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Multivariate spatial-statistical models are often used when modelingenvironmental and socio-demographic processes. The most commonly used modelsfor multivariate spatial covariances assume both stationarity and symmetry forthe cross-covariances, but these assumptions are rarely tenable in practice. Inthis article we introduce a new and highly flexible class of nonstationary andasymmetric multivariate spatial covariance models that are constructed bymodeling the simpler and more familiar stationary and symmetric multivariatecovariances on a warped domain. Inspired by recent developments in theunivariate case, we propose modeling the warping function as a composition of anumber of simple injective warping functions in a deep-learning framework.Importantly, covariance-model validity is guaranteed by construction. Weestablish the types of warpings that allow for cross-covariance symmetry andasymmetry, and we use likelihood-based methods for inference that arecomputationally efficient. The utility of this new class of models is shownthrough two data illustrations: a simulation study on nonstationary data and anapplication on ocean temperatures at two different depths.

Modeling Nonstationary and Asymmetric Multivariate Spatial Covariances via Deformations