Asymptotic behavior of the likelihood function of covariance matrices of spatial Gaussian processes

Ralf Zimmermann*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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.

Original languageEnglish
Article number494070
JournalJournal of Applied Mathematics
Volume2010
Number of pages17
ISSN1110-757X
DOIs
Publication statusPublished - 2010
Externally publishedYes

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