Combining regional climate model output via a multivariate Markov random field model

Conditional autoregressive (CAR) models, and the more general Markov random field models, are

excellent tools for analyzing data laid out on spatial lattices. These lattices may be regular, such

as the grids associated with images, remote-sensing data, climate models, etc., or irregular, such as

U.S. census divisions (counties, tracts, or block-groups) or other administrative units. Besag (1974)

laid out the basic framework for Markov random field models. For random variables y1,...,yn observed

at n locations on the lattice structure, the collection of conditional distributions f(yi|y−i), i = 1,...,n

(where y−i refers to all random variables except the ith one) can be combined under certain regularity

conditions to form a joint distribution f(y1,...,yn). Rue and Held (2006) can be consulted for an

excellent exposition of the theory of Markov random fields; see also the reviews in the texts by Cressie

(1993), Banerjee et al. (2004), and Schabenberger and Gotway (2005). CAR models are Gaussian

Markov random fields. It should be noted that they are not the same as the so-called simultaneously

specified autoregressive (SAR) models and are actually a more general construc