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Multivariate Gaussian Random Fields Using Systems of Stochastic Partial Differential Equations

In this paper a new approach for constructing \emph{multivariate} Gaussian random fields (GRFs) using systems of stochastic partial differential equations (SPDEs) has been introduced and applied to simulated data and real data. By solving a system of SPDEs, we can construct multivariate GRFs. On the theoretical side, the notorious requirement of non-negative definiteness for the covariance matrix of the GRF is satisfied since the constructed covariance matrices with this approach are automatically symmetric positive definite. Using the approximate stochastic weak solutions to the systems of SPDEs, multivariate GRFs are represented by multivariate Gaussian \emph{Markov} random fields (GMRFs) with sparse precision matrices. Therefore, on the computational side, the sparse structures make it possible to use numerical algorithms for sparse matrices to do fast sampling from the random fields and statistical inference. Therefore, the \emph{big-n} problem can also be partially resolved for these models. These models out-preform existing multivariate GRF models on a commonly used real dataset.