Self-exciting Point Process Model Research Articles

Modeling Terror Attacks with Self-Exciting Point Processes and Forecasting the Number of Terror Events.

Rampant terrorism poses a serious threat to the national security of many countries worldwide, particularly due to separatism and extreme nationalism. This paper focuses on the development and application of a temporal self-exciting point process model to the terror data of three countries: the US, Turkey, and the Philippines. To account for occurrences with the same time-stamp, this paper introduces the order mark and reward term in parameter selection. The reward term considers the triggering effect between events in the same time-stamp but different order. Additionally, this paper provides comparisons between the self-exciting models generated by day-based and month-based arrival times. Another highlight of this paper is the development of a model to predict the number of terror events using a combination of simulation and machine learning, specifically the random forest method, to achieve better predictions. This research offers an insightful approach to discover terror event patterns and forecast future occurrences of terror events, which may have practical application towards national security strategies.

Drug Treatment Effect Model Based on MODWT and Hawkes Self-Exciting Point Process.

In precision medicine, especially in the pharmacodynamic area, the lack of an adequate long-term drug effect monitoring model leads to a quite low robustness to the instant drug treatment. Modelling the effect of drug based on the monitoring variables is essential to measure the drug benefit and its side effect preciously. In order to model the complex drug behavior in the context of time series, a sin function is selected to describe the basic trend of heart rate variable that is medically monitored. A Hawkes self-exciting point process model is chosen to describe the effect caused by multiple and sequential drug usage at different time points. The model considers the time lag between the drug given time and the drug effect during the whole drug emission period. A cumulative Gamma distribution is employed to describe the time lag effect. Simulation results demonstrate the established model effectively when describing the baseline trend and the drug effect with low noise levels, where the maximal overlap discrete wavelet transformation is utilized for the information decomposition in the frequency zone. The real data of the variables heart rate and drug liquemin from a medical database is analyzed. Instead of the original time series, scale variable s4 is selected according to the Granger cointegration test. The results show that the model accurately characterizes the cumulative drug effect with the Pearson correlation test value as 0.22, which is more significant for the value under 0.1. In the future, the model can be extended to more complicated scenarios through taking into account multiple monitoring variables and different kinds of drugs.

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.

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.

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.

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.

Scalable high-resolution forecasting of sparse spatiotemporal events with kernel methods: A winning solution to the NIJ “Real-Time Crime Forecasting Challenge”

We propose a generic spatiotemporal event forecasting method which we developed for the National Institute of Justice’s (NIJ) Real-Time Crime Forecasting Challenge (National Institute of Justice (2017)). Our method is a spatiotemporal forecasting model combining scalable randomized Reproducing Kernel Hilbert Space (RKHS) methods for approximating Gaussian processes with autoregressive smoothing kernels in a regularized supervised learning framework. While the smoothing kernels capture the two main approaches in current use in the field of crime forecasting, kernel density estimation (KDE) and self-exciting point process (SEPP) models, the RKHS component of the model can be understood as an approximation to the popular log-Gaussian Cox Process model. For inference, we discretize the spatiotemporal point pattern and learn a log-intensity function using the Poisson likelihood and highly efficient gradient-based optimization methods. Model hyperparameters including quality of RKHS approximation, spatial and temporal kernel lengthscales, number of autoregressive lags and bandwidths for smoothing kernels as well as cell shape, size and rotation, were learned using cross validation. Resulting predictions significantly exceeded baseline KDE estimates and SEPP models for sparse events.

Crime prediction by data-driven Green’s function method

We develop an algorithm that forecasts cascading events, by employing a Green’s function scheme on the basis of the self-exciting point process model. This method is applied to open data of 10 types of crimes happened in Chicago. It shows a good prediction accuracy superior to or comparable to the standard methods which are the expectation–maximization method and prospective hotspot maps method. We find a cascade influence of the crimes that has a long-time, logarithmic tail; this result is consistent with an earlier study on burglaries. This long-tail feature cannot be reproduced by the other standard methods. In addition, a merit of the Green’s function method is the low computational cost in the case of high density of events and/or large amount of the training data.

Diffusion on Social Media Platforms: A Point Process Model for Interaction among Similar Content

Social media platforms disseminate a massive volume of user-generated content, some of which convey similar and overlapping information. We study how the diffusion of a given piece of content (called a cascade) is influenced by the diffusion of other cascades carrying similar content (called parallel cascades). We theorize that the diffusion of a cascade can be inhibited or amplified by that of parallel cascades containing similar content. To study this phenomenon, we formulate a generalized version of the self-exciting point process model and showcase a novel approach to evaluating the parallel diffusion of similar social media content. We estimate the model using Twitter data. We observe that, on average, the diffusion of a cascade is inhibited by the concurrent diffusion of parallel cascades with similar content. We further identify an asymmetry among content producers as the diffusion of content contributed by those with larger networks is more likely to be amplified by the diffusion of similar content. Our study underscores the importance of accounting for content similarity as failing to do so may overestimate assessments of a cascade’s diffusion. Our results also suggest that smaller, individual social media content contributors should avoid publishing repetitive content and channel their efforts towards developing novel content, while this is not a concern for larger content contributors.

Space–time inhomogeneous background intensity estimators for semi-parametric space–time self-exciting point process models

Histogram maximum likelihood estimators of semi-parametric space–time self-exciting point process models via expectation–maximization algorithm can be biased when the background process is inhomogeneous. We explore an alternative estimation method based on the variable bandwidth kernel density estimation (KDE) and EM algorithm. The proposed estimation method involves expanding the semi-parametric models by incorporating an inhomogeneous background process in space and time and applying the variable bandwidth KDE to estimate the background intensity function. Using an example, we show how the variable bandwidth KDE can be estimated this way. Two simulation examples based on residual analysis are designed to evaluate and validate the ability of our methods to recover the background intensity function and parametric triggering intensity function.

Handling missing data in self-exciting point process models

Self-exciting point processes have been applied to a wide variety of applications to understand event rates and clustering as a function of time and space. Typically, estimation procedures require a full temporal history of the data and do not handle cases where some of the history of the process is missing and unobserved. However, in many applications data collection is non-persistent resulting in known intervals of time where events of the process are unobserved. Motivated by these situations, a Bayesian estimation procedure for self-exciting point processes with missing histories is developed. The method naturally handles the missing data mechanism probabilistically through a specific step and is demonstrated on simulated data and a real conflict monitoring data where records over a period of time have been lost.

A Review of Self-Exciting Spatio-Temporal Point Processes and Their Applications

Self-exciting spatio-temporal point process models predict the rate of events as a function of space, time, and the previous history of events. These models naturally capture triggering and clustering behavior, and have been widely used in fields where spatio-temporal clustering of events is observed, such as earthquake modeling, infectious disease, and crime. In the past several decades, advances have been made in estimation, inference, simulation, and diagnostic tools for self-exciting point process models. In this review, I describe the basic theory, survey related estimation and inference techniques from each field, highlight several key applications, and suggest directions for future research.

Self-Exciting Point Processes with Spatial Covariates: Modelling the Dynamics of Crime

SummaryCrime has both varying patterns in space, related to features of the environment, economy and policing, and patterns in time arising from criminal behaviour, such as retaliation. Serious crimes may also be presaged by minor crimes of disorder. We demonstrate that these spatial and temporal patterns are generally confounded, requiring analyses to take both into account, and propose a spatiotemporal self-exciting point process model that incorporates spatial features, near repeat and retaliation effects, and triggering. We develop inference methods and diagnostic tools, such as residual maps, for this model, and through extensive simulation and crime data obtained from Pittsburgh, Pennsylvania, demonstrate its properties and usefulness.

Self-exciting point process models for political conflict forecasting

In 2008, the Defense Advanced Research Project Agency commissioned a database known as the Integrated Crisis Early Warning System to serve as the foundation for models capable of detecting and predicting increases in political conflict worldwide. Such models, by signalling expected increases in political conflict, would help inform and prepare policymakers to react accordingly to conflict proliferation both domestically and internationally. Using data from the Integrated Crisis Early Warning System, we construct and test a self-exciting point process, or Hawkes process, model to describe and predict amounts of domestic, political conflict; we focus on Colombia and Venezuela as examples for this model. By comparing the accuracy of fitted models to the observed data, we find that we are able to closely describe occurrences of conflict in each country. Thus, using this model can allow policymakers to anticipate relative increases in the amount of domestic political conflict following major events.

Seismic: A Self-Exciting Point Process Model for Predicting Tweet Popularity using Hashtags

Seismic: A Self-Exciting Point Process Model for Predicting Tweet Popularity using Hashtags

Improving the Robustness and Accuracy of Crime Prediction with the Self-Exciting Point Process Through Isotropic Triggering

The self-exciting point process (SEPP) is a model of the spread of crime in space and time, incorporating background and triggering processes. It shows promising predictive performance and forms the basis of a popular commercial software package, however few detailed case studies describing the application of the SEPP to crime data exist in the scientific literature. Using open crime data from the City of Chicago, USA, we apply the SEPP to crime prediction of assaults and burglaries in nine distinct geographical regions of the city. The results indicate that the algorithm is not robust to certain features of the data, generating unrealistic triggering functions in various cases. A simulation study is used to demonstrate that this outcome is associated with a reduction in predictive accuracy. Analysing the second-order spatial properties of the data demonstrates that the failures in the algorithm are correlated with anisotropy. A modified version of the SEPP model is developed in which triggering is non-directional. We show that this provides improved robustness, both in terms of the triggering structure and the predictive accuracy.

Modeling E-mail Networks and Inferring Leadership Using Self-Exciting Point Processes

ABSTRACTWe propose various self-exciting point process models for the times when e-mails are sent between individuals in a social network. Using an expectation–maximization (EM)-type approach, we fit these models to an e-mail network dataset from West Point Military Academy and the Enron e-mail dataset. We argue that the self-exciting models adequately capture major temporal clustering features in the data and perform better than traditional stationary Poisson models. We also investigate how accounting for diurnal and weekly trends in e-mail activity improves the overall fit to the observed network data. A motivation and application for fitting these self-exciting models is to use parameter estimates to characterize important e-mail communication behaviors such as the baseline sending rates, average reply rates, and average response times. A primary goal is to use these features, estimated from the self-exciting models, to infer the underlying leadership status of users in the West Point and Enron networks. Supplementary materials for this article are available online.

Nonparametric Estimation for Self-Exciting Point Processes—A Parsimonious Approach

There is ample evidence that in applications of self-exciting point-process models, the intensity of background events is often far from constant. If a constant background is imposed that assumption can reduce significantly the quality of statistical analysis, in problems as diverse as modeling the after-shocks of earthquakes and the study of ultra-high frequency financial data. Parametric models can be used to alleviate this problem, but they run the risk of distorting inference by misspecifying the nature of the background intensity function. On the other hand, a purely nonparametric approach to analysis leads to problems of identifiability; when a nonparametric approach is taken, not every aspect of the model can be identified from data recorded along a single observed sample path. In this article, we suggest overcoming this difficulty by using an approach based on the principle of parsimony, or Occam’s razor. In particular, we suggest taking the point-process intensity to be either a constant or to have maximum differential entropy, in cases where there is not sufficient empirical evidence to suggest that the background intensity function is more complex than those models. This approach is seldom, if ever, used for nonparametric function estimation in other settings, not least because in those cases more data are typically available. However, our “ontological parsimony” argument is appropriate in the context of self-exciting point-process models. Supplementary materials are available online.

A Point Process Framework for Predicting Popularity of Content in Online Social Networks

In this paper, we extend previous models to predict the popularity of user-generated content for online social networks (OSNs). We focus on Twitter as the most popular and widely used micro-blogging online social network, and specifically measure the popularity of a brand’s tweet by analyzing the time-series path of its subsequent activities (i.e. retweets, replies and marks as favorite) using multivariate marked Hawkes process models. Self and mutually exciting point process models as a multivariate class are adopted to simulate popularity growth patterns of brand post contents on Twitter. While previous self-exciting point process models aggregate the three types of events into one stream of information and therefore ignore exciting effects among different types of events, this study focuses on incorporating the event type into the predictive models for content popularity, explicitly looking at the effects of various types of users’ activities on Twitter. We aim to develop a mutually exciting point process model in order to incorporate such effects among different types of events. Our results suggest that the mutual-exciting point process model outperforms the self-exciting point process model to predict the popularity of online content, as it is able to capture mutual excitement effects and cross interactions between a sequence of events from one type to another. These results are useful in developing and executing a plan for online activities by the brand owners.

Modelling interregional links in electricity price spikes

Abnormally high price spikes in spot electricity markets represent a significant risk to market participants. As such, a literature has developed that focuses on forecasting the probability of such spike events, moving beyond simply forecasting the level of price. Many univariate time series models have been proposed to deal with spikes within an individual market region. This paper is the first to develop a multivariate self-exciting point process model for dealing with price spikes across connected regions in the Australian National Electricity Market. The importance of the physical infrastructure connecting the regions on the transmission of spikes is examined. It is found that spikes are transmitted between the regions, and the size of spikes is influenced by the available transmission capacity. It is also found that improved risk estimates are obtained when inter-regional linkages are taken into account.