Regularized estimation for highly multivariate spatial Gaussian random fields

2026-04-08

AI Summary

This study addresses the challenges of high computational complexity and spurious dependencies among irrelevant variables in estimating covariance structures for high-dimensional multivariate spatial Gaussian random fields. The authors propose a sparse estimation framework based on LASSO penalization, which, for the first time, incorporates sparse regularization into the Cholesky factor of the multivariate Matérn covariance matrix. This approach simultaneously ensures positive semi-definiteness and automatically identifies conditionally independent variable pairs. A composite likelihood-based objective function is formulated and efficiently optimized via a projected block coordinate descent algorithm. Empirical results demonstrate that the method accurately recovers the underlying sparse dependence structure, substantially reduces estimation error, and successfully enables spatial prediction on a geochemical dataset comprising 36 variables observed at 3,998 spatial locations—a task infeasible for conventional methods.

Abstract

Estimating covariance parameters for multivariate spatial Gaussian random fields is computationally challenging, as the number of parameters grows rapidly with the number of variables, and likelihood evaluation requires operations of order $\mathcal{O}((np)^3)$. In many applications, however, not all cross-dependencies between variables are relevant, suggesting that sparse covariance structures may be both statistically advantageous and practically necessary. We propose a LASSO-penalized estimation framework that induces sparsity in the Cholesky factor of the multivariate Matérn correlation matrix, enabling automatic identification of uncorrelated variable pairs while preserving positive semidefiniteness. Estimation is carried out via a projected block coordinate descent algorithm that decomposes the optimization into tractable subproblems, with constraints enforced at each iteration through appropriate projections. Regularization parameter selection is discussed for both the likelihood and composite likelihood approaches. We conduct a simulation study demonstrating the ability of the method to recover sparse correlation structures and reduce estimation error relative to unpenalized approaches. We illustrate our procedure through an application to a geochemical dataset with $p = 36$ variables and $n = 3998$ spatial locations, showing the practical impact of the method and making spatial prediction feasible in a setting where standard approaches fail entirely.

Problem

• multivariate spatial Gaussian random fields
• covariance estimation
• high dimensionality
• sparsity
• computational complexity

Innovation

• LASSO regularization
• Cholesky factor sparsity
• multivariate Matérn covariance
• projected block coordinate descent
• high-dimensional spatial statistics