Considering covariates in the covariance structure of spatial processes

In spatial statistics one usually assumes that observations are partial realizations of a stochastic process {Y(x), x is an element of R-C}, where commonly C = 2, and the components of the location vector x are geographical coordinates. Frequently, it is assumed that Y(.) follows a Gaussian process (GP) with stationary covariance structure. In this setting the usual aim is to make spatial interpolation to unobserved locations of interest, based on observed values at monitored locations. This interpolation is heavily based on the specification of the mean and covariance structure of the GP. In environmental problems the assumption of stationary covariance structures is commonly violated due to local influences in the covariance structure of the process. We propose models which relax the assumption of stationary GP by accounting for covariate information in the covariance structure of the process. Usually at each location x, covariates related to Y(.) are also observed. We initially propose the use of covariates to allow the latent space model of Sampson and Guttorp to be of dimension C>2. Then we discuss a particular case of the latent space model by using a representation projected down from C dimensions to 2 in order to model the 2D correlation structure better. Inference is performed under the Bayesian paradigm, and Markov chain Monte Carlo methods are used to obtain samples from the resultant posterior distributions under each model. As illustration of the proposed models, we analyse solar radiation in British Columbia, and mean temperature in Colorado. Copyright (C) 2011 John Wiley & Sons, Ltd.