Bayesian covariance modeling of multivariate spatial random fields

Abstract: In this work we present full Bayesian inference for a new flexible nonseparable class of cross-covariance functions for multivariate spatial data. A Bayesian test is proposed for separability of covariance functions which is much more interpretable than parameters related to separability. Spatial models have been increasingly applied in several areas, such as environmental science, climate science and agriculture. These data are usually available in space, time and possibly for several processes. In this context the modeling of dependence is crucial for correct uncertainty quantification and reliable predictions. In particular, for multivariate spatial data we need to specify a valid cross-covariance function, which defines the dependence between the components of a response vector for all locations in the spatial domain. However, cross-covariance functions are not easily specified and the computational burden is a limitation for model complexity. In this work, we propose a nonseparable covariance function that is based on the convex combination of separable covariance functions and on latent dimensions representation of the vector components. The covariance structure proposed is valid and flexible. We simulate four different scenarios for different degrees of separability and compute the posterior probability of separability. It turns out that the posterior probability is much easier to interpret than actual model parameters. We illustrate our methodology with a weather dataset from Cear\'a, Brazil.