Interpreting the correlation structure of spatial models for multivariate lattice data

There is an increasing interest on studying multivariate spatial problems, in which several attributes are measured in the same spatial location. The increasing collection of large geographical databases is stimulating the work with this type of data. The complexity of the phenomenon under study has led to the adoption of a hierarchical model with many layers of latent underlying random effects structured as a spatially oriented neighborhood graph. The large number of latent random parameters naturally leads to a Bayesian approach which models these underlying parameters by imposing stochastic smoothing restrictions. The joint prior distribution is determined by a set of conditional distributions based on intuitively simple geographical neighborhood relationships. However, the implied correlation structure is typically unknown in analytical terms and empirical calculations show that it is complex and counter-intuitive in some cases. In this work, we deal with a flexible model for multivariate measurements that takes into account the cross-correlations between variables and their spatial locations. Explicit expressions for the prior and posterior covariance matrices of the underlying parameters were obtained. We provide intuitive interpretation for the covariance matrices showing how they are connected with the topological structure of the three-dimensional neighborhood graph connecting geographical areas and variables. We give an illustrative example that shows how this interpretation is useful to understand the correlations between spatial multivariate variables.