Practical Maximum Pseudolikelihood for Spatial Point Patterns

We describe a technique for computing approximate maximum pseudolikelihood estimates of the parameters of a spatial point process. The method is an extension of Berman and Turner's [7] device for maximising the likelihoods of inhomogeneous spatial Poisson processes. For a very wide class of spatial point process models the likelihood is intractable, while the pseudolikelihood [8] is known explicitly, except for the computation of an integral over the sampling region. Approximating this integral by a finite sum in a special way yields an approximate pseudolikelihood which is formally equivalent to the (weighted) likelihood of a loglinear model with Poisson responses. This can be maximised using standard statistical software for generalised linear or additive models, provided the conditional intensity of the process takes an `exponential family' form. Using this approach we are able to fit rapidly a wide variety of spatial point process models of Gibbs type, incorporating spatial trends,..