Spatial inference for nonlattice data using Markov random fields
Linda Werner
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2004
ISSN 1404028X

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 nonlattice locations
we get a reasonable model also at nonlattice 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 N x N 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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