Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes

Ralf Zimmermann

Journal of Applied Mathematics, 2010, vol. 2010, issue 1

Abstract: The covariance structure of spatial Gaussian predictors (aka Kriging predictors) is generally modeled by parameterized covariance functions; the associated hyperparameters in turn are estimated via the method of maximum likelihood. In this work, the asymptotic behavior of the maximum likelihood of spatial Gaussian predictor models as a function of its hyperparameters is investigated theoretically. Asymptotic sandwich bounds for the maximum likelihood function in terms of the condition number of the associated covariance matrix are established. As a consequence, the main result is obtained: optimally trained nondegenerate spatial Gaussian processes cannot feature arbitrary ill-conditioned correlation matrices. The implication of this theorem on Kriging hyperparameter optimization is exposed. A nonartificial example is presented, where maximum likelihood‐based Kriging model training is necessarily bound to fail.

Date: 2010
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https://doi.org/10.1155/2010/494070

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2010:y:2010:i:1:n:494070

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