A Spatio-Temporal Dirichlet Process Mixture Model on Linear Networks for Crime Data

TL;DR

This work tackles crime clustering by modeling events on a city’s street network rather than in Euclidean space. It introduces a spatio-temporal Dirichlet process mixture on a linear network, with cluster centers on the network and component densities given by a network-aware spatial kernel $K_S$ and a Gaussian temporal kernel $K_T$, enabling automatic discovery of space-time hotspots. A finite truncation of the DP and a convolution kernel estimator account for network geometry, and fully Bayesian MCMC yields posterior uncertainty for cluster locations and intensities. Post-processing addresses label switching, and model assessment compares theoretical and empirical proportions to validate fit, showing improvements over planar DP models. Applied to Valencia (robbery and theft, 2018–2019), the method identifies seasonal hotspots within about 1.1 km and 3 months, reveals higher crime near amenities, and detects a positive dependence between the two crime types, with risk boundaries offering a practical tool for targeted interventions.

Abstract

Analyzing crime events is crucial to understand crime dynamics and it is largely helpful for constructing prevention policies. Point processes specified on linear networks can provide a more accurate description of crime incidents by considering the geometry of the city. We propose a spatio-temporal Dirichlet process mixture model on a linear network to analyze crime events in Valencia, Spain. We propose a Bayesian hierarchical model with a Dirichlet process prior to automatically detect space-time clusters of the events and adopt a convolution kernel estimator to account for the network structure in the city. From the fitted model, we provide crime hotspot visualizations that can inform social interventions to prevent crime incidents. Furthermore, we study the relationships between the detected cluster centers and the city's amenities, which provides an intuitive explanation of criminal contagion.

Figures (9)

• Figure 1: Locations of (a) robbery incidences, (b) theft incidences, and (c) amenities.
• Figure 2: Estimated first-order spatial intensity of (a) robbery and (b) theft by quarters and corresponding crime counts in the parentheses.
• Figure 3: Estimated $K$-functions for (a) robbery and (b) theft, illustrated over time and distance dimensions. Gray colors are simulation envelopes generated from homogeneous Poisson processes, with the lower surface representing the minimum and the upper one representing the maximum K-functions. White colors indicate K-functions from the observed crime data. Areas where the observed $K$-function exceeds the simulation envelopes are highlighted in white.
• Figure 4: Detected cluster centers and their offspring by quarters of (a) robbery and (b) theft. The red-outlined dots indicate detected cluster centers, while the surrounding dots of the same color represent their offspring belonging to the same cluster.
• Figure 5: Posterior means of the density of the observed cases of (a) robbery and (b) theft by quarter. Black dots represent the observed crime events during each period, while red triangles indicate the detected cluster centers.