Self-exciting Point Process Research Articles

Modeling Long-Range Dynamic Correlations of Words in Written Texts with Hawkes Processes

It has been clarified that words in written texts are classified into two groups called Type-I and Type-II words. The Type-I words are words that exhibit long-range dynamic correlations in written texts while the Type-II words do not show any type of dynamic correlations. Although the stochastic process of yielding Type-II words has been clarified to be a superposition of Poisson point processes with various intensities, there is no definitive model for Type-I words. In this study, we introduce a Hawkes process, which is known as a kind of self-exciting point process, as a candidate for the stochastic process that governs yielding Type-I words; i.e., the purpose of this study is to establish that the Hawkes process is useful to model occurrence patterns of Type-I words in real written texts. The relation between the Hawkes process and an existing model for Type-I words, in which hierarchical structures of written texts are considered to play a central role in yielding dynamic correlations, will also be discussed.

Inference of Intensity-Based Models for Load-Sharing Systems With Damage Accumulation

To model damage accumulation for load-sharing systems, two models given by intensity functions of self-exciting point processes are proposed: a model with additive damage accumulation and a model with multiplicative damage accumulation. Both models include the model without damage accumulation as a special case. For both models, the likelihood functions are derived and maximum likelihood estimators and likelihood ratio tests are given in a scale-invariant version and a scale-dependent version. Furthermore, a Bayesian approach using Markov chain Monte Carlo methods for posterior computation is provided. The frequentist and Bayesian methods are applied to a data set of failures of tension wires of concrete beams where a significant damage accumulation effect is confirmed by both additive and multiplicative damage accumulation models. This is all the more remarkable as a simulation study indicates that the tests for an existing damage accumulation effect are rather conservative. Moreover, prediction intervals for the failure times of the tension wires in a new experiment are given, which improve former prediction intervals derived without damage accumulation. The simulation study considers a scenario with a fixed time horizon and one with fixed numbers of failed components of the systems.

Self-exciting point process modelling of crimes on linear networks

Although there are recent developments for the analysis of first and second-order characteristics of point processes on networks, there are very few attempts in introducing models for network data. Motivated by the analysis of crime data in Bucaramanga (Colombia), we propose a spatiotemporal Hawkes point process model adapted to events living on linear networks. We first consider a non-parametric modelling strategy, for which we follow a non-parametric estimation of both the background and the triggering components. Then we consider a semi-parametric version, including a parametric estimation of the background based on covariates, and a non-parametric one of the triggering effects. Our model can be easily adapted to multi-type processes. Our network model outperforms a planar version, improving the fitting of the self-exciting point process model.

Non-Markovian Inverse Hawkes Processes

Hawkes processes are a class of self-exciting point processes with a clustering effect whose jump rate is determined by its past history. They are generally regarded as continuous-time processes and have been widely applied in a number of fields, such as insurance, finance, queueing, and statistics. The Hawkes model is generally non-Markovian because its future development depends on the timing of past events. However, it can be Markovian under certain circumstances. If the exciting function is an exponential function or a sum of exponential functions, the model can be Markovian with a generator of the model. In contrast to the general Hawkes processes, the inverse Hawkes process has some specific features and self-excitation indicates severity. Inverse Markovian Hawkes processes were introduced by Seol, who studied some asymptotic behaviors. An extended version of inverse Markovian Hawkes processes was also studied by Seol. With this paper, we propose a non-Markovian inverse Hawkes process, which is a more general inverse Hawkes process that features several existing models of self-exciting processes. In particular, we established both the law of large numbers (LLN) and Central limit theorems (CLT) for a newly considered non-Markovian inverse Hawkes process.

Parameter Estimation of Binned Hawkes Processes

A key difficulty that arises from real event data is imprecision in the recording of event time-stamps. In many cases, retaining event times with a high precision is expensive due to the sheer volume of activity. Combined with practical limits on the accuracy of measurements, binned data is common. In order to use point processes to model such event data, tools for handling parameter estimation are essential. Here we consider parameter estimation of the Hawkes process, a type of self-exciting point process that has found application in the modeling of financial stock markets, earthquakes and social media cascades. We develop a novel optimization approach to parameter estimation of binned Hawkes processes using a modified Expectation-Maximization algorithm, referred to as Binned Hawkes Expectation Maximization (BH-EM). Through a detailed simulation study, we demonstrate that existing methods are capable of producing severely biased and highly variable parameter estimates and that our novel BH-EM method significantly outperforms them in all studied circumstances. We further illustrate the performance on network flow (NetFlow) data between devices in a real large-scale computer network, to characterize triggering behavior. These results highlight the importance of correct handling of binned data. Supplementary materials for this article are available online.

Separating acoustic signal into underlying behaviors with self-exciting point process models

In animal communication, signals can arise endogenously or in response to cues, such as signals by conspecifics. When one signal serves dual functions, such as in birds that use the same song for mate attraction and territorial defense, the underlying reason for a vocalization cannot be determined without direct observations, and even then, may be hard to discern. We present an inhomogeneous, self-exciting point process model to estimate the underlying reasons for why an individual initiates a signal. In our application of these models, endogenous signals are assumed to arise at a constant rate, but each signal can also instigate (“self-excite”) additional signals by conspecific individuals. When applied to bullfrog (Rana catesbeiana) calls and ovenbird (Seiurus aurocapilla) songs, our model performs as well as a homogeneous point process model typically used to describe count data, while providing additional detail on the underlying motivations for signals. Although we apply the models to acoustic signals, our model can be applied to any self-exciting process and can be extended to include spatiotemporal dynamics in signals.

An ephemerally self-exciting point process

AbstractAcross a wide variety of applications, the self-exciting Hawkes process has been used to model phenomena in which the history of events influences future occurrences. However, there may be many situations in which the past events only influence the future as long as they remain active. For example, a person spreads a contagious disease only as long as they are contagious. In this paper, we define a novel generalization of the Hawkes process that we call theephemerally self-exciting process. In this new stochastic process, the excitement from one arrival lasts for a randomly drawn activity duration, hence the ephemerality. Our study includes exploration of the process itself as well as connections to well-known stochastic models such as branching processes, random walks, epidemics, preferential attachment, and Bayesian mixture models. Furthermore, we prove a batch scaling construction of general, marked Hawkes processes from a general ephemerally self-exciting model, and this novel limit theorem both provides insight into the Hawkes process and motivates the model contained herein as an attractive self-exciting process in its own right.

Hawkes Processes Framework With a Gamma Density As Excitation Function: Application to Natural Disasters for Insurance.

Hawkes processes are temporal self-exciting point processes. They are well established in earthquake modelling or finance and their application is spreading to diverse areas. Most models from the literature have two major drawbacks regarding their potential application to insurance. First, they use an exponentially-decaying form of excitation, which does not allow a delay between the occurrence of an event and its excitation effect on the process and does not fit well on insurance data consequently. Second, theoretical results developed from these models are valid only when time of observation tends to infinity, whereas the time horizon for an insurance use case is of several months or years. In this paper, we define a complete framework of Hawkes processes with a Gamma density excitation function (i.e. estimation, simulation, goodness-of-fit) instead of an exponential-decaying function and we demonstrate some mathematical properties (i.e. expectation, variance) about the transient regime of the process. We illustrate our results with real insurance data about natural disasters in Luxembourg.

Limit theorems for a discrete-time marked Hawkes process

Hawkes process is a self-exciting point process with wide applications in many fields, such as finance, seismology, and ecology. Hawkes processes are defined for the continuous-time setting. However, data is often recorded in a discrete-time or aggregated scheme. To model the temporal process in aggregated way, oftentimes a discrete-time type Hawkes process is more desirable. In this paper, we study the limit theorems for a discrete-time marked Hawkes process first proposed in Xu et al. (2020).

Modeling and predicting Chinese stock downside risks via Gaussian mixture models and marked self-exciting point process

The downside risks in stock markets may lead to huge losses and bring profound impact on businesses and governments. In order to model and predict these risks in Chinese stock markets, we propose using Gaussian mixture model to fit the risks within a given threshold, while fitting the extreme risks exceeding the threshold with a marked self-exciting point process. In the simulation study, we establish the consistencies of estimators and evaluate the performances of our proposed models by the accuracy of predicted VaR. Finally we apply our proposed models to analyze the behaviors of four real data sets. In order to illustrate our proposed approach can obtain an improved estimate for the VaR, CAViaR and the GARCH-EVT models are chosen for comparative analysis. The analysis shows that our proposed model have better performances than its competitors.

Common Growth Patterns for Regional Social Networks: A Point Process Approach

In this paper, we study macroscopic growth dynamics of social network link formation. Rather than focusing on one particular dataset, we find invariant behavior in regional social networks that are geographically concentrated. Empirical findings suggest that the startup phase of a regional network can be modeled by a self-exciting point process. After the startup phase ends, the growth of the links can be modeled by a non-homogeneous Poisson process with a constant rate across the day but varying rates from day to day, plus a nightly inactive period when local users are expected to be asleep. Conclusions are drawn based on analyzing four different datasets, three of which are regional and a non-regional one is included for contrast.

A self-exciting point process to study multicellular spatial signaling patterns

Multicellular organisms rely on spatial signaling among cells to drive their organization, development, and response to stimuli. Several models have been proposed to capture the behavior of spatial signaling in multicellular systems, but existing approaches fail to capture both the autonomous behavior of single cells and the interactions of a cell with its neighbors simultaneously. We propose a spatiotemporal model of dynamic cell signaling based on Hawkes processes-self-exciting point processes-that model the signaling processes within a cell and spatial couplings between cells. With this cellular point process (CPP), we capture both the single-cell pathway activation rate and the magnitude and duration of signaling between cells relative to their spatial location. Furthermore, our model captures tissues composed of heterogeneous cell types with different bursting rates and signaling behaviors across multiple signaling proteins. We apply our model to epithelial cell systems that exhibit a range of autonomous and spatial signaling behaviors basally and under pharmacological exposure. Our model identifies known drug-induced signaling deficits, characterizes signaling changes across a wound front, and generalizes to multichannel observations.

Recursive computation of the Hawkes cumulants

We propose a recursive method for the computation of the cumulants of self-exciting point processes of Hawkes type, based on standard combinatorial tools such as Bell polynomials. This closed-form approach is easier to implement on higher-order cumulants in comparison with existing methods based on differential equations, tree enumeration or martingale arguments. The results are corroborated by Monte Carlo simulations, and also apply to the computation of joint cumulants generated by multidimensional self-exciting processes.

Modelling Burglary in Chicago using a self-exciting point process with isotropic triggering

Self-exciting point processes have been proposed as models for the location of criminal events in space and time. Here we consider the case where the triggering function is isotropic and takes a non-parametric form that is determined from data. We pay special attention to normalisation issues and to the choice of spatial distance measure, thereby extending the current methodology. After validating these ideas on synthetic data, we perform inference and prediction tests on public domain burglary data from Chicago. We show that the algorithmic advances that we propose lead to improved predictive accuracy.

Accelerating the estimation of renewal Hawkes self-exciting point processes

The renewal Hawkes process is a nascent point process model that generalizes the Hawkes process. Although it has shown strong application potential, fitting the renewal Hawkes process to data remains a challenging task, especially on larger datasets. This article tackles this challenge by providing two approaches that significantly reduce the time required to fit renewal Hawkes processes. Since derivative-based methods for optimization, in general, converge faster than derivative-free methods, our first approach is to derive algorithms for evaluating the gradient and Hessian of the log-likelihood function and then use a derivative-based method, such as the Newton–Raphson method, in maximizing the likelihood, instead of the derivative-free method currently being used. Our second approach is to seek linear time algorithms that produce accurate approximations to the likelihood function, and then directly optimize the approximation to the log-likelihood function. Our simulation experiments show that the Newton–Raphson method reduces the computational time by about 30%. Furthermore, the approximate likelihood methods produce equally accurate estimates compared to the methods based on the exact likelihood and are about 20–40 times faster on datasets with about 10,000 events. We conclude with an analysis of price changes of several currencies relative to the US Dollar.

An extension of Hawkes processes with ephemeral nearest effects

Hawkes processes have been widely studied in both theory and applications because of their self-exciting effects, but there are still some questions and room left for exploring the self-exciting point processes. In this paper, an ephemeral self-exciting Hawkes process is developed, which contains 1-memory and 0-memory self-exciting point processes. After giving the definition of the ephemeral self-exciting Hawkes process, some explanations on this new point process, properties, declustering of the occurrences of events, positive and negative self-exciting effects are presented. Moments of this point process are derived as well in the paper by using the Laplace transforms and the elementary method. Some special cases are considered and the computation methods of moments are also presented by the difference method and inverse Laplace transforms, respectively. Combining dynamics of the self-exciting point process and the delayed renewal process is an important feature of the proposed ephemeral self-exciting Hawkes process, which may shed light on the related studies in the future.

On exploring feature representation learning of items to forecast their rise and fall in social media

User-item interactions in social media provide a rich dataset for wide applications such as viral marketing and recommender systems. Post retweeting behaviors and venue check-in events by users are the most representative. While existing studies predict items’ rise and fall, i.e., tweet popularity and venue closure detection, using hand-crafted features, this paper aims at exploring feature representation learning to improve prediction performance. We target at two essential time-series classification tasks on social media, including Shutdown Risk Prediction (SRP) of venues and Tweet Popularity Prediction (TPP) of posts. We study how feature representation learning of items can benefit both SRP and TPP tasks. The main idea is to learn item embedding vectors as features in item-item graphs constructed from time series of check-in events and retweeting behaviors. The learned features are used together with manually-defined features to enlarge the representation capability. In the TPP task, we also propose a pattern-aware self-exciting point process (PSEISMIC) model to generate time-series features. Experiments conducted on Instagram, Foursquare, and Twitter datasets exhibit promising performance of jointly utilizing learned and extracted features in both tasks. PSEISMIC can also further boost TPP accuracy. The major contribution of this work is three-fold. First, we propose to jointly deal with SRP and TPP under the same framework of feature extraction and learning. Second, we show that feature presentation learning of items can benefit these two prediction tasks with time series data. Third, by incorporating time series patterns, the proposed PSEISMIC further improves the performance of popularity prediction.

Fast estimation of multivariate spatiotemporal Hawkes processes and network reconstruction

We present a fast, accurate estimation method for multivariate Hawkes self-exciting point processes widely used in seismology, criminology, finance and other areas. There are two major ingredients. The first is an analytic derivation of exact maximum likelihood estimates of the nonparametric triggering density. We develop this for the multivariate case and add regularization to improve stability and robustness. The second is a moment-based method for the background rate and triggering matrix estimation, which is extended here for the spatiotemporal case. Our method combines them together in an efficient way, and we prove the consistency of this new approach. Extensive numerical experiments, with synthetic data and real-world social network data, show that our method improves the accuracy, scalability and computational efficiency of prevailing estimation approaches. Moreover, it greatly boosts the performance of Hawkes process-based models on social network reconstruction and helps to understand the spatiotemporal triggering dynamics over social media.

Network Self-Exciting Point Processes To Measure Health Impacts of COVID-19

The assessment of the health impacts of the COVID-19 pandemic requires the consideration of mobility networks. To this aim, we propose to augment spatio-temporal point process models with mobility network covariates. We show how the resulting model can be employed to predict contagion patterns and to help important decisions such as the distribution of vaccines. The application of the proposed methodology to 27 European countries shows that human mobility, along with vaccine doses and government policies, are significant predictors of the number of new COVID-19 reported infections and are therefore key variables for decision making.

Is Gang Violent Crime More Contagious than Non-Gang Violent Crime?

Gangs are thought to enhance participation in violence. It is expected then that gang-related violent crimes trigger additional crimes in a contagious manner, above and beyond what is typical for non-gang violent crime. This paper uses a multivariate self-exciting point process model to estimate the extent of contagious spread of violent crime for both gang-related and non-gang aggravated assaults and homicides in recent data from Los Angeles. The degree of contagious cross-triggering between gang-related and non-gang violent crime is also estimated. Gang-related violence triggers twice as many offspring events as non-gang violence and there is little or no cross-triggering. Gang-related offspring events are significantly more lethal than non-gang offspring events, but no more lethal than non-contagious background gang crimes. Contagious spread of gang-related violent crime is different from contagion in non-gang violence. The results support crime prevention policies that target the disruption of gang retaliations.