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

Abstract

Summary Past studies have shown that crime events are often clustered. This study proposes a spatiotemporal Hawkes‐type point process model, which includes a background component with daily and weekly periodization, and a clustering component that is triggered by previous events. We generalize the non‐parametric stochastic reconstruction method so that we can estimate each component in the background rate and the triggering response that appears in the model conditional intensity: the background rate includes a daily and a weekly periodicity, a separable spatial component and a long‐term background trend. Two relaxation coefficients are introduced to stabilize and secure the estimation process. This model is used to describe the occurrences of violence or robbery cases in Castellón, Spain, during 2 years. The results show that robbery crime is highly influenced by daily life rhythms, revealed by its daily and weekly periodicity, and that about 3% of such crimes can be explained by clustering. Further diagnostic analysis shows that the model could be improved by considering the following ingredients: the daily occurrence patterns are different between weekends and working days; in the city centre, robbery activity shows different temporal patterns, in both weekly periodicity and long‐term trend, from other suburb areas.

A Non-Parametric Hawkes Process Model of Primary and Secondary Accidents on a UK Smart Motorway

“…This derivation follows closely that in Zhuang and Mateu (2019), with an additional restriction on ρ i,j , and h being a univariate function. One then smooths this initial estimate using kernel functions, with more details on this and all other model components in the supplementary material.…”

“…Our methodology is based on the work of Zhuang and Mateu (2019), where it is shown how to construct the conditional intensity of a Hawkes process in a non-parametric way by applying kernel smoothing to the observed data on crime. We modify this model to make it applicable to traffic data.…”

“…First, we adapt the work in Zhuang and Mateu (2019) to reconstruct a one-dimensional and unidirectional spatial triggering function. To begin, we state that for a spatiotemporal point process N p with conditional intensity function λ(t, x) and a predictable process f(t, x) ≥ 0, over a time domain [T 1 , T 2 ] and space domain S, we can write:…”

Estimation, diagnostics, and extensions of nonparametric Hawkes processes with kernel functions

“…This positive feedback finally yields a trivial solution with ̂(t) = 0 . To avoid this positive feedback, Zhuang et al (2002) and Zhuang and Mateu (2019) introduce relaxation coefficients to prevent positive feedback. Instead of (43), they use as the conditional intensity function, and the estimates for and g become…”

“…We can improve the above estimates, as well as the histogram estimate of the excitation terms, by introducing kernel smoothing with a correction of the edge effect. The details are available in Zhuang and Mateu (2019) and omitted here. Zhuang and Mateu (2019) analyze robbery crimes in Castellon, Spain from 2012 to 2013 and disentangle the different background components using the method described in this subsection, as in Fig.…”