In a wide range of applications, dependence on smoothly-varying covariates leads spatial point count intensities to feature positive correlation for nearby locations. In applications where the points are `marked' with individual attributes, the distributions for points with varying attributes may also differ. We introduce a class of hierarchical spatial regression models for relating marked point process intensities to location-specific covariates and to individual-specific attributes, and for modeling the remaining intensity variation that arises from dependence on unobserved or unreported covariates. The magnitude of residual intensity variation is a measure of how completely the covariates explain the observed variations in point intensities.
The models, extending those recently introduced by Wolpert and Ickstadt, treat the point patterns as doubly-stochastic Poisson random fields with a random inhomogeneous Poisson intensity given by a spatial mixture of gamma or other infinitely-divisible independent-increment random fields. Bayesian prior distributions for a third level of hierarchy are elicited to reflect beliefs about the homogeneity, continuity, and similar features of the intensity.
Inference is based on posterior and predictive distributions computed using Markov chain Monte Carlo methods featuring data augmentation and an efficient method for sampling from independent-increment random fields. The models are illustrated in an application to a four-dimensional spatial regression analysis of origin/destination trip data from the 1994/95 METRO survey of Portland, Oregon.