Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes
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 s...
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| Format: | Article |
| Language: | English |
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Wiley
2010-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2010/494070 |
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| author | Ralf Zimmermann |
| author_facet | Ralf Zimmermann |
| author_sort | Ralf Zimmermann |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-8f8b613b2a304ce0821bf8ab7b49296d |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2010-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-8f8b613b2a304ce0821bf8ab7b49296d2025-02-03T07:25:48ZengWileyJournal of Applied Mathematics1110-757X1687-00422010-01-01201010.1155/2010/494070494070Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian ProcessesRalf Zimmermann0German Aerospace Center (DLR), Lilienthalplatz 7, 38108 Braunschweig, GermanyThe 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.http://dx.doi.org/10.1155/2010/494070 |
| spellingShingle | Ralf Zimmermann Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes Journal of Applied Mathematics |
| title | Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes |
| title_full | Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes |
| title_fullStr | Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes |
| title_full_unstemmed | Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes |
| title_short | Asymptotic Behavior of the Likelihood Function of Covariance Matrices of Spatial Gaussian Processes |
| title_sort | asymptotic behavior of the likelihood function of covariance matrices of spatial gaussian processes |
| url | http://dx.doi.org/10.1155/2010/494070 |
| work_keys_str_mv | AT ralfzimmermann asymptoticbehaviorofthelikelihoodfunctionofcovariancematricesofspatialgaussianprocesses |