Some recent work on multivariate Gaussian Markov random fields

Some recent work on conditional formulation of multivariate Gaussian Markov random fields is presented. The focus is on model constructions by compatible conditionals and coregionalization. Special attention is given to multivariate generalizations of univariate models. Beginning with univariate model constructions, a survey of key approaches to formulating multivariate extensions is presented. Two challenges in the formulation and implementation of multivariate models are highlighted: (1) entanglement of spatial and non-spatial components, and (2) enforcement for positivity condition. Managing the two challenges by decomposition, separation, and constrained parameterization is discussed. Also highlighted is the challenge of flexible modeling of (conditional) cross-spatial dependencies and, in particular, asymmetric cross-spatial dependencies. Interpretation of asymmetric cross-spatial dependencies is also discussed. A coregionalization framework which connects and unifies the various lines of model development is presented. The framework enables a systematic development of a broad range of models via linear and spatially varying coregionalization, respectively, with extensions to locally adaptive models. Formulation of multivariate models over variable-specific lattices is discussed. Selected models are illustrated with examples of Bayesian multivariate and spatiotemporal disease mapping. Potential applications of coregionalization models in imaging analysis, covariance modeling, dimension reduction, and latent variable analysis are briefly mentioned.