Investigating mesh‐based approximation methods for the normalization constant in the log Gaussian Cox process likelihood

Abstract

The log Gaussian Cox process (LGCP) is a frequently applied method for modeling point pattern data. The normalization constant of the LGCP likelihood involves an integral over a latent field. That integral is computationally expensive, making it troublesome to perform inference with standard methods. The so‐called stochastic partial differential equation–integrated nested Laplace approximation (SPDE‐INLA) framework enables fast approximate inference for a range of hierarchical models, where a key component is to approximate the latent field by a triangulated mesh. Recent research has made it possible to fit LGCP models with this framework using an approximate integration method to compute the integral. We carefully describe several alternative variants of that approximate integration method and derive an analytical formula for the integral in question, which actually is exact under the triangular mesh assumption used by SPDE‐INLA. We compare the different integration strategies through a comprehensive simulation study and find that the analytical formula is often more accurate, but not always. Among the approximate integration methods, we recommend a simple extension to a method implemented in an R‐package for fitting LGCP models.