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

Zimmermann, Ralf (2010) Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes. Journal of Applied Mathematics, 2010, pp. 1-17. Hindawi Publishing Corporation. DOI: 10.1155/2010/494070 ISSN 1110-757X

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Official URL: http://dx.doi.org/10.1155/2010/494070

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.

Item URL in elib:https://elib.dlr.de/68506/
Document Type:Article
Additional Information:Article ID: 494070
Title:Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes
Authors:
AuthorsInstitution or Email of AuthorsAuthors ORCID iD
Zimmermann, Ralfralf.zimmermann (at) dlr.deUNSPECIFIED
Date:December 2010
Journal or Publication Title:Journal of Applied Mathematics
Refereed publication:Yes
Open Access:Yes
Gold Open Access:Yes
In SCOPUS:Yes
In ISI Web of Science:No
Volume:2010
DOI :10.1155/2010/494070
Page Range:pp. 1-17
Editors:
EditorsEmail
Marcellan, F.UNSPECIFIED
Publisher:Hindawi Publishing Corporation
ISSN:1110-757X
Status:Published
Keywords:spatial Gaussian process · Kriging interpolation · design and analysis of computer experiments · best linear unbiased estimator · hyper-parameter training · maximum likelihood estimation · condition number
HGF - Research field:Aeronautics, Space and Transport
HGF - Program:Aeronautics
HGF - Program Themes:Aircraft Research (old)
DLR - Research area:Aeronautics
DLR - Program:L AR - Aircraft Research
DLR - Research theme (Project):L - Simulation & Validation (old)
Location: Braunschweig
Institutes and Institutions:Institute of Aerodynamics and Flow Technology > C²A²S²E - Center for Computer Applications in AeroSpace Science and Engineering
Deposited By: Zimmermann, Dr. Ralf
Deposited On:19 Jan 2011 11:46
Last Modified:24 Jun 2015 03:03

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