Bayesian Markov random field modeling for spatial data

Linda Werner

Abstract:: Spatial data sets are common in the environmental sciences. In this study we suggest a hierarchical model for a spatial stochastic field. The main focus of this article is to approximate a stochastic field with a Gaussian Markov Random Field (GMRF) to exploit the computational advantages of the Markov field, concerning predictions etc. The variation of the stochastic field is modelled as a linear trend plus micro-variation in the form of a GMRF defined on a lattice. To estimate model parameters we adopt a Bayesian perspective, and use Monte Carlo integration with samples; from Markov Chain simulations.
: Our methods do not demand lattice or near-lattice data, but are developed for a general spatial data set, leaving the lattice to be specified by the modeller. The model selection problem that comes with the artificial grid is in this article addressed with cross-validation, but we also suggest other alternatives. Finally we apply the methods to a data set of elemental composition of forest soil

Key words:: Gaussian Markov random fields, Markov chain Monte Carlo, non-lattice data, soil data, elemental composition