A semi-parametric spatiotemporal Hawkes-type point process modelwith periodic background for crime data
Inst. Statist. Math.
Univ. Jaume I of Castellón, Spain

Jiancang Zhuang
Jorge Mateu

Past studies have shown that crime events are often clustered (Mohler et al., 2011). This
study proposes a spatiotemporal Hawkes-type point process model, which includes a backgroundcomponent with daily and weekly periodization, and a clustering component that is triggeredby previous events. Its conditional intensity function has a form of1Pr {N ([t, t + ∆) × (x, x + ∆x ) × (y, y + ∆y )) = 1 | Ht }∆t ,∆x ,∆y ↓0 ∆t ∆x ∆yZ t− Z Zg(t − s) h(x − u, y − v) N (du × dv × ds),
= µ0 µt (t) µd (t) µw (t) µb (x, y) + A

λ(t, x, y) ≡lim−∞

S

where Ht denotes the σ-algebra generated by the observational history of the process N before
time t but not including t, A and µ0 are relaxation coefficients to be estimated, the averagevalues of background trend µt (t), daily periodicity µd (t), weekly periodicity µw (t) and spatial
background rate µb (x, y) are all standardized to 1, and the temporal response g and spatial
response h are p.d.f.s.
We generalize the nonparametric stochastic reconstruction method developed by Zhuang
et al. (2004) and Zhuang (2006) so that we can estimate each component in the background rate
and the triggering response that appears in the model conditional intensity: the backgroundrate includes a daily and a weekly periodicity, a separable spatial component, and a long-term
background trend, and the triggering response. The new stochastic reconstruction method
developed in this study fits better to crime data and is simple to understand and to estimate,
without requiring much prior knowledge of the studied phenomena. Using this method, we verify
the existence of periodic components and the triggered effect in the process of the studied crime
phenomena. Another key point is that we introduce the relaxation coefficients to stabilize andfasten the estimation process. In the estimation procedures, these two relaxation parameters
are estimated by MLE when the background components and the excitation response functions
are given. We have developed an iterative algorithm to estimate the background and clustering
components and the relaxation parameters simultaneously.
The above model and method are applied the data of the occurrences of violence or robbery
related to crimes in the city of Castellon, Spain, during two years. The results show that, as asocial phenomenon, this kind of crime is highly influenced by the daily life rhythms, revealed byits daily and weekly periodicity, and that about 3% of such crimes can be explained by clustering.
Further diagnostic residual analysis show that the model could be improved by considering thefollowing ingredients: (1) Background dominates the whole process while the clustering effectonly contributes about 3%. (2) The periodicity effect is strong in the background. (3) Residual
analysis shows that crime activity is different during weekends from working days. (4) Downtown
has different characteristics in crime activities from suburb regions.

References
Mohler, G. O., Short, M. B., Brantingham, P. J., Schoenberg, F. P., and Tita, G. E. (2011). Self-exciting point processmodeling of crime. Journal of the American Statistical Association, 106(493):100–108.
Zhuang, J. (2006). Second-order residual analysis of spatiotemporal point processes and applications in model evaluation.
Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68(4):635–653.
Zhuang, J., Ogata, Y., and Vere-Jones, D. (2004). Analyzing earthquake clustering features by using stochastic reconstruction. Journal of Geophysical Research, 109(3):B05301.