The last decade has witnessed substantial developments in scalable models for univariate spatial processes, but such methods for multivariate spatial processes, especially when the number of outcomes is moderately large, are limited in comparison. In this work, we extend scalable modeling strategies for a single process to multivariate processes. We pursue Bayesian inference which is attractive for full uncertainty quantification of the latent spatial process. Our approach exploits distribution theory for the Matrix-Normal distribution, which we use to construct scalable versions of a hierarchical linear model of coregionalization (LMC) and spatial factor models that deliver inference over a high-dimensional parameter space including the latent spatial process.

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