Linear-Cost Covariance Functions for Gaussian Random Fields

TL;DR

This paper introduces a new class of covariance functions for Gaussian random fields that enable linear-cost computations for large datasets, significantly improving efficiency in sampling, kriging, and likelihood evaluation.

Contribution

The authors propose hierarchical covariance functions that lead to matrices with structures allowing linear and near-linear computational complexity for key Gaussian field operations.

Findings

Sampling and likelihood evaluation scale as O(n).

Kriging scales as O(log n) after preprocessing.

Numerical experiments demonstrate efficiency on datasets with over two million observations.

Abstract

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.