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Likelihood approximation with hierarchical matrices for large spatial datasets

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  • Litvinenko, Alexander
  • Sun, Ying
  • Genton, Marc G.
  • Keyes, David E.

Abstract

The unknown parameters (variance, smoothness, and covariance length) of a spatial covariance function can be estimated by maximizing the joint Gaussian log-likelihood function. To overcome cubic complexity in the linear algebra, the discretized covariance function is approximated in the hierarchical (H-) matrix format. The H-matrix format has a log-linear computational cost and O(knlogn) storage, where the rank k is a small integer, and n is the number of locations. The H-matrix technique can approximate general covariance matrices (also inhomogeneous) discretized on a fairly general mesh that is not necessarily axes-parallel, and neither the covariance matrix itself nor its inverse has to be sparse. It is investigated how the H-matrix approximation error influences the estimated parameters. Numerical examples with Monte Carlo simulations, where the true values of the unknown parameters are given, and an application to soil moisture data with unknown parameters are presented. The C, C++ codes and data are freely available.

Suggested Citation

  • Litvinenko, Alexander & Sun, Ying & Genton, Marc G. & Keyes, David E., 2019. "Likelihood approximation with hierarchical matrices for large spatial datasets," Computational Statistics & Data Analysis, Elsevier, vol. 137(C), pages 115-132.
    • Handle: RePEc:eee:csdana:v:137:y:2019:i:c:p:115-132
    DOI: 10.1016/j.csda.2019.02.002
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    Cited by:

    1. Kelly R. Moran & Matthew W. Wheeler, 2022. "Fast increased fidelity samplers for approximate Bayesian Gaussian process regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(4), pages 1198-1228, September.
    2. Acosta, Jonathan & Alegría, Alfredo & Osorio, Felipe & Vallejos, Ronny, 2021. "Assessing the effective sample size for large spatial datasets: A block likelihood approach," Computational Statistics & Data Analysis, Elsevier, vol. 162(C).
    3. Jialuo Liu & Tingjin Chu & Jun Zhu & Haonan Wang, 2022. "Large spatial data modeling and analysis: A Krylov subspace approach," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 1115-1143, September.
    4. Huang Huang & Sameh Abdulah & Ying Sun & Hatem Ltaief & David E. Keyes & Marc G. Genton, 2021. "Competition on Spatial Statistics for Large Datasets," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 26(4), pages 580-595, December.
    5. Guinness, Joseph, 2022. "Nonparametric spectral methods for multivariate spatial and spatial–temporal data," Journal of Multivariate Analysis, Elsevier, vol. 187(C).

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