Inter-event Time Distribution Research Articles

Renewal Theory as a Universal Quantitative Framework to Characterize Phase Singularity Regeneration in Mammalian Cardiac Fibrillation.

Despite a century of research, no clear quantitative framework exists to model the fundamental processes responsible for the continuous formation and destruction of phase singularities (PS) in cardiac fibrillation. We hypothesized PS formation/destruction in fibrillation could be modeled as self-regenerating Poisson renewal processes, producing exponential distributions of interevent times governed by constant rate parameters defined by the prevailing properties of each system. PS formation/destruction were studied in 5 systems: (1) human persistent atrial fibrillation (n=20), (2) tachypaced sheep atrial fibrillation (n=5), (3) rat atrial fibrillation (n=4), (5) rat ventricular fibrillation (n=11), and (5) computer-simulated fibrillation. PS time-to-event data were fitted by exponential probability distribution functions computed using maximum entropy theory, and rates of PS formation and destruction (λf/λd) determined. A systematic review was conducted to cross-validate with source data from literature. In all systems, PS lifetime and interformation times were consistent with underlying Poisson renewal processes (human: λf, 4.2%/ms±1.1 [95% CI, 4.0-5.0], λd, 4.6%/ms±1.5 [95% CI, 4.3-4.9]; sheep: λf, 4.4%/ms [95% CI, 4.1-4.7], λd, 4.6%/ms±1.4 [95% CI, 4.3-4.8]; rat atrial fibrillation: λf, 33%/ms±8.8 [95% CI, 11-55], λd, 38%/ms [95% CI, 22-55]; rat ventricular fibrillation: λf, 38%/ms±24 [95% CI, 22-55], λf, 46%/ms±21 [95% CI, 31-60]; simulated fibrillation λd, 6.6-8.97%/ms [95% CI, 4.1-6.7]; R2≥0.90 in all cases). All PS distributions identified through systematic review were also consistent with an underlying Poisson renewal process. Poisson renewal theory provides an evolutionarily preserved universal framework to quantify formation and destruction of rotational events in cardiac fibrillation.

Analysis of human behavior statistics law based on WeChat Moment

WeChat is one of the most popular social platforms where users share their feelings and opinions in Moments, Wechat postings that are usually in forms of text, pictures and videos, etc. This paper analyze the Moment posting behavior of WeChat users by obtaining 10196 Moments of 628 users. According to the result, both the number of commits and the inter-event time conform to the power-law distribution in a continuous period of time, revealing a fact that the majority of Moments are posted by a small portion of users. Then our study on individual activity unveils a relatively strong positive correlation between user activity and the exponent of inter-event time distribution, which agrees with previous studies. Additionally, the paper points out that WeChat users have a low-burstiness, low-memory posting pattern, which is slightly different from that of other online systems owing to Wechat users’ generally high activity. This paper also studied the effects of sleep on user behavior. Our findings have significance in understanding the human dynamic mechanism of Wechat users.

Modeling of Extreme Rainfall Recurrence Using Conditional Probability Approach

Forecasts for the occurrence of extreme rainfall are related to time lapse estimates until the next extreme rain event. The accuracy of these estimates depends on the choice of interevent time distributions of extreme rain events. The aim of this research is to estimate the probability of time recurrence of extreme rainfall. In this research, we observed the time between extreme rain events distributed by Weibull and Pareto. Let T be the time interval for the next extreme rain and it is assumed that there has been no extreme rain at time interval (0, t0]. To analyze the occurrence of the next extreme rain event, the concept of conditional probability P(Δt0|t0) is used. P(Δt0|t0) is the probability of extreme rain occurrence during the time interval Δt0. The results show that by using three models, we found that the next extreme rain event is not depending on the time interval Δt0 since the last extreme rain event. Other results show that for case studies of extremerain events in South Sulawesi it was found that the time of occurrence of extreme rainfall was influenced by parameter value for each model.

Interevent time distribution, burst, and hybrid percolation transition

Understanding of a hybrid percolation transitions (HPTs) induced by cluster coalescence, exhibiting a jump in the giant cluster size and a critical behavior of finite clusters, is fundamental and intriguing. Here, we uncover the underlying mechanism using the so-called restricted-random network model, in which clusters are ranked by size and partitioned into small- and large-cluster sets. As clusters are merged and their rankings are updated, they may move back and forth across the set boundary. The intervals of these crossings exhibit a self-organized critical (SOC) behavior with two power-law exponents. During this process, a bump is formed and eliminated in the cluster size distribution, characterizing the criticality of the HPT. This SOC behavior is in contrast to the critical branching process, which governs the avalanche dynamics of the HPT in the pruning process. Finally, we find that a burst of such crossing events occurs and signals the upcoming abrupt transition.

Copula-based algorithm for generating bursty time series.

Dynamical processes in various natural and social phenomena have been described by a series of events or event sequences showing non-Poissonian, bursty temporal patterns. Temporal correlations in such bursty time series can be understood not only by heterogeneous interevent times (IETs) but also by correlations between IETs. Modeling and simulating various dynamical processes requires us to generate event sequences with a heavy-tailed IET distribution and memory effects between IETs. For this, we propose a Farlie-Gumbel-Morgenstern copula-based algorithm for generating event sequences with correlated IETs when the IET distribution and the memory coefficient between two consecutive IETs are given. We successfully apply our algorithm to the cases with heavy-tailed IET distributions. We also compare our algorithm to the existing shuffling method to find that our algorithm outperforms the shuffling method for some cases. Our copula-based algorithm is expected to be used for more realistic modeling of various dynamical processes.

Analytically solvable autocorrelation function for weakly correlated interevent times.

Long-term temporal correlations observed in event sequences of natural and social phenomena have been characterized by algebraically decaying autocorrelation functions. Such temporal correlations can be understood not only by heterogeneous interevent times (IETs) but also by correlations between IETs. In contrast to the role of heterogeneous IETs on the autocorrelation function, little is known about the effects due to the correlations between IETs. To rigorously study these effects, we derive an analytical form of the autocorrelation function for the arbitrary IET distribution in the case with weakly correlated IETs, where the Farlie-Gumbel-Morgenstern copula is adopted for modeling the joint probability distribution function of two consecutive IETs. Our analytical results are confirmed by numerical simulations for exponential and power-law IET distributions. For the power-law case, we find a tendency of the steeper decay of the autocorrelation function for the stronger correlation between IETs. Our analytical approach enables us to better understand long-term temporal correlations induced by the correlations between IETs.

Characteristics of earthquake clusters: Application to western Corinth Gulf (Greece)

The western Corinth Gulf, one of the most seismically active areas in Greece with frequent earthquake sequences and a dense seismological network, is an ideal region for investigating the properties of earthquake clusters. We study 47 earthquake clusters identified using a high-resolution earthquake catalog and the CURATE clustering algorithm. Based on the skewness and kurtosis of their moment release history, 31 sequences are successfully distinguished as earthquake swarms and 13 as mainshock-aftershock sequences (three sequences could not be classified). The relationship between skewness and kurtosis leads to a parabola where mainshock–aftershock sequences are located to the right part (high values of skewness and kurtosis), whereas earthquake swarms to the center–left (low values of skewness and kurtosis). This discrimination is validated by taking into consideration the occurrence time of the largest event in the sequence, as well as the difference in magnitude between the largest event and other larger events. Twelve clusters, which consist of several burst-like repeaters, show evidence of fluid driven activity with diffusivity ranging from 0.5 to 1.8 m2/s for the triggering front and 0.1 to 0.4 m2/s for the back front. The observed temporal evolution of earthquake swarms compared to that expected from the ETAS stochastic model shows that they have distinct values for the model parameters α and μ. Finally, interevent time distributions for four clusters having at least 100 events show that a Lognormal distribution fits best in all cases, suggesting a degree of triggering between subsequent events.

A Khmaladze-transformed test of fit with ML estimation in the presence of recurrent events

This article provides a goodness-of-fit test for the distribution function or the survival function in a recurrent event setting, when the inter-event time parametric structure is estimated from the observed data. Of concern is the null hypothesis that the inter-event time distribution is absolutely continuous and belongs to a parametric family , where the q-dimensional parameter space is neither known nor specified. We proposed a Khmaladze martingale-transformed type of test (Khmaladze, 1981), adapted to recurrent events. The test statistic combines two likelihood sources of estimation to form a parametric empirical process: (1) a product-limit nonparametric maximum likelihood estimator (NPMLE; Peña et al., 2001a) that is a consistent estimator of F, say, and (2) a point process likelihood estimator (Jacod, 1974/1975). These estimators are combined to construct a Kolmogorov-Smirnov (KS) type of test (Kolmogorov 1933; Smirnov, 1933). Empirical process and martingale weak convergence frameworks are utilized for theoretical derivations and motivational justification of the proposed transformation. A simulation study is conducted for performance assessment, and the test is applied to a problem investigated by Proschan (1963) on air-conditioning failure in a fleet of Boeing 720 jets.

Quantifying postural sway dynamics using burstiness and interevent time distributions

We propose an approach for analysing the dynamics of human postural sway using measures applied to study inhomogeneous temporal processes. Basically, we defined zero-crossings of center of pressure (COP) trajectories as events, obtained the sequence of interevent times and investigated the mean interevent time, the burstiness coefficient and the full functional form of the interevent time distribution. We applied this approach to data obtained for healthy young adults during quiet standing, under bipedal and unipedal conditions. We found that the proposed COP-based measures are able to detect differences between bipedal and unipedal postural sway temporal patterns, including the presence (or absence) of burstiness. We discussed the potential of this analysis to quantify temporal patterns of postural sway.

Statistical dynamics of early creep stages in disordered materials

When materials are loaded below their short-term strength over extended periods, a slow time-dependent process known as creep deformation takes place. During creep deformation, the structural properties of a material evolve as a function of time. By means of a generic coarse-grained mesoscopic elastoplastic model which envisages deformation as a sequence of stochastically activated discrete events, we study the creep deformation of disordered materials. We find that the structural evolution of the material during creep modifies not only the average material properties but also changes the statistics of those properties. We analyze the emergence of correlations in the strain localization and deformation activity patterns, the variation of the event rate and the evolution of the inter-event time distribution. We find that the event rate follows the Omori law of aftershocks, which is the discrete counterpart of Andrade's transient creep law, and that the exponent of these laws only depends on the microstructural heterogeneity. Finally, we find during the initial stages of transient creep a transition from Poisson distributed inter-event times towards a non-trivial power law distribution.

Human dynamics in near-miss accidents resulting from unsafe behavior of construction workers

Near misses, as a common result of unsafe behavior and/or unsafe conditions, are precursors of accidents in the construction industry. Given the complexity of near misses, seeking regularities in time series of near misses is almost impossible. We present a human dynamics model based on near-miss data to reveal common dynamic features of near-miss accidents that result from unsafe behavior of construction workers. To validate the model, we conducted an empirical case study based on unsafe behavior related to near misses from a database in the Early Warning System for Safety Risk Management in Wuhan Metro Construction, created by the Huazhong University of Science and Technology. Heavy-tailed distribution and approximate power-law distribution are observed in the inter-event time series of near misses, rather than being randomly distributed in time in current models. The strong burst and weak memory phenomena were observed in the whole, annual, and categorized inter-event time distributions of near misses. Moreover, the results showed a non-trivial, monotonous increase of the power-law exponent with the activity of different construction site in near misses, thus indicating that heterogeneity and complexity exist in general. We defined a new safety performance metric that is more reliable than the number of near misses and validated this performance metric by using expert assessment results of metro construction sites. Furthermore, we proposed further research prospects on human dynamics in construction safety behavior.

Non-extensive statistical analysis of acoustic emissions series recorded during the uniaxial compression of brittle rocks

Non-extensive statistical physics (NESP) provides the theoretical tool to investigate complex systems that appear in different length-scales and violate the classical Boltzmann–Gibbs statistics. In the present study, we apply the concept of NESP to analyze the Acoustic Emissions (AE) series recorded during uniaxial compression of brittle rocks, such as marble and sandstone specimens. In both types of rock samples, the cumulative distribution of inter-event times (interval times between successive AE hits) are well described by the q-exponential function. The variations of the q-index of Tsallis entropy and of the βq parameter with the applied stress follow distinct behavior in each type of rock, indicating in both cases the entrance of the system to the critical stage of fracture. The analysis of the experimental data suggests that NESP is suitable for describing AE activity in rock samples under compression, in a similar way to Earth’s seismic activity.

Burstiness in activity-driven networks and the epidemic threshold

We study the effect of heterogeneous temporal activations on epidemic spreading in temporal networks. We focus on the susceptible-infected-susceptible model on activity-driven networks with burstiness. By using an activity-based mean-field approach, we derive a closed analytical form for the epidemic threshold for arbitrary activity and inter-event time distributions. We show that, as expected, burstiness lowers the epidemic threshold while its effect on prevalence is twofold. In low-infective systems burstiness raises the average infection probability, while it weakens epidemic spreading for high infectivity. Our results can help clarify the conflicting effects of burstiness reported in the literature. We also discuss the scaling properties at the transition, showing that they are not affected by burstiness.

Fitting procedure for the two-state Batch Markov modulated Poisson process

The Batch Markov Modulated Poisson Process (BMMPP) is a subclass of the versatile Batch Markovian Arrival Process (BMAP) which has been proposed for the modeling of dependent events occurring in batches (such as group arrivals, failures or risk events). This paper focuses on exploring the possibilities of the BMMPP for the modeling of real phenomena involving point processes with group arrivals. The first result in this sense is the characterization of the two-state BMMPP with maximum batch size equal to K, the BMMPP2(K), by a set of moments related to the inter-event time and batch size distributions. This characterization leads to a sequential fitting approach via a moments matching method. The performance of the novel fitting approach is illustrated on both simulated and a real teletraffic data set, and compared to that of the EM algorithm. In addition, as an extension of the inference approach, the queue length distributions at departures in the queueing system BMMPP/M/1 is also estimated.

Distribution of interevent avalanche times in disordered and frustrated spin systems

Hysteresis loops and the associated avalanche statistics of spin systems, such as the random-field Ising and Edwards-Anderson spin-glass models, have been extensively studied. A particular focus has been on self-organized criticality, manifest in power-law distributions of avalanche sizes. Considerably less work has been done on the statistics of the times between avalanches. This paper considers this issue, generalizing the work of Nampoothiri et al. [Phys. Rev. E 96, 032107 (2017)] in one space dimension to higher space dimensions. In addition to the interevent statistics of all avalanches, we also consider what happens when events are restricted to those exceeding a certain threshold size. Doing so raises the possibility of altering the definition of time to count the number of small events between the large ones, which provides for an analog to the concept of natural time introduced by the geophysics community with the goal of predicting patterns in seismic events. We analyze the distribution of time and natural time intervals both in the case of models that include only nearest-neighbor interactions, as well as models with (sparse) long-range couplings.

Estimating the earthquake occurrence rates in Corinth Gulf (Greece) through Markovian arrival process modeling

ABSTRACTThe Markovian Arrival Process (MAP) is applied as a candidate model to describe the time-varying earthquake activity in Corinth Gulf, Greece. To the best of our knowledge, this is the first attempt to study the earthquake temporal evolution with the specific class of MAPs. A complete catalogue is used for the earthquake temporal distribution investigation, along with data sets of different magnitude cutoffs. The study area is divided into its western and eastern subareas, and possible variations in the earthquake occurrence times were sought. Hidden states of MAPs correspond to different levels of seismicity, and hence various numbers of states are examined. Akaike and Bayes information criteria are implemented for identifying the best model, and comparison to the most known and broadly accepted theoretical interevent time distributions is provided. In all cases, the fitted MAPs with phase type distributed intearrival times outperform the models with other distributions. Important indicators of the underlying Markov process are computed, and the earthquake frequency is approximated by the counting process. The analysis demonstrates high index of burstiness for the earthquake generation in the eastern part, i.e. long quiescent periods alternate with short ones of intense seismic activity.

Time-statistical laws of workers’ unsafe behavior in the construction industry: A case study

The construction industry is extremely high risk, and the unsafe behavior of workers is thought to be a critical factor in that risk level. Considering the limitation that existing studies merely analyze the relationship between unsafe behavior and the time factor, this paper uses a case study to explore two aspects of the time-statistical laws of workers’ unsafe behavior: (i) the characteristics of interevent time distributions and (ii) association rules for different worker types. First, workers’ unsafe acts at one metro construction site are collected and classified. Second, interevent time distributions of workers’ unsafe behavior from different types are analyzed via a human dynamics approach. Finally, a rule mining database is built, from which association rules concerning unsafe behavior, worker type and construction phase are determined using the Apriori algorithm. The results indicate that the interevent time distributions are fat tailed and show that workers’ unsafe behavior has the characteristics of burstiness and memory. Furthermore, the strong rules ‘construction phase → unsafe behavior’ exist for different worker types. The research presented in this paper can facilitate a better understanding of workers’ unsafe behavioral patterns and can accordingly be used to control frequent and widespread unsafe acts among different types of workers in different construction phases.

Degree-correlations in a bursting dynamic network model

We propose a mathematical description of a dynamic network model in which the number of links fluctuates over time according to the degree-preferences of nodes. More specifically, we consider the minimal case of a bipartite directed network where we have two groups of nodes, each group has nodes with a given capability to bear links. One group is composed of nodes that create as many links as possible, the generators. The other group is composed of nodes which delete as many links as possible, i.e., the destroyers. We provide here a novel analytical formulation of the evolution of the system through coupled master equations for the two interacting populations, recovering the steady state degree distributions and a new analytic description of the transient dynamics to the equilibrium. Further, fluctuations are shown to be connected to a peak in degree correlation at a critical point of the system corresponding to equal-size populations of generators and destroyers. We investigate the nature of the neighbor connectivity and the temporal assortativity of the network, noticing that degree correlation are anomalously large at criticality and that they are not a pointwise characterization of the system in time but they emerge as an aggregate temporal property. Moreover, we examine the bursty dynamics of the network as a temporal property where the system heterogeneously evolves over time alternating between periods of low and high connectivity displaying a heavy-tailed distribution in the inter-event times distributions. Finally, we introduce a generalization of the model in which intermittent states can control the velocity of the network’s evolution. We will also provide examples of possible economic applications of the present network model.

Complexity of the Yellowstone Park Volcanic Field Seismicity in Terms of Tsallis Entropy.

The Yellowstone Park volcanic field is one of the most active volcanic systems in the world, presenting intense seismic activity that is characterized by several earthquake swarms over the last decades. In the present work, we focused on the spatiotemporal properties of the recent earthquake swarms that occurred on December–January 2008–2009 and the 2010 Madison Plateau swarm, using the approach of Non Extensive Statistical Physics (NESP). Our approach is based on Tsallis entropy, and is used in order to describe the behavior of complex systems where fracturing and strong correlations exist, such as in tectonic and volcanic environments. This framework is based on the maximization of the non-additive Tsallis entropy Sq, introducing the q-exponential function and the entropic parameter q that expresses the degree of non-extentivity of the system. The estimation of the q-parameters could be used as a correlation degree among the events in the spatiotemporal evolution of seismicity. Using the seismic data provided by University of Utah Seismological Stations (UUSS), we analyzed the inter-event time (T) and distance (r) distribution of successive earthquakes that occurred during the two swarms, fitting the observed data with the q-exponential function, resulting in the estimation of the Tsallis entropic parameters qT, qr for the inter-event time and distance distributions, respectively. Furthermore, we studied the magnitude-frequency distribution of the released earthquake energies E as formulated in the frame of NESP, which results in the estimation of the qE parameter. Our analysis provides the triplet (qE, qT, qr) that describes the magnitude-frequency distribution and the spatiotemporal scaling properties of each of the studied earthquake swarms. In addition, the spatial variability of qE throughout the Yellowstone park volcanic area is presented and correlated with the existence of the regional hydrothermal features.

Hierarchical burst model for complex bursty dynamics.

Temporal inhomogeneities observed in various natural and social phenomena have often been characterized in terms of scaling behaviors in the autocorrelation function with a decaying exponent γ, the interevent time distribution with a power-law exponent α, and the burst size distributions. Here the interevent time is defined as a time interval between two consecutive events in the event sequence, and the burst size denotes the number of events in a bursty train detected for a given time window. To understand such temporal scaling behaviors implying a hierarchical temporal structure, we devise a hierarchical burst model by assuming that each observed event might be a consequence of the multilevel causal or decision-making process. By studying our model analytically and numerically, we confirm the scaling relation α+γ=2, established for the uncorrelated interevent times, despite of the existence of correlations between interevent times. Such correlations between interevent times are supported by the stretched exponential burst size distributions, for which we provide an analytic argument. In addition, by imposing conditions for the ordering of events, we observe an additional feature of log-periodic behavior in the autocorrelation function. Our modeling approach for the hierarchical temporal structure can help us better understand the underlying mechanisms behind complex bursty dynamics showing temporal scaling behaviors.