced by 1-year of Kundalini Yoga meditation treatment. Taken together, these results suggest that OCD alters the dynamic attractor of the brain's resting state and hint at a possible novel neurophysiological characterization of this psychiatric disorder and how this therapy can possibly modulate brain function.
... Then, we can conclude that the underlying dynamics is out of equilibrium without performing any further analysis. Such a remark is especially relevant in the recent debate on the possibility to deduce the non-equilibrium character of a system from partial observation [8], in particular recalling that the time-series of a scalar Gaussian process (in our case ω(t)) is always symmetric under time-reversal [40,41]. ...
We derive a Thermodynamic Uncertainty Relation bounding the mean squared displacement of a Gaussian process with memory, driven out of equilibrium by unbalanced thermal baths and/or by external forces. Our bound is tighter with respect to previous results and also holds at finite time. We apply our findings to experimental and numerical data for a many-body interacting granular fluid, characterized by regimes of anomalous diffusion. In some cases, our relation can distinguish between equilibrium and non-equilibrium behavior, a non-trivial inference task, particularly for Gaussian processes.
... Now pour attention to another sequence, S 1 ¼ fp Zþ1 ; Á Á Á ; p i ; p iþ1 ; Á Á Á ; p N g. The analysis is conducted by utilizing one of the special features of the Markov chain: time reversibility [3,5,17,20]. ...
Stochastic Point Location problem considering that a learning entity (i.e. mechanisms, algorithm, etc) attempts to locate a certain point by interaction with a stochastic environment is encountered widely in Machine Learning. A conventional technique is to sample the search space into discrete points and perform a random walk. Nevertheless, the random walk is confined to the neighboring point. In this paper, an extended version of the random walk-based triple level algorithm is introduced to overcome the aforementioned defect. Specifically, the proposed algorithm exploits the multi-Markovian switching to generalize the random walk concerning adjacent nodes to intermittent nodes. Hence, the whole approach could be regarded as the Markov chain, and its transform matrix could be constructed, followed by a rigorous mathematical pf procedure of the convergence. The experimental results demonstrate the effectiveness and efficiency of the proposed algorithm, showing its abilities of stronger stability, a higher precision, and a faster speed in comparison with the counterparts available in open literatures.
... Based on this argument, it seems that some form of data transformation is often required before classical ARMA modeling (Box and Cox (1964)). However, sometimes simple monotonic transformations do not correct asymmetry (Weiss (1975)). Thus, new approaches have been developed to model non-normal time series with ARMAtype dependency structure. ...
... A time reversible process is also stationary (Lawrance, 1991). If a process x(t) is Gaussian (i.e., all its finite dimensional distributions are multivariate normal) then it is reversible (Weiss, 1975 ...
Monte Carlo control methods, in which a control action, possibly expressed in terms of a parametric relationship, is tested by stochastic simulation and subsequently optimized by a global optimization procedure, are promising for complex hydrosystems with nonlinear dynamics. In particular, they have proved powerful for the management of large reservoir systems, where simulation and optimization are performed on a time scale of the order of a month. A basic requirement of these methods is a proper technique for stochastic generation of hydrological inputs, respecting characteristic behaviours of hydrological processes, such as seasonality, intermittence, long term persistence and roughness (fractality). However, most control problems in hydrosystems require time scales of study much finer than monthly, e.g., hourly or even finer. Examples are the control of spillway gates and of hydropower turbines. On fine time scales, another behaviour of hydrological processes becomes important and necessary to reproduce: time irreversibility. As common stochastic techniques produce time series whose properties are symmetric in time, a new stochastic simulation method is presented, which is capable of generating sequences with the required properties related to time irreversibility. The method is generic as it can reproduce any marginal distribution, covariance structure and irreversibility index, and can work both in simulation and forecast mode.
... On the one hand, reversibility implies stationarity [29]. On the other hand, linear Gaussian random processes and static nonlinear transformations of such processes are reversible, and significant time irreversibility excludes Gaussian linear processes or linear ARMA models as possible generating dynamics, implying instead nonlinear dynamics, (linear or nonlinear) non-Gaussian, [29][30][31][32]. The asymmetry under time reversal of some system variable's statistical properties provides a quantitative estimate of the thermodynamic entropy production Σ t of the system generating the activity, even when the details of the system are unknown [33][34][35]. ...
Characterising brain activity at rest is of paramount importance to our understanding both of general principles of brain functioning and of the way brain dynamics is affected in the presence of neurological or psychiatric pathologies. We measured the time-reversal symmetry of spontaneous electroencephalographic brain activity recorded from three groups of patients and their respective control group under two experimental conditions (eyes open and closed). We evaluated differences in time irreversibility in terms of possible underlying physical generating mechanisms. The results showed that resting brain activity is generically time-irreversible at sufficiently long time scales, and that brain pathology is generally associated with a reduction in time-asymmetry, albeit with pathology-specific patterns. The significance of these results and their possible dynamical aetiology are discussed. Some implications of the differential modulation of time asymmetry by pathology and experimental condition are examined.
... , x n } and {x n , . . . , x 1 } have the same joint probability distributions (i.e. for a given s > 0, P(X (t), X (t +s)) = P(X (t +s), X (t))), then a stationary process X (t) is defined as time reversible [1] and it is invariant under the reversal arrow of time. Gaussian linear processes are taken to be this kind of reversible processes. ...
As a fundamental property of nonlinear time series, temporal irreversibility (TI)is a hot topic in various fields. Among these studies, many methods and measures have been developed. However, only a few studies contribute to the performance comparison of these methods and measures. Taking the daily mean air temperature from observations over China as an example, the performance of five well developed measures was compared in this study. Three of them are directly related to air temperature increments with A defined from sign ratio, G from magnitude ratio and E from skewness, and the other two measures are L 2 from directed horizontal visibility graph (DHVG for short)and L 1 from consecutive increasing and decreasing steps (CIDS for short). The results show that five measures reach almost consistent conclusions to TI of daily mean air temperature from observations over China, but with different statistical significance and sensitivity to low-frequency trend and measured noise. They also show that the higher TI occurs over southern China, and lower TI over northeast China and southwest China close to Basin of Sichuan and Yungui-Tibetan Plateau. Further studies indicate that this dominated TI behavior is closely related to the number difference between extreme decreasing temperature variations and extreme increasing temperature variations. When conditional threshold exceeds certain critical values, this marked TI can be well quantified by the extreme decreasing but not increasing temperature variations. At last, combining all aspects considered, measure from CIDS is the best among these five TI measures due to its robustness to low-frequency trend and measured noise.
... • Two reversible stochastic processes, namely a time series of values drawn from a Gaussian distribution N (0, 1), and an Ornstein-Uhlenbeck process, a mean-reverting linear Gaussian process T [41]. • Two dissipative chaotic maps, respectively, a logistic map (defined as x n+1 = ax n (1 − x n ), with a = 4.0) and a Henon map (x n+1 = 1 + y n − ax 2 t , y n+1 = bx t , with a = 1.4 and b = 0.3). ...
Time irreversibility, i.e., the lack of invariance of the statistical properties of a system under time reversal, is a fundamental property of all systems operating out of equilibrium. Time reversal symmetry is associated with important statistical and physical properties and is related to the predictability of the system generating the time series. Over the past fifteen years, various methods to quantify time irreversibility in time series have been proposed, but these can be computationally expensive. Here, we propose a new method, based on permutation entropy, which is essentially parameter-free, temporally local, yields straightforward statistical tests, and has fast convergence properties. We apply this method to the study of financial time series, showing that stocks and indices present a rich irreversibility dynamics. We illustrate the comparative methodological advantages of our method with respect to a recently proposed method based on visibility graphs, and discuss the implications of our results for financial data analysis and interpretation.
... Considering space transformations, observe that a path X from a stationary Gaussian autoregressive process is always statistically symmetric with respect to parity and reversion transformations, so this process is always time reversible for even or odd variables [27,28]; time irreversibility arises when non-stationarity is present. ...
We use the definition of statistical symmetry as the invariance of a probability distribution under a given transformation and apply the concept to the underlying probability distribution of stochastic processes. To measure the degree of statistical asymmetry, we take the Kullback–Leibler divergence of a given probability distribution with respect to the corresponding transformed one and study it for the Gaussian autoregressive process using transformations on the temporal correlations’ structure. We then illustrate the employment of this notion as a time series analysis tool by measuring local statistical asymmetries of foreign exchange market price data for three transformations that capture distinct autocorrelation behaviors of the series—independence, non-negative correlations and Markovianity—obtaining a characterization of price movements in terms of each statistical symmetry.
... , x n } and X ′ = {x n , x n−1 , . . . , x 1 } have completely the same joint probability distribution [20]. That is to say, reversible time series occur equally and the join distribution p(X ) and p(X ′ ) are coincident. ...
We propose a method to improve the measure of real-valued time series irreversibility which contains two tools: the directed horizontal visibility graph and the Kullback–Leibler divergence. The degree of time irreversibility is estimated by the Kullback–Leibler divergence between the in and out degree distributions presented in the associated visibility graph. In our work, we reframe the in and out degree distributions by encoding them with different embedded dimensions used in calculating permutation entropy(PE). With this improved method, we can not only estimate time series irreversibility efficiently, but also detect time series irreversibility from multiple dimensions. We verify the validity of our method and then estimate the amount of time irreversibility of series generated by chaotic maps as well as global stock markets over the period 2005–2015. The result shows that the amount of time irreversibility reaches the peak with embedded dimension d=3 under circumstances of experiment and financial markets.
... A time series is said to be reversible if its probabilistic properties are invariant with respect to time reversal (Diks et al., 1995). For example, a simple Gaussian random walk is timereversal invariant (Weiss, 1975). Practical implementations of temporal asymmetry measures use, for example, the difference between the probability density functions of the original and time-reversed series, or of their corresponding variances (Zumbach, 2009(Zumbach, , 2012, or temporal-based correlation measures over different temporal windows, or between the past and future data blocks of the same temporal length (Zamparo et al., 2013), or by using Granger causality (Winkler et al., 2016). ...
Optogenetically evoked local field potential (LFP) recorded from the medial prefrontal cortex (mPFC) of mice during basal conditions and following a systemic cocaine administration were analyzed. Blue light stimuli were delivered to mPFC through a fiber optic every 2 s and each trial was repeated 100 times. As in the previous study, we used a surrogate data method to check that nonlinearity was present in the experimental LFPs and only used the last 1.5 s of steady activity to measure the LFPs phase resetting induced by the brief 10 ms light stimulus. We found that the steady dynamics of the mPFC in response to light stimuli could be reconstructed in a three-dimensional phase space with topologically similar “8”-shaped attractors across different animals. Therefore, cocaine did not change the complexity of the recorded nonlinear data compared to the control case. The phase space of the reconstructed attractor is determined by the LFP time series and its temporally shifted versions by a multiple of some lag time. We also compared the change in the attractor shape between cocaine-injected and control using (1) dendrogram clustering and (2) Frechet distance. We found about 20% overlap between control and cocaine trials when classified using dendrogram method, which suggest that it may be possible to describe mathematically both data sets with the same model and slightly different model parameters. We also found that the lag times are about three times shorter for cocaine trials compared to control. As a result, although the phase space trajectories for control and cocaine may look similar, their dynamics is significantly different.
... .; XðÀt n Þg have the same joint probability distribution for every n, and every t 1 ; . . .; t n , then the process is reversible (symmetry in time) (Weiss, 1975). A stationary random process without this property is not reversible and is said to be directional. ...
We provide empirical evidence of directionality in high-frequency multivariate time series of the five largest U.S. banks between 1999 and 2017. The directionality is more apparent during crisis periods than during noncrisis periods, and it has only a low association with volatility. We use directionality and volatility as a regime-switching criterion between two-regime threshold vector autoregressive (TVAR) models for forecasting share prices. We compare the forecasting performances using mean relative error squared, and a weighted average of the forecasting error, with weights based on the estimated conditional variance, for individual model components and as a group. We have demonstrated that moving directionality can provide early warning of increased volatility and crisis periods, and has potential for improving one-step ahead forecasts using TVAR(1) models.
... Statistically speaking, if {X(t 1 ), X(t 2 ), · · · , X(t n )} have same joint probability distributions to {X(−t 1 ), X(−t 2 ), · · · , X(−t n )} for every t and n and to {X(t −1+m ), X(t −2+m ), · · · , X(t −n+m )}, for every n and m, the process is time reversible [3][4][5]. Moreover, if X = {x 1 , x 2 , · · · , x t } is time reversible or directional symmetry, the probability distributions of the X τ m = {x t , x t+τ , · · · , x t+(m−1)τ } for all m and τ are symmetrical [6], which allow us to measure time irreversibility or directionality from multi-dimensional and higher order differential time series. ...
Permutation relative entropy is proposed to quantify time irreversibility in our nonlinear dynamics analysis of electroencephalogram (EEG). Ordinal patterns in multi-dimension phase space of time series are symbolized, and the probabilistic divergences of all symmetric ordinal pairs are measured by relative entropy as time irreversibility. Analyzing multi-channel EEG from 18 healthy volunteers and 18 epileptic patients (all in their seize-free intervals), the derived relative entropy of symmetrical ordinal patterns shows advantages to other time irreversible parameters and significantly distinguish two kinds of brain electric signals (the epileptic have lower temporal asymmetries than the healthy). Test results prove that it is effective to quantity time irreversibility by measuring probabilistic divergence of symmetrical ordinal patterns and validate our hypothesis that epilepsy has lasting impacts on brain' nonlinear dynamics, leading to a decline in brain signals directional asymmetry or time irreversibility, and the losing temporal asymmetries stemming from our findings may contribute to the preclinical diagnosis of epilepsy.
... , x n } is called statistically time-reversible if S − = {x (−1) , x (−2) , . . . , x (−n) } has the same joint distribution with S [48]. This merely im- plies that both S and S − series are statistically equally probable, and consequently, any independent and identical distributed random variable is a reversible sequence. ...
The behavior of stock prices has been thoroughly studied throughout the last century, and contradictory results have been reported in the corresponding literature. In this paper, a network theoretical approach is provided to investigate how crises affected the behavior of US stock prices. We analyze high frequency data from S&P500 via the Horizontal Visibility Graph method, and find that all major crises that took place worldwide in the last twenty years, affected significantly the behavior of the price-index. Nevertheless, we observe that each of those crises impacted the index in a different way and magnitude. Interestingly, our results suggest that the predictability of the price-index series increases during the periods of crises.
... Both of these behaviors, periodicity and uncorrelated randomness, represent loss of multiscale information, which is interpreted as reflecting a breakdown of physiologic control [1]. In 1975, the concept of "time irreversibility" was introduced [2] to refer to the loss of consistency in a time series if it is read backwards in time. The capacity of living beings for self-organization is related to the single direction of energy flows [3] and the loss of this capacity is related to age and ageing [4]. ...
Purpose
To analyze the complexity of the gait through the measurement of Multiscale Time Irreversibility in accelerometry signals obtained at the hip of healthy subjects and patients with intermittent claudication in order to differentiate the two situations.
Methods
Ten healthy elderly subjects (age 60.2 ± 4.8 years; height 173.6 ± 6.6 cm; weight 88.9 ± 11.3 kg); and 12 patients with peripheral arterial disease (age 63.1 ± 5.4 years, height 168.6 ± 6.5 cm; weight 81.2 ± 15.9 kg) walked at a comfortable, freely-chosen pace for 10 min in an open circuit wearing a triaxial accelerometer on each hip. The Asymmetry Index was calculated (scales 1–20) from the accelerometry series on the axes X, Y and Z for each hip using the simplified algorithm proposed by M. Costa.
Results
A lower asymmetry can be seen in the group of patients on the Y axis of both legs with respect to the control group (p = 0.001). Comparing one leg with the other, only the patients showed a difference on the Z axis (p = 0.04) with less asymmetry in the claudicant leg (0.29 ± 0.15 vs. 0.55 ± 0.49).
Conclusions
The analysis of Multiscale Time Irreversibility through the Asymmetry Index is useful for the study of human gait and can reveal behavior that allows pathological situations to be distinguished.
Keywords
Human gait
Time irreversibility
Complexity
Nonlineal dynamics
Multiscale
... A time series S = {x 1 , x 2 ,. .. , x n } is called statistically time-reversible if S − = {x (−1) , x (−2) ,. .. , x (−n) } has the same joint distribution with S[48]. This merely implies that both S and S − series are statistically equally probable, and consequently, 125 any independent and identical distributed random variable is a reversible sequence. ...
... Asymmetric IIS shape has been associated with epilepsy pathology 43 . Temporal asymmetry implies underlying nonlinear generators 44 , which in turn have been shown to characterize epileptogenic brain areas 45,46 . One test of whether SRC overall favors pro-or anti-epileptogenic mechanisms would involve experiments that interrupt or enhance post-seizure sleep. ...
Objective:
Local field potentials (LFPs) arise from synchronous activation of millions of neurons, producing seemingly consistent waveform shapes and relative synchrony across electrodes. Interictal spikes (IISs) are LFPs associated with epilepsy that are commonly used to guide surgical resection. Recently, changes in neuronal firing patterns observed in the minutes preceding seizure onset were found to be reactivated during postseizure sleep, a process called seizure-related consolidation (SRC), due to similarities with learning-related consolidation. Because IISs arise from summed neural activity, we hypothesized that changes in IIS shape and relative synchrony would be observed in the minutes preceding seizure onset and would be reactivated preferentially during postseizure slow-wave sleep (SWS).
Methods:
Scalp and intracranial recordings were obtained continuously across multiple days from clinical macroelectrodes implanted in patients undergoing treatment for intractable epilepsy. Data from scalp electrodes were used to stage sleep. Data from intracranial electrodes were used to detect IISs using a previously established algorithm. Partial correlations were computed for sleep and wake periods before and after seizures as a function of correlations observed in the minutes preceding seizures. Magnetic resonance imaging (MRI) and computed tomography (CT) scans were co-registered with electroencephalography (EEG) to determine the location of the seizure-onset zone (SOZ).
Results:
Changes in IIS shape and relative synchrony were observed on a subset of macroelectrodes minutes before seizure onset, and these changes were reactivated preferentially during postseizure SWS. Changes in synchrony were greatest for pairs of electrodes where at least one electrode was located in the SOZ.
Significance:
These data suggest preseizure changes in neural activity and their subsequent reactivation occur across a broad spatiotemporal scale: from single neurons to LFPs, both within and outside the SOZ. The preferential reactivation of seizure-related changes in IISs during postseizure SWS adds to a growing body of literature suggesting that pathologic neural processes may utilize physiologic mechanisms of synaptic plasticity.
... This distance is based on a stationary ergodic process, which relies on the following three (3) considerations: 1. Any reversible process is stationary, and any time reversal of a reversible process is also sta- tionary [13][14][15]. 2. If {X n ,n 2 R} is stationary, then the time-reversed process f ~ X n ; n 2 Rg is also stationary. ...
The problem of controlling stationarity involves an important aspect of forecasting, in which a time series is analyzed in terms of levels or differences. In the literature, non-parametric stationary tests, such as the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) tests, have been shown to be very important; however, they are affected by problems with the reliability of lag and sample size selection. To date, no theoretical criterion has been proposed for the lag-length selection for tests of the null hypothesis of stationarity. Their use should be avoided, even for the purpose of so-called 'confirmation'. The aim of this study is to introduce a new method that measures the distance by obtaining each numerical series from its own time-reversed series. This distance is based on a novel stationary ergodic process, in which the stationary series has reversible symmetric features, and is calculated using the Dynamic Time-warping (DTW) algorithm in a self-correlation procedure. Furthermore, to establish a stronger statistical foundation for this method, the F-test is used as a statistical control and is a suggestion for future statistical research on resolving the problem of a sample of limited size being introduced. Finally, as described in the theoretical and experimental documentation, this distance indicates the degree of non-stationarity of the times series.
... , X t1 } have the same joint probability distribution. For the proof see [10]. What we are interested in is the following specification to the AR(1) and MA(1) cases. ...
We investigate the relative information efficiency of financial markets by measuring the entropy of the time series of high frequency data. Our tool to measure efficiency is the Shannon entropy, applied to 2-symbol and 3-symbol discretisations of the data. Analysing 1-minute and 5-minute price time series of 55 Exchange Traded Funds traded at the New York Stock Exchange, we develop a methodology to isolate true inefficiencies from other sources of regularities, such as the intraday pattern, the volatility clustering and the microstructure effects. The first two are modelled as multiplicative factors, while the microstructure is modelled as an ARMA noise process. Following an analytical and empirical combined approach, we find a strong relationship between low entropy and high relative tick size and that volatility is responsible for the largest amount of regularity, averaging 62% of the total regularity against 18% of the intraday pattern regularity and 20% of the microstructure.
Information processing in the human brain can be modeled as a complex dynamical system operating out of equilibrium with multiple regions interacting nonlinearly. Yet, despite extensive study of the global level of nonequilibrium in the brain, quantifying the irreversibility of interactions among brain regions at multiple levels remains an unresolved challenge. Here, we present the Directed Multiplex Visibility Graph Irreversibility framework, a method for analyzing neural recordings using network analysis of time-series. Our approach constructs directed multilayer graphs from multivariate time-series where information about irreversibility can be decoded from the marginal degree distributions across the layers, which each represents a variable. This framework is able to quantify the irreversibility of every interaction in the complex system. Applying the method to magnetoencephalography recordings during a long-term memory recognition task, we quantify the multivariate irreversibility of interactions between brain regions and identify the combinations of regions which showed higher levels of nonequilibrium in their interactions. For individual regions, we find higher irreversibility in cognitive versus sensorial brain regions while for pairs, strong relationships are uncovered between cognitive and sensorial pairs in the same hemisphere. For triplets and quadruplets, the most nonequilibrium interactions are between cognitive–sensorial pairs alongside medial regions. Combining these results, we show that multilevel irreversibility offers unique insights into the higher-order, hierarchical organization of neural dynamics from the perspective of brain network dynamics.
Permutation time irreversibility is an important method to quantify the nonequilibrium characteristics; however, ordinal pattern is a coarse-graining alternative and cannot accurately represent detailed structural information. In this paper, a fuzzy permutation time irreversibility (fpTIR) is proposed by measuring the difference among vector elements based on a negative exponential function. Amplitude permutation of vector is constructed and its membership degree is calculated, then the probability distribution difference of the forward and backward sequences is measured for fpTIR. As a comparison, Shannon entropy is calculated as the average amount of information in the fuzzy permutation probability distribution, i.e., fuzzy permutation entropy (fPEn), to measure the complexity of the system. According to the surrogate theory, mode series are generated by logistic, Henon, and first-order autoregressive systems to verify the fpTIR, which is then applied to analyze heart rates of congestive heart failure, healthy elderly and healthy young subjects from PhysioNet database. Results suggest that fpTIR effectively measures the nonequilibrium characteristic of system and improves the accuracy of heart rate analysis. Since fpTIR and fPEn are different in analyzing probability distributions, they have discrepancies in chaotic series and even opposite results in the heart rate signals, where the results of fpTIR are consistent with theory of complexity loss in aging and disease. In conclusion, fpTIR not only accurately characterizes the structure of sequences and enhances the effect of the nonequilibrium analysis of complex systems, but also provides a new perspective and theoretical basis for exploring complex systems from the perspectives of nonequilibrium dynamics and entropy complexity.
This book provides a comprehensive and self-contained overview of recent progress in nonequilibrium statistical mechanics, in particular, the discovery of fluctuation relations and other time-reversal symmetry relations. The significance of these advances is that nonequilibrium statistical physics is no longer restricted to the linear regimes close to equilibrium, but extends to fully nonlinear regimes. These important new results have inspired the development of a unifying framework for describing both the microscopic dynamics of collections of particles, and the macroscopic hydrodynamics and thermodynamics of matter itself. The book discusses the significance of this theoretical framework in relation to a broad range of nonequilibrium processes, from the nanoscale to the macroscale, and is essential reading for researchers and graduate students in statistical physics, theoretical chemistry and biological physics.
By locating the running maxima and minima of a time series, and measuring the current deviation from them, it is possible to generate processes that are relevant for the analysis of the business cycle and for characterizing bull and bear phases in financial markets. First, the measurement of the time distance from the running peak originates a first order Markov chain, whose characteristics can be used for testing time reversibility of economic dynamics and specific types of asymmetries in financial markets. Secondly, the gap processes can be combined to provide a nonparametric measure of the growth cycle. The paper derives the time series properties of the gap process and other related processes that arise from the same measurement context, and proposes new nonparametric tests of time reversibility and new measures of the output gap.
We analyse the European Center for Medium‐Range Weather Forecasts reanalysis temperature time series over China and find that daily mean temperature at 850 hPa pressure level warms gradually and cools rapidly, which is known as the asymmetry property. Previous studies pointed out that front events contribute to this asymmetry property, but only presented indirect evidences. Here, we confirm this conjecture with more convincing and direct evidences. The time series of front events are obtained over China by an improved objective front detection algorithm. The high Pearson correlation between time series of monthly temperature asymmetry measure and time series of monthly front events over some specific regions indicates they are closely related to each other and front events indeed contribute to temperature asymmetry. We discover that the North China Plain is a representative region where front events contribute to temperature asymmetry significantly. Further diagnostic analysis by temperature tendency equation shows that the asymmetry of daily mean temperature series is due to the asymmetry between the frequency and intensity (mainly the intensity of meridional nonlinear advection term) of cold and warm fronts in North China Plain.
Intrioution. The heart rate variability is based on measuring (time) intervals between R-peaks (of RR-intervals) of an electrocardiogram and plotting a rhythmogram on their basis with its subsequent analysis by various mathematical methods. Using nonlinear methods in HRV and ECG analysis has proven to be very advantageous. Time irreversibility is a fundamental parameter of a system, it defines justification and necessity of applying nonlinear methods for analysis of a system’s dynamics.
Objective. We propose an algorithm for testing the probability of a time series' irreversibility, showing its effectiveness in the process of HRV analysis. In this article, complexity of HRV will be described by two parameters: entropy EnRE [18] and correlation dimension D2 [19]. Naturally, the chosen parameters EnRE and D2 in no way can be used for comprehensive description of complexity of HRV, but we will be able to tress the necessary sufficiency of such an approach.
Materials and methods. We used long-term HRV records by Massachusetts Institute of Technology – Boston’s Beth Israel Hospital (MIT-BIH) from [15], a free-access, on-line archive of physiological signals for Normal Sinus Rhythm (NSR) RR Interval, Congestive Heart Failure (CHF) RR Interval and Atrial Fibrillation (AF) Databases [16]. In [17], we have developed a special modification to the classic Mann-Whitney (MW) U-test in order to use the test for comparison of Time Series with an equal number of elements N – Time Series MW M-test. Here the new statistical -test was proposed for finding the probability of time series' irreversibility.
Conclusion. In this article, we propose a statistical -test for assessment of probability of irreversibility of time series. It has been shown that the new statistical -test accurately identifies times series reversibility and irreversibility in known cases of synthetic data. For long-term HRV records of MIT-BIH database for NSR, CHF and AF groups, we have compared values of z-score, which statistically defines the limit of irreversibility of time series, and values of HRV complexity indicators: entropy EnRE [18] and correlation dimension D2 [19]. We have noted the following:
HRV is time irreversible nonlinear dynamic process, with the exception of AF episodes;
nonlinear indicators of HRV complexity – entropy EnRE and correlation dimension D2 – have been analyzed, and there is a conclusive difference between NSR and analyzed pathological states;
analyzed time series have been presented in D2-z-EnRE phase space, and their reliable separability has been shown. It can be stated that the analyzed D2-z-EnRE phase space is sufficient for research of nonlinear HRV events in this case.
In heart period (HP) variability (HPV) recordings the percentage of negative HP variations tends to be greater than that of positive ones and this pattern is referred to as HPV asymmetry (HPVA). HPVA has been studied in several experimental conditions in healthy and pathological populations, but its origin is unclear. The baroreflex (BR) exhibits an asymmetric behavior as well given that it reacts more importantly to positive than negative arterial pressure (AP) variations. We tested the hypothesis that the BR asymmetry (BRA) is a HPVA determinant over spontaneous fluctuations of HP and systolic AP (SAP). We studied 100 healthy subjects (age from 21 to 70 yrs, 54 males) comprising 20 subjects in each age decade. Electrocardiogram and noninvasive AP were recorded for 15 minutes at rest in supine position (REST) and during active standing (STAND). The HPVA was evaluated via Porta's index and Guzik's index, while the BRA was assessed as the difference, and normalized difference, between BR sensitivities computed over positive and negative SAP variations via the sequence method applied to HP and SAP variability. HPVA significantly increased during STAND and decreased progressively with age. BRA was not significantly detected both at REST and during STAND. However, we found a significant positive association between BRA and HPVA markers during STAND persisting even within the age groups. This study supports the use of HPVA indexes as descriptors of BRA and identified a challenge soliciting the BR response like STAND to maximize the association between HPVA and BRA markers.
We investigate the relative information efficiency of financial markets by measuring the entropy of the time series of high frequency data. Our tool to measure efficiency is the Shannon entropy, applied to 2-symbol and 3-symbol discretisations of the data. Analysing 1-min and 5-min price time series of 55 Exchange Traded Funds traded at the New York Stock Exchange, we develop a methodology to isolate residual inefficiencies from other sources of regularities, such as the intraday pattern, the volatility clustering and the microstructure effects. The first two are modelled as multiplicative factors, while the microstructure is modelled as an ARMA noise process. Following an analytical and empirical combined approach, we find a strong relationship between low entropy and high relative tick size and that volatility is responsible for the largest amount of regularity, averaging 62% of the total regularity against 18% of the intraday pattern regularity and 20% of the microstructure.
The study of sleep has continued to garner increased attention. However, most studies assume stationarity of sleep electroencephalogram (EEG) signals, whereas they are typically nonlinear and nonstationary. Little work has focused on the time irreversibility of sleep EEG signals. Hence, the aim of this work is to reveal the temporally irreversible structures of rapid-eye-movement (REM) and non-REM sleep using a visibility algorithm, which is robust to nonstationarity and finite-size effect. Results show that the temporal structure of non-REM sleep is more irreversible than that of REM sleep. The degree of irreversibility is highest in slow-wave sleep. Moreover, statistical analysis suggests that aging is the major factor that affects the irreversibility of sleep signals, while gender and body mass index contribute insignificantly. The dominant role of slow oscillations on the irreversible structures of the sleep signals is also indicated.
Directionality can be seen in many stationary time series from various disciplines, but it is overlooked when fitting linear models with Gaussian errors. Moreover, we cannot rely on distinguishing directionality by comparing a plot of a time series in time order with a plot in reverse time order. In general, a statistical measure is required to detect and quantify directionality. There are several quite different qualitative forms of directionality, and we distinguish: rapid rises followed by slow recessions; rapid increases and rapid decreases from the mean followed by slow recovery towards the mean; directionality above or below some threshold; and intermittent directionality. The first objective is to develop a suite of statistical measures that will detect directionality and help classify its nature. The second objective is to demonstrate the potential benefits of detecting directionality. We consider applications from business, environmental science, finance and medicine. Time series data are collected from many processes, both natural and anthropogenic, by a wide range of organizations, and directionality can easily be monitored as part of routine analysis. We suggest that doing so may provide new insights to the processes.
Time irreversibility, i.e. the lack of invariance of the statistical properties of a system under time reversal, is a fundamental property of all systems operating out of equilibrium. Time reversal symmetry is associated with important statistical and physical properties and is related to the predictability of the system generating the time series. Over the past fifteen years, various methods to quantify time irreversibility in time series have been proposed, but these can be computationally expensive. Here we propose a new method, based on permutation entropy, which is essentially parameter-free, temporally local, yields straightforward statistical tests, and has fast convergence properties. We apply this method to the study of financial time series, showing that stocks and indices present a rich irreversibility dynamics. We illustrate the comparative methodological advantages of our method with respect to a recently proposed method based on visibility graphs, and discuss the implications of our results for financial data analysis and interpretation.
In this paper, the wearable device is used to remotely monitor the physiological information for reducing and avoiding the possibility of accidents by real-time analysis algorithm. The mean value and standard deviation of QT quantified parameters(QI1 and QI2) of analysis algorithm is 2.75±2.16 and 8.25±1.09 in the normal case, respectively. In the abnormal case, the QI1 and QI2 is-10±4.3 and 14.3±4.19, respectively. The normal and abnormal case can be distinguished by selected threshold of 0 and 10 for QI1 and QI2. It will be feasible for future clinical application.
Time irreversibility is an important property of nonequilibrium dynamic systems. A visibility graph approach was recently proposed, and this approach is generally effective to measure time irreversibility of time series. However, its result may be unreliable when dealing with high-dimensional systems. In this work, we consider the joint concept of time irreversibility and adopt the phase-space reconstruction technique to improve this visibility graph approach. Compared with the previous approach, the improved approach gives a more accurate estimate for the irreversibility of time series, and is more effective to distinguish irreversible and reversible stochastic processes. We also use this approach to extract the multiscale irreversibility to account for the multiple inherent dynamics of time series. Finally, we apply the approach to detect the multiscale irreversibility of financial time series, and succeed to distinguish the time of financial crisis and the plateau. In addition, Asian stock indexes away from other indexes are clearly visible in higher time scales. Simulations and real data support the effectiveness of the improved approach when detecting time irreversibility.
Recent work has provided ample evidence that nonlinear methods of time series analysis potentially allow for detecting periods of anomalous dynamics in paleoclimate proxy records that are otherwise hidden to classical statistical analysis. Following upon these ideas, in this study, we systematically test a set of Late Holocene terrestrial paleoclimate records from Northern Europe for indications of intermittent periods of time- irreversibility during which the data are incompatible with a stationary linear-stochastic process. Our analysis reveals that the onsets of both the Medieval Climate Anomaly and the Little Ice Age, the end of the Roman Warm Period, and the Late Antique Little Ice Age have been characterized by such dynamical anomalies. These findings may indicate qualitative changes in the dominant regime of interannual climate variability in terms of large-scale atmospheric circulation patterns, ocean-atmosphere interactions, and external forcings affecting the climate of the North Atlantic region.
Time reversibility is a fundamental hypothesis in time series. In this paper, Gini-based equivalents for time series concepts that enable to construct a Gini-based test for time reversibility under merely first-order moment assumptions are developed. The key idea is that the relationship between two variables using Gini (as measured by Gini autocorrelations and partial autocorrelations) can be measured in two directions, which are not necessarily equal. This implies a built-in capability to discriminate between looking at forward and backward directions in time series. The Gini creates two bi-directional Gini autocorrelations (and partial autocorrelations), looking forward and backward in time, which are not necessarily equal. The difference between them may assist in identifying models with underlying heavy-tailed and non-normal innovations. Gini-based test and Gini-based correlograms, which serve as visual tools to examine departures from the symmetry assumption, are constructed. Simulations are used to illustrate the suggested Gini-based framework and to validate the statistical test. An application to a real data set is presented.
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