Representing spatial dependence: an exploration of shared structure between two models

Spatial dependence poses challenges in the identification and estimation of statistical quantities. Many models exist that account for spatial dependence in some form, but most such models rely on an adjacency matrix to describe the relatedness among locations. However, adjacency matrices are fairly rigid ways to represent dependence: rather than estimating the strengths of relationships among locations, they only permit estimation of the global presence of one spatial structure. Before developing more adaptive spatial models, it would be useful to categorize existing representations of spatial structure in spatial models. In this work, I examine the relationship between Poisson conditional autoregressive (CAR) models, which use adjacency matrices to represent spatial dependence, and Hawkes process models, which use a spatiotemporal kernel to represent spatial dependence. I derive conditions under which Poisson CAR models and Hawkes process models are equivalent, indicating that they share a similar representation of spatial structure. By beginning to characterize current methods of representing spatial structure, this work sets the stage for the development of more flexible spatial models.