HLIBCov: Parallel Hierarchical Matrix Approximation of Large Covariance Matrices and Likelihoods with Applications in Parameter Identification

TL;DR

The paper introduces HLIBCov, a parallel hierarchical matrix software for efficiently estimating parameters of large covariance matrices in Gaussian models, demonstrated on a dataset with 2 million locations.

Contribution

It presents a novel implementation of hierarchical matrix techniques for fast covariance matrix approximation and likelihood computation in large-scale spatial statistics.

Findings

Achieves log-linear computational complexity for large covariance matrices.

Successfully estimates parameters in a dataset with 2 million locations.

Provides efficient algorithms for Cholesky, determinant, inverse, and quadratic form computations.

Abstract

We provide more technical details about the HLIBCov package, which is using parallel hierarchical ($\H$-) matrices to identify unknown parameters of the covariance function (variance, smoothness, and covariance length). These parameters are estimated by maximizing the joint Gaussian log-likelihood function. The HLIBCov package approximates large dense inhomogeneous covariance matrices with a log-linear computational cost and storage requirement. We explain how to compute the Cholesky factorization, determinant, inverse and quadratic form in the H-matrix format. To demonstrate the numerical performance, we identify three unknown parameters in an example with 2,000,000 locations on a PC-desktop.