Alexandra Mello Schmidt and Anthony O'Hagan
University of Sheffield
Publication details: Journal of the Royal Statistical Society, Series B, 65, 745-758, 2003.
In geostatistics it is common practice to assume that the underlying spatial process is stationary and isotropic, that is the spatial distribution is unchanged when the origin of the index set is translated and the process is stationary under rotations about the origin. However in environmental problems, it is not very realistic to make such assumptions since local influences in the correlation structure of the spatial process may be clearly found in the data.
This paper proposes a Bayesian model wherein the main aim is to address the anisotropy problem. Following Sampson & Guttorp (JASA, 1992), we define the correlation function of the spatial process by reference to a latent space, denoted by D, where stationarity and isotropy hold. The space where the gauged monitoring sites lie is denoted by G. We adopt a Bayesian approach in which the mapping between G space and D space is represented by an unknown function d(.). A Gaussian process prior distribution is defined for d(.). Unlike the Sampson & Guttorp approach, the mapping of both gauged and ungauged sites is handled in a single framework, and predictive inferences take explicit account of uncertainty in the mapping. Monte Carlo Markov Chain (MCMC) methods are used to obtain samples from the posterior distributions. Three examples are discussed, two simulated data sets and the solar radiation data set also analysed by Sampson & Guttorp.
Keywords: Anisotropy; Gaussian Process; Kriging; Markov Chain Monte Carlo; Prediction.