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Mathematical Problems in Engineering
Volume 2016, Article ID 6723410, 11 pages
Research Article

An Improved Approach for Estimating the Hyperparameters of the Kriging Model for High-Dimensional Problems through the Partial Least Squares Method

1Department of Aerospace Engineering, University of Michigan, 1320 Beal Avenue, Ann Arbor, MI 48109, USA
2ONERA, 2 Avenue Édouard Belin, 31055 Toulouse, France
3SNECMA, Rond-Point René Ravaud-Réau, 77550 Moissy-Cramayel, France
4Institut Clément Ader, CNRS, ISAE-SUPAERO, Université de Toulouse, 10 Avenue Edouard Belin, 31055 Toulouse Cedex 4, France

Received 31 December 2015; Revised 10 May 2016; Accepted 24 May 2016

Academic Editor: Erik Cuevas

Copyright © 2016 Mohamed Amine Bouhlel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


During the last years, kriging has become one of the most popular methods in computer simulation and machine learning. Kriging models have been successfully used in many engineering applications, to approximate expensive simulation models. When many input variables are used, kriging is inefficient mainly due to an exorbitant computational time required during its construction. To handle high-dimensional problems (100+), one method is recently proposed that combines kriging with the Partial Least Squares technique, the so-called KPLS model. This method has shown interesting results in terms of saving CPU time required to build model while maintaining sufficient accuracy, on both academic and industrial problems. However, KPLS has provided a poor accuracy compared to conventional kriging on multimodal functions. To handle this issue, this paper proposes adding a new step during the construction of KPLS to improve its accuracy for multimodal functions. When the exponential covariance functions are used, this step is based on simple identification between the covariance function of KPLS and kriging. The developed method is validated especially by using a multimodal academic function, known as Griewank function in the literature, and we show the gain in terms of accuracy and computer time by comparing with KPLS and kriging.