An explicit link between Gaussian fields and Gaussian Markov random fields; The SPDE approach

Continuously indexed Gaussian fields (GFs) is the most important ingredient in spatial statistical modelling and geo-statistics. The specification through the covariance function gives an intuitive interpretation of its properties. On the computational side, GFs are hampered with the "big-n" problem, since the cost of factorising dense matrices is cubic in the dimension. Although the computational power today is all-time-high, this fact seems still to be a computational bottleneck in applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the involved precision matrix sparse which enables the use of numerical algorithms for sparse matrices, that for fields in R^2 only use the square-root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parametrisation. In this paper, we show that using an approximate stochastic weak solution to (linear) stochastic partial differential equations (SPDEs), we can, for some GFs in the Matérn class, provide an explicit link, for any triangulation of R^d, between GFs and GMRFs. The consequence is that we can take the best from the two worlds and do the modelling using GFs but do the computations using GMRFs. Perhaps more importantly, our approach generalises to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere.