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Communications in Statistics - Simulation and Computation
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Design of Experiment in Global Sensitivity Analysis Based on ANOVA High-Dimensional Model Representation
Xiaodi Wanga; Yincai Tanga; Xueping Chenb; Yingshan Zhanga a School of Finance and Statistics, East China Normal University, Shanghai, China b Department of Mathematics, Jiangsu Teachers University of Technology, Changzhou, China Online publication date: 11 June 2010
To cite this Article Wang, Xiaodi , Tang, Yincai , Chen, Xueping and Zhang, Yingshan(2010) 'Design of Experiment in
Global Sensitivity Analysis Based on ANOVA High-Dimensional Model Representation', Communications in Statistics Simulation and Computation, 39: 6, 1183 — 1195 To link to this Article: DOI: 10.1080/03610918.2010.484122 URL: http://dx.doi.org/10.1080/03610918.2010.484122
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Communications in Statistics—Simulation and Computation® , 39: 1183–1195, 2010 Copyright © Taylor & Francis Group, LLC ISSN: 0361-0918 print/1532-4141 online DOI: 10.1080/03610918.2010.484122
Design of Experiment in Global Sensitivity Analysis Based on ANOVA High-Dimensional Model Representation XIAODI WANG1 , YINCAI TANG1 , XUEPING CHEN2 , AND YINGSHAN ZHANG1 1
School of Finance and Statistics, East China Normal University, Shanghai, China 2 Department of Mathematics, Jiangsu Teachers University of Technology, Changzhou, China This article reviews global sensitivity analysis based on ANOVA high-dimensional model representation. To overcome the computational difficulties and explore the use of design of experiment (DOE) in global sensitivity analysis, two methods are presented. If the form of the objective function f is known, we use DOE to estimate the global sensitivity indices instead of Monte Carlo simulation. Otherwise, we use the observed values of the experiments to do global sensitivity analysis. These methods are easy to implement and can reduce the computational cost. An example is given to show the feasibility of replacing Monte Carlo (MC) or quasi-Monte Carlo (quasi-MC) simulation by design of experiment. Keywords Design of experiment; Global sensitivity analysis; Global sensitivity indices; Monte Carlo algorithm; Orthogonal decomposition. Mathematics Subject Classification 62-XXX.
1. Introduction Global sensitivity analysis has been used in physical, finance and chemical areas (Douglass and Stolarski, 1989; Feng et al., 2004; Wang, 2003). Its main purpose is to investigate the structure of a multivariate function f with input variables x1 xm . (Efron and Stein, 1981; Saltelli et al., 1999; Stein, 1987; Takemura, 1983; Whitney, 1943a,b). If m is large, the model is a high-dimensional model. Two high-dimensional model representations have been investigated, the ANOVA high-dimensional model representation (ANOVA HDMR) and the finite difference high-dimensional model representation (FD HDMR). This article focuses on the ANOVA HDMR. By orthogonal decomposition technique, the Received January 8, 2010; Accepted April 1, 2010 Address correspondence to Xiaodi Wang, School of Finance and Statistics, East China Normal University, Shanghai 200241, China; E-mail:
[email protected]
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function f can be decomposed into summands of different dimensions. This decomposition is called the ANOVA HDMR of f Since the ANOVA model representation is too complex, it is necessary to select a simplified function to approximate it. With the reasonable assumption that small subsets of input variables have the main impact upon the output, which is often used in highdimensional models, two kinds of approximation methods have been discussed. One method, proposed by Rabitz, is to delete the summands with dimensions larger than a certain order L (L < m). The second method is to identify the non influential factors first, and then fix them with certain values. Sobolm (1993) proposed global sensitivity indices to measure the importance of the input variables and their interactions and access the “approximation error”. However, the calculation of these indices involves multidimensional integrals. Monte Carlo or quasi-Monte Carlo simulation can be used. But this will result in too much computational or experimental cost. Thus, obtaining the global sensitivity indices is the main difficulty in global sensitivity analysis based on ANOVA HDMR. Design of experiment is an effective method for data analysis in statistics, and it has a broad variety of applications. In this article, by sampling survey theory, we establish an relationship between DOE and Monte Carlo simulation. Thus, the global sensitivity indices can be estimated using DOE, and the estimation error can be calculated. This method can reduce the sample size, and solve the difficulty in global sensitivity analysis based on ANOVA HDMR. The article proceeds as follows. Section 2 introduces the global sensitivity analysis based on ANOVA HDMR. Basic concepts and theorems are presented in this section, and an example is given to illustrated the theory. In Sec. 3, the methods of DOE in global sensitivity analysis are presented. Conclusions are drawn in Sec. 4.
2. ANOVA HDMR 2.1. Orthogonal Decomposition and Global Sensitivity Indices Assume that fx1 xm is square integrable. The ANOVA HDMR of fx1 xm can be written as: fx1 xm = f0 +
fi xi +
i
fij xi xj + · · · + f12m x1 x2 xm
(1)
i
