Inter-event Time Distribution Research Articles

Analysis of microseismicity associated with the 2017 seismic swarm near the Aegean coast of NW Turkey

A thorough spatiotemporal analysis of the intense seismic activity that took place near the Aegean coast of NW Turkey during January–March 2017 was conducted, aiming to identify its causative relation to the regional seismotectonic properties. In this respect, absolute and relative locations are paired and a catalog consisting of 2485 events was compiled. Relative locations are determined with high accuracy using the double-difference technique and differential times both from phase pick data and from cross-correlation measurements. The spatial distribution of the relocated events revealed a south-dipping causative fault along with secondary and smaller antithetic segments. Spatially, the seismicity started at the westernmost part and migrated with time to the easternmost part of the activated area. Temporally, two distinctive periods are observed, namely an early period lasting 1 month and a second period which includes the largest events in the sequence. The investigation of the interevent time distribution revealed a triggering mechanism, whereas the ETAS parameters show a strong external force (μ > 1), which might be attributed to the existence of the Tuzla geothermal field.

Heterogeneous characters modeling of instant message services users' online behavior.

Research on temporal characteristics of human dynamics has attracted much attentions for its contribution to various areas such as communication, medical treatment, finance, etc. Existing studies show that the time intervals between two consecutive events present different non-Poisson characteristics, such as power-law, Pareto, bimodal distribution of power-law, exponential distribution, piecewise power-law, et al. With the occurrences of new services, new types of distributions may arise. In this paper, we study the distributions of the time intervals between two consecutive visits to QQ and WeChat service, the top two popular instant messaging services in China, and present a new finding that when the value of statistical unit T is set to 0.001s, the inter-event time distribution follows a piecewise distribution of exponential and power-law, indicating the heterogeneous character of IM services users’ online behavior in different time scales. We infer that the heterogeneous character is related to the communication mechanism of IM and the habits of users. Then we develop a combination model of exponential model and interest model to characterize the heterogeneity. Furthermore, we find that the exponent of the inter-event time distribution of the same service is different in two cities, which is correlated with the popularity of the services. Our research is useful for the application of information diffusion, prediction of economic development of cities, and so on.

Point Process Models for Quasi-Periodic Volcanic Earthquakes

Low frequency or long period (LP) earthquakes are a common phenomenon at active volcanoes, and are ubiquitous at persistently active andesitic and dacitic subduction zone volcanoes. At these systems, LP earthquakes provide critical information regarding the state of volcanic unrest, and their occurrence rates are key data on which eruption forecasts are based. Point process modeling of volcanic earthquake occurrence allows potential insights into the underlying physical processes driving unrest, and quantitative, probabilistic, eruption forecasts, for example, through application of the Failure Forecast Method (FFM). However, unlike high-frequency volcano-tectonic (VT) earthquakes, which are typically random or clustered in time, LPs are more commonly quasi-periodic or \textit{anti-clustered}. Consequently, the existing Poisson point process methods used to model occurrence rates of VT earthquakes are unlikely to be optimal for LP data. Here we evaluate the performance of candidate inhomogeneous point process formulations of the FFM for quasi-periodic LP data, based on four different inter-event time distributions: exponential (for Poisson), gamma, inverse Gaussian, and Weibull. Using example LP data recorded before a large explosion at Tungurahua volcano, Ecuador, we examine how well these models explain the observed data, and the quality of retrospective forecasts of eruption time. We use a Markov chain Monte Carlo approach to estimate parameter posterior distributions within a Bayesian framework. Goodness-of-fit is assessed using Quantile-Quantile and Kolmogorov-Smirnov methods, and results are benchmarked against those obtained from idealized synthetic datasets. Inverse Gaussian and gamma models were both found to fit the data well, with the inverse Gaussian model slightly outperforming the gamma model. However, retrospective forecasting analysis shows that the gamma model performs best, with the initial preference for the inverse Gaussian model controlled by catalog incompleteness late in the sequence. The gamma model fits the data significantly better than the Poisson model, and simulations show it produces better forecasts for highly periodic data. Simulations also show that forecast precision increases with the degree of periodicity of the earthquake process using the gamma model, and so should be better for LP earthquakes than VTs. These results provide a new framework for point process modeling of volcanic earthquake time series, and verification of alternative models.

Aftershock sequences and seismic-like organization of acoustic events produced by a single propagating crack

Brittle fractures of inhomogeneous materials like rocks, concrete, or ceramics are of two types: Nominally brittle and driven by the propagation of a single dominant crack or quasi-brittle and resulting from the accumulation of many microcracks. The latter goes along with acoustic noise, whose analysis has revealed that events form aftershock sequences obeying characteristic laws reminiscent of those in seismology. Yet, their origin lacks explanation. Here we show that such a statistical organization is not only specific to the multi-cracking situations of quasi-brittle failure and seismology, but also rules the acoustic events produced by a propagating crack. This simpler situation has permitted us to relate these laws to the overall scale-free distribution of inter-event time and energy and to uncover their selection by the crack speed. These results provide a comprehensive picture of how acoustic events are organized upon material failure in the most fundamental of fracture states: single propagating cracks.

Limits of the memory coefficient in measuring correlated bursts.

Temporal inhomogeneities in event sequences of natural and social phenomena have been characterized in terms of interevent times and correlations between interevent times. The inhomogeneities of interevent times have been extensively studied, while the correlations between interevent times, often called correlated bursts, are far from being fully understood. For measuring the correlated bursts, two relevant approaches were suggested, i.e., memory coefficient and burst size distribution. Here a burst size denotes the number of events in a bursty train detected for a given time window. Empirical analyses have revealed that the larger memory coefficient tends to be associated with the heavier tail of the burst size distribution. In particular, empirical findings in human activities appear inconsistent, such that the memory coefficient is close to 0, while burst size distributions follow a power law. In order to comprehend these observations, by assuming the conditional independence between consecutive interevent times, we derive the analytical form of the memory coefficient as a function of parameters describing interevent time and burst size distributions. Our analytical result can explain the general tendency of the larger memory coefficient being associated with the heavier tail of burst size distribution. We also find that the apparently inconsistent observations in human activities are compatible with each other, indicating that the memory coefficient has limits to measure the correlated bursts.

Short-ranged memory model with preferential growth.

In this work we introduce a variant of the Yule-Simon model for preferential growth by incorporating a finite kernel to model the effects of bounded memory. We characterize the properties of the model combining analytical arguments with extensive numerical simulations. In particular, we analyze the lifetime and popularity distributions by mapping the model dynamics to corresponding Markov chains and branching processes, respectively. These distributions follow power laws with well-defined exponents that are within the range of the empirical data reported in ecologies. Interestingly, by varying the innovation rate, this simple out-of-equilibrium model exhibits many of the characteristics of a continuous phase transition and, around the critical point, it generates time series with power-law popularity, lifetime and interevent time distributions, and nontrivial temporal correlations, such as a bursty dynamics in analogy with the activity of solar flares. Our results suggest that an appropriate balance between innovation and oblivion rates could provide an explanatory framework for many of the properties commonly observed in many complex systems.

Stochastic modelling of earthquake interoccurrence times in Northwest Himalaya and adjoining regions

ABSTRACTThis work concentrates on the distribution of earthquake interevent times in northwest Himalaya and its adjacent regions. We consider 12 time-dependent probability distributions for analysis. The maximum likelihood estimation and Fisher information matrix-based methods are used to estimate model parameters and their respective uncertainties. Results from three model selection criteria suggest that the best fit arises from exponentiated Weibull, exponentiated exponential, Weibull and gamma distributions. An intermediate fit comes from exponentiated Rayleigh, and lognormal distributions, whereas the remaining distributions exhibit a poor fit to the seismic interoccurrence times of present catalogue. Using exponentiated Weibull model, it is observed that the estimated cumulative probability of magnitude 6.0 or higher event in the northwest Himalaya reaches 0.92–0.95 after about 20–23 (2019–2022) years since the last event in 1999. The conditional probability, for an elapsed time of 19 years (i.e. 2018), reaches 0.90–0.95 after about 20–25 (2038–2043) years from now. A series of conditional probability curves is also presented to understand the recent and future earthquake hazard in the study region. This temporal evolution of seismic interevent times may provide important clues to the underlying physical mechanism of earthquake genesis in the northwest Himalaya region.

A Gillespie Algorithm for Non-Markovian Stochastic Processes

The Gillespie algorithm provides statistically exact methods for simulating stochastic dynamics modelled as interacting sequences of discrete events including systems of biochemical reactions or earthquake occurrences, networks of queuing processes or spiking neurons, and epidemic and opinion formation processes on social networks. Empirically, the inter-event times of various phenomena obey long-tailed distributions. The Gillespie algorithm and its variants either assume Poisson processes (i.e., exponentially distributed inter-event times), use particular functions for time courses of the event rate, or work for non-Poissonian renewal processes, including the case of long-tailed distributions of inter-event times, but at a high computational cost. In the present study, we propose an innovative Gillespie algorithm for renewal processes on the basis of the Laplace transform. The algorithm makes use of the fact that a class of point processes is represented as a mixture of Poisson processes with different event rates. The method is applicable to multivariate renewal processes whose survival function of inter-event times is completely monotone. It is an exact algorithm and works faster than a recently proposed Gillespie algorithm for general renewal processes, which is exact only in the limit of infinitely many processes. We also propose a method to generate sequences of event times with a tunable amount of positive correlation between inter-event times. We demonstrate our algorithm with exact simulations of epidemic processes on networks, finding that a realistic amount of positive correlation in inter-event times only slightly affects the epidemic dynamics.

Dual-induced multifractality in online viewing activity.

Although recent studies have found that the long-term correlations relating to the fat-tailed distribution of inter-event times exist in human activity and that these correlations indicate the presence of fractality, the property of fractality and its origin have not been analyzed. We use both detrended fluctuation analysis and multifractal detrended fluctuation analysis to analyze the time series in online viewing activity separating from Movielens and Netflix. We find long-term correlations at both the individual and communal levels and that the extent of correlation at the individual level is determined by the activity level. These long-term correlations also indicate that there is fractality in the pattern of online viewing. We first find a multifractality that results from the combined effect of the fat-tailed distribution of inter-event times (i.e., the times between successive viewing actions of individuals) and the long-term correlations in online viewing activity and verify this finding using three synthesized series. Therefore, it can be concluded that the multifractality in online viewing activity is caused by both the fat-tailed distribution of inter-event times and the long-term correlations and that this enlarges the generic property of human activity to include not just physical space but also cyberspace.

Research Spotlights

The first of the two papers appearing in this issue's Research Spotlights section is “A Gillespie Algorithm for Non-Markovian Stochastic Processes,” by Naoki Masuda and Luis E. C. Rocha. This paper is concerned with stochastic dynamical modeling of interacting sequences of discrete events. Various applications include modeling of earthquake occurrences, modeling of networks of spiking neurons, and modeling of epidemic formation on networks. In this paper, the authors propose the Laplace Gillespie algorithm, which is distinguishable from previous variants of the Gillespie algorithm as a computationally efficient approach applicable to realistic, non-Poissonian renewal processes while being accurate for an arbitrary number of ongoing renewal processes. The easy-to-digest five-step algorithm is presented first. This is followed by presentation of the necessary and sufficient conditions for a renewal process to be simulated by the Laplace Gillespie algorithm and a long list of examples of distributions of interevent times for which the algorithm can be used. The authors demonstrate their new approach by performing exact simulations of an epidemic process. The codes used in the numerical simulations section are made available in the supplementary materials to readers who want to test this algorithm on their own application. The work in the second paper, “The Ghosts of Departed Quantities in Switches and Transitions,” by Mike R. Jeffrey, represents an addition to the literature on piecewise-dynamical systems theory. The author offers a derivation of and motivation for a new uniform switching model that allows for the study of dynamics at discontinuities. The key feature of the model is the inclusion of a term that is hidden outside the so-called switching layer. This term represents the ghosts, or artifacts, due to the transition, and it requires careful attention. The author shows how analyzing the hidden dynamics inside this layer from what is encoded in this term is extremely important to understanding local and global behavior of solutions. In one example, for instance, the author shows that even determining whether a system will cross through a switch cannot be done without considering the effects of nonlinearity at the switch. The paper concludes by offering the reader suggestions for future research and by making an appeal for the development of numerical solution methods aimed at “capturing the ghosts of switching” through the proposed model.

Modeling correlated bursts by the bursty-get-burstier mechanism.

Temporal correlations of time series or event sequences in natural and social phenomena have been characterized by power-law decaying autocorrelation functions with decaying exponent γ. Such temporal correlations can be understood in terms of power-law distributed interevent times with exponent α and/or correlations between interevent times. The latter, often called correlated bursts, has recently been studied by measuring power-law distributed bursty trains with exponent β. A scaling relation between α and γ has been established for the uncorrelated interevent times, while little is known about the effects of correlated interevent times on temporal correlations. In order to study these effects, we devise the bursty-get-burstier model for correlated bursts, by which one can tune the degree of correlations between interevent times, while keeping the same interevent time distribution. We numerically find that sufficiently strong correlations between interevent times could violate the scaling relation between α and γ for the uncorrelated case. A nontrivial dependence of γ on β is also found for some range of α. The implication of our results is discussed in terms of the hierarchical organization of bursty trains at various time scales.

Are triggering rates of labquakes universal? Inferring triggering rates from incomplete information

The acoustic emission activity associated with recent rock fracture experiments under different conditions has indicated that some features of event-event triggering are independent of the details of the experiment and the materials used and are often even indistinguishable from tectonic earthquakes. While the event-event triggering rates or aftershock rates behave pretty much identical for all rock fracture experiments at short times, this is not the case for later times. Here, we discuss how these differences can be a consequence of the aftershock identification method used and show that the true aftershock rates might have two distinct regimes. Specifically, tests on a modified Epidemic Type Aftershock Sequence model show that the model rates cannot be correctly inferred at late times based on temporal information only if the activity rates or the branching ratio are high. We also discuss both the effect of the two distinct regimes in the aftershock rates and the effect of the background rate on the inter-event time distribution. Our findings should be applicable for inferring event-event triggering rates for many other types of triggering and branching processes as well.

A mathematical model for the occurrence of historical events

A mathematical model was proposed for the frequency distribution of historical inter-event time τ. A basic ingredient was constructed by assuming the significance of a newly occurring historical event depending on the magnitude of a preceding event, the decrease of its significance by oblivion during the successive events, and an independent Poisson process for the occurrence of the event. The frequency distribution of τ was derived by integrating the basic ingredient with respect to all social fields and to all stake holders. The function of such a distribution was revealed as the forms of an exponential type, a power law type or an exponential-with-a-tail type depending on the values of constants appearing in the ingredient. The validity of this model was studied by applying it to the two cases of Modern China and Northern Ireland Troubles, where the τ-distribution varies depending on the different countries interacting with China and on the different stage of history of the Troubles, respectively. This indicates that history is consisted from many components with such different types of τ-distribution, which are the similar situation to the cases of other general human activities.

Global regionalized seismicity in view of Non-Extensive Statistical Physics

In the present work we study the distribution of Earth’s shallow seismicity on different seismic zones, as occurred from 1981 to 2011 and extracted from the Centroid Moment Tensor (CMT) catalog. Our analysis is based on the subdivision of the Earth’s surface into seismic zones that are homogeneous with regards to seismic activity and orientation of the predominant stress field. For this, we use the Flinn–Engdahl regionalization (FE) (Flinn and Engdahl, 1965), which consists of fifty seismic zones as modified by Lombardi and Marzocchi (2007). The latter authors grouped the 50 FE zones into larger tectonically homogeneous ones, utilizing the cumulative moment tensor method, resulting into thirty-nine seismic zones. In each one of these seismic zones we study the distribution of seismicity in terms of the frequency–magnitude distribution and the inter-event time distribution between successive earthquakes, a task that is essential for hazard assessments and to better understand the global and regional geodynamics. In our analysis we use non-extensive statistical physics (NESP), which seems to be one of the most adequate and promising methodological tools for analyzing complex systems, such as the Earth’s seismicity, introducing the q-exponential formulation as the expression of probability distribution function that maximizes the Sq entropy as defined by Tsallis, (1988). The qE parameter is significantly greater than one for all the seismic regions analyzed with value range from 1.294 to 1.504, indicating that magnitude correlations are particularly strong. Furthermore, the qT parameter shows some temporal correlations but variations with cut-off magnitude show greater temporal correlations when the smaller magnitude earthquakes are included. The qT for earthquakes with magnitude greater than 5 takes values from 1.043 to 1.353 and as we increase the cut-off magnitude to 5.5 and 6 the qT value ranges from 1.001 to 1.242 and from 1.001 to 1.181 respectively, presenting a significant decrease. Our findings support the ideas of universality within the Tsallis approach to describe Earth’s seismicity and present strong evidence ontemporal clustering and long-range correlations of seismicity in each of the tectonic zonesanalyzed.

Regularity of a renewal process estimated from binary data.

Assessment of the regularity of a sequence of events over time is important for clinical decision-making as well as informing public health policy. Our motivating example involves determining the effect of an intervention on the regularity of HIV self-testing behavior among high-risk individuals when exact self-testing times are not recorded. Assuming that these unobserved testing times follow a renewal process, the goals of this work are to develop suitable methods for estimating its distributional parameters when only the presence or absence of at least one event per subject in each of several observation windows is recorded. We propose two approaches to estimation and inference: a likelihood-based discrete survival model using only time to first event; and a potentially more efficient quasi-likelihood approach based on the forward recurrence time distribution using all available data. Regularity is quantified and estimated by the coefficient of variation (CV) of the interevent time distribution. Focusing on the gamma renewal process, where the shape parameter of the corresponding interevent time distribution has a monotone relationship with its CV, we conduct simulation studies to evaluate the performance of the proposed methods. We then apply them to our motivating example, concluding that the use of text message reminders significantly improves the regularity of self-testing, but not its frequency. A discussion on interesting directions for further research is provided.

The temporal event graph

Temporal networks are increasingly being used to model the interactions of complex systems. Most studies require the temporal aggregation of edges (or events) into discrete time steps to perform analysis. In this article we describe a static, lossless, and unique representation of a temporal network, the temporal event graph (TEG). The TEG describes the temporal network in terms of both the inter-event time and two-event temporal motif distributions. By considering these distributions in unison we provide a new method to characterise the behaviour of individuals and collectives in temporal networks as well as providing a natural decomposition of the network. We illustrate the utility of the TEG by providing examples on both synthetic and real temporal networks.

Temporal dynamics of online petitions.

Online petitions are an important avenue for direct political action, yet the dynamics that determine when a petition will be successful are not well understood. Here we analyze the temporal characteristics of online-petition signing behavior in order to identify systematic differences between popular petitions, which receive a high volume of signatures, and unpopular ones. We find that, in line with other temporal characterizations of human activity, the signing process is typically non-Poissonian and non-homogeneous in time. However, this process exhibits anomalously high memory for human activity, possibly indicating that synchronized external influence or contagion play and important role. More interestingly, we find clear differences in the characteristics of the inter-event time distributions depending on the total number of signatures that petitions receive, independently of the total duration of the petitions. Specifically, popular petitions that attract a large volume of signatures exhibit more variance in the distribution of inter-event times than unpopular petitions with only a few signatures, which could be considered an indication that the former are more bursty. However, petitions with large signature volume are less bursty according to measures that consider the time ordering of inter-event times. Our results, therefore, emphasize the importance of accounting for time ordering to characterize human activity.

Non-stationarity and internal correlations of the occurrence process of mining-induced seismic events

A point process, e.g., the seismic process, is potentially predictable when it is non-stationary, internally correlated or both. In this paper, an analysis of the occurrence process of mining-induced seismic events from Rudna copper mine in Poland is presented. Stationarity and internal correlation are investigated in complete seismic time series and segmentally in subseries demonstrating relatively stable seismicity rates. It is shown that the complete seismic series are non-stationary; however, most of their shorter subseries become stationary. In the stationary subseries, the distribution of interevent time is closer to the exponential distribution, which is characteristic for the Poisson process. However, in most of these subseries, the differences between the interevent time and Poisson distributions are still significant, revealing correlations among seismic events.

Analyzing the posting behaviors in news forums with incremental inter-event time

Online human behaviors are widely discussed in various fields. Three key factors, named priority, interest and memory are found crucial in human behaviors. Existing research mainly focuses on the identified and active users. However, the anonymous users and the inactive ones exist widely in news forums, whose behaviors do not receive enough attention. They cannot offer abundant postings like the others. It requires us to study posting behaviors of all the users including anonymous ones, identified ones, active ones and inactive ones in news forums only at the collective level. In this paper, the memory effects of the posting behaviors in news forums are investigated at the collective level. On the basis of the incremental inter-event time, a new model is proposed to describe the posting behaviors at the collective level. The results on twelve actual news events demonstrate the good performance of our model to describe the posting behaviors at the collective level in news forums. In addition, we find the symmetric incremental inter-event time distribution and the similar posting patterns in different durations.

Statistical analysis of the 2012–2013 Torreperogil–Sabiote seismic series, Spain

The study of earthquake swarms and their characteristics can improve our understanding of the transient processes that provoke seismic crises. The spatio-temporal process of the energy release is often linked with changes of statistical properties, and thus, seismicity parameters can help to reveal the underlying mechanism in time and space domains. Here, we study the Torreperogil–Sabiote 2012–2013 seismic series (southern Spain), which was relatively long lasting, and it was composed by more than 2000 events. The largest event was a magnitude 3.9 event which occurred on February 5, 2013. It caused slight damages, but it cannot explain the occurrence of the whole seismic crises which was not a typical mainshock–aftershock sequence. To shed some light on this swarm occurrence, we analyze the change of statistical properties during the evolution of the sequence, in particular, related to the magnitude and interevent time distributions. Furthermore, we fit a modified version of the epidemic type aftershock sequence (ETAS) model in order to investigate changes of the background rates and the trigger potential. Our results indicate that the sequence was driven by an aseismic transient stressing rate and that the system passes after the swarm occurrence to a new forcing regime with more typical tectonic characteristics.