Bayesian inference in Gaussian model-based geostatistics

P.J. Ribeiro Jr. & P.J. Diggle

Abstract

The term geostatistics refers to a collection of methods used in the analysis of a particular kind of spatial data, in which measured values $Y_i$ at spatial locations $u_i$ can be regarded as noisy observations from an underlying process in continuous space. In particular, in a geostatistical analysis spatial interpolation or smoothing of the observed values is often carried out by a procedure known as kriging. In its basic form, kriging involves the construction of a linear predictor for an unobserved value of the process, and the form of this linear predictor is chosen with reference to the covariance structure of the data as estimated by a data-analytic tool known as the variogram. Often, no explicit underlying stochastic model is declared.

In this text, we adopt a model-based approach to this class of problems, by which we mean that we start with an explict stochastic model and derive associated methods of parameter estimation, interpolation and smoothing by applying general statistical principles to the observed data under the assumed model. In particular, we use hierarchical spatial linear models whose components are Gaussian stochastic processes with specified parametric covariance structure, and Bayesian methods of inference with independent priors for the separate model parameters.

We present results using this model-based approach, and compare them with classical geostatistical solutions. We derive posterior distributions for model parameters, and predictive distributions for values of the underlying spatial process, taking into account different degrees of parameter uncertainty including uncertainty about some or all of the covariance parameters. We provide a catalogue of posterior and predictive distributions for particular combinations of prior choices and degrees of parameter uncertainty. We discuss computational aspects of the implementation, including non-iterative Monte Carlo inference. Finally, we give illustrative analyses of simulated data.

Keywords: Bayesian inference, geostatistics, kriging, linear mixed models, spatial prediction.

Technical Report, ST-99-08
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1-4YF
e-mail: p.ribeiro@lancaster.ac.uk

http://www.maths.lancs.ac.uk/~ribeiro/

Last modified: Thu May 4 19:58:48 BST 2000