Multivariate Gaussian Random Fields Using Systems of Stochastic Partial Differential Equations

TL;DR

This paper introduces a novel method for constructing multivariate Gaussian random fields using systems of stochastic partial differential equations, ensuring positive definiteness and enabling efficient computation with sparse matrices.

Contribution

The paper presents a new SPDE-based approach for multivariate GRFs that guarantees positive definiteness and leverages sparse matrices for computational efficiency.

Findings

Models outperform existing multivariate GRF models on real data

Constructed covariance matrices are automatically positive definite

Sparse precision matrices enable fast sampling and inference

Abstract

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.