  The Markov chain random field (MCRF) model/theory provides a non-linear
spatial Bayesian updating solution at the neighborhood nearest data level for
simulating categorical spatial variables. In the MCRF solution, the spatial
dependencies among nearest data and the central random variable is a
probabilistic directed acyclic graph that conforms to a neighborhood-based
Bayesian network on spatial data. By selecting different neighborhood sizes and
structures, applying the spatial conditional independence assumption to nearest
neighbors, or incorporating ancillary information, one may construct specific
MCRF models based on the MCRF general solution for various application
purposes. Simplified MCRF models based on assuming the spatial conditional
independence of nearest data involve only spatial transition probabilities, and
one can implement them easily in sequential simulations. In this article, we
prove the spatial Bayesian network characteristic of MCRFs, and test the
optimal neighborhoods under the spatial conditional independence assumption.
The testing results indicate that the quadrantal (i.e., one nearest datum per
quadrant) neighborhood is generally the best choice for the simplified MCRF
solution, performing better than other sectored neighborhoods and non-sectored
neighborhoods with regard to simulation accuracy and pattern rationality.
