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 device [1] 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 [2, 3] is known explicitly, except for the computation of an integral over the sampling region. Approximating this integral by a finite sum yields an approximate pseudolikelihood which is formally equivalent to the 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, interaction between points, dependence on spatial covariates, and mark information.

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