Conditionally linear models for non-homogeneous spatial random fields

We consider parsimonious representations of non-homogeneous spatial random fields. We focus on processes that can be represented as linear combinations of basis functions. As the basis functions are allowed to depend on unknown parameters, we identify such models with conditionally linear processes. We present a detailed description of an approach that uses discrete process convolutions with spatially varying, compactly supported kernels. We discuss the similarities and differences between this approach and the predictive Gaussian process approach. We also discuss the problem of obtaining decompositions of a spatial random field, as well as spatio-temporal extensions of our spatial models.