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

Abstract

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.

Insider Threat Detection via Hierarchical Neural Temporal Point Processes

“…A temporal point process (TPP) is a stochastic process composed of a time series of events that occur in continuous time [16]. The temporal point process is widely used for modeling the sequence data with time information, such as health-care analysis, earthquakes and aftershocks modeling and social network analysis [9], [17], [18]. The traditional methods of temporal point processes usually make parametric assumptions about how the observed events are generated, e.g., by Poisson processes or self-exciting point processes.…”

“…In literature, the marked temporal point process (MTPP) is a general mathematical framework to model the event time and type information of a sequence. It has been widely used for predicting the earthquakes and aftershocks [9]. The traditional MTPP models make assumptions about how the events occur, which may be violated in reality.…”

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

“…Simulations from self-exciting point process models have proved useful both for examining inference and for the simulation studies that are discussed in Sections 2 and 4. Various simulation algorithms for self-exciting point processes have been discussed by Reinhart (2018), section 3.3. We chose to use an algorithm that was introduced by Zhuang et al (2004) for earthquake aftershock sequence models, which is fast and efficient for our model structure.…”

“…We adapted the expectation-maximization procedure to fit our extended model. The expectation-maximization procedure for self-exciting processes was first described by Veen and Schoenberg (2008) and follows the general procedure that was described by Reinhart (2018), section 3.1. A latent variable u i is introduced for each event i, indicating whether the event came from the background process (u i = 0) or was triggered by a previous event j (u i = j).…”

A Semiparametric Spatiotemporal Hawkes-Type Point Process Model with Periodic Background for Crime Data

“…In other words, once an event occurs in the process, no matter whether it is a background event or an event excited by others, it excites a process of its own direct offspring according to some probability rules. Many powerful tools have been developed for the Hawkes process, such as stochastic declustering, stochastic reconstruction, the expectation-maximization algorithm, first-and higher order residuals, and Bayesian analysis, as well as the theories that are associated with the asymptotic properties (see the review by Reinhart (2018)) The most common tools to predict crimes include 'hotspotting' (e.g. Bowers et al (2004), Ratcliffe (2004) and Levine (2017)), 'near repeats' (e.g.…”