Likelihood Approximation With Hierarchical Matrices For Large Spatial Datasets

TL;DR

This paper introduces an efficient method for estimating parameters of large spatial datasets by approximating covariance matrices with hierarchical matrices, significantly reducing computational complexity.

Contribution

The authors develop a hierarchical matrix approach to approximate covariance matrices, enabling scalable likelihood estimation for large spatial datasets.

Findings

Efficient covariance matrix approximation with $O(kn \, \log n)$ complexity.

Applicable to inhomogeneous covariance functions and irregular meshes.

Validated through Monte Carlo simulations and soil moisture data analysis.

Abstract

We use available measurements to estimate the unknown parameters (variance, smoothness parameter, and covariance length) of a covariance function by maximizing the joint Gaussian log-likelihood function. To overcome cubic complexity in the linear algebra, we approximate the discretized covariance function in the hierarchical (H-) matrix format. The H-matrix format has a log-linear computational cost and storage $O(kn \log n)$, where the rank $k$ is a small integer and $n$ is the number of locations. The H-matrix technique allows us to work with general covariance matrices in an efficient way, since H-matrices can approximate inhomogeneous covariance functions, with a fairly general mesh that is not necessarily axes-parallel, and neither the covariance matrix itself nor its inverse have to be sparse. We demonstrate our method with Monte Carlo simulations and an application to soil moisture data. The C, C++ codes and data are freely available.