# Log Gaussian Cox processes on the sphere

## TL;DR

This paper extends the theory of log Gaussian Cox processes to the sphere, providing existence conditions, theoretical properties, and applications to galaxy sky positions with model checking and inhomogeneity considerations.

## Contribution

It introduces LGCPs on the sphere, establishes their existence and properties, and applies them to astrophysical data with model validation techniques.

## Key findings

- LGCPs on the sphere can model galaxy positions effectively.
- The paper provides sufficient conditions for LGCP existence on the sphere.
- Model checking via thinning procedures is discussed and evaluated.

## 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.