Inter-event Time Distribution Research Articles

Temporal scaling theory for bursty time series with clusters of arbitrarily many events.

Long-term temporal correlations in time series in a form of an event sequence have been characterized using an autocorrelation function that often shows a power-law decaying behavior. Such scaling behavior has been mainly accounted for by the heavy-tailed distribution of interevent times, i.e., the time interval between two consecutive events. Yet, little is known about how correlations between consecutive interevent times systematically affect the decaying behavior of the autocorrelation function. Empirical distributions of the burst size, which is the number of events in a cluster of events occurring in a short time window, often show heavy tails, implying that arbitrarily many consecutive interevent times may be correlated with each other. In the present study, we propose a model for generating a time series with arbitrary functional forms of interevent time and burst size distributions. Then, we analytically derive the autocorrelation function for the model time series. In particular, by assuming that the interevent time and burst size are power-law distributed, we derive scaling relations between power-law exponents of the autocorrelation function decay, interevent time distribution, and burst size distribution. These analytical results are confirmed by numerical simulations. Our approach helps to rigorously and analytically understand the effects of correlations between arbitrarily many consecutive interevent times on the decaying behavior of the autocorrelation function.

Unsupervised clustering of catalogue-driven features for characterizing temporal evolution of labquake stress

SUMMARY Earthquake forecasting poses significant challenges, especially due to the elusive nature of stress states in fault systems. To tackle this problem, we use features derived from seismic catalogues obtained from acoustic emission (AE) signals recorded during triaxial stick-slip experiments on natural fractures in three Westerly granite samples. We extracted 47 physically explainable features from AE data that described spatio-temporal evolution of stress and damage in the vicinity of the fault surface. These features are then subjected to unsupervised clustering using the K-means method, revealing three distinct stages with a proper agreement with the temporal evolution of stress. The recovered stages correspond to the mechanical behaviour of the rock, characterized as initial stable (elastic) deformation, followed by a transitional stage leading to an unstable deformation prior to failure. Notably, AE rate, clustering-localization features, fractal dimension, b-value, interevent time distribution, and correlation integral are identified as significant features for the unsupervised clustering. The systematically evolving stages can provide valuable insights for characterizing preparatory processes preceding earthquake events associated with geothermal activities and waste-water injections. In order to address the upscaling issue, we propose to use the most important features and, in case of normalization challenge, removing non-universal features, such as AE rate. Our findings hold promise for advancing earthquake prediction methodologies based on laboratory experiments and catalogue-driven features.

Constructing temporal networks with bursty activity patterns

Human social interactions tend to vary in intensity over time, whether they are in person or online. Variable rates of interaction in structured populations can be described by networks with the time-varying activity of links and nodes. One of the key statistics to summarize temporal patterns is the inter-event time, namely the duration between successive pairwise interactions. Empirical studies have found inter-event time distributions that are heavy-tailed, for both physical and digital interactions. But it is difficult to construct theoretical models of time-varying activity on a network that reproduce the burstiness seen in empirical data. Here we develop a spanning-tree method to construct temporal networks and activity patterns with bursty behavior. Our method ensures any desired target inter-event time distributions for individual nodes and links, provided the distributions fulfill a consistency condition, regardless of whether the underlying topology is static or time-varying. We show that this model can reproduce burstiness found in empirical datasets, and so it may serve as a basis for studying dynamic processes in real-world bursty interactions.

Statistics of the number of renewals, occupation times, and correlation in ordinary, equilibrium, and aging alternating renewal processes.

The renewal process is a point process where an interevent time between successive renewals is an independent and identically distributed random variable. Alternating renewal process is a dichotomous process and a slight generalization of the renewal process, where the interevent time distribution alternates between two distributions. We investigate statistical properties of the number of renewals and occupation times for one of the two states in alternating renewal processes. When both means of the interevent times are finite, the alternating renewal process can reach an equilibrium. On the other hand, an alternating renewal process shows aging when one of the means diverges. We provide analytical calculations for the moments of the number of renewals, occupation time statistics, and the correlation function for several case studies in the interevent-time distributions. We show anomalous fluctuations for the number of renewals and occupation times when the second moment of interevent time diverges. When the mean interevent time diverges, distributional limit theorems for the number of events and occupation times are shown analytically. These are known as the Mittag-Leffler distribution and the generalized arcsine law in probability theory.

Amplitude and inter-event time statistics for the island volcanoes Stromboli, Mount Etna, Yasur, and Whakaari

Detailed analyses of past major and minor seismo-volcanic events can help to understand the eruptive behavior of volcanoes and the underlying physical and chemical processes. Catalogs of these eruptions and, specifically, seismo-volcanic events may be generated using continuous seismic recordings at stations in the proximity of volcanoes. Here, we apply a recently-developed automated approach Adaptive-Window Volcanic Event Selection Analysis Module (AWESAM) to seismic data from Stromboli (Italy), Mount Etna (Italy), Yasur (Vanuatu) and Whakaari (New Zealand). We perform an inter-event time analysis to identify characteristic patterns in the events’ recurrence time and the volcanic activity. Using this identical approach for all volcanoes, we were able to discover that despite their differing types and activity, they exhibit similar statistical behaviors. For Whakaari, we noticed a bimodal inter-event time distribution for large events. Since this observation is based on single station data, further in-depth investigations are needed once more data is available in future. We also derive a new amplitude-frequency relationship from seismo-volcanic events. With this relation, we can confirm a change in slope for large events at Stromboli, which is based on 10 years of data. Additionally, we apply a classification model to events from Stromboli to differentiate between low-period (LP) events and high-frequency (HF) events and found an alternating behavior in the frequency of these events before and after the two paroxysms in 2019.

Non-Markovian epidemic spreading on temporal networks

Many empirical studies have revealed that the occurrences of contacts associated with human activities are non-Markovian temporal processes with a heavy tailed inter-event time distribution. Besides, there has been increasing empirical evidence that the infection and recovery rates are time-dependent. However, we lack a comprehensive framework to analyze and understand non-Markovian contact and spreading processes on temporal networks. In this paper, we propose a general formalism to study non-Markovian dynamics on non-Markovian temporal networks. We find that, under certain conditions, non-Markovian dynamics on temporal networks are equivalent to Markovian dynamics on static networks. Interestingly, this result is independent of the underlying network topology.

Continuous-flow simulation of manufacturing systems with assembly/disassembly machines, multiple loops and general layout

Performance evaluation methods are important to design and control manufacturing systems. Approximate analytical methods are fast, but they may be limited by the restrictive assumptions on the system. On the contrary, simulation has not specific limitations in its applicability, but the time to model and analyse a manufacturing system can increase as the level of detail addressed by the model increases. The main contribution of this study is presenting a computationally efficient methodology to simulate single-part continuous-flow manufacturing systems with assembly/disassembly machines, multiple loops, general layout and general inter-event time distributions. By using graph theory, a new method is presented to identify the machines causing slowdown, blocking and starvation in a general layout and determine the time before the occurrence of a state transition for each machine and the time before the fulfilment or depletion of each buffer. By advancing the time clock to the next event-time accordingly, the number of discrete events needed to be simulated is decreased compared to a discrete-event simulation with discrete flow of parts. As a result, the proposed method is on average 15 times faster than DES methods in the analysis of discrete-flow systems, and 110 times faster on average in the analysis of continuous-flow systems. The low computational time of the proposed method allows to simulate systems under general assumptions and in a very short time.

Acoustic Emissions in Rock Deformation and Failure: New Insights from Q-Statistical Analysis

We propose a new statistical analysis of the Acoustic Emissions (AE) produced in a series of triaxial deformation experiments leading to fractures and failure of two different rocks, namely, Darley Dale Sandstone (DDS) and AG Granite (AG). By means of q-statistical formalism, we are able to characterize the pre-failure processes in both types of rocks. In particular, we study AE inter-event time and AE inter-event distance distributions. Both of them can be reproduced with q-exponential curves, showing universal features that are observed here for the first time and could be important in order to understand more in detail the dynamics of rock fractures.

Complexity of Recent Earthquake Swarms in Greece in Terms of Non-Extensive Statistical Physics.

Greece exhibits the highest seismic activity in Europe, manifested in intense seismicity with large magnitude events and frequent earthquake swarms. In the present work, we analyzed the spatiotemporal properties of recent earthquake swarms that occurred in the broader area of Greece using the Non-Extensive Statistical Physics (NESP) framework, which appears suitable for studying complex systems. The behavior of complex systems, where multifractality and strong correlations among the elements of the system exist, as in tectonic and volcanic environments, can adequately be described by Tsallis entropy (Sq), introducing the Q-exponential function and the entropic parameter q that expresses the degree of non-additivity of the system. Herein, we focus the analysis on the 2007 Trichonis Lake, the 2016 Western Crete, the 2021-2022 Nisyros, the 2021-2022 Thiva and the 2022 Pagasetic Gulf earthquake swarms. Using the seismicity catalogs for each swarm, we investigate the inter-event time (T) and distance (D) distributions with the Q-exponential function, providing the qT and qD entropic parameters. The results show that qT varies from 1.44 to 1.58, whereas qD ranges from 0.46 to 0.75 for the inter-event time and distance distributions, respectively. Furthermore, we describe the frequency-magnitude distributions with the Gutenberg-Richter scaling relation and the fragment-asperity model of earthquake interactions derived within the NESP framework. The results of the analysis indicate that the statistical properties of earthquake swarms can be successfully reproduced by means of NESP and confirm the complexity and non-additivity of the spatiotemporal evolution of seismicity. Finally, the superstatistics approach, which is closely connected to NESP and is based on a superposition of ordinary local equilibrium statistical mechanics, is further used to discuss the temporal patterns of the earthquake evolution during the swarms.

The role of three-dimensional fault interactions in creating complex seismic sequences

A physics-based earthquake simulator should reproduce first-order empirical power-law behaviors of magnitudes and clustering. These laws have emerged spontaneously in either discrete or low-dimension continuum simulations without power-law or stochastic heterogeneity. We show that the same emergence can occur in 3-D continuum simulations with fault interactions and rate-and-state friction. Our model approximates a strike-slip fault system as three en echelon faults. Simulations show spatio-temporally clustered earthquake sequences exhibiting characteristic Gutenberg-Richter scaling as well as empirical inter-event time distribution. The Gutenberg-Richter scaling occurs only in partial ruptures that result from fault interactions. With fault interactions, partial ruptures emerge when seismogenic width W over characteristic nucleation length L∞ is larger than 16.24, but none occur without fault interaction. The mainshock recurrence times of individual faults remain quasi-periodic. The system mainshock recurrence time is a combination of short-term Omori-type decay and Brownian passage time. Higher W/L∞ increase short-term clustering probability to at most 30%. These results indicate that physics-based multi-cycle models adequately reflect observed statistical signatures and show practical potential for long-term hazard assessment and medium-term forecasting.

A Non-Extensive Statistical Physics View of the Temporal Properties of the Recent Aftershock Sequences of Strong Earthquakes in Greece

Greece is one of Europe’s most seismically active areas. Seismic activity in Greece has been characterized by a series of strong earthquakes with magnitudes up to Mw = 7.0 over the last five years. In this article we focus on these strong events, namely the Mw6.0 Arkalochori (27 September 2021), the Mw6.3 Elassona (3 March 2021), the Mw7.0 Samos (30 October 2020), the Mw5.1 Parnitha (19 July 2019), the Mw6.6 Zakynthos (25 October 2018), the Mw6.5 Kos (20 July 2017) and the Mw6.1 Mytilene (12 June 2017) earthquakes. Based on the probability distributions of interevent times between the successive aftershock events, we investigate the temporal evolution of their aftershock sequences. We use a statistical mechanics model developed in the framework of Non-Extensive Statistical Physics (NESP) to approach the observed distributions. NESP provides a strictly necessary generalization of Boltzmann–Gibbs statistical mechanics for complex systems with memory effects, (multi)fractal geometries, and long-range interactions. We show how the NESP applicable to the temporal evolution of recent aftershock sequences in Greece, as well as the existence of a crossover behavior from power-law (q ≠ 1) to exponential (q = 1) scaling for longer interevent times. The observed behavior is further discussed in terms of superstatistics. In this way a stochastic mechanism with memory effects that can produce the observed scaling behavior is demonstrated. To conclude, seismic activity in Greece presents a series of significant earthquakes over the last five years. We focus on strong earthquakes, and we study the temporal evolution of aftershock sequences of them using a statistical mechanics model. The non-extensive parameter q related with the interevent times distribution varies between 1.62 and 1.71, which suggests a system with about one degree of freedom.

On the Structure of the Intermittency of Rainfall

Quantification of rainfall intermittency via. interevent time distribution, series of continuous wet spells (burst size) and variability in interevent times between rainfall events is essential for planning and management of water resources and hydrologic extremes. However, their structure, quantification and association with long-term climatology are less explored. In this paper, a complex system-based measure – burstiness – is used to quantify the variability of interevent times across six meteorologically homogenous zones of India. It is observed that burstiness is related to the burst size as well as long-term rainfall climatology. The existence of unimodal and bimodal structure in burstiness distribution reveals the uniqueness and differences in the rainfall patterns. The differences in sensitivity of rainfall to burstiness highlight the role of interplay between climate-landscape and reveals the importance of the intermittent structure of rainfall. The study provides an approach to model intermittency by preserving the temporal structure of the interevent time distribution.

Burst and Memory-aware Transformer: capturing temporal heterogeneity.

Burst patterns, characterized by their temporal heterogeneity, have been observed across a wide range of domains, encompassing event sequences from neuronal firing to various facets of human activities. Recent research on predicting event sequences leveraged a Transformer based on the Hawkes process, incorporating a self-attention mechanism to capture long-term temporal dependencies. To effectively handle bursty temporal patterns, we propose a Burst and Memory-aware Transformer (BMT) model, designed to explicitly address temporal heterogeneity. The BMT model embeds the burstiness and memory coefficient into the self-attention module, enhancing the learning process with insights derived from the bursty patterns. Furthermore, we employed a novel loss function designed to optimize the burstiness and memory coefficient values, as well as their corresponding discretized one-hot vectors, both individually and jointly. Numerical experiments conducted on diverse synthetic and real-world datasets demonstrated the outstanding performance of the BMT model in terms of accurately predicting event times and intensity functions compared to existing models and control groups. In particular, the BMT model exhibits remarkable performance for temporally heterogeneous data, such as those with power-law inter-event time distributions. Our findings suggest that the incorporation of burst-related parameters assists the Transformer in comprehending heterogeneous event sequences, leading to an enhanced predictive performance.

Inter-event time distribution pattern of earthquakes and its spatial, temporal, and magnitude sensitivity: the case of Turkey

Inter-event time distribution pattern of earthquakes and its spatial, temporal, and magnitude sensitivity: the case of Turkey

Forecasting tectonic tremor activity using a renewal process model

In many tectonically active regions of the world, a variety of slow deformation phenomena have been discovered and collectively termed slow earthquakes. Tectonic tremor is the high-frequency component of slow earthquakes and can be analyzed to monitor the overall slow deformation process, both spatially and temporally. Although tectonic tremor activity is complex, it does possess some characteristic patterns, such as spatial segmentation, a quasi-periodic recurrence, migration, and tidal modulation. These features are helpful for forecasting future activity if they are properly modeled in a quantitative manner. Here, we propose a stochastic renewal process to standardize and forecast tectonic tremor activity in the Nankai subduction zone, southwest Japan, using a 12.5-year tremor catalog that is divided into a 10-year estimation period and 2.5-year forecasting period. We group the tremor events into small rectangular 10-km regions and observe that the distribution of inter-event times is nearly bimodal, with the short and long inter-event times representing the characteristic times of nearby tremor interactions and long-term stress accumulation processes, respectively. Therefore, as the probabilistic distribution for the renewal process, we adopt a mixture distribution of log-normal and Brownian passage time distributions for the short and long inter-event times, respectively. The model parameters are successfully estimated for 72% of the entire tremor zone using a maximum likelihood method. This standard model can be used to extract anomalous tremor activity, such as that associated with long-term slow-slip events. We derive a scaling relationship between two characteristic times, the relative plate motion, episodicity of tremor activity, and tremor duration by characterizing the spatial differences in tremor activity. We confirm that the model can forecast the occurrence of the next tremor event at a given reference time for a certain prediction interval. This study can serve as a first step for implementing more complex models to improve the space–time forecasting of slow earthquakes.

On the Patterns and Scaling Properties of the 2021–2022 Arkalochori Earthquake Sequence (Central Crete, Greece) Based on Seismological, Geophysical and Satellite Observations

The 27 September 2021 damaging mainshock (Mw6.0) close to Arkalochori village is the strongest earthquake that was recorded during the instrumental period of seismicity in Central Crete (Greece). The mainshock was preceded by a significant number of foreshocks that lasted nearly four months. Maximum ground subsidence of about 18 cm was estimated from InSAR processing. The aftershock sequence is located in an almost NE-SW direction and divided into two main clusters, the southern and the northern ones. The foreshock activity, the deformation area, and the strongest aftershocks are located within the southern cluster. Based on body-wave travel times, a 3-D velocity model was developed, while using combined space and ground-based geodetic techniques, the co-seismic ground deformation is presented. Moreover, we examined the co-seismic static stress changes with respect to the aftershocks’ spatial distribution during the major events of the foreshocks, the Mw = 6.0 main event as well as the largest aftershock. Both the foreshock and the aftershock sequences obey the scaling law for the frequency-magnitude distribution as derived from the framework of non-extensive statistical physics (NESP). The aftershock production rate decays according to the modified Omori scaling law, exhibiting various Omori regimes due to the generation of secondary aftershock sequences. The analysis of the inter-event time distribution, based on NESP, further indicates asymptotic power-law scaling and long-range correlations among the events. The spatiotemporal evolution of the aftershock sequence indicates triggering by co-seismic stress transfer, while its slow migration towards the outer edges of the area of the aftershocks, related to the logarithm of time, further indicates a possible afterslip.

Scaling laws of inter-event time casualties of narco-war in Mexico

From the official report of narco-war-related casualties in Mexico from December 2006 to September 2011, we show that the inter-event time distribution (calculated for a range of minimum sizes) approximately obeys simple scaling laws similar to violent conflicts in Iraq (2003–2005), Afghanistan (2008–2010) and Northern Ireland (1969–2001). Furthermore, normalizing deaths by population municipalities (the smallest Mexican political entities) yields even better fitting results.

Implications of the statistics of seismicity recorded within the Groningen gas field

Abstract We reanalysed the induced seismicity data from the Groningen gas reservoir. We used the well-maintained induced event catalogue of the KNMI. The distributions of seismic moments and interevent times show a power law behaviour over several decades, and we find that upon increasing the magnitude threshold, these distributions remained scale-invariant. Because of this scale-invariance, we can put a constraint on the average loading of the elastic energy within the reservoir, which upon reaching a critical value gives rise to the seismic events. We find that the elastic energy roughly increases proportional to time. We also propose a new machine learning approach for declustering the seismic events, separating correlated from independent events. We find that only few events are truly independent, i.e. exponentially distributed over time. There are also few aftershocks following an Omori-type power law. The bulk of events presents a Gamma distribution for interevent times. This gives us an indication that the rigidity in the reservoir is high, but whether this results in overall correlated events should be settled with physics-based arguments rather than statistical ones.

Burstiness of Human Physical Activities and Their Characterization

Human behavior is known to be heterogeneous and fluctuates temporally. Many studies have focused on the fluctuation of Inter-Event Times (IETs), the times between one action and the next in various types of human behavior. The existence of this fluctuation cannot be explained by a stationary random Poisson process and is called burstiness. In this study, we collected human physical activity data while specifying the age of the subjects and their situations (e.g., children's play and adults' housework) and analyzed their event time series. We confirmed that burstiness both in children and adults. Especially, burstiness in the physical activity of 2-3 year children is, for the first time as far as we know, observed. We also confirmed that each activity situation has its own specific IETs distribution characteristics. Our results may provide important clues to identifying the mechanisms of burstiness in human physical activities.

Poisson Behavior Leads to Bias When Testing for Periodicity in the Paleoseismic Record of Large Earthquakes

Abstract A Poisson process giving rise to earthquakes that occur randomly in time has become a de facto null hypothesis when assessing the periodicity of large (Mw>∼6–7) earthquakes in the paleoseismic record. This implies an exponential distribution of inter-event times (IETs) and therefore an abundance of IETs that are very short relative to the mean value. As such, the Poisson model posits that large ruptures occurring in rapid succession should be relatively common. Below some threshold IET defined by site specific conditions, however, these short IET earthquakes are unlikely to be recorded as distinct events in the paleoseismic record. This article presents the results of simple Monte Carlo simulations that quantify the potential effects of truncation of short IETs on the apparent periodicity of large earthquakes generated by a Poisson process. Results indicate that this truncation effect results in chronologies that appear systematically more periodic than the original sequence of events. The magnitude of this discrepancy depends primarily on the ratio of the minimum preserved and mean IETs in addition to the number of events in the chronology. As such, previous statistical analyses that have assessed for periodicity in the paleoseismic record of large earthquakes likely incorporated a bias in favor of apparent periodicity by employing Poisson behavior as a null hypothesis. This bias can be corrected if the minimum preserved IET can be determined or reasonably assumed for a particular paleoseismic record or site.