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Gaussian Processes (GP) is a staple in the toolkit of a spatial statistician.Well-documented computing roadblocks in the analysis of large geospatialdatasets using Gaussian Processes have now been successfully mitigated viaseveral recent statistical innovations. Nearest Neighbor Gaussian Processes(NNGP) has emerged as one of the leading candidates for such massive-scalegeospatial analysis owing to their empirical success. This article reviews theconnection of NNGP to sparse Cholesky factors of the spatial precision(inverse-covariance) matrices. Focus of the review is on these sparse Choleskymatrices which are versatile and have recently found many diverse applicationsbeyond the primary usage of NNGP for fast parameter estimation and predictionin the spatial (generalized) linear models. In particular, we discussapplications of sparse NNGP Cholesky matrices to address multifacetedcomputational issues in spatial bootstrapping, simulation of large-scalerealizations of Gaussian random fields, and extensions to non-parametric meanfunction estimation of a Gaussian Process using Random Forests. We also reviewa sparse-Cholesky-based model for areal (geographically-aggregated) data thataddresses interpretability issues of existing areal models. Finally, wehighlight some yet-to-be-addressed issues of such sparse Choleskyapproximations that warrants further research.

Sparse nearest neighbor Cholesky matrices in spatial statistics