# Spatial inference for non-lattice data using Markov Random fields

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## Abstract

This thesis deals with how computationally effective lattice models could be used for inference of data with a continuous spatial index. The fundamental idea is to approximate a Gaussian field with a Gaussian Markov random field (GMRF) on a lattice. Using a bilinear interpolation at non-lattice locations we get a reasonable model also at non-lattice locations. We can thus exploit the computational benefits of a lattice model even for data with continuous spatial index. In Paper A, a GMRF model is used in a Bayesian approach for prediction of a spatial random field. A hierarchical parametric model is setup, and inference is made by Markov Chain Monte Carlo simulations. In this way we obtain predictors and estimated prediction uncertainties as well as estimates of model parameters. The spatial correlation is modelled as a GMRF on a lattice which is interpolated between lattice points. The methods are tested on a data set of Calcium content in forest soils of southern Sweden. In Paper B, we develop a methodology for kriging large data sets. By approximating a full Gaussian model with an interpolated GMRF the kriging weights can be calculated with less computation. For n observations and a full model, calculation of the kriging weights requires inversion of an n x n covariance matrix. Approximating the model with a GMRF defined on an N x N lattice, the computations can be reduced to inversion of an NxN band limited matrix. For large data sets the full n x n matrix might not be possible to invert, and the GMRF approximation is then not only time saving, but is what makes it possible to perform kriging with the full data set.

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