  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.
