Log Gaussian Cox processes on the sphere

Abstract

A log Gaussian Cox process (LGCP) is a doubly stochastic construction consisting of a Poisson point process with a random log-intensity given by a Gaussian random field. Statistical methodology have mainly been developed for LGCPs defined in the $d$-dimensional Euclidean space. This paper concerns the case of LGCPs on the $d$-dimensional sphere, with $d=2$ of primary interest. We discuss the existence problem of such LGCPs, provide sufficient existence conditions, and establish further useful theoretical properties. The results are applied for the description of sky positions of galaxies, in comparison with previous analysis based on a Thomas process, using simple estimation procedures and making a careful model checking. We account for inhomogeneity in our models, and as the model checking is based on a thinning procedure which produces homogeneous/isotropic LGCPs, we discuss its sensitivity.