Understanding London Crime with Log-Gaussian Cox Processes

Abstract

It is well understood that crime is clustered in both space and time. This is recognized by the emergence of so-called hot-spots, which are often short-lived. There are a num- ber of theories in criminology that provide an explanation for this, although crime is a complex phenomenon and matching theories with empirical evidence is an ongoing task. Whilst standard regression techniques can be used, the assumptions are typi- cally violated because of spatio-temporal autocorrelation, resulting in poor predictive performance. Recent work has advocated the use of Gaussian process models for their flexibility and tractability, although a Gaussian assumption is only an appropriate approximation in regions of high crime. Instead, a log-Gaussian Cox process (LGCP) is proposed as a model of criminal activity in space and time. We adopt a Bayesian approach to provide a full quantification of uncertainty. The resulting posterior distri- bution is intractable and poses significant computational challenges that we overcome by the use of Laplace approximation or Markov Chain Monte Carlo methods. In either case, by assuming a grid structure, we use Kronecker methods to accelerate the linear algebra procedures involved. We demonstrate the proposed methodology with London crime data for 2016, using socio-economic covariates derived from census data.