Spatial Modelling and Scaling Error

Much of spatial modelling is a generalisation of the one-dimensional time series methods of the preceding chapter to two or more dimensions. Whereas time series methods discretise time, spatial models usually discretise space in the form of a raster of adjoining cells or a lattice of connected nodes. And Gaussian probability distributions feature equally prominently in both fields. An important class is that of Gaussian Markov Random Fields (GMRFs) which generalise AR(1) models to multiple dimensions, and which includes common spatial model types such as Conditional Autoregressive (CAR) and Simultaneous Autoregressive (SAR) models, which are characterised by different covariance functions. In AR(1) models, the system state \(z(t)\) is determined by the value of the state one time step earlier. In GMRF, which are defined on a lattice, the state \(z(s)\) is determined by the values in the neighbourhood of s, denoted as \(N_s\) . The neighbourhood consists of the nodes that have a direct link (or ‘edge’) to s. So if we want to predict the value of \(z(s)\) at some point in our space of interest, and the values of all \(s' \in N_s\) are known, then nodes further away can provide no additional information. We say that every \(z(s)\) is conditionally independent from nodes outside its neighbourhood \(N_s\) . But note that if we were to write down the covariance matrix \(\Sigma \) for all points in our lattice, we would see a non-zero covariance for every pair of points irrespective of whether they are in each other’s neighbourhood or not. We already saw that in the AR(1) models of the preceding chapter where every \(z(t)\) was calculated from its immediate predecessor \(z(t-1)\) , but where the covariance function of the equivalent GP extended to all times. However, the precision matrix \(\Omega = \Sigma ^{-1}\) of our GMRF, i.e. the inverse of the covariance matrix, only has non-zero values on its diagonal and for those \(\Omega [i,j]\) where \(s_i\) and \(s_j\) are in each other’s neighbourhood. So the nodes and edges of the lattice structure tell us immediately where the non-zero values in \(\Omega \) are. We already made use of this useful property of the precision matrix when we discussed graphical modelling (Chap. 16 ).