Linear-Cost Covariance Functions for Gaussian Random Fields

• 成果类型: Article
• 署名作者: Chen, Jie; Stein, Michael L.
• 署名单位: International Business Machines (IBM); IBM USA; Rutgers University System; Rutgers University New Brunswick
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2021.1919122
• 发表日期: 2023
• 页码: 147-164
• 关键词: fixed-domain asymptotics; fast multipole method; cholesky factorization; parameter-estimation; PROCESS MODELS; matrix; approximation; likelihood; efficient; algorithms

摘要

Gaussian random fields (GRF) are a fundamental stochastic model for spatiotemporal data analysis. An essential ingredient of GRF is the covariance function that characterizes the joint Gaussian distribution of the field. Commonly used covariance functions give rise to fully dense and unstructured covariance matrices, for which required calculations are notoriously expensive to carry out for large data. In this work, we propose a construction of covariance functions that result in matrices with a hierarchical structure. Empowered by matrix algorithms that scale linearly with the matrix dimension, the hierarchical structure is proved to be efficient for a variety of random field computations, including sampling, kriging, and likelihood evaluation. Specifically, with n scattered sites, sampling and likelihood evaluation has an O(n) cost and kriging has an O(log n) cost after preprocessing, particularly favorable for the kriging of an extremely large number of sites (e.g., predicting on more sites than observed). We demonstrate comprehensive numerical experiments to show the use of the constructed covariance functions and their appealing computation time. Numerical examples on a laptop include simulated data of size up to one million, as well as a climate data product with over two million observations.