A Bayesian Approach and Vecchia Grouping for Estimating Spatial Covariances in Large Datasets

ABSTRACT Covariance matrices play a central role in the modeling, inference, and prediction of spatial data, but computations involving large covariance matrices become prohibitively expensive as the number of locations increases. This paper considers Bayesian nonparametric estimation of nonstationary spatial covariance structures for large datasets. The proposed approach is built on a Vecchia‐type conditional representation of an ordered Gaussian process, which induces a sparse precision matrix and sparse Cholesky factorization and allows the high‐dimensional model to be expressed as a sequence of Bayesian linear regressions. To further improve scalability, we apply a grouping strategy over the ordered locations, which removes weak dependencies, produces a block‐structured sparse approximation, and substantially reduces computational cost. Distance‐based ordering is also used to improve the quality of the approximation. Simulation studies show that, compared with the ungrouped version, the grouped approach yields favorable posterior interval performance while requiring less computational effort. In addition, real‐data analysis demonstrates that the proposed method can serve as a computationally efficient approximation to the Kidd and Katzfuss‐based posterior covariance structure in large‐scale settings.