On new families of anisotropic spatial log-Gaussian Cox processes

Abstract

Cox processes are natural models for point process phenomena that are environmentally driven, but much less natural for phenomena driven primarily by interactions amongst the points. The class of log-Gaussian Cox processes (LGCPs) has an elegant simplicity, and one of its attractive features is the tractability of the multivariate normal distribution carries over, to some extent, to the associated Cox process. In the statistical analysis of spatial point patterns, it is often assumed isotropy because of a simpler interpretation and ease of analysis. However, there are many cases in which the spatial structure depends on the direction. In this paper, we introduce new families of anisotropic spatial LGCPs that are useful to model spatial anisotropic point patterns that exhibit a degree of clustering. We propose classes of families consisting of elliptical and non-elliptical models. The former can be reduced to isotropic forms after some rotations, while the latter family goes beyond this property. We derive analytical forms for the covariance of the associated random field, and some second-order characteristics. A moment-based estimation procedure is followed to make inference on the parameters that control the degree of anisotropy. The estimation procedure is evaluated through a simulation study under a variety of scenarios and various degrees of anisotropy. Our methodology is illustrated on two real datasets of earthquakes in South America and the Mediterranean Europe.