Flexible multivariate spatio-temporal Hawkes process models of terrorism

Abstract

We develop flexible multivariate spatio-temporal Hawkes process models to analyze patterns of terrorism. Previous applications of point process methods to political violence data mainly utilize temporal Hawkes process models, neglecting spatial variation in these attack patterns. This limits what can be learned from these models as any effective counter-terrorism strategy requires knowledge on both when and where attacks are likely to occur. Even the existing work on spatio-temporal Hawkes processes imposes restrictions on the triggering function that are not well-suited for terrorism data. Therefore, we generalize the structure of the spatio-temporal triggering function considerably, allowing for nonseprability, nonstatitionarity, and cross-triggering (i.e., across the groups). To demonstrate the utility of our models, we analyze two samples of real-world terrorism data: Afghanistan (2002Afghanistan ( -2013 for univariate analysis and Nigeria (2009-2017) for bivariate analysis. Jointly, these two studies demonstrate that our models outperform standard Hawkes process models, besting widely-used alternatives in overall model fit and revealing spatio-temporal patterns that are, by construction, masked in these models (e.g., increasing dispersion in cross-triggering over time).

On minimum contrast method for multivariate spatial point processes

“…In this section, we apply our methods to the point pattern data of 2014 terrorism attacks in Nigeria that are considered in Zhu et al (2021) and Jun and Cook (2022).…”

“…A multivariate spatial point pattern is comprised of a random number of events at random locations in a given spatial region D ⊂ R d (where d is usually set to 2 or 3), where each event belongs to one of a finite number of distinct "types" (Cox and Lewis, 1972). These "presence only" data have become increasingly common, with multivariate spatial point patterns now found in a variety of disciplines including epidemiology (Lawson, 2012), neuroscience (Baddeley et al, 2014), ecology (Waagepetersen et al, 2016), meteorology (Jun et al, 2019), and political science (Jun and Cook, 2022). Therefore, to provide insights of the spatial arrangement of multi-type events that appears in many real life applications, it is important to develop an effective statistical method to analyze the multivariate spatial point process-the underlying stochastic mechanism which generates the (observed) multivariate spatial point pattern.…”

Flexible Spatio-Temporal Hawkes Process Models for Earthquake Occurrences

“…We can also observe that the triggering effect is diminishing as time passes due to the structure assumed for g. Hawkes processes have a wide range of applications including crime occurrences (Mohler et al, 2011;Zhu and Xie, 2022), terrorism (Jun and Cook, 2022), social media (Yuan et al, 2019), and infectious disease such as COVID-19 (Browning et al, 2021). For earthquakes applications, Ogata (1988) suggested the Epidemic-Type Aftershock Sequence (ETAS) model, which is a temporal Hawkes process for earthquake occurrences.…”

Capturing the Spatio-Temporal Diffusion Effects of Armed Conflict: A Non-parametric Smoothing Approach

“…We utilize a set of exponential decay functions with varying decay rates in our smoothing basis, in order establish an implicit bias towards fitting an exponentially decaying effect of past conflict over both time and space (as so far observed in the literature (Mueller et al, 2022)), while still allowing for other patterns if applicable. As a result, our proposed model has similarities to a spatio-temporal Hawkes process (Reinhart, 2018;Jun and Cook, 2022), a selfexciting point-process with an underlying Poisson intensity, that typically employs exponentially decaying triggering functions across time and space, which is increasingly used in epidemiology (Meyer et al, 2012;Schoenberg et al, 2019) and to model the dynamics of crime (Mohler, 2014;Reinhart, 2018;Reinhart and Greenhouse, 2018). Note, that both the temporal as well as spatial resolution of conflict data is limited.…”