Analysis of massive marked point patterns with stochastic partial differential equations

In this paper we describe a novel approach to modelling marked point patterns based on recent computational developments for Bayesian inference. We use the flexible class of log-Gaussian Cox Processes to model the intensity of the different observed point patterns. We propose several types of models to account for spatial variability and provide a modelling framework that allows for a common spatial component to all point processes (regardless of the mark) and also for a mark-specific spatial components. In this way, we provide a method of assessing whether all processes share a common spatial distribution or there are specific features. In order to fit these models, we have resorted to the Integrated Nested Laplace Approximation (INLA) method and the Stochastic Partial Differential Equation (SPDE) approach. This defines a connection between point process and geostatistics, as we model a point pattern by means of a continuous spatial process. Our new approach to spatial modelling is applied to a massive dataset on the occurrence of tornados in the United States. We have divided the tornados in the 1950–2013 period according to their magnitude and fitted our proposed models.