论空间协方差矩阵排序对 Matérn 参数 Tile 低秩估计的影响
On the Impact of Spatial Covariance Matrix Ordering on Tile Low-Rank Estimation of Matérn Parameters
AI总结：
本文研究了在空间统计中生成协方差矩阵前对地理空间位置进行排序的算法对 Tile 低秩近似（TLR）的压缩质量和 Matérn 参数估计准确性的影响，并分析了各排序算法的优缺点以指导实践。
摘要：
空间统计建模与预测涉及生成和处理一个 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会变得不可行。因此，将并行处理与近似方法相结合可以成为应对这一挑战的优雅解决方案，其依赖于将矩阵视为一组小 Tile 而非完整结构的并行求解器。每个处理单元可以处理单个 Tile，从而实现更好的性能。近似操作也可以在 Tile 级别进行，以获得更好的压缩效果和更快的执行速度。Tile 低秩（TLR）近似作为一种基于 Tile 的近似算法，近期已被用于空间统计应用。然而，TLR 算法的质量主要依赖于矩阵元素的排序。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，因为该效率高度依赖于每个 Tile 的单独秩。因此，本文旨在研究一些现有排序算法在生成空间协方差矩阵之前对地理空间位置进行排序的准确性和性能。此外，我们强调了每种排序算法在空间统计应用背景下的优缺点，并为从业者提供了如何谨慎选择排序算法的提示。我们在多种排序算法和相关性设置下，使用 TLR 近似评估了压缩质量以及 Matérn 协方差函数统计参数估计的准确性。
英文摘要：
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 Matérn covariance function using TLR approximation under various ordering algorithms and settings of correlations.
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