Abstract
We provide more technical details about the HLIBCov package, which is using parallel hierarchical (H-) matrices to:•Approximate large dense inhomogeneous covariance matrices with a log-linear computational cost and storage requirement.•Compute matrix-vector product, Cholesky factorization and inverse with a log-linear complexity.•Identify unknown parameters of the covariance function (variance, smoothness, and covariance length).These unknown parameters are estimated by maximizing the joint Gaussian log-likelihood function. To demonstrate the numerical performance, we identify three unknown parameters in an example with 2,000,000 locations on a PC-desktop.
Highlights
This paper is complementary to the paper [2]
In this paper we use parallel hierarchical (H-) matrices for approx45 imating dense covariance matrices and the joint Gaussian log-likelihood with computational complexity O(kαnlogα n/p), where p is the number of cores, n is the number of measurements; k n is the maximal rank, used in the hierarchical matrix, which defines the quality of the approximation; and α = 1 or 2
The Matern form of spatial correlations was introduced into statistics as a flexible parametric class [51], with one parameter determining the smoothness of the underlying spatial random field [32]
Summary
This paper is complementary to the paper [2]. Novelty of this work. We estimate unknown parameters of the covariance function. Let n be the number of spatial measurements Z located irregularly across a given geographical region at locations s := {s1,...,sn} ∈ Rd, d ≥ 1. These data are frequently modeled as a realization from a stationary Gaussian spatial random field. To infer unknown parameters θ we maximize the Gaussian log-likelihood function: L(θ). In this work we extend the applicability of HLIBpro to dense covariance matrices and log-likelihood functions
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