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Cross Validation and Maximum Likelihood estimations of hyper-parameters of Gaussian processes with model misspecification

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  • Bachoc, François

Abstract

The Maximum Likelihood (ML) and Cross Validation (CV) methods for estimating covariance hyper-parameters are compared, in the context of Kriging with a misspecified covariance structure. A two-step approach is used. First, the case of the estimation of a single variance hyper-parameter is addressed, for which the fixed correlation function is misspecified. A predictive variance based quality criterion is introduced and a closed-form expression of this criterion is derived. It is shown that when the correlation function is misspecified, the CV does better compared to ML, while ML is optimal when the model is well-specified. In the second step, the results of the first step are extended to the case when the hyper-parameters of the correlation function are also estimated from data.

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  • Bachoc, François, 2013. "Cross Validation and Maximum Likelihood estimations of hyper-parameters of Gaussian processes with model misspecification," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 55-69.
    • Handle: RePEc:eee:csdana:v:66:y:2013:i:c:p:55-69
    DOI: 10.1016/j.csda.2013.03.016
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    1. Zhang, Hao, 2004. "Inconsistent Estimation and Asymptotically Equal Interpolations in Model-Based Geostatistics," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 250-261, January.
    2. Hao Zhang & Dale L. Zimmerman, 2005. "Towards reconciling two asymptotic frameworks in spatial statistics," Biometrika, Biometrika Trust, vol. 92(4), pages 921-936, December.
    3. Andrianakis, Ioannis & Challenor, Peter G., 2012. "The effect of the nugget on Gaussian process emulators of computer models," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 4215-4228.
    4. Marrel, Amandine & Iooss, Bertrand & Van Dorpe, François & Volkova, Elena, 2008. "An efficient methodology for modeling complex computer codes with Gaussian processes," Computational Statistics & Data Analysis, Elsevier, vol. 52(10), pages 4731-4744, June.
    5. Paulo, Rui & García-Donato, Gonzalo & Palomo, Jesús, 2012. "Calibration of computer models with multivariate output," Computational Statistics & Data Analysis, Elsevier, vol. 56(12), pages 3959-3974.
    6. Ying, Zhiliang, 1991. "Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process," Journal of Multivariate Analysis, Elsevier, vol. 36(2), pages 280-296, February.
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    Cited by:

    1. François Bachoc & Emile Contal & Hassan Maatouk & Didier Rullière, 2017. "Gaussian processes for computer experiments," Post-Print hal-01665936, HAL.
    2. Acharki, Naoufal & Bertoncello, Antoine & Garnier, Josselin, 2023. "Robust prediction interval estimation for Gaussian processes by cross-validation method," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).
    3. Cousin, Areski & Maatouk, Hassan & Rullière, Didier, 2016. "Kriging of financial term-structures," European Journal of Operational Research, Elsevier, vol. 255(2), pages 631-648.
    4. Gerber, Florian & Nychka, Douglas W., 2021. "Parallel cross-validation: A scalable fitting method for Gaussian process models," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).
    5. Aikaterini P. Kyprioti & Alexandros A. Taflanidis & Norberto C. Nadal-Caraballo & Madison O. Campbell, 2021. "Incorporation of sea level rise in storm surge surrogate modeling," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 105(1), pages 531-563, January.
    6. Bachoc, François & Lagnoux, Agnès & Nguyen, Thi Mong Ngoc, 2017. "Cross-validation estimation of covariance parameters under fixed-domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 160(C), pages 42-67.
    7. Jize Zhang & Alexandros A. Taflanidis & Norberto C. Nadal-Caraballo & Jeffrey A. Melby & Fatimata Diop, 2018. "Advances in surrogate modeling for storm surge prediction: storm selection and addressing characteristics related to climate change," Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards, Springer;International Society for the Prevention and Mitigation of Natural Hazards, vol. 94(3), pages 1225-1253, December.
    8. Bachoc, François & Bevilacqua, Moreno & Velandia, Daira, 2019. "Composite likelihood estimation for a Gaussian process under fixed domain asymptotics," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    9. Auffray, Yves & Barbillon, Pierre & Marin, Jean-Michel, 2014. "Bounding rare event probabilities in computer experiments," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 153-166.
    10. Kleijnen, Jack P.C. & Mehdad, E., 2013. "Conditional simulation for efficient global optimization," Other publications TiSEM 52e4860d-9887-4a63-b19a-7, Tilburg University, School of Economics and Management.
    11. Binois, M. & Ginsbourger, D. & Roustant, O., 2015. "Quantifying uncertainty on Pareto fronts with Gaussian process conditional simulations," European Journal of Operational Research, Elsevier, vol. 243(2), pages 386-394.
    12. Betancourt, José & Bachoc, François & Klein, Thierry & Idier, Déborah & Pedreros, Rodrigo & Rohmer, Jérémy, 2020. "Gaussian process metamodeling of functional-input code for coastal flood hazard assessment," Reliability Engineering and System Safety, Elsevier, vol. 198(C).
    13. Lee, Dongjin & Kramer, Boris, 2023. "Multifidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos-Kriging," Reliability Engineering and System Safety, Elsevier, vol. 235(C).
    14. Cousin, Areski & Maatouk, Hassan & Rullière, Didier, 2016. "Kriging of financial term-structures," European Journal of Operational Research, Elsevier, vol. 255(2), pages 631-648.
    15. Bachoc, François, 2014. "Asymptotic analysis of the role of spatial sampling for covariance parameter estimation of Gaussian processes," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 1-35.

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