Autoregressive Mixture Models for Dynamic Spatial Poisson Processes: Application to Tracking Intensity of Violent Crime

This article develops a set of tools for smoothing and prediction with dependent point event patterns. The methodology is motivated by the problem of tracking weekly maps of violent crime events, but is designed to be straightforward to adapt to a wide variety of alternative settings. In particular, a Bayesian semiparametric framework is introduced for modeling correlated time series of marked spatial Poisson processes. The likelihood is factored into two independent components: the set of total integrated intensities and a series of process densities. For the former it is assumed that Poisson intensities are realizations from a dynamic linear model. In the latter case, a novel class of dependent stick-breaking mixture models are proposed to allow nonparametric density estimates to evolve in discrete time. This, a simple and flexible new model for dependent random distributions, is based on autoregressive time series of marginally beta random variables applied as correlated stick-breaking proportions. The approach allows for marginal Dirichlet process priors at each time and adds only a single new correlation term to the static model specification. Sequential Monte Carlo algorithms are described for on-line inference with each model component, and marginal likelihood calculations form the basis for inference about parameters governing temporal dynamics. Simulated examples are provided to illustrate the methodology, and we close with results for the motivating application of tracking violent crime in Cincinnati.