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February 2021 A General Framework for Vecchia Approximations of Gaussian Processes
Matthias Katzfuss, Joseph Guinness
Statist. Sci. 36(1): 124-141 (February 2021). DOI: 10.1214/19-STS755

Abstract

Gaussian processes (GPs) are commonly used as models for functions, time series, and spatial fields, but they are computationally infeasible for large datasets. Focusing on the typical setting of modeling data as a GP plus an additive noise term, we propose a generalization of the Vecchia (J. Roy. Statist. Soc. Ser. B 50 (1988) 297–312) approach as a framework for GP approximations. We show that our general Vecchia approach contains many popular existing GP approximations as special cases, allowing for comparisons among the different methods within a unified framework. Representing the models by directed acyclic graphs, we determine the sparsity of the matrices necessary for inference, which leads to new insights regarding the computational properties. Based on these results, we propose a novel sparse general Vecchia approximation, which ensures computational feasibility for large spatial datasets but can lead to considerable improvements in approximation accuracy over Vecchia’s original approach. We provide several theoretical results and conduct numerical comparisons. We conclude with guidelines for the use of Vecchia approximations in spatial statistics.

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Matthias Katzfuss. Joseph Guinness. "A General Framework for Vecchia Approximations of Gaussian Processes." Statist. Sci. 36 (1) 124 - 141, February 2021. https://doi.org/10.1214/19-STS755

Information

Published: February 2021
First available in Project Euclid: 21 December 2020

MathSciNet: MR4194207
Digital Object Identifier: 10.1214/19-STS755

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.36 • No. 1 • February 2021
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