8.2 Stochastic partial differntial equation approach

I do not have understand it but:

• A GRF with a Matern covariance matrix can be expressed as a solution to predict the variable of interest.

• the parameters of Matèrn covariance and SPDE are coupled (INLA default is smouthness of 1/2 ie exponential cov.)

We can approximate SPDE using the Finite Element method: divide $D$ into a set of of non-intersecting triangles.

That bring us to a mesh with $n$ nodes and $n$ basis function (each function decrease when going away of the node). That allows use to go from a continuous Gaussian field to a discrete indexed Gaussian markov random field.

$$x(s) = \sum^n_{k = 1} \psi_k(s)x_k$$

$x_k$ are the zero mean gaussian distributed weights

$x = (x_1, ..., x_n) \sim N(0, Q^{-1}(\tau,k))$ (N is a joint distribution) give an approximation of $x(s)$