Investigating Clustering and Violence Interruption in Gang-Related Violent Crime Data Using Spatial–Temporal Point Processes With Covariates

Abstract

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.Keywords Nonparametric estimation • L 2 regularization • Point processes • Social network • Cumulants Fast estimation of Hawkes processes.

Beach-level 24-hour forecasts of Florida red tide-induced respiratory irritation

“…Hawkes processes have been used to model a remarkable range of phenomena. From earthquakes (Ogata, 1988;Türkyilmaz et al, 2013), to gang related violence (Mohler et al, 2011;Park et al, 2021) to econometrics (Bacry et al, 2013;Errais et al, 2010). This type of model performs well in situations where there is evidence for a clustering of events in time.…”

Beach-level 24-hour forecasts of Florida red tide-induced respiratory irritation

“…Hawkes processes have been used to model a remarkable range of phenomena. From earthquakes [26,27], to gang related violence [28,29] to econometrics [30,31]. This type of model performs well in situations where there is evidence for a clustering of events in time.…”

ETAS Space–Time Modeling of Chile Triggered Seismicity Using Covariates: Some Preliminary Results

“…In particular, based on the methodology proposed by [15], we suggest the use of a specific branching-type model for earthquake description (the ETAS model) in a regression-oriented version, accounting also for external covariates, related to the depth of events and to some GPS measurements. For further examples on applications incorporating the external information in self-exciting models, see [16][17][18][19].…”