Recent technological advances have enabled researchers in a variety of fields to collect accurately geocoded data for several variables simultaneously. In many cases it may be most appropriate to jointly model these multivariate spatial processes without constraints on their conditional relationships. When data have been collected on a regular lattice, the multivariate conditionally autoregressive (MCAR) models are a common choice. However, inference from these MCAR models relies heavily on the pre-specified neighborhood structure and often assumes a separable covariance structure. Here, we present a multivariate spatial model using a spectral analysis approach that enables inference on the conditional relationships between the variables that does not rely on a pre-specified neighborhood structure, is non-separable, and is computationally efficient. Covariance and cross-covariance functions are defined in the spectral domain to obtain computational efficiency. The resulting pseudo posterior inference on the correlation matrix allows for quantification of the conditional dependencies. A comparison is made with an MCAR model that is shown to be highly sensitive to the choice of neighborhood. The approaches are illustrated for the toxic element arsenic and four other soil elements whose relative concentrations were measured on a microscale spatial lattice. Understanding conditional relationships between arsenic and other soil elements provides insights for mitigating pervasive arsenic poisoning in drinking water in southern Asia and elsewhere.
"Bayesian Spectral Modeling for Multivariate Spatial Distributions of Elemental Concentrations in Soil." Bayesian Anal. 13 (1) 1 - 28, March 2018. https://doi.org/10.1214/16-BA1034