Tony O'Hagan - Academic pages - Abstracts

## Bayesian Inference for Nonstationary Spatial
Covariance Structure via Spatial Deformations

Alexandra Mello Schmidt and Anthony O'Hagan

*University of Sheffield*

**Publication details: **
*Journal of the Royal Statistical Society, Series B*, **65**, 745-758, 2003.

### Abstract

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.

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Updated: 25 January 2004
Maintained by: Tony O'Hagan