ABSTRACT
Chapter 5 builds progressively from simple to general methods for estimating spatially correlated variables. To do so, it introduces basis expansion using splines, and emphasizes tensor splines to implement basis expansion for two spatial coordinates. It then introduces generalized additive models (GAMs), and briefly outlines how generalized linear mixed models (GLMMs) and GAMs can both estimate the wiggliness associated with a tensor spline. Then, using point-count data for bald eagles in western North America, it compares four different ways of constructing the covariance or precision matrix for a two-dimensional spatial variable in a GLMM. Specifically, it first extends time-series methods to define a conditional distribution for log-densities on a two-dimensional grid, and then extends this by jointly constructing the gridded spatial covariance. It then contrasts this with either a conditional autoregressive process or the stochastic partial differential equation (SPDE) method for specifying the precision matrix for over an irregular spatial domain. In each case, the chapter shows that these specifications result in similar spatial basis functions, and also similar estimates of bald-eagle densities in the case study example.
