Spatial statistical modeling and prediction involve generating and manipulating an n*n symmetric positive definite covariance matrix, where n denotes the number of spatial 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 to this challenge by relying on parallel solvers that deal with the matrix as a set of small tiles instead of the full structure. Each processing unit can process a single tile, allowing better performance. The approximation can also be performed at the tile level for better compression and faster execution. The Tile Low-Rank (TLR) approximation, a tile-based approximation algorithm, has recently been used in spatial statistics applications. However, the quality of TLR algorithms mainly relies on ordering the matrix elements. This order can impact the compression quality and, therefore, the efficiency of the underlying linear solvers, which highly depends on the individual ranks of each tile. Thus, herein, we aim to investigate the accuracy and performance of some existing ordering algorithms that are used to order the geospatial locations before generating the spatial covariance matrix. Furthermore, we highlight the pros and cons of each ordering algorithm in the context of spatial statistics applications and give hints to practitioners on how to choose the ordering algorithm carefully. We assess the quality of the compression and the accuracy of the statistical parameter estimates of the Mat\'ern covariance function using TLR approximation under various ordering algorithms and settings of correlations.

翻译：空间统计建模与预测涉及生成并处理一个 n*n 的对称正定协方差矩阵，其中 n 表示空间位置的数量。然而，当 n 较大时，使用传统方法处理该协方差矩阵将变得不可行。因此，将并行处理与近似方法相结合，可以优雅地解决这一挑战，其核心在于采用并行求解器将矩阵视为一组小瓦片而非完整结构进行处理。每个处理单元可独立处理单个瓦片，从而获得更优性能；近似操作也可在瓦片层级进行，以实现更高压缩比和更快的执行速度。瓦片低秩（TLR）近似作为一种基于瓦片的近似算法，近年来已被应用于空间统计领域。然而，TLR算法的质量主要依赖于矩阵元素的排序方式——这种排序会影响压缩质量，进而影响底层线性求解器的效率，而该效率高度依赖于每个瓦片的独立秩。因此，本文旨在研究在生成空间协方差矩阵前用于排序地理空间位置的若干现有排序算法的准确性与性能。此外，我们将在空间统计应用背景下阐明每种排序算法的优缺点，并为实践者提供如何审慎选择排序算法的指导建议。通过采用不同排序算法和相关性设置下的TLR近似方法，我们评估了马特恩协方差函数的压缩质量与统计参数估计的准确性。