On the impact of spatial covariance matrix ordering on tile low-rank estimation of Matérn parameters

Spatial statistical modeling involves processing an $n\times n$ symmetric positive definite covariance matrix, where $n$ denotes the number of locations. However, when $n$ is large, processing this covariance matrix using traditional methods becomes prohibitive. Thus, coupling parallel processing with approximation can be an elegant solution by relying on parallel solvers that deal with the matrix as a set of small tiles instead of the full structure. The approximation can also be performed at the tile level for better compression and faster execution. The tile low-rank (TLR) approximation has recently been used to compress the covariance matrix, which mainly relies on ordering the matrix elements, which can impact the compression quality and the efficiency of the underlying solvers. This work investigates the accuracy and performance of location-based ordering algorithms. We highlight the pros and cons of each ordering algorithm and give practitioners hints on carefully choosing the ordering algorithm for TLR approximation. We assess the quality of the compression and the accuracy of the statistical parameter estimates of the Matérn covariance function using TLR approximation under various ordering algorithms and settings of correlations through simulations on irregular grids. Our conclusions are supported by an application to daily soil moisture data in the Mississippi Basin area.