Sparse Cholesky matrices in spatial statistics

Gaussian Processes (GP) is a staple in the toolkit of a spatial statistician. Well-documented computing roadblocks in the analysis of large geospatial datasets using Gaussian Processes have now been successfully mitigated via several recent statistical innovations. Nearest Neighbor Gaussian Processes (NNGP) has emerged as one of the leading candidates for such massive-scale geospatial analysis owing to their empirical success. This article reviews the connection of NNGP to sparse Cholesky factors of the spatial precision (inverse-covariance) matrices. Focus of the review is on these sparse Cholesky matrices which are versatile and have recently found many diverse applications beyond the primary usage of NNGP for fast parameter estimation and prediction in the spatial (generalized) linear models. In particular, we discuss applications of sparse NNGP Cholesky matrices to address multifaceted computational issues in spatial bootstrapping, simulation of large-scale realizations of Gaussian random fields, and extensions to non-parametric mean function estimation of a Gaussian Process using Random Forests. We also review a sparse-Cholesky-based model for areal (geographically-aggregated) data that addresses interpretability issues of existing areal models. Finally, we highlight some yet-to-be-addressed issues of such sparse Cholesky approximations that warrants further research.