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Variable Selection for Kriging in Computer Experiments Journal of Quality Technology A Quarterly Journal of Methods, Applications and Related Topics ISSN: 0022-4065 (Print) 2575-6230 (Online) Journal homepage: https://www.tandfonline.com/loi/ujqt20 Variable selection for kriging in computer experiments Hengzhen Huang, Dennis K. J. Lin, Min-Qian Liu & Qiaozhen Zhang To cite this article: Hengzhen Huang, Dennis K. J. Lin, Min-Qian Liu & Qiaozhen Zhang (2020) Variable selection for kriging in computer experiments, Journal of Quality Technology, 52:1, 40-53, DOI: 10.1080/00224065.2019.1569959 To link to this article: https://doi.org/10.1080/00224065.2019.1569959 Published online: 15 Mar 2019. Submit your article to this journal Article views: 114 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=ujqt20 JOURNAL OF QUALITY TECHNOLOGY 2020, VOL. 52, NO. 1, 40–53 https://doi.org/10.1080/00224065.2019.1569959 Variable selection for kriging in computer experiments Hengzhen Huanga, Dennis K. J. Linb, Min-Qian Liuc, and Qiaozhen Zhangc aCollege of Mathematics and Statistics, Guangxi Normal University, Guilin, China; bDepartment of Statistics, The Pennsylvania State University, University Park, Pennsylvania; cSchool of Statistics and Data Science, LPMC & KLMDASR, Nankai University, Tianjin, China ABSTRACT KEYWORDS An efficient variable selection technique for kriging in computer experiments is proposed. Bayesian variable selection; Kriging models are popularly used in the analysis of computer experiments. The conven- experimental design; tional kriging models, the ordinary kriging, and universal kriging could lead to poor predic- Gaussian process; Gibbs sampler; prediction tion performance because of their prespecified mean functions. Identifying an appropriate mean function for kriging is a critical issue. In this article, we develop a Bayesian variable- selection method for the mean function and the performance of the proposed method can be guaranteed by the convergence property of Gibbs sampler. A real-life application on pis- ton design from the computer experiment literature is used to illustrate its implementation. The usefulness of the proposed method is demonstrated via the practical example and some simulative studies. It is shown that the proposed method compares favorably with the existing methods and performs satisfactorily in terms of several important measurements relevant to variable selection and prediction accuracy. 1. Introduction GP while keeping the mean function as a constant. Their strategy can be used as a screening tool in the As science and technology have advanced to a higher early stage of experimentation. Following the spirit of level, computer experiments are becoming increasingly Welch et al. (1992), Li and Sudjianto (2005) developed prevalent surrogates for physical experiments because a penalized likelihood method and Linkletter et al. of their economy. A computer experiment, however, (2006) developed a Bayesian method to screen varia- can still be time consuming and costly. One primary bles with significant impact on the GP. After the goal of a computer experiment is to build an inexpen- screening stage, a more elaborate kriging model with sive metamodel that approximates the original expen- a complex mean function is desirable. sive computer model well. The kriging model first The focus here is on the next stage—how to appro- proposed by Sacks et al. (1989) in computer experi- priately select variables (terms) for the mean function ments is desirable because of its convenience, flexibil- of kriging. It is noteworthy that the aim of selecting ity, and broad generality. the mean function of kriging is to establish an initial A kriging model mainly contains two parts: a mean model with high quality, which would facilitate function and a stationary Gaussian process (GP). advanced analysis such as the sensitivity analysis, Typically, a kriging model either uses a constant as model calibration, model validation, and heuristic the mean function (known as the ordinary kriging, optimization. The popular initial model is the OK OK) or assumes some prespecified variables in the model, and sometimes a UK model with some prespe- mean function (known as the universal kriging, UK). cified variables in the mean function is adopted as the A computer experiment usually contains a large num- initial model. Some literature (Hung 2011; Joseph, ber of input variables, so it is important to identify Hung, and Sudjianto 2008; Martin and Simpson those (relatively few) variables having significant 2005), however, has shown that both the OK and UK impact on the response. The idea of variable-selection can be underperformed in terms of prediction accur- for computer experiments is initialized by Welch et al. acy when strong trends exist. The main reason is that (1992). In their landmark work, attention was focused a constant mean in the OK may be inadequate to cap- on selecting variables with significant impact on the ture the overall trend, while the prespecified mean CONTACT Min-Qian Liu [email protected] School of Statistics and Data Science, Nankai University, Tianjin 300071, P. R. China; Hengzhen Huang [email protected] College of Mathematics and Statistics, Guangxi Normal University, Guilin, 541004, China. ß 2019 American Society for Quality JOURNAL OF QUALITY TECHNOLOGY 41 function in the UK may contain some redundant pre- simulation studies are conducted in Section 5 to fur- dictors. As a result, identifying an appropriate mean ther demonstrate the usefulness of the SSBK. Finally, function is crucial for building an accurate kriging concluding remarks and further discussions are given model. The existing literature suggests two major in Section 6. approaches to accomplish this purpose: the blind krig- ing (BK) method, which uses a Bayesian forward 2. A hierarchical kriging model selection procedure (Joseph, Hung, and Sudjianto 2008); and the penalized blind kriging (PBK) method, In this section, we first describe the UK model, then a which uses the penalized likelihood approach for vari- hierarchical setting for variable selection in which the able selection (Hung 2011). These methods first iden- UK model details are presented. The hierarchical set- tify mean functions as a variable selection problem ting introduced here is the same as in Le and Zidek and then the built initial models can effectively reduce (2006), with an exception that the idea of variable the prediction error. The BK and PBK also recom- selection is incorporated. To make a better presenta- mended that one should treat the mean function and tion, a concise description without loss of important the GP as an integration rather than dealing with elements is given below. Note that the hierarchical them separately. Both methods, however, suffer from setting introduced here is different from that of € their own limitations: the theoretical properties of Han and Gortz (2012) because multi-fidelity experi- the BK method are difficult to derive, while the ments are beyond the scope of this article. As alluded appealing oracle property of the PBK method only to earlier, however, our method can be used to estab- holds when the design forms a complete lattice. lish high-quality initial models for this kind When the experimental resources are limited, a of experiment. complete lattice design is typically infeasible. In this article, we will focus on variable selection for the 2.1. The universal kriging model mean function. Although there are some universal ; Rp approaches to simultaneously select the active varia- Suppose the input vector x 2X X is the design ; :::; bles for both the GP and the mean function (see, for region with p distinct factors and C¼ff1ðxÞ fkðxÞg example, Marrel et al. 2008), as will be seen, these is the set containing all candidate variables (not universal approaches provide a preliminary basis for including the intercept) for the mean function, where selecting the mean function. k is the number of candidate variables that are func- This article develops a new variable selection tions (e.g., linear and quadratic effects and two-factor method, the stochastic searched blind kriging (SSBK), interactions) of the p factors. The UK model consid- for identifying the proper mean function of kriging. ered here at the screening stage for the mean function The SSBK is a Bayesian approach, which allows the assumes that the output of a computer code YðxÞ is number of candidate variables to be much larger than represented by the number of runs. The SSBK fully accounts for the Xk l ; spatial correlation that is inherent to a kriging model YðÞx ¼ ifiðÞx þ ZðÞx [1] when selecting the mean function. It is shown that i¼1 l l ; :::; l T correctly accounting for the correlation has an impact where ¼ð 1 kÞ is the vector of unknown on the variable-selection result. Moreover, the pro- regression parameters and ZðxÞ is a centered station- posed method is rational and flexible in the sense that ary GP. Given any two points xi and xj, the covari- ; it possesses the Markov convergence property under a ance of YðxiÞ and YðxjÞ is given by cov½YðxiÞ ; r2 ; ; wide class of experimental designs and, hence, does YðxjÞ ¼ cov½ZðxiÞ ZðxjÞ ¼ Rðxi xjÞ, where Rðxi xjÞ 2 not suffer from the limitations of the BK and is the correlation function and r is the process vari- PBK methods. ance. The most commonly used correlation function The remainder of this article is organized as fol- is perhaps the power exponential correlation function: p lows. Section 2 introduces a hierarchical kriging Y au ; qjxiuÀxjuj ; model that provides the basis for the SSBK. Section 3 RðÞxi xj ¼ u [2] presents a step-by-step algorithm for the SSBK and u¼1 ; <q < describes its implementation details. In Section 4,a where xiu is the uth element of xi 0 u 1 and real-life application on piston design analysis 0<au 2; u ¼ 1; :::; p. Throughout this article, we (Hoffman et al.
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