Return interval distribution of extreme events and long-term memory

Extreme events in the Higgs oscillator: A dynamical study and forecasting approach

Many dynamical systems exhibit unexpected large amplitude excursions in the chronological progression of a state variable. In the present work, we consider the dynamics associated with the one-dimensional Higgs oscillator, which is realized through gnomonic projection of a harmonic oscillator defined on a spherical space of constant curvature onto a Euclidean plane, which is tangent to the spherical space. While studying the dynamics of such a Higgs oscillator subjected to damping and an external forcing, various bifurcation phenomena, such as symmetry breaking, period doubling, and intermittency crises are encountered. As the driven parameter increases, the route to chaos takes place via intermittency crisis, and we also identify the occurrence of extreme events due to the interior crisis. The study of probability distribution also confirms the occurrence of extreme events. Finally, we train the long short-term memory neural network model with the time-series data to forecast extreme events.

Long-term memory induced correction to Arrhenius law

The Kramers escape problem is a paradigmatic model for the kinetics of rare events, which are usually characterized by Arrhenius law. So far, analytical approaches have failed to capture the kinetics of rare events in the important case of non-Markovian processes with long-term memory, as occurs in the context of reactions involving proteins, long polymers, or strongly viscoelastic fluids. Here, based on a minimal model of non-Markovian Gaussian process with long-term memory, we determine quantitatively the mean FPT to a rare configuration and provide its asymptotics in the limit of a large energy barrier E. Our analysis unveils a correction to Arrhenius law, induced by long-term memory, which we determine analytically. This correction, which we show can be quantitatively significant, takes the form of a second effective energy barrierE ' < E and captures the dependence of rare event kinetics on initial conditions, which is a hallmark of long-term memory. Altogether, our results quantify the impact of long-term memory on rare event kinetics, beyond Arrhenius law.

Extreme rotational events in a forced-damped nonlinear pendulum

Since Galileo's time, the pendulum has evolved into one of the most exciting physical objects in mathematical modeling due to its vast range of applications for studying various oscillatory dynamics, including bifurcations and chaos, under various interests. This well-deserved focus aids in comprehending various oscillatory physical phenomena that can be reduced to the equations of the pendulum. The present article focuses on the rotational dynamics of the two-dimensional forced-damped pendulum under the influence of the ac and dc torque. Interestingly, we are able to detect a range of the pendulum's length for which the angular velocity exhibits a few intermittent extreme rotational events that deviate significantly from a certain well-defined threshold. The statistics of the return intervals between these extreme rotational events are supported by our data to be spread exponentially at a specific pendulum's length beyond which the external dc and ac torque are no longer sufficient for a full rotation around the pivot. The numerical results show a sudden increase in the size of the chaotic attractor due to interior crisis, which is the source of instability that is responsible for triggering large amplitude events in our system. We also notice the occurrence of phase slips with the appearance of extreme rotational events when the phase difference between the instantaneous phase of the system and the externally applied ac torque is observed.

Extreme events in a complex network: Interplay between degree distribution and repulsive interaction

The role of topological heterogeneity in the origin of extreme events in a network is investigated here. The dynamics of the oscillators associated with the nodes are assumed to be identical and influenced by mean-field repulsive interactions. An interplay of topological heterogeneity and the repulsive interaction between the dynamical units of the network triggers extreme events in the nodes when each node succumbs to such events for discretely different ranges of repulsive coupling. A high degree node is vulnerable to weaker repulsive interactions, while a low degree node is susceptible to stronger interactions. As a result, the formation of extreme events changes position with increasing strength of repulsive interaction from high to low degree nodes. Extreme events at any node are identified with the appearance of occasional large-amplitude events (amplitude of the temporal dynamics) that are larger than a threshold height and rare in occurrence, which we confirm by estimating the probability distribution of all events. Extreme events appear at any oscillator near the boundary of transition from rotation to libration at a critical value of the repulsive coupling strength. To explore the phenomenon, a paradigmatic second-order phase model is used to represent the dynamics of the oscillator associated with each node. We make an annealed network approximation to reduce our original model and, thereby, confirm the dual role of the repulsive interaction and the degree of a node in the origin of extreme events in any oscillator associated with a node.

A multiplex analysis of phonological and orthographic networks

The study of natural language using a network approach has made it possible to characterize novel properties ranging from the level of individual words to phrases or sentences. A natural way to quantitatively evaluate similarities and differences between spoken and written language is by means of a multiplex network defined in terms of a similarity distance between words. Here, we use a multiplex representation of words based on orthographic or phonological similarity to evaluate their structure. We report that from the analysis of topological properties of networks, there are different levels of local and global similarity when comparing written vs. spoken structure across 12 natural languages from 4 language families. In particular, it is found that differences between the phonetic and written layers is markedly higher for French and English, while for the other languages analyzed, this separation is relatively smaller. We conclude that the multiplex approach allows us to explore additional properties of the interaction between spoken and written language.

Analysis of inter-transaction time fluctuations in the cryptocurrency market

We analyze tick-by-tick data representing major cryptocurrencies traded on some different cryptocurrency trading platforms. We focus on such quantities like the inter-transaction times, the number of transactions in time unit, the traded volume, and volatility. We show that the inter-transaction times show long-range power-law autocorrelations. These lead to multifractality expressed by the right-side asymmetry of the singularity spectra f ( α ) indicating that the periods of increased market activity are characterized by richer multifractality compared to the periods of quiet market. We also show that neither the stretched exponential distribution nor the power-law-tail distribution is able to model universally the cumulative distribution functions of the quantities considered in this work. For each quantity, some data sets can be modeled by the former and some data sets by the latter, while both fail in other cases. An interesting, yet difficult to account for, observation is that parallel data sets from different trading platforms can show disparate statistical properties.

Inferring long memory using extreme events

Many natural and physical processes display long memory and extreme events. In these systems, the measured time series is invariably contaminated by noise and/or missing data. As the extreme events display a large deviation from the mean behavior, noise and/or missing data do not affect the extreme events as much as it affects the typical values. Since the extreme events also carry the information about correlations in the full-time series, we can use them to infer the correlation properties of the latter. In this work, we construct three modified time series using only the extreme events from a given time series. We show that the correlations in the original time series and in the modified time series are related, as measured by the exponent obtained from the detrended fluctuation analysis technique. Hence, the correlation exponents for a long memory time series can be inferred from its extreme events alone. We demonstrate this approach for several empirical time series.

Clustering of extreme events in time series generated by the fractional Ornstein-Uhlenbeck equation

We analyze the time clustering phenomenon in sequences of extremes of time series generated by the fractional Ornstein-Uhlenbeck (fO-U) equation as the source of long-term correlation. We used the percentile-based definition of extremes based on the crossing theory or run theory, where a run is a sequence of L contiguous values above a given percentile. Thus, a sequence of extremes becomes a point process in time, being the time of occurrence of the extreme the starting time of the run. We investigate the relationship between the Hurst exponent related to the time series generated by the fO-U equation and three measures of time clustering of the corresponding extremes defined on the base of the 95th percentile. Our results suggest that for persistent pure fractional Gaussian noise, the sequence of the extremes is clusterized, while extremes obtained by antipersistent or Markovian pure fractional Gaussian noise seem to behave more regularly or Poissonianly. However, for the fractional Ornstein-Uhlenbeck equation, the clustering of extremes is evident even for antipersistent and Markovian cases. This is a result of short range correlations caused by differential and drift terms. The drift parameter influences the extremes clustering effect-it drops with increasing value of the parameter.

Extreme events in stochastic transport on networks

Extreme events are emergent phenomena in multi-particle transport processes on complex networks. In practice, such events could range from power blackouts to call drops in cellular networks to traffic congestion on roads. All the earlier studies of extreme events on complex networks had focused only on the nodal events. If random walks are used to model the transport process on a network, it is known that degree of the nodes determines the extreme event properties. In contrast, in this work, it is shown that extreme events on the edges display a distinct set of properties from that of the nodes. It is analytically shown that the probability for the occurrence of extreme events on an edge is independent of the degree of the nodes linked by the edge and is dependent only on the total number of edges on the network and the number of walkers on it. Further, it is also demonstrated that non-trivial correlations can exist between the extreme events on the nodes and the edges. These results are in agreement with the numerical simulations on synthetic and real-life networks.

An overview of the extremal index

For a wide class of stationary time series, extreme value theory provides limiting distributions for rare events. The theory describes not only the size of extremes but also how often they occur. In practice, it is often observed that extremes cluster in time. Such short-range clustering is also accommodated by extreme value theory via the so-called extremal index. This review provides an introduction to the extremal index by working through a number of its intuitive interpretations. Thus, depending on the context, the extremal index may represent (i) the loss of independently and identically distributed degrees of freedom, (ii) the multiplicity of a compound Poisson point process, and (iii) the inverse mean duration of extreme clusters. More recently, the extremal index has also been used to quantify (iv) recurrences around unstable fixed points in dynamical systems.

Analysis of fluctuations in the first return times of random walks on regular branched networks

The first return time (FRT) is the time it takes a random walker to first return to its original site, and the global first passage time (GFPT) is the first passage time for a random walker to move from a randomly selected site to a given site. We find that in finite networks, the variance of FRT, Var(FRT), can be expressed as Var(FRT) = 2⟨FRT⟩⟨GFPT⟩ - ⟨FRT⟩² - ⟨FRT⟩, where ⟨·⟩ is the mean of the random variable. Therefore a method of calculating the variance of FRT on general finite networks is presented. We then calculate Var(FRT) and analyze the fluctuation of FRT on regular branched networks (i.e., Cayley tree) by using Var(FRT) and its variant as the metric. We find that the results differ from those in such other networks as Sierpinski gaskets, Vicsek fractals, T-graphs, pseudofractal scale-free webs, (u, v) flowers, and fractal and non-fractal scale-free trees.

Extremes of fractional noises: A model for the timings of arrhythmic heart beats in post-infarction patients

We have analyzed symbol sequences of heart beat annotations obtained from 24-h electrocardiogram recordings of 184 post-infarction patients (from the Cardiac Arrhythmia Suppression Trial database, CAST). In the symbol sequences, each heart beat was coded as an arrhythmic or as a normal beat. The symbol sequences were analyzed with a model-based approach which relies on two-parametric peaks over the threshold (POT) model, interpreting each premature ventricular contraction (PVC) as an extreme event. For the POT model, we explored (i) the Shannon entropy which was estimated in terms of the Lempel-Ziv complexity, (ii) the shape parameter of the Weibull distribution that best fits the PVC return times, and (iii) the strength of long-range correlations quantified by detrended fluctuation analysis (DFA) for the two-dimensional parameter space. We have found that in the frame of our model the Lempel-Ziv complexity is functionally related to the shape parameter of the Weibull distribution. Thus, two complementary measures (entropy and strength of long-range correlations) are sufficient to characterize realizations of the two-parametric model. For the CAST data, we have found evidence for an intermediate strength of long-range correlations in the PVC timings, which are correlated to the age of the patient: younger post-infarction patients have higher strength of long-range correlations than older patients. The normalized Shannon entropy has values in the range 0.5<h_LZ<1.0 which indicates a high degree of randomness in the PVC timings. For the CAST and the model data, the ranges of both measures were found to be in good accordance. The correlation between the age and the persistence strength found for the CAST data could be explained as a change of model parameters.

Scaling laws for diffusion on (trans)fractal scale-free networks

Fractal (or transfractal) features are common in real-life networks and are known to influence the dynamic processes taking place in the network itself. Here, we consider a class of scale-free deterministic networks, called (u, v)-flowers, whose topological properties can be controlled by tuning the parameters u and v; in particular, for u > 1, they are fractals endowed with a fractal dimension df, while for u = 1, they are transfractal endowed with a transfractal dimension d̃_f. In this work, we investigate dynamic processes (i.e., random walks) and topological properties (i.e., the Laplacian spectrum) and we show that, under proper conditions, the same scalings (ruled by the related dimensions) emerge for both fractal and transfractal dimensions.

Finite-size effects on return interval distributions for weakest-link-scaling systems

The Weibull distribution is a commonly used model for the strength of brittle materials and earthquake return intervals. Deviations from Weibull scaling, however, have been observed in earthquake return intervals and the fracture strength of quasibrittle materials. We investigate weakest-link scaling in finite-size systems and deviations of empirical return interval distributions from the Weibull distribution function. Our analysis employs the ansatz that the survival probability function of a system with complex interactions among its units can be expressed as the product of the survival probability functions for an ensemble of representative volume elements (RVEs). We show that if the system comprises a finite number of RVEs, it obeys the κ-Weibull distribution. The upper tail of the κ-Weibull distribution declines as a power law in contrast with Weibull scaling. The hazard rate function of the κ-Weibull distribution decreases linearly after a waiting time τ(c) ∝ n(1/m), where m is the Weibull modulus and n is the system size in terms of representative volume elements. We conduct statistical analysis of experimental data and simulations which show that the κ Weibull provides competitive fits to the return interval distributions of seismic data and of avalanches in a fiber bundle model. In conclusion, using theoretical and statistical analysis of real and simulated data, we demonstrate that the κ-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.

Copulas and time series with long-ranged dependencies

We review ideas on temporal dependencies and recurrences in discrete time series from several areas of natural and social sciences. We revisit existing studies and redefine the relevant observables in the language of copulas (joint laws of the ranks). We propose that copulas provide an appropriate mathematical framework to study nonlinear time dependencies and related concepts-like aftershocks, Omori law, recurrences, and waiting times. We also critically argue, using this global approach, that previous phenomenological attempts involving only a long-ranged autocorrelation function lacked complexity in that they were essentially monoscale.

Manipulation of extreme events on scale-free networks

Extreme events taking place on networks are not uncommon. We show that it is possible to manipulate the extreme event occurrence probabilities and distribution over the nodes of a scale-free network by tuning the nodal capacity. This can be used to reduce the number of extreme event occurrences. However, monotonic nodal capacity enhancements, beyond a point, do not lead to any substantial reduction in the number of extreme events. We point out the practical implication of this result for network design in the context of reducing extreme event occurrences.

Fertility heterogeneity as a mechanism for power law distributions of recurrence times

We study the statistical properties of recurrence times in the self-excited Hawkes conditional Poisson process, the simplest extension of the Poisson process that takes into account how the past events influence the occurrence of future events. Specifically, we analyze the impact of the power law distribution of fertilities with exponent α, where the fertility of an event is the number of triggered events of first generation, on the probability distribution function (PDF) f(τ) of the recurrence times τ between successive events. The other input of the model is an exponential law quantifying the PDF of waiting times between an event and its first generation triggered events, whose characteristic time scale is taken as our time unit. At short-time scales, we discover two intermediate power law asymptotics, f(τ)~τ(-(2-α)) for τ<<τ(c) and f(τ)~τ(-α) for τ(c)<<τ<<1, where τ(c) is associated with the self-excited cascades of triggered events. For 1<<τ<<1/ν, we find a constant plateau f(τ)=/~const, while at long times, 1/ν</~τ, f(τ)=/~e(-ντ) has an exponential tail controlled by the arrival rate ν of exogenous events. These results demonstrate a novel mechanism for the generation of power laws in the distribution of recurrence times, which results from a power law distribution of fertilities in the presence of self-excitation and cascades of triggering.

Scaling properties and universality of first-passage-time probabilities in financial markets

Financial markets provide an ideal frame for the study of crossing or first-passage time events of non-Gaussian correlated dynamics, mainly because large data sets are available. Tick-by-tick data of six futures markets are herein considered, resulting in fat-tailed first-passage time probabilities. The scaling of the return with its standard deviation collapses the probabilities of all markets examined--and also for different time horizons--into single curves, suggesting that first-passage statistics is market independent (at least for high-frequency data). On the other hand, a very closely related quantity, the survival probability, shows, away from the center and tails of the distribution, a hyperbolic t(-1/2) decay typical of a Markovian dynamics, albeit the existence of memory in markets. Modifications of the Weibull and Student distributions are good candidates for the phenomenological description of first-passage time properties under certain regimes. The scaling strategies shown may be useful for risk control and algorithmic trading.

Clustering of extreme and recurrent events in deterministic chaotic systems

We study the nontrivial clustering properties of extreme or recurrent events in the context of deterministic chaotic systems. We find that correlations between return times of such events can depend nonmonotonically on the threshold used to define the events, which leads to counterintuitive behavior. In particular, the distribution of the conditional return intervals can indicate clustering as well as repelling of extreme events for the same condition but different thresholds-in sharp contrast to what has been observed for stochastic processes with long-range correlations as well as for independent and identically distributed random variables. This has important implications for the time-dependent hazard assessment of extreme events, indicating that possible threshold dependencies should always be taken into account.

Extreme events on complex networks

A wide spectrum of extreme events ranging from traffic jams to floods take place on networks. Motivated by these, we employ a random walk model for transport and obtain analytical and numerical results for the extreme events on networks. They reveal an unforeseen, and yet a robust, feature: small degree nodes of a network are more likely to encounter extreme events than the hubs. Further, we also study the recurrence time distribution and scaling of the probabilities for extreme events. These results suggest a revision of design principles and can be used as an input for designing the nodes of a network so as to smoothly handle extreme events.

Distribution of first-return times in correlated stationary signals

We present an analytical expression for the first return time (FRT) probability density function of a stationary correlated signal. Precisely, we start by considering a stationary discrete-time Ornstein-Uhlenbeck (OU) process with exponential decaying correlation function. The first return time distribution for this process is derived by adopting a well-known formalism typically used in the study of the FRT statistics for nonstationary diffusive processes. Then, by a subordination approach, we treat the case of a stationary process with power-law tail correlation function and diverging correlation time. We numerically test our findings, obtaining in both cases a good agreement with the analytical results. We notice that neither in the standard OU nor in the subordinated case a simple form of waiting time statistics, like stretched-exponential or similar, can be obtained while it is apparent that long time transient may shadow the final asymptotic behavior.

Universal patterns in sound amplitudes of songs and music genres

We report a statistical analysis of more than eight thousand songs. Specifically, we investigated the probability distribution of the normalized sound amplitudes. Our findings suggest a universal form of distribution that agrees well with a one-parameter stretched Gaussian. We also argue that this parameter can give information on music complexity, and consequently it helps classify songs as well as music genres. Additionally, we present statistical evidence that correlation aspects of the songs are directly related to the non-Gaussian nature of their sound amplitude distributions.

Beyond word frequency: bursts, lulls, and scaling in the temporal distributions of words

BACKGROUND

Zipf's discovery that word frequency distributions obey a power law established parallels between biological and physical processes, and language, laying the groundwork for a complex systems perspective on human communication. More recent research has also identified scaling regularities in the dynamics underlying the successive occurrences of events, suggesting the possibility of similar findings for language as well.

METHODOLOGY/PRINCIPAL FINDINGS

By considering frequent words in USENET discussion groups and in disparate databases where the language has different levels of formality, here we show that the distributions of distances between successive occurrences of the same word display bursty deviations from a Poisson process and are well characterized by a stretched exponential (Weibull) scaling. The extent of this deviation depends strongly on semantic type -- a measure of the logicality of each word -- and less strongly on frequency. We develop a generative model of this behavior that fully determines the dynamics of word usage.

CONCLUSIONS/SIGNIFICANCE

Recurrence patterns of words are well described by a stretched exponential distribution of recurrence times, an empirical scaling that cannot be anticipated from Zipf's law. Because the use of words provides a uniquely precise and powerful lens on human thought and activity, our findings also have implications for other overt manifestations of collective human dynamics.

Scaling and memory in the return intervals of energy dissipation rate in three-dimensional fully developed turbulence

We study the statistical properties of return intervals r between successive energy dissipation rates above a certain threshold Q in three-dimensional fully developed turbulence. We find that the distribution function P(Q)(r) scales with the mean return interval R(Q) as P(Q)(r)=R(Q)(-1)f(r/R(Q)) for R(Q) is an element of [50,500], where the scaling function f(x) has two power-law regimes. The scaling behavior is statistically validated by the Cramér-von Mises criterion. The return intervals are short-term and long-term correlated and possess multifractal nature. The Hurst index of the return intervals decays exponentially against R(Q), predicting that rare extreme events with R(Q)-->infinity are also long-term correlated with the Hurst index H(infinity)=0.639. These phenomenological findings have potential applications in risk assessment of extreme events at very large R(Q).

Extreme value statistics and return intervals in long-range correlated uniform deviates

We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider (e.g., maximum relative to minimum) are such that the reference point from which the maximum is measured is itself a random quantity. We analytically calculate the limiting distributions for independent and identically distributed random variables, and use these as a reference point for correlated cases. The distributions are different from that of the maximum itself (i.e., a Weibull distribution), reflecting the fact that the distribution of the reference point either dominates over or convolves with the distribution of the maximum. The functional form of the limiting distributions is unaffected by correlations, although the convergence is slower. We show that our findings can be directly generalized to a wide class of stochastic processes. We also analyze return interval distributions, and compare them to recent conjectures of their functional form.