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Likelihood Approximation With Hierarchical Matrices For Large Spatial Datasets

8 September 2017
A. Litvinenko
Ying Sun
M. Genton
David E. Keyes
ArXiv (abs)PDFHTML
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(knlog⁡n)O(kn \log n)O(knlogn), where the rank kkk is a small integer and nnn 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.

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