Nearest‐neighbor sparse Cholesky matrices in spatial statistics

Abstract

Gaussian process (GP) is a staple in the toolkit of a spatial statistician. Well‐documented computing roadblocks in the analysis of large geospatial datasets using GPs have now largely been mitigated via several recent statistical innovations. Nearest neighbor Gaussian process (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) 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 nonparametric mean function estimation of a GP 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 warrant further research. This article is categorized under: Algorithms and Computational Methods > Algorithms Algorithms and Computational Methods > Numerical Methods

Scalable Predictions for Spatial Probit Linear Mixed Models Using Nearest Neighbor Gaussian Processes

“…Letting denote the column of , and the -dimensional vector with 1 at the position and 0’s elsewhere, one can solve for as where trsolve denotes solution of a triangular linear system. As A is strictly lower triangular with at-most entries per row, the linear system in (7) can be solved in flops (see Saha and Datta, 2018a ; Datta, 2021 , for the algorithm). Repeating this for .…”

“…where trsolve denotes solution of a triangular linear system. As A is strictly lower triangular with at-most O(1) entries per row, the linear system in ( 7) can be solved in O(n) flops (see Saha and Datta, 2018a;Datta, 2021, for the algorithm). Repeating this for j = 1, .…”

The Matérn Model: A Journey through Statistics, Numerical Analysis and Machine Learning

“…Perhaps the most successful approximation is Vecchia's method [163], which has attracted a remarkable amount of attention in recent times [inc. 147,48,49,67,47]. The Vecchia approximation can be used with any correlation model and its basic idea is is to replace (13) with a product of Gaussian conditional distributions, in which each conditional distribution involves only a small subset of the data.…”

“…• Vecchia's approximation [163] and its extensions [e.g. 48,67,83,47] imply a sparse approximation of of ch(Σ −1 n ) and are often applied to the Matérn model, although they can be applied to any covariance model. One potential limitation of these method is that they depend on an ordering of the variables and the choice of conditioning sets which determines the Cholesky sparsity pattern [see 67].…”