Mathematical Problems in Engineering

Volume 2016, Article ID 6723410, 11 pages

http://dx.doi.org/10.1155/2016/6723410

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

^{1}Department of Aerospace Engineering, University of Michigan, 1320 Beal Avenue, Ann Arbor, MI 48109, USA^{2}ONERA, 2 Avenue Édouard Belin, 31055 Toulouse, France^{3}SNECMA, Rond-Point René Ravaud-Réau, 77550 Moissy-Cramayel, France^{4}Institut 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.

#### Abstract

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.

#### 1. Introduction

During the last years, the kriging model [1–4], which is referred to as the Gaussian process model [5], has become one of the most popular methods in computer simulation and machine learning. It is used as a substitute of high-fidelity codes representing physical phenomena and aims to reduce the computational time of a particular process. For instance, the kriging model is used successfully in several optimization problems [6–11]. Kriging is not well adapted to high-dimensional problem, principally due to large matrix inversion problems. In fact, the kriging model becomes much time consuming when a large number of input variables are used since a large number of sampling points are required. Indeed, it is recommended in [12] to use sampling points, with the number of dimensions, for obtaining a good accuracy of the kriging model. As a result, we need to increase the size of the kriging covariance matrix which becomes computationally very expensive to invert. Moreover, this inversion’s problem induces difficulty in the classical hyperparameters estimation through the maximization of the likelihood function.

A recent method, called KPLS [13], is developed to reduce computational time which uses, during a construction of the kriging model, the dimensional reduction method “Partial Least Squares” (PLS). This method is able to reduce the number of hyperparameters of a kriging model, such that their number becomes equal to the number of principal components retained by the PLS method. The KPLS method is thus able to rapidly build a kriging model for high-dimensional problems (100+) while maintaining a good accuracy. However, it has been shown in [13] that the KPLS model is less accurate than the kriging model in many cases, in particular for multimodal functions.

In this paper, we propose an extra step that supplements [13] in order to improve its accuracy. Under hypothesis that kernels used for building the KPLS model are of exponential type with the same form (all Gaussian kernels, e.g.), we choose the hyperparameters found by the KPLS model as an initial point to optimize the likelihood function of a conventional kriging model. In fact, this approach is performed by identifying the covariance function of the KPLS model as a covariance function of a kriging model. The fact of considering the identified kriging model, instead of the KPLS model, leads to extending the search space where the hyperparameters are defined and thus to making the resulting model more flexible than the KPLS model.

This paper is organized in 3 main sections. In Section 2, we present a review of the KPLS model. In Section 3, we discuss our new approach under the hypothesis needed for its applicability. Finally, numerical results are shown to confirm the efficiency of our method followed by a summary of what we have achieved.

#### 2. Construction of KPLS

In this section, we introduce the notation and describe the theory behind the construction of the KPLS model. Assume that we have evaluated a cost deterministic function of points () with , and we denote by the matrix . For simplicity, is considered to be a hypercube expressed by the product between intervals of each direction space; that is, , where with for . Simulating these inputs gives the outputs with , for .

##### 2.1. Construction of the Kriging Model

For building the kriging model, we assume that the deterministic response is realization of a stochastic process [14–17]:The presented formula, with an unknown constant, corresponds to ordinary kriging [8] which is a particular case of universal kriging [15]. The stochastic term is considered as realization of a stationary Gaussian process with and a covariance function, also called kernel function, given bywhere is the process variance and is the correlation function between and . However, the correlation function depends on hyperparameters which are considered to be known. We also denote the vector as and the correlation matrix as . We use to denote the prediction of the true function . Under the hypothesis above, the best linear unbiased predictor for , given the observations , iswhere denotes an -vector of ones andIn addition, the estimation of is given byMoreover, ordinary kriging provides an estimate of the variance of the prediction, which is given by

Note that the assumption of a known covariance function with known parameters is unrealistic in reality and they are often unknown. For this reason, the covariance function is typically chosen from among a parametric family of kernels. In this work, only the covariance functions of exponential type are considered, in particular the Gaussian kernel. Indeed, the Gaussian kernel is the most popular kernel in kriging metamodels of simulation models, which is given by We note that the parameters , for , can be interpreted as measuring how strongly the input variables , respectively, affect the output . If is very large, the kernel given by (7) tends to zero and thus leads to a low correlation. In fact, we see in Figure 1 how the correlation curve rapidly varies from a point to another when .