Sparse nearest neighbor Cholesky matrices in spatial statistics

Abstract

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 largely been mitigated via several recent statistical innovations. Nearest Neighbor Gaussian Processes (NNGP) has emerged as one of the leading candidate for such massive-scale geospatial analysis owing to their empirical success. This articles reviews the connection of NNGP to sparse Cholesky factors of the spatial precision (inverse-covariance) matrix. 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 long-established interpretability issues of existing areal models. Finally, we highlight some yet-to-be-addressed issues of such sparse Cholesky approximations that warrants further research.

Vecchia-approximated Deep Gaussian Processes for Computer Experiments

“…Foreshadowing a DGP application in Section 3, one may define B i (W ), µ i (W ) and σ 2 i (W ) identically but with w/W in place of x/X. With this representation, we convert a large n × n matrix inversion (O(n 3 )) into n-many m × m matrix inversions (O(nm 3 )), a significant improvement if m n. The details of Vecchia-GPs, including numerous options for orderings, conditioning sets, hyperparameterizations, and computational considerations, are spread across multiple works (e.g., Katzfuss et al, 2020a;Katzfuss and Guinness, 2021;Guinness, 2018Datta et al, 2016;Datta, 2021;Finley et al, 2019). These specifications, along with software implementations (e.g.…”

“…See Heaton et al (2019) and Liu et al (2020a) for thorough reviews. Here we are drawn to a family of methods that leverage "Vecchia" approximation (Vecchia, 1988), which imposes a structure that generates a sparse Cholesky factorization of the precision matrix (Katzfuss et al, 2020a;Datta, 2021). When appropriately scaled or generalized (Stein et al, 2004;Stroud et al, 2017;Datta et al, 2016;Katzfuss and Guinness, 2021), and matched with sparse-matrix and multi-core computing facilities, Vecchia-GPs dominate competitors (Katzfuss et al, 2020b) on the frontier of accuracy, UQ, and speed.…”