基于SVR的元建模及其在稳健参数设计中的应用
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摘要
为了应对日益激烈的全球竞争,制造业如何以高质量、低成本、短周期获得竞争优势,已成为工业界和学术界极为关注的问题。从现代质量工程的观点来看,质量是设计和制造出来的,产生质量问题的根本原因是波动。质量设计作为减少波动的有效途径,被广泛地应用于产品/过程的设计阶段。经典的试验设计是质量设计的一个重要手段,但其在工业生产领域的应用过程中存在一些不足,如,试验成本高,研制周期长,实现难度大等。针对以上不足,国际上一些学者提出了计算机试验设计,其标志性文献为1989年Sacks等发表的两篇文章,它奠定了计算机试验设计的基础,对计算机试验设计的发展产生了深远影响。
本文在已有的各种元模型构建技术的基础上,以基于元模型的稳健参数设计为研究对象,综合运用系统建模、仿真试验与实证研究等方法,系统地研究了基于非半正定核的支持向量回归机(supprot vector regression, SVR)的构建技术、基于梯度信息的SVR的构建技术、组合元模型的构建技术以及基于单个元模型特别是组合元模型的稳健参数设计。本文的主要研究内容归纳总结如下：
首先,研究了单个元模型的构建技术。本部分重点研究了基于非半正定核的SVR构建技术和基于梯度信息的SVR的构建技术。一方面,针对现有的求解大型SVR的最小最优化(sequential minimal optimization, SMO)算法无法解决核函数为非半正定这一问题,将SVR的原始规划问题进行展开并求解其KKT(Karush-Kuhn-Tucker)条件,减少了需要考虑的Lagrange乘子数目,避免了大量繁琐的判别条件,简化了算法的实现。通过经典的测试函数及鲍鱼数据验证了基于非半正定核SVR算法的有效性。另一方面,针对小样本情形下SVR回归效果不理想这一问题,通过修改目标函数及约束条件,将梯度信息引入到传统的SVR的构建中,重新构造了决策函数。采用了三个基准函数对元模型进行了验证,提出的元模型比传统的SVR在回归精度上有明显的改进。
其次,研究了组合元模型的构建技术。根据样本集来选择单个元模型会加大选用一个并不合适的元模型的概率,针对此问题,综合运用了简单算术平均的方法及递归的思想,以预测均方误差为算法的停机准则,逐步将单个元模型的算术平均模型去替代备选元模型中最差的元模型,以达到在弱化不理想的单个元模型的权重的同时加强理想的单个元模型的权重的目的。分别采用二维、三维、六维及八维的数据对元模型进行了验证,提出的组合方法有效地屏蔽了不理想的单个元模型的负面影响,并且不随样本集变化而变化明显,具有较高的稳健性。
再次,研究了单个元模型特别是组合元模型在稳健参数设计中的应用。针对基于双响应曲面模型的稳健参数设计中所采用的多项式模型对样本数据(特别是方差数据)拟合精度低这一问题,提出了将SVR模型、径向基函数模型、Kriging模型特别是组合元模型应用于双响应曲面的策略。先将SVR模型、径向基函数模型、Kriging模型以及组合元模型去近似均值响应及方差响应,然后再求解基于元模型的随机优化问题,最后得到最优因子搭配水平。以打印墨水为例进行了验证分析,所得的结果与已有的采用多项式模型进行优化的结果相近,说明本文方法的有效性,同时,提出的方法得到更小的均方误差,说明本文方法的优越性。
本文通过研究单个元模型的构建技术、组合元模型的构建技术以及基于单个元模型特别是组合元模型的稳健参数设计方法,进一步丰富了稳健参数设计的研究内涵。最后,本文指出了可进一步研究的问题。
In order to deal with the increasing global competition, industrial and academic circles focus on how to have an advantage over their competitors by high quality, low cost, and short development cycle. On the view of modern quality engineering, quality originates from designing and manufacturing. Variation is the basic factor influencing the quality of products. Quality design, which is an useful tool to reduce variation, is widely employed in the design phase of product/process. Classic design of experiment is an important mean for quality design. Nevertheless, there are some shortcomings, such as, high experimental cost, time-consuming, being difficult to implement, and so on, in its application in industrial areas. Considering these inefficiencies above, some foreign scholars proposed the design of computer experiment, the typical literature of which are those two papers published by Sacks. These two papers laid the foundation for design of computer experiment, and have deep influence on it.
Based on several metamodeling techniques, regarding the robust parameter design based on metamodel as its research object, and using systematic modeling, simulation techniques, and empirical research, this paper systematically studied support vector regression (SVR) based on non-positive semi-definite (non-PSD) kernels, SVR based on gradient information, ensemble of surrogates, and the robust parameter design (RPD) based on metamodeling.
Firstly, the stand-alone surrogates are studied. SVR based on non-PSD kernels and SVR based on gradient information are studied in this part of the paper. On the one hand, considering the traditional sequential minimal optimization (SMO) algorithms, which is often used to solve the optimization problem in SVR, can not deal with the SVR metamodel with non-PSD, the original quadratic programming is spreaded and the Karush-Kuhn-Tucker (KKT) condition is solved in this part. After employing this strategy, the number of the Lagrange multipliers which is needed to consider is diminished, and the tedious judgments are avoided, therefore, the implementation of solving the optimization problem is simplified. The classic test functions and the abalone data are used to test the efficiency of this algorithm. On the other hand, considering the bad performance of the traditional SVR with small samples, the gradient information around the samples are added into the construction of the SVR metamodel after changing the objective function and constrained functions, then the decision function is reconstructed. Three benchmark functions are employed to test the improved SVR metamodel, and the results of the experiment show that the proposed method has better prediction accuracy than the traditional ones.
Secondly, the ensemble of surrogates is studied. Considering the choice of metamodel is highly depended on the set of the samples which is used to construct the metamodels, the simple arithmetic average method and the philosophy of recursion are employed, and the root mean square error (RMSE) of prediction is adopted as the stop criterion of the algorithm. After continually replacing the worst candidate stand-alone metamodel with the arithmetic average model, the weight of the best stand-alone metamodel is raised while that of the worst one is reduced.2-dim,3-dim, and 6-dim test functions, and 8-dim abalone data are used to verify the performance of ensemble technique. The results show that the ensemble of surrogates efficiently kicks out the negative effects of the improper stand-alone metamodels, and the performance of the ensemble of surrogates does not vary apparently with samples. Therefore, ensemble of surrogates to a certain extent is a robust model.
Thirdly, the RPD based on the stand-alone metamodel and the ensemble of surrogates is studied. Considering the dual response surface model in RPD are highly depended on the metamodeling, the SVR metamodel, Kriging metamodel, and radius basis function(RBF) metamodel, and especially the ensemble of surrogates made from these three metamodels are applied into the dual response surface in RPD. Above all, the SVR metamodel, Kriging metamodel, and RBF metamodel are constructed respectively, and then the ensemble of surrogates is also constructed using the above-mentioned metamodels. The next, the mean response and the variation response are built using these stand-alone metamodels and the ensemble of surrogates respectively. In addition, the random optimization process is taken, and the best recommended setting is obtained. The printing ink experiment is employed to test the performances of these stand-alone metamodels and the ensemble of surrogates. These metamodels proposed in this paper have similar recommended settings to the previous polynomial models, which indicates the effectiveness of these metamodels, and the mean square errors (MSEs) with SVR and ensemble of surrogates are lower than the traditional polynomial models, which indicates the superiority of our proposed metamodels.
The stand-alone metamodels, the ensemble of surrogates, and the RPD based on the metamodels are studied systematically, which extends the scientific connotation of RPD. Finally, this paper points out the topics for further study.
本文在已有的各种元模型构建技术的基础上,以基于元模型的稳健参数设计为研究对象,综合运用系统建模、仿真试验与实证研究等方法,系统地研究了基于非半正定核的支持向量回归机(supprot vector regression, SVR)的构建技术、基于梯度信息的SVR的构建技术、组合元模型的构建技术以及基于单个元模型特别是组合元模型的稳健参数设计。本文的主要研究内容归纳总结如下：
首先,研究了单个元模型的构建技术。本部分重点研究了基于非半正定核的SVR构建技术和基于梯度信息的SVR的构建技术。一方面,针对现有的求解大型SVR的最小最优化(sequential minimal optimization, SMO)算法无法解决核函数为非半正定这一问题,将SVR的原始规划问题进行展开并求解其KKT(Karush-Kuhn-Tucker)条件,减少了需要考虑的Lagrange乘子数目,避免了大量繁琐的判别条件,简化了算法的实现。通过经典的测试函数及鲍鱼数据验证了基于非半正定核SVR算法的有效性。另一方面,针对小样本情形下SVR回归效果不理想这一问题,通过修改目标函数及约束条件,将梯度信息引入到传统的SVR的构建中,重新构造了决策函数。采用了三个基准函数对元模型进行了验证,提出的元模型比传统的SVR在回归精度上有明显的改进。
其次,研究了组合元模型的构建技术。根据样本集来选择单个元模型会加大选用一个并不合适的元模型的概率,针对此问题,综合运用了简单算术平均的方法及递归的思想,以预测均方误差为算法的停机准则,逐步将单个元模型的算术平均模型去替代备选元模型中最差的元模型,以达到在弱化不理想的单个元模型的权重的同时加强理想的单个元模型的权重的目的。分别采用二维、三维、六维及八维的数据对元模型进行了验证,提出的组合方法有效地屏蔽了不理想的单个元模型的负面影响,并且不随样本集变化而变化明显,具有较高的稳健性。
再次,研究了单个元模型特别是组合元模型在稳健参数设计中的应用。针对基于双响应曲面模型的稳健参数设计中所采用的多项式模型对样本数据(特别是方差数据)拟合精度低这一问题,提出了将SVR模型、径向基函数模型、Kriging模型特别是组合元模型应用于双响应曲面的策略。先将SVR模型、径向基函数模型、Kriging模型以及组合元模型去近似均值响应及方差响应,然后再求解基于元模型的随机优化问题,最后得到最优因子搭配水平。以打印墨水为例进行了验证分析,所得的结果与已有的采用多项式模型进行优化的结果相近,说明本文方法的有效性,同时,提出的方法得到更小的均方误差,说明本文方法的优越性。
本文通过研究单个元模型的构建技术、组合元模型的构建技术以及基于单个元模型特别是组合元模型的稳健参数设计方法,进一步丰富了稳健参数设计的研究内涵。最后,本文指出了可进一步研究的问题。
In order to deal with the increasing global competition, industrial and academic circles focus on how to have an advantage over their competitors by high quality, low cost, and short development cycle. On the view of modern quality engineering, quality originates from designing and manufacturing. Variation is the basic factor influencing the quality of products. Quality design, which is an useful tool to reduce variation, is widely employed in the design phase of product/process. Classic design of experiment is an important mean for quality design. Nevertheless, there are some shortcomings, such as, high experimental cost, time-consuming, being difficult to implement, and so on, in its application in industrial areas. Considering these inefficiencies above, some foreign scholars proposed the design of computer experiment, the typical literature of which are those two papers published by Sacks. These two papers laid the foundation for design of computer experiment, and have deep influence on it.
Based on several metamodeling techniques, regarding the robust parameter design based on metamodel as its research object, and using systematic modeling, simulation techniques, and empirical research, this paper systematically studied support vector regression (SVR) based on non-positive semi-definite (non-PSD) kernels, SVR based on gradient information, ensemble of surrogates, and the robust parameter design (RPD) based on metamodeling.
Firstly, the stand-alone surrogates are studied. SVR based on non-PSD kernels and SVR based on gradient information are studied in this part of the paper. On the one hand, considering the traditional sequential minimal optimization (SMO) algorithms, which is often used to solve the optimization problem in SVR, can not deal with the SVR metamodel with non-PSD, the original quadratic programming is spreaded and the Karush-Kuhn-Tucker (KKT) condition is solved in this part. After employing this strategy, the number of the Lagrange multipliers which is needed to consider is diminished, and the tedious judgments are avoided, therefore, the implementation of solving the optimization problem is simplified. The classic test functions and the abalone data are used to test the efficiency of this algorithm. On the other hand, considering the bad performance of the traditional SVR with small samples, the gradient information around the samples are added into the construction of the SVR metamodel after changing the objective function and constrained functions, then the decision function is reconstructed. Three benchmark functions are employed to test the improved SVR metamodel, and the results of the experiment show that the proposed method has better prediction accuracy than the traditional ones.
Secondly, the ensemble of surrogates is studied. Considering the choice of metamodel is highly depended on the set of the samples which is used to construct the metamodels, the simple arithmetic average method and the philosophy of recursion are employed, and the root mean square error (RMSE) of prediction is adopted as the stop criterion of the algorithm. After continually replacing the worst candidate stand-alone metamodel with the arithmetic average model, the weight of the best stand-alone metamodel is raised while that of the worst one is reduced.2-dim,3-dim, and 6-dim test functions, and 8-dim abalone data are used to verify the performance of ensemble technique. The results show that the ensemble of surrogates efficiently kicks out the negative effects of the improper stand-alone metamodels, and the performance of the ensemble of surrogates does not vary apparently with samples. Therefore, ensemble of surrogates to a certain extent is a robust model.
Thirdly, the RPD based on the stand-alone metamodel and the ensemble of surrogates is studied. Considering the dual response surface model in RPD are highly depended on the metamodeling, the SVR metamodel, Kriging metamodel, and radius basis function(RBF) metamodel, and especially the ensemble of surrogates made from these three metamodels are applied into the dual response surface in RPD. Above all, the SVR metamodel, Kriging metamodel, and RBF metamodel are constructed respectively, and then the ensemble of surrogates is also constructed using the above-mentioned metamodels. The next, the mean response and the variation response are built using these stand-alone metamodels and the ensemble of surrogates respectively. In addition, the random optimization process is taken, and the best recommended setting is obtained. The printing ink experiment is employed to test the performances of these stand-alone metamodels and the ensemble of surrogates. These metamodels proposed in this paper have similar recommended settings to the previous polynomial models, which indicates the effectiveness of these metamodels, and the mean square errors (MSEs) with SVR and ensemble of surrogates are lower than the traditional polynomial models, which indicates the superiority of our proposed metamodels.
The stand-alone metamodels, the ensemble of surrogates, and the RPD based on the metamodels are studied systematically, which extends the scientific connotation of RPD. Finally, this paper points out the topics for further study.
引文
[1]Gu L. A Comparison of Polynomial Based Regression Models in Vehicle Safety Analysis[A]. ASME Design Engineering Technical Conferences[C]. Pittsburgh, PA:ASME,2001
[2]Koch P N, Simpson T W, Allen J K, Mistree F. Statistical Approximations for Multidisciplinary Design Optimization:The Problem of Size[J]. Journal of Aircraf,1999,36(1):275-286
[3]Myers R H. Response Surface Methodology-Current Status and Future Directions (with Discussion)[J]. Journal of Quality Technology,1997,31(1):30-44
[4]Myers R H, Montgomery D C, Vining G G, Borror C M, Kowalski S M. Response Surface Methodology:A Retrospective and Literature Survey[J]. Journal of Quality Technology,2004,36(1):53-77
[5]Cresssie N. Spatial Prediction and Ordinary Kriging[J]. Mathematical Geology, 1988,20(4):405-421
[6]Toal D J J, Bressloff N W, Keane A J. Kriging Hyperparameter Tuning Strategies[J]. AIAA Journal,2008,46(5):1240-1252
[7]Joseph V R, Hung Y, Sudjianto A. Blind Kriging:A New Method for Developing Metamodels[J]. Journal of Mechanical Design,2008,130(3):1-8
[8]Dyn N, Levin D, Rippa S. Numerical Procedures for Surface Fitting of Scattered Data by Radial Basis Functions [J]. SIAM Journal of Scientific and Statistical Computing,1986,7(2):639-659
[9]Fang K T, Li R, Sudjianto A. Design and Modeling for Computer Experiments [M]. New York:CRC Press,2006
[10]Friedman J H. Multivariate Adaptive Regressive Splines [J]. The annals of statistics,1991,19(1):1-67
[11]De Boor C, Ron A. On Multivariate Polynomial Interpolation[J]. Constructive Approximation,1990,6:287-302.
[12]Langley P, Simon H A. Applications of Machine Learning and Rule Induction[J]. Communications of the ACM,1995,38(11):55-64
[13]Chen W. A Robust Concept Exploration Method for Configuring Complex System[D]. Ph.D. Dissertation Thesis, Atlanta, GA,1995
[14]Anderson-Cook C M, Prewitt K. Some Guidelines for Using Nonparametric Methods for Modeling Data from Response Surface Designs[J]. Journal of Modern Applied Statistical Methods,2005,4(1):106-119
[15]Pickle S M, Robinson T J, Birch J B, Anderson-Cook C M. A Semi-Parametric Approach to Robust Parameter Design[J]. Journal of Statistical Planning and Inference,2008,138:114-131
[16]Vining G G, Bohn L. Response Surfaces for the Mean and Variance Using a Nonparametric Approach [J]. Journal of Quality Technology,1998,30:282-291
[17]Wan W, Birch J B. A Semiparametric Technique for the Multi-Response Optimization Problem[J]. Quality and Reliability Engineering International,2011, 27(1):47-59
[18]Mays J, Birch J B, Einsporn R. An Overview of Model-Robust Regression[J]. J. Statist. Comput. Simulation,2000,66:79-100
[19]Wong P C, Bergeron R D. Scientific Visualization-Overviews, Methodologies and Techniques[M]. Los Alamitos, CA:IEEE Computer Society Press,1997
[20]Kodiyalam S, Yang R J, Gu L. High Performance Computing and Surrogate Modeling for Rapid Visualization with Multidisciplinary Optimization [J]. AIAA Journal,2004,42(11):2347-2235
[21]Ligetti C, Simpson T W. Metamodel-Driven Design Optimization Using Integrative Graphical Design Interfaces:Results from a Job-Shop Manufacturing Simulation Experiment [J]. Transactions of the ASME, Journal of Computing and Information Science in Engineering,2005,5(1):8-17
[22]Forrester A I J, Jones D R. Global Optimization of Deceptive Functions with Sparse Sampling[A].12th AIAA/ISSMO multidisciplinary analysis and optimization conference[C]. Victoria, British Colombia:AIAA,2008
[23]Sacks J, Schiller S B, Welch W J. Designs for Computer Experiments [J]. Technometrics 1989,31(1):41-47
[24]Sacks J, Welch W J, Mitchell T J, Wynn H P. Design and Analysis of Computer Experiments [J]. Statistical science,1989,4(4):409-435
[25]Andrews D W K, Whang Y-J. Additive Interactive Regression Models: Circumvention of the Curse of Dimensionality [J]. Econometric Theory,1990,6: 466-479
[26]Friedman J H, Stuetzle W. Projection Pursuit Regression[J]. Journal of the American statistical Association,1981,76(372):817-823
[27]Chen Z. Fitting Multivariate Regression Functions by Interaction Spline Models[J]. Journal of the Royal Statistical Society,1993,55(2):473-491
[28]Leoni N, Amon C H. Bayesian Surrogates for Integrating Numerical, Analytical, and Experimental Data:Application to Inverse Heat Transfer in Wearable Computers[J]. Components and Packaging Technology,2000,23(1):23-32
[29]Apley D W, Liu J, Chen W. Understanding the Effects of Model Uncertainty in Robust Design with Computer Experiments [J]. Journal of Consumer Research, 2006,128:945-958
[30]刘东雷,申长雨,刘春太,辛勇,孙玲,伍晓宇.基于响应曲面法与改进遗传算法的rhcm成型工艺优化[J].机械工程学报,2011,47(14)：54-61
[31]张烘州,明伟伟,安庆龙,陈明,戎斌,韩冰.响应曲面法在表面粗糙度预测模型及参数优化中的应用[J].上海交通大学学报,2011,44(4)：447-451
[32]赵媚,潘尔顺,郭瑜,孙志礼.基于双响应曲面法的稳健参数设计[J].工业工程与管理,2011,15(1)：87-91
[33]Kleijnen J P C, Beers W V, Nieuwenhuyse I V. Expected Improvement in Efficient Global Optimization through Bootstrapped Kriging[J]. Journal of Global Optimization
[34]Cressie N, Kang E L. High-Resolution Digital Soil Mapping:Kriging for Very Large Datasets [J]. Progress in Soil Science,2010,1(1):49-36
[35]柳强,王成恩.基于kriging模型的复杂产品管线敷设顺序粒子群优化[J].机械工程学报,2011,47(13)：140-146
[36]黄尊地,梁习锋,钟睦.基于kriging模型的挡风墙优化设计[J].中南大学学报(自然科学版),2011,42(7)：2152-2155
[37]容江磊,谷正气,杨易,江涛,尹郁琦.基于kriging模型的跑车尾翼断面形状的气动优化[J].中国机械工程,2011,22(2)：243-247
[38]陈志英,任远,白广忱,高阳.粒子群优化的kriging近似模型及其在可靠性分析中的应用[J].航空动力学报,2011,26(7)：1522-1530
[39]An J, Owen A. Quasi-Regression[J]. Journal of Complexity,2001,17(4):588-607
[40]Jiang T, Owen A B. Quasi-Regression with Shrinkage[J]. Mathematics and Computers in Simulation,2003,62(2-3):231-241
[41]Chen V C P, Ruppert D, Shoemaker C A. Applying Experimental Design and Regression Splines to High-Dimensional Continuous-State Stochastic Dynamic Programming[J]. Operations Research,1999,47(1):38-53
[42]Papadrakakis M, Lagaros M, Tsompanakis Y. Structural Optimization Using Evolution Strategies and Neural Networks [J]. Computer Methods in Applied Mechanics and Engineering,1998,156(1-4):309-333
[43]Martin J D, Simpson T W. Use of Kriging Models to Approximate Deterministic Computer Models[J]. AIAA Journal,2005,43(4):853-863
[44]Kleijnen J P C, van Beers W. Kriging for Interpolation in Random Simulation[J]. Journal of the Operational Research,2003,54:255-262
[45]Jones B, Johnson R T. Design and Analysis for the Gaussian Process Model[J]. Quality and Reliability Engineering International,2009,25:515-550
[46]Han G, Santner T J, Notz W I, Bartel D L. Prediction for Computer Experiments Having Quantitative and Qualitative Input Variables[J]. Technometrics,2009,51: 278-288
[47]Taddy M A, Lee H K H, Gray G A, Griffin J D. Bayesian Guided Pattern Search for Robust Local Optimization[J]. Technometrics,2009,51:389-401
[48]Bayarri M J, Berger J O, Kennedy M C, Kottas A, Paulo R, Sacks J, Cafeo J A, Lin C H, Tu J. Predicting Vehicle Crashworthiness:Validation of Computer Models for Functional and Hierarchical Data[J]. Journal of the American Statistical Association,2009,104:929-943
[49]Jin R, Chen W, Simpson T W. Comparative Studies of Metamodeling Techniques under Multiple Modeling Criteria[J]. Structural and Multidisciplinary Optimization,2001,23(1):1-13
[50]Wang L, Beeson D. A Comparison of Metamodeling Methods Using Practical Industry Requirements [A]. The 47th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference[C]. Newport, Rhode Island, USA: AIAA,2006
[51]Chen V C P, Tsui K-L, Barton R R, Meckesheimer M. A Review on Design, Modeling and Applications of Computer Experiments [J]. IIE Transactions,2006, 38:273-291
[52]Meckesheimer M, Barton R R, Simpson T W, Booker A J. Computationally Inexpensive Metamodel Assessment Strategies[J]. AIAA Journal,2002,40(10): 2053-2060
[53]Kennedy M C, O'Hagan A. Bayesian Calibration of Computer Models[J]. Journal of the Royal Statistical Society,2001,63(3):525-464
[54]Hooker G. Discovering Additive Structure in Black Box Functions[A]. Proceedings of the tenth ACM SIGKDD international conference on knowledge discovery and data mining[C]. Seattle, WA, USA:ACM,2004
[55]Owen A B. Assessing Linearity in High Dimensions[J]. The Annals of Statistics, 2000,28(1):1-19
[56]Kleijnen J P C. Handbook of Computational Statistics:Concepts and Fundamentals [M]. Heidelberg, Germany. Springer-Verlag,2004
[57]Giunta A A, Dudley J M, Narducci R, Grossman B, Haftka R T, Mason W H, Watson L T. Noisy Aerodynamic Response and Smooth Approximations in Hsct Design[A].5th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization[C]. Washington, DC:AIAA,1994
[58]Madsen J I, Shyy W, Haftka R T. Response Surface Techniques for Diffuser Shape Optimization[J]. AIAA Journal,2000,38(9):1512-1518
[59]Jin R, Du X, Chen W. The Use of Metamodeling Techniques for Optimization under Uncertainty [J]. Structural and Multidisciplinary Optimization,2003,25(2): 99-116
[60]Van Beers W, Kleijnen J P C. Kriging Interpolation in Simulation:A Survey [A]. Proceedings of the 2004 Winter Simulation Conference[C]. Piscataway, NJ:IEEE, 2004
[61]邓乃扬,田英杰.数据挖掘的新方法-支持向量机[M].北京：科学出版社,2004
[62]Morris M D, Mitchell T J, Ylvisaker D. Bayesian Design and Analysis of Computer Experiments:Use of Derivatives in Surface Prediction[J]. Technometrics,1993,35(3):243-255
[63]Toropov V V, Filatov A A. Multi-Parameter Structural Optimization Using Fem and Multipoint Approximation[J]. Structural and Multidisciplinary Optimization, 1993,6:7-14
[64]Wang L P, Grandhi R V, Canfield R A. Multivariate Hermite Approximation for Design Optimization[J]. International Journal for Numerical Methods in Engineering,1996,39:787-803
[65]Rasmussen J. Nonlinear Programming by Cumulative Approximation Refinement[J]. Structural and Multidisciplinary Optimization,1998,15:1-7
[66]Shin Y S, Grandhi R V. A Global Structural Optimization Technique Using an Interval Method[J]. Structural and Multidisciplinary Optimization,2001,22: 351-363
[67]李清,李伟明,徐大丰.基于元模型的企业模型表达[J].清华大学学报(自然科学版),2008,48(7)：1209-1212
[68]张昆仑,刘新亮,郭波.基于高斯过程元模型的产品设计时间估计方法[J].计算机集成制造系统,2011,17(1)：18-22
[69]张伟,周建军,李建涛,朱一凡,李群.基于仿真元模型的探索性分析方法研究[J].系统仿真学报,2011,23(7)：1336-1341
[70]严建援,李凯,张路.一种基于语法的供应链流程定义元模型[J].管理科学学报,2010,13(6)：33-43
[71]Clarke S M, Griebsch J H, Simpson T W. Analysis of Support Vector Regression for Approximation of Complex Engineering Analyses[J]. Journal of Mechanical Design,2005,127(6):1077-1087
[72]Zhang P. Model Selection Via Multifold Cross Validation[J]. Statistica Sinica, 2003,13:783-809
[73]Roecker E B. Prediction Error and Its Estimation for Subset-Selected Models[J]. Technometrics,1991,33(4):459-468
[74]Utans J, Moody J. Selecting Neural Network Architectures Via the Prediction Risk:Application Tocorporate Bond Rating Prediction[A]. Proceedings of the IEEE 1st International Conference on AI Applications on Wall Street[C]. New York:IEEE,1991
[75]Leblanc M, Tibshirani R. Combining Estimates in Regression and Classification[J]. Journal of theAmerican Statistical Association,1996,91(436): 1641-1650
[76]Yang Y. Regression with Multiple Candidate Models:Selecting or Mixing?[J]. Statistica Sinica,2003,13:783-809
[77]Meckesheimer M. A Framework for Metamodel-Based Design:Subsystem Metamodel Assessment and Implementation Issues[D]. Pennsylvania,2001
[78]Mitchell T J, Morris M D. Bayesian Design and Analysis of Computer Experiments:Two Examples[J]. Statistica Sinica,1992,2:359-379
[79]Lin Y. An Efficient Robust Concept Exploration Method and Sequential Exploratory Experimental Design[D]. Atlanta,2004
[80]Simpson T W, Mauery T M, Korte J J, Mistree F. Kriging Metamodels for Global Approximation in Simulation-Based Multidisciplinary Design Optimizatio[J]. AIAA Journal,2001,39(12):2233-2241
[81]Mack Y, Goel T, Shyy W, Haftka R T, Queipo N V. Multiple Surrogates for Shape Optimization for Bluff-Body Facilitated Mixing[A].43rd AIAA aerospace sciences meeting and exhibit[C]. Reno, Nevada:AIAA,2005
[82]Glaz B, Goel T, Liu L, Friedmann P P, Haftka R T. Application of a Weighted Average Surrogateapproach to Helicopter Rotor Blade Vibration Reduction[A]. 48th AIAA/ASME/ASCHE/AHS/ASC Structures, Structural Dynamics & Materials Conference[C]. Honolulu, HI:AIAA,2007
[83]Samad A, Kim K, Goel T, Haftka R T, Shyy W. Multiple Surrogate Modeling for Axial Compressor Blade Shape Optimization[J]. Journal of Propulsion and Power, 2008,24(2):302-310
[84]Sanchez E, Pintos S, Queipo N V. Toward an Optimal Ensemble of Kernel-Based Approximations with Engineering Applications [J]. Structural and Multidisciplinary Optimization,2008,36(3):247-261
[85]Viana F A C, Haftka R T. Using Multiple Surrogates for Minimization of the Rms Error in Metamodeling[A]. ASME Design Engineering Technical Conferences-Design Automation Conference[C]. NY:ASME,2008
[86]李建平,王维平,周敏,吴强.多拟合法凸线性组合元模型及其应用[A].2007系统仿真技术及其应用学术会议[C].中国安徽合肥：中国自动化学会系统仿真专业委员会、中国系统仿真学会仿真应用专业委员会,2007
[87]唐小我,曹长修.递归等权组合预测方法研究[J].电子科技大学学报,1992,21(5)：545-550
[88]Taguchi G. Introduction to Quality Engineering[M]. New York:White Plains, 1986
[89]Taguchi G Systems of Experimental Design:Engineering Methods to Optimize Quality and Minimize Cost[M]. New York:White Plains,1987
[90]Myers R H, Khuri A I, Vining G. Response Surface Alternatives to the Taguchi Robust Parameter Design Approach [J]. American Statistician,1992,46:131-139
[91]Myers R H, Montgomery D C, Anderson-Cook C M. Response Surface Methodology:Process and Product Optimization Using Designed Experiments; Third Edition[M]. New York:Wiley,2009
[92]Vining G G, Myers R H. Combining Taguchi and Response Surface Philosophies: A Dual Response Approach[J]. Journal of Quality Technology,1990,22:38-45
[93]Del Castillo E, Montgomery D C. A Nonlinear Programming Solution to the Dual Response Problem [J]. Journal of Quality Technology,1993,25:199-204
[94]Lin D K J, Tu W. Dual Response Surface Optimization[J]. Journal of Quality Technology,1995,27:34-39
[95]Ames A E, Mattucci N, Macdonald S, Szonyi G, Hawkins D M. Quality Loss Functions for Optimization across Multiple Response Surfaces[J]. Quality and Reliability Engineering International,1997,29:339-346
[96]Antony J. Multi-Response Optimization in Industrial Experiments Using Taguchi's Quality Loss Function and Principal Component Analysis[J]. Quality and Reliability Engineering International,2000,16(1):3-8
[97]Kim K, Lin D K J. Dual Response Surface Optimization:A Fuzzy Modeling Approach[J]. Journal of Quality Technology,1998,30:1-10
[98]Tong L, Su C. Optimizing Multi-Response Problems in the Taguchi Method by Fuzzy Multiple Attribute Decision Making[J]. Quality and Reliability Engineering International,1997,13(1):25-34
[99]Kumar P, Barua P B, Gaindhar J L. Quality Optimization (Multi-Characteristics) through Taguchi's Technique and Utility Concept[J]. Quality and Reliability Engineering International,2000,16(6):475-485
[100]Tang L C, Xu K. Unified Approach for Dual Response Surface Optimization [J]. Journal of Quality Technology,2002,34:437-447
[101]Kim Y J, Cho B R. Development of Priority-Based Robust Design[J]. Journal of Quality Technology,2002,14:355-363
[102]Koksoy O, Doganaksoy N. Joint Optimization of Mean and Standard Deviation in Response Surface Experimentation [J]. Journal of Quality Technology,2003, 35(3):239-252
[103]Lam S W, Tang L C. A Graphical Approach to the Dual Response Robust Design Problems[A]. Reliability and maintainability symposium. Proceedings. Annual[C]. Reno, NV:RAMS,2005
[104]Koksoy O, Yalcinoz T. A Hopfield Neural Network Approach to the Dual Response Problem[J]. Quality and Reliability Engineering International,2005,21: 595-603
[105]赵媚,潘尔顺,郭瑜,孙志礼.基于双响应曲面法的稳健参数设计[J].工业工程与管理,2010,15(1)：87-91
[106]何桢,王晶,李湧范.基于改进的距离函数法的多响应稳健参数设计 [J].天津大学学报,2010,43(7)：644-649
[107]汪建均,马义中,汪新.广义线性模型的贝叶斯分析及稳健参数设计应用[J].系统工程,2009,27(4)：71-77
[108]崔庆安,何桢,车建国.一种基于支持向量机的非参数双响应曲面法[J].天津大学学报,2006,39(8)：1008-1014
[109]Powell M J D. Radial Basis Functions for Multivariable Interpolation:A Review[A]. Proceedings of the IMA Conference on Algorithms for the Approximation of Functions and Data[C]. London:IMA,1987
[110]McDonald D B, Grantham W J, Tabor W L, Murphy M J. Response Surface Model Development for Global/Local Optimization Using Radial Basis Functions[A]. the 8th AIAA Symposium on Multidisciplinary Analysis and Optimization[C]. Long Beach, CA:AIAA,2000
[111]Bishop C M. Neural Networks for Pattern Recognition[M]. New York:Oxford University Press,1995
[112]Zerpa L, Queipo N V, Pintos S, Salager J. An Optimization Methodology of Alkaline-Surfactant-Polymer Flooding Processes Using Field Scale Numerical Simulation and Multiple Surrogates [J]. Journal of Petroleum Science and Engineering,2005,47:197-208
[113]Goel T, Haftka R T, Shyy W, Queipo N V. Ensemble of Surrogates[J]. Structural and Multidisciplinary Optimization,2007,33(3):199-216
[114]Acar E, Rais-Rohani M. Ensemble of Metamodels with Optimized Weight Factors[J]. Structural and Multidisciplinary Optimization,2009,37:279-294
[115]Viana F A C, Haftka R T, Steffen V. Multiple Surrogate:How Cross-Validation Errors Can Help Us to Obtain the Best Predictor [J]. Structrual and Multidiscplinary Optimization,2009,39:439-457
[116]Vapnik V N. The Nature of Statistical Learning Theory[M]. New York: Springer-Verlag,1995
[117]Vapnik V N. Estimation of Dependences Based on Empirical Data[M]. NY: Springer-Verlag,1982
[118]Osuna E, Freund R, Girosi F. Improved Training Algorithm for Support Vector Machines [A]. Neural Networks and Signal Processing VII-proceedings of the 1997 IEEE Workshop[C]. New York:IEEE,1997
[119]Osuna E. Support Vector Machines:Training and Applications[M]. Cambridge, MA:Massachusetts Institute of Technology,1998
[120]Smola A J, Scholkopf B. (1998). A Tutorial on Support Vector Regression. In: Royal Holloway College
[121]Keerthi S, Shevade S, Bhattacharyya C, Murthy K. Improvements to Smo Algorithm for Svm Regression[J]. IEEE Transaction on Neural Networks,2000, 11:1188-1193
[122]Flake G W, Lawrence S. Efficient Svm Regression Training with Smo[J]. Machine Learning,2002,46:271-290
[123]Lin H-T, Lin C-J. (2003). A Study on Sigmoid Kernels for Svm and the Training of Non-Psd Kernels by Smo-Type Methods. In. TaiPei, Taiwan:National Taiwan University
[124]Platt J C. Fast Training of Support Vector Machines Using Sequential Minimal Optimization. Advances in Kernel Methods:Support Vector Machines[M]. Cambridge MA:MIT Press,1998
[125]Lin C-J. Asymptotic Convergence of an Smo Algorithm without Any Assumptions [J]. IEEE Transactions on Neural Networks,2002,13(1):248--250
[126]Palagi L, Sciandrone M. On the Convergence of a Modified Version of Svmlight Algorithm[J]. Optimization Methods and Software,2005,20(2-3):315-332
[127]Ong C S, Mary X, Canu S. Learning with Non-Positive Kernels[A]. Proceedings of ICML[C]. New York:ACM,2004
[128]Kwok J T, Tsang I W. Linear Dependency between E and the Input Noise in E-Support Vector Regression[J]. IEEE Transaction on Neural Networks,2003, 14(3):544-553
[129]Wang S-t, Zhu J-g, Chung F L, Lin Q, Hu D-w. Theoretically Optimal Parameter Choices for Support Vector Regression Machines Huber-Svr and Norm_R-Svr with Noisy Input[J]. Soft Computing,2005,9(10):732-741
[130]周晓剑,马义中,朱嘉钢.Smo算法的简化及其在非正定核条件下的应用[J].计算机研究与发展,2010,47(11)：1962-1969
[131]周晓剑,马义中,朱嘉钢,刘利平,汪建均.求解非半正定核huber-支持向量回归机问题的序列最小最优化算法研究[J].控制理论与应用,2010,27(9)：1178-1184
[132]周晓剑,马义中.两种求解非正定核laplace-Svr的smo算法[J].控制与决策,2009,24(11)：1657-1662
[133]Sellar R S, Batill S M, Renaud J E. Concurrent Subspace Optimization Using Gradient-Based Neural Network Response Surface Mappings [A].6th AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization conference[C]. Bellevue, Washington:AIAA,1996
[134]Liu W, Batill S. Gradient-Enhanced Neural Network Response Surface Approximations[A].8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization conference[C]. Long Beach, CA: AIAA,2000
[135]Van K F, Vervenne K. Gradient-Enhanced Response Surface Building[J]. Structural and Multidisciplinary Optimization,2004,27:337-351
[136]Kim C, Wang S, Choi K K. Efficient Response Surface Modeling by Using Moving Least-Squares Method and Sensitivity[J]. AIAA Journal,2005,43(11): 2404-2411
[137]Chung H-S, Alonso J J. Using Gradients to Construct Cokriging Approximation Models for High-Dimensional Design Optimization Problems[A].40th AIAA Aerospace Sciences Meeting and Exhibit[C]. Reno, Nevada:AIAA,2002
[138]Liu W, Batill S M. Gradient-Enhanced Response Surface Approximations Using Kriging Models[A].9th AIAA/ISSMO Symposium and Exhibit on Multidisciplinary Analysis and Optimization conference[C]. Atlanta, GA:AIAA, 2002
[139]Alexander I J, Forrester A J K. Recent Advances in Surrogate-Based Optimization[J]. Progress in Aerospace Sciences,2009,45:50-79
[140]Barthelemy J, Hall L E. Automatic Differentiation as a Tool in Engineering Design[A].4th AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization conference[C]. Cleveland, Ohio:AIAA,1992
[141]Giles M B, Pierce N A. An Introduction to the Adjoint Approach to Design[J]. Flow, Turbulence and Combustion,2000,65(3-4):393-415
[142]Sobieszczanski-Sobieski J, Haftka. Multidisciplinary Aerospace Design Optimization:Survey of Recent Developments Industrial Experiments [J]. Structural and Multidisciplinary Optimization,1997,14(1):1-23
[143]Giunta A A, Watson L T. A Comparison of Approximation Modeling Techniques: Polynomial Versus Interpolating Models[A]. Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis & Optimization, AIAA-98-4758[C]. St. Louis, MO:AIAA,1998
[144]Simpson T W, Peplinski J D, Koch P N, Allen J K. Meta-Models for Computer Based Engineering Design:Survey and Recommendations [J]. Engineering with computers,2001,17:128-150
[145]Li W, Padula S. Approximation Methods for Conceptual Design of Complex Systems[M]. Brentwood, TN:Nashboro Press,2005
[146]Queipo N V, Haftka R T, Shyy W, Goel T, Vaidyanathan R, Tucker P K. Surrogate-Based Analysis and Optimization [J]. Progress in Aerospace Sciences, 2005,41:1-28
[147]Papila N, Shyy W, Griffin L W, Dorney D J. Shape Optimization of Supersonic Turbines Using Response Surface and Neural Network Methods [J]. Journal of Propulsion and Power,2001,18:509-518
[148]Shyy W, Papila N, Vaidyanathan R, Tucker P K. Global Design Optimization for Aerodynamics and Rocket Propulsion Components[J]. Progress in Aerospace Sciences 2001,37:59-118
[149]Vaidyanathan R, Tucker P K, Papila N, Shyy W. Cfd Based Design Optimization for a Single Element Rocket Injector[J]. Journal of Propulsion and Power,2004,20(4):705-717
[150]Stander N, Roux W, Giger M, Redhe M, Fedorova N, Haarhoff J. A Comparison of Meta-Modeling Techniques for Crashworthiness Optimization[A]. Proceedings of the 10th AIAA/ISSMO multidisciplinary analysis and optimization conference[C]. Albany, NY:AIAA,2004
[151]Fang H, Rais-Rohani M, Liu Z, Horstemeyer M F. A Comparative Study of Metamodeling Methods for Multi-Objective Crashworthiness Optimization [J]. Computers & Structures,2005,83:2121-2136
[152]Viana F A C, Haftka R T. Using Multiple Surrogates for Metamodeling[A]. ASMO-UK/ISSMO International Conference on Engineering Design Optimization[C]. Bath, UK:ISSMO,2008
[153]Yang Y. Regression with Multiple Candidate Models:Selecting or Mixing[J]. Statistic Sinica,2003,13(5):783-809
[154]Shaibu A B, Cho B R. Another View of Dual Response Surface Modeling and Optimization in Robust Parameter Design[J]. The International Journal of Advanced Manufacturing Technology 2009,41:631-641
[155]Owen A B. Orthogonal Arrays for Computer Experiments, Integration, and Visualization[J]. Statistica Sinica,1992,2:439-452
[156]Fang K T, Lin D K J, Winker P, Zhang Y. Uniform Design:Theory and Application[J]. Technometrics,2000,39(3):237-248
[157]McKay M D, Bechman R J, Conover W J. A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code[J]. Technometrics,1979,21(2):239-245
[158]Zhou X J, Ma Y Z, Li X F. Ensemble of Surrogates with Recursive Arithmetic Average[J]. Structrual and Multidiscplinary Optimization,2011,44(5):651-671
[159]Box G E P, Draper N R. Expirical Model Building and Response Surface[M]. New York:Wiley,1987
[160]Copeland K A F, Nelson P R. Dual Response Optimization Via Direct Function Minimization[J]. Journal of Quality Technology,1996,28:331-336
[161]Koch P N, Yang R J, Gu L. Design for Six Sigma through Robust Optimization[J]. Structural and Multidisciplinary Optimization,2004,26(3-4):235-248
[162]Wang G G, Simpson T W. Fuzzy Clustering Based Hierarchical Metamodeling for Space Reduction and Design Optimization[J]. Journal of Engineering Optimization,2004,36(3):313-335
[2]Koch P N, Simpson T W, Allen J K, Mistree F. Statistical Approximations for Multidisciplinary Design Optimization:The Problem of Size[J]. Journal of Aircraf,1999,36(1):275-286
[3]Myers R H. Response Surface Methodology-Current Status and Future Directions (with Discussion)[J]. Journal of Quality Technology,1997,31(1):30-44
[4]Myers R H, Montgomery D C, Vining G G, Borror C M, Kowalski S M. Response Surface Methodology:A Retrospective and Literature Survey[J]. Journal of Quality Technology,2004,36(1):53-77
[5]Cresssie N. Spatial Prediction and Ordinary Kriging[J]. Mathematical Geology, 1988,20(4):405-421
[6]Toal D J J, Bressloff N W, Keane A J. Kriging Hyperparameter Tuning Strategies[J]. AIAA Journal,2008,46(5):1240-1252
[7]Joseph V R, Hung Y, Sudjianto A. Blind Kriging:A New Method for Developing Metamodels[J]. Journal of Mechanical Design,2008,130(3):1-8
[8]Dyn N, Levin D, Rippa S. Numerical Procedures for Surface Fitting of Scattered Data by Radial Basis Functions [J]. SIAM Journal of Scientific and Statistical Computing,1986,7(2):639-659
[9]Fang K T, Li R, Sudjianto A. Design and Modeling for Computer Experiments [M]. New York:CRC Press,2006
[10]Friedman J H. Multivariate Adaptive Regressive Splines [J]. The annals of statistics,1991,19(1):1-67
[11]De Boor C, Ron A. On Multivariate Polynomial Interpolation[J]. Constructive Approximation,1990,6:287-302.
[12]Langley P, Simon H A. Applications of Machine Learning and Rule Induction[J]. Communications of the ACM,1995,38(11):55-64
[13]Chen W. A Robust Concept Exploration Method for Configuring Complex System[D]. Ph.D. Dissertation Thesis, Atlanta, GA,1995
[14]Anderson-Cook C M, Prewitt K. Some Guidelines for Using Nonparametric Methods for Modeling Data from Response Surface Designs[J]. Journal of Modern Applied Statistical Methods,2005,4(1):106-119
[15]Pickle S M, Robinson T J, Birch J B, Anderson-Cook C M. A Semi-Parametric Approach to Robust Parameter Design[J]. Journal of Statistical Planning and Inference,2008,138:114-131
[16]Vining G G, Bohn L. Response Surfaces for the Mean and Variance Using a Nonparametric Approach [J]. Journal of Quality Technology,1998,30:282-291
[17]Wan W, Birch J B. A Semiparametric Technique for the Multi-Response Optimization Problem[J]. Quality and Reliability Engineering International,2011, 27(1):47-59
[18]Mays J, Birch J B, Einsporn R. An Overview of Model-Robust Regression[J]. J. Statist. Comput. Simulation,2000,66:79-100
[19]Wong P C, Bergeron R D. Scientific Visualization-Overviews, Methodologies and Techniques[M]. Los Alamitos, CA:IEEE Computer Society Press,1997
[20]Kodiyalam S, Yang R J, Gu L. High Performance Computing and Surrogate Modeling for Rapid Visualization with Multidisciplinary Optimization [J]. AIAA Journal,2004,42(11):2347-2235
[21]Ligetti C, Simpson T W. Metamodel-Driven Design Optimization Using Integrative Graphical Design Interfaces:Results from a Job-Shop Manufacturing Simulation Experiment [J]. Transactions of the ASME, Journal of Computing and Information Science in Engineering,2005,5(1):8-17
[22]Forrester A I J, Jones D R. Global Optimization of Deceptive Functions with Sparse Sampling[A].12th AIAA/ISSMO multidisciplinary analysis and optimization conference[C]. Victoria, British Colombia:AIAA,2008
[23]Sacks J, Schiller S B, Welch W J. Designs for Computer Experiments [J]. Technometrics 1989,31(1):41-47
[24]Sacks J, Welch W J, Mitchell T J, Wynn H P. Design and Analysis of Computer Experiments [J]. Statistical science,1989,4(4):409-435
[25]Andrews D W K, Whang Y-J. Additive Interactive Regression Models: Circumvention of the Curse of Dimensionality [J]. Econometric Theory,1990,6: 466-479
[26]Friedman J H, Stuetzle W. Projection Pursuit Regression[J]. Journal of the American statistical Association,1981,76(372):817-823
[27]Chen Z. Fitting Multivariate Regression Functions by Interaction Spline Models[J]. Journal of the Royal Statistical Society,1993,55(2):473-491
[28]Leoni N, Amon C H. Bayesian Surrogates for Integrating Numerical, Analytical, and Experimental Data:Application to Inverse Heat Transfer in Wearable Computers[J]. Components and Packaging Technology,2000,23(1):23-32
[29]Apley D W, Liu J, Chen W. Understanding the Effects of Model Uncertainty in Robust Design with Computer Experiments [J]. Journal of Consumer Research, 2006,128:945-958
[30]刘东雷,申长雨,刘春太,辛勇,孙玲,伍晓宇.基于响应曲面法与改进遗传算法的rhcm成型工艺优化[J].机械工程学报,2011,47(14)：54-61
[31]张烘州,明伟伟,安庆龙,陈明,戎斌,韩冰.响应曲面法在表面粗糙度预测模型及参数优化中的应用[J].上海交通大学学报,2011,44(4)：447-451
[32]赵媚,潘尔顺,郭瑜,孙志礼.基于双响应曲面法的稳健参数设计[J].工业工程与管理,2011,15(1)：87-91
[33]Kleijnen J P C, Beers W V, Nieuwenhuyse I V. Expected Improvement in Efficient Global Optimization through Bootstrapped Kriging[J]. Journal of Global Optimization
[34]Cressie N, Kang E L. High-Resolution Digital Soil Mapping:Kriging for Very Large Datasets [J]. Progress in Soil Science,2010,1(1):49-36
[35]柳强,王成恩.基于kriging模型的复杂产品管线敷设顺序粒子群优化[J].机械工程学报,2011,47(13)：140-146
[36]黄尊地,梁习锋,钟睦.基于kriging模型的挡风墙优化设计[J].中南大学学报(自然科学版),2011,42(7)：2152-2155
[37]容江磊,谷正气,杨易,江涛,尹郁琦.基于kriging模型的跑车尾翼断面形状的气动优化[J].中国机械工程,2011,22(2)：243-247
[38]陈志英,任远,白广忱,高阳.粒子群优化的kriging近似模型及其在可靠性分析中的应用[J].航空动力学报,2011,26(7)：1522-1530
[39]An J, Owen A. Quasi-Regression[J]. Journal of Complexity,2001,17(4):588-607
[40]Jiang T, Owen A B. Quasi-Regression with Shrinkage[J]. Mathematics and Computers in Simulation,2003,62(2-3):231-241
[41]Chen V C P, Ruppert D, Shoemaker C A. Applying Experimental Design and Regression Splines to High-Dimensional Continuous-State Stochastic Dynamic Programming[J]. Operations Research,1999,47(1):38-53
[42]Papadrakakis M, Lagaros M, Tsompanakis Y. Structural Optimization Using Evolution Strategies and Neural Networks [J]. Computer Methods in Applied Mechanics and Engineering,1998,156(1-4):309-333
[43]Martin J D, Simpson T W. Use of Kriging Models to Approximate Deterministic Computer Models[J]. AIAA Journal,2005,43(4):853-863
[44]Kleijnen J P C, van Beers W. Kriging for Interpolation in Random Simulation[J]. Journal of the Operational Research,2003,54:255-262
[45]Jones B, Johnson R T. Design and Analysis for the Gaussian Process Model[J]. Quality and Reliability Engineering International,2009,25:515-550
[46]Han G, Santner T J, Notz W I, Bartel D L. Prediction for Computer Experiments Having Quantitative and Qualitative Input Variables[J]. Technometrics,2009,51: 278-288
[47]Taddy M A, Lee H K H, Gray G A, Griffin J D. Bayesian Guided Pattern Search for Robust Local Optimization[J]. Technometrics,2009,51:389-401
[48]Bayarri M J, Berger J O, Kennedy M C, Kottas A, Paulo R, Sacks J, Cafeo J A, Lin C H, Tu J. Predicting Vehicle Crashworthiness:Validation of Computer Models for Functional and Hierarchical Data[J]. Journal of the American Statistical Association,2009,104:929-943
[49]Jin R, Chen W, Simpson T W. Comparative Studies of Metamodeling Techniques under Multiple Modeling Criteria[J]. Structural and Multidisciplinary Optimization,2001,23(1):1-13
[50]Wang L, Beeson D. A Comparison of Metamodeling Methods Using Practical Industry Requirements [A]. The 47th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference[C]. Newport, Rhode Island, USA: AIAA,2006
[51]Chen V C P, Tsui K-L, Barton R R, Meckesheimer M. A Review on Design, Modeling and Applications of Computer Experiments [J]. IIE Transactions,2006, 38:273-291
[52]Meckesheimer M, Barton R R, Simpson T W, Booker A J. Computationally Inexpensive Metamodel Assessment Strategies[J]. AIAA Journal,2002,40(10): 2053-2060
[53]Kennedy M C, O'Hagan A. Bayesian Calibration of Computer Models[J]. Journal of the Royal Statistical Society,2001,63(3):525-464
[54]Hooker G. Discovering Additive Structure in Black Box Functions[A]. Proceedings of the tenth ACM SIGKDD international conference on knowledge discovery and data mining[C]. Seattle, WA, USA:ACM,2004
[55]Owen A B. Assessing Linearity in High Dimensions[J]. The Annals of Statistics, 2000,28(1):1-19
[56]Kleijnen J P C. Handbook of Computational Statistics:Concepts and Fundamentals [M]. Heidelberg, Germany. Springer-Verlag,2004
[57]Giunta A A, Dudley J M, Narducci R, Grossman B, Haftka R T, Mason W H, Watson L T. Noisy Aerodynamic Response and Smooth Approximations in Hsct Design[A].5th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization[C]. Washington, DC:AIAA,1994
[58]Madsen J I, Shyy W, Haftka R T. Response Surface Techniques for Diffuser Shape Optimization[J]. AIAA Journal,2000,38(9):1512-1518
[59]Jin R, Du X, Chen W. The Use of Metamodeling Techniques for Optimization under Uncertainty [J]. Structural and Multidisciplinary Optimization,2003,25(2): 99-116
[60]Van Beers W, Kleijnen J P C. Kriging Interpolation in Simulation:A Survey [A]. Proceedings of the 2004 Winter Simulation Conference[C]. Piscataway, NJ:IEEE, 2004
[61]邓乃扬,田英杰.数据挖掘的新方法-支持向量机[M].北京：科学出版社,2004
[62]Morris M D, Mitchell T J, Ylvisaker D. Bayesian Design and Analysis of Computer Experiments:Use of Derivatives in Surface Prediction[J]. Technometrics,1993,35(3):243-255
[63]Toropov V V, Filatov A A. Multi-Parameter Structural Optimization Using Fem and Multipoint Approximation[J]. Structural and Multidisciplinary Optimization, 1993,6:7-14
[64]Wang L P, Grandhi R V, Canfield R A. Multivariate Hermite Approximation for Design Optimization[J]. International Journal for Numerical Methods in Engineering,1996,39:787-803
[65]Rasmussen J. Nonlinear Programming by Cumulative Approximation Refinement[J]. Structural and Multidisciplinary Optimization,1998,15:1-7
[66]Shin Y S, Grandhi R V. A Global Structural Optimization Technique Using an Interval Method[J]. Structural and Multidisciplinary Optimization,2001,22: 351-363
[67]李清,李伟明,徐大丰.基于元模型的企业模型表达[J].清华大学学报(自然科学版),2008,48(7)：1209-1212
[68]张昆仑,刘新亮,郭波.基于高斯过程元模型的产品设计时间估计方法[J].计算机集成制造系统,2011,17(1)：18-22
[69]张伟,周建军,李建涛,朱一凡,李群.基于仿真元模型的探索性分析方法研究[J].系统仿真学报,2011,23(7)：1336-1341
[70]严建援,李凯,张路.一种基于语法的供应链流程定义元模型[J].管理科学学报,2010,13(6)：33-43
[71]Clarke S M, Griebsch J H, Simpson T W. Analysis of Support Vector Regression for Approximation of Complex Engineering Analyses[J]. Journal of Mechanical Design,2005,127(6):1077-1087
[72]Zhang P. Model Selection Via Multifold Cross Validation[J]. Statistica Sinica, 2003,13:783-809
[73]Roecker E B. Prediction Error and Its Estimation for Subset-Selected Models[J]. Technometrics,1991,33(4):459-468
[74]Utans J, Moody J. Selecting Neural Network Architectures Via the Prediction Risk:Application Tocorporate Bond Rating Prediction[A]. Proceedings of the IEEE 1st International Conference on AI Applications on Wall Street[C]. New York:IEEE,1991
[75]Leblanc M, Tibshirani R. Combining Estimates in Regression and Classification[J]. Journal of theAmerican Statistical Association,1996,91(436): 1641-1650
[76]Yang Y. Regression with Multiple Candidate Models:Selecting or Mixing?[J]. Statistica Sinica,2003,13:783-809
[77]Meckesheimer M. A Framework for Metamodel-Based Design:Subsystem Metamodel Assessment and Implementation Issues[D]. Pennsylvania,2001
[78]Mitchell T J, Morris M D. Bayesian Design and Analysis of Computer Experiments:Two Examples[J]. Statistica Sinica,1992,2:359-379
[79]Lin Y. An Efficient Robust Concept Exploration Method and Sequential Exploratory Experimental Design[D]. Atlanta,2004
[80]Simpson T W, Mauery T M, Korte J J, Mistree F. Kriging Metamodels for Global Approximation in Simulation-Based Multidisciplinary Design Optimizatio[J]. AIAA Journal,2001,39(12):2233-2241
[81]Mack Y, Goel T, Shyy W, Haftka R T, Queipo N V. Multiple Surrogates for Shape Optimization for Bluff-Body Facilitated Mixing[A].43rd AIAA aerospace sciences meeting and exhibit[C]. Reno, Nevada:AIAA,2005
[82]Glaz B, Goel T, Liu L, Friedmann P P, Haftka R T. Application of a Weighted Average Surrogateapproach to Helicopter Rotor Blade Vibration Reduction[A]. 48th AIAA/ASME/ASCHE/AHS/ASC Structures, Structural Dynamics & Materials Conference[C]. Honolulu, HI:AIAA,2007
[83]Samad A, Kim K, Goel T, Haftka R T, Shyy W. Multiple Surrogate Modeling for Axial Compressor Blade Shape Optimization[J]. Journal of Propulsion and Power, 2008,24(2):302-310
[84]Sanchez E, Pintos S, Queipo N V. Toward an Optimal Ensemble of Kernel-Based Approximations with Engineering Applications [J]. Structural and Multidisciplinary Optimization,2008,36(3):247-261
[85]Viana F A C, Haftka R T. Using Multiple Surrogates for Minimization of the Rms Error in Metamodeling[A]. ASME Design Engineering Technical Conferences-Design Automation Conference[C]. NY:ASME,2008
[86]李建平,王维平,周敏,吴强.多拟合法凸线性组合元模型及其应用[A].2007系统仿真技术及其应用学术会议[C].中国安徽合肥：中国自动化学会系统仿真专业委员会、中国系统仿真学会仿真应用专业委员会,2007
[87]唐小我,曹长修.递归等权组合预测方法研究[J].电子科技大学学报,1992,21(5)：545-550
[88]Taguchi G. Introduction to Quality Engineering[M]. New York:White Plains, 1986
[89]Taguchi G Systems of Experimental Design:Engineering Methods to Optimize Quality and Minimize Cost[M]. New York:White Plains,1987
[90]Myers R H, Khuri A I, Vining G. Response Surface Alternatives to the Taguchi Robust Parameter Design Approach [J]. American Statistician,1992,46:131-139
[91]Myers R H, Montgomery D C, Anderson-Cook C M. Response Surface Methodology:Process and Product Optimization Using Designed Experiments; Third Edition[M]. New York:Wiley,2009
[92]Vining G G, Myers R H. Combining Taguchi and Response Surface Philosophies: A Dual Response Approach[J]. Journal of Quality Technology,1990,22:38-45
[93]Del Castillo E, Montgomery D C. A Nonlinear Programming Solution to the Dual Response Problem [J]. Journal of Quality Technology,1993,25:199-204
[94]Lin D K J, Tu W. Dual Response Surface Optimization[J]. Journal of Quality Technology,1995,27:34-39
[95]Ames A E, Mattucci N, Macdonald S, Szonyi G, Hawkins D M. Quality Loss Functions for Optimization across Multiple Response Surfaces[J]. Quality and Reliability Engineering International,1997,29:339-346
[96]Antony J. Multi-Response Optimization in Industrial Experiments Using Taguchi's Quality Loss Function and Principal Component Analysis[J]. Quality and Reliability Engineering International,2000,16(1):3-8
[97]Kim K, Lin D K J. Dual Response Surface Optimization:A Fuzzy Modeling Approach[J]. Journal of Quality Technology,1998,30:1-10
[98]Tong L, Su C. Optimizing Multi-Response Problems in the Taguchi Method by Fuzzy Multiple Attribute Decision Making[J]. Quality and Reliability Engineering International,1997,13(1):25-34
[99]Kumar P, Barua P B, Gaindhar J L. Quality Optimization (Multi-Characteristics) through Taguchi's Technique and Utility Concept[J]. Quality and Reliability Engineering International,2000,16(6):475-485
[100]Tang L C, Xu K. Unified Approach for Dual Response Surface Optimization [J]. Journal of Quality Technology,2002,34:437-447
[101]Kim Y J, Cho B R. Development of Priority-Based Robust Design[J]. Journal of Quality Technology,2002,14:355-363
[102]Koksoy O, Doganaksoy N. Joint Optimization of Mean and Standard Deviation in Response Surface Experimentation [J]. Journal of Quality Technology,2003, 35(3):239-252
[103]Lam S W, Tang L C. A Graphical Approach to the Dual Response Robust Design Problems[A]. Reliability and maintainability symposium. Proceedings. Annual[C]. Reno, NV:RAMS,2005
[104]Koksoy O, Yalcinoz T. A Hopfield Neural Network Approach to the Dual Response Problem[J]. Quality and Reliability Engineering International,2005,21: 595-603
[105]赵媚,潘尔顺,郭瑜,孙志礼.基于双响应曲面法的稳健参数设计[J].工业工程与管理,2010,15(1)：87-91
[106]何桢,王晶,李湧范.基于改进的距离函数法的多响应稳健参数设计 [J].天津大学学报,2010,43(7)：644-649
[107]汪建均,马义中,汪新.广义线性模型的贝叶斯分析及稳健参数设计应用[J].系统工程,2009,27(4)：71-77
[108]崔庆安,何桢,车建国.一种基于支持向量机的非参数双响应曲面法[J].天津大学学报,2006,39(8)：1008-1014
[109]Powell M J D. Radial Basis Functions for Multivariable Interpolation:A Review[A]. Proceedings of the IMA Conference on Algorithms for the Approximation of Functions and Data[C]. London:IMA,1987
[110]McDonald D B, Grantham W J, Tabor W L, Murphy M J. Response Surface Model Development for Global/Local Optimization Using Radial Basis Functions[A]. the 8th AIAA Symposium on Multidisciplinary Analysis and Optimization[C]. Long Beach, CA:AIAA,2000
[111]Bishop C M. Neural Networks for Pattern Recognition[M]. New York:Oxford University Press,1995
[112]Zerpa L, Queipo N V, Pintos S, Salager J. An Optimization Methodology of Alkaline-Surfactant-Polymer Flooding Processes Using Field Scale Numerical Simulation and Multiple Surrogates [J]. Journal of Petroleum Science and Engineering,2005,47:197-208
[113]Goel T, Haftka R T, Shyy W, Queipo N V. Ensemble of Surrogates[J]. Structural and Multidisciplinary Optimization,2007,33(3):199-216
[114]Acar E, Rais-Rohani M. Ensemble of Metamodels with Optimized Weight Factors[J]. Structural and Multidisciplinary Optimization,2009,37:279-294
[115]Viana F A C, Haftka R T, Steffen V. Multiple Surrogate:How Cross-Validation Errors Can Help Us to Obtain the Best Predictor [J]. Structrual and Multidiscplinary Optimization,2009,39:439-457
[116]Vapnik V N. The Nature of Statistical Learning Theory[M]. New York: Springer-Verlag,1995
[117]Vapnik V N. Estimation of Dependences Based on Empirical Data[M]. NY: Springer-Verlag,1982
[118]Osuna E, Freund R, Girosi F. Improved Training Algorithm for Support Vector Machines [A]. Neural Networks and Signal Processing VII-proceedings of the 1997 IEEE Workshop[C]. New York:IEEE,1997
[119]Osuna E. Support Vector Machines:Training and Applications[M]. Cambridge, MA:Massachusetts Institute of Technology,1998
[120]Smola A J, Scholkopf B. (1998). A Tutorial on Support Vector Regression. In: Royal Holloway College
[121]Keerthi S, Shevade S, Bhattacharyya C, Murthy K. Improvements to Smo Algorithm for Svm Regression[J]. IEEE Transaction on Neural Networks,2000, 11:1188-1193
[122]Flake G W, Lawrence S. Efficient Svm Regression Training with Smo[J]. Machine Learning,2002,46:271-290
[123]Lin H-T, Lin C-J. (2003). A Study on Sigmoid Kernels for Svm and the Training of Non-Psd Kernels by Smo-Type Methods. In. TaiPei, Taiwan:National Taiwan University
[124]Platt J C. Fast Training of Support Vector Machines Using Sequential Minimal Optimization. Advances in Kernel Methods:Support Vector Machines[M]. Cambridge MA:MIT Press,1998
[125]Lin C-J. Asymptotic Convergence of an Smo Algorithm without Any Assumptions [J]. IEEE Transactions on Neural Networks,2002,13(1):248--250
[126]Palagi L, Sciandrone M. On the Convergence of a Modified Version of Svmlight Algorithm[J]. Optimization Methods and Software,2005,20(2-3):315-332
[127]Ong C S, Mary X, Canu S. Learning with Non-Positive Kernels[A]. Proceedings of ICML[C]. New York:ACM,2004
[128]Kwok J T, Tsang I W. Linear Dependency between E and the Input Noise in E-Support Vector Regression[J]. IEEE Transaction on Neural Networks,2003, 14(3):544-553
[129]Wang S-t, Zhu J-g, Chung F L, Lin Q, Hu D-w. Theoretically Optimal Parameter Choices for Support Vector Regression Machines Huber-Svr and Norm_R-Svr with Noisy Input[J]. Soft Computing,2005,9(10):732-741
[130]周晓剑,马义中,朱嘉钢.Smo算法的简化及其在非正定核条件下的应用[J].计算机研究与发展,2010,47(11)：1962-1969
[131]周晓剑,马义中,朱嘉钢,刘利平,汪建均.求解非半正定核huber-支持向量回归机问题的序列最小最优化算法研究[J].控制理论与应用,2010,27(9)：1178-1184
[132]周晓剑,马义中.两种求解非正定核laplace-Svr的smo算法[J].控制与决策,2009,24(11)：1657-1662
[133]Sellar R S, Batill S M, Renaud J E. Concurrent Subspace Optimization Using Gradient-Based Neural Network Response Surface Mappings [A].6th AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization conference[C]. Bellevue, Washington:AIAA,1996
[134]Liu W, Batill S. Gradient-Enhanced Neural Network Response Surface Approximations[A].8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization conference[C]. Long Beach, CA: AIAA,2000
[135]Van K F, Vervenne K. Gradient-Enhanced Response Surface Building[J]. Structural and Multidisciplinary Optimization,2004,27:337-351
[136]Kim C, Wang S, Choi K K. Efficient Response Surface Modeling by Using Moving Least-Squares Method and Sensitivity[J]. AIAA Journal,2005,43(11): 2404-2411
[137]Chung H-S, Alonso J J. Using Gradients to Construct Cokriging Approximation Models for High-Dimensional Design Optimization Problems[A].40th AIAA Aerospace Sciences Meeting and Exhibit[C]. Reno, Nevada:AIAA,2002
[138]Liu W, Batill S M. Gradient-Enhanced Response Surface Approximations Using Kriging Models[A].9th AIAA/ISSMO Symposium and Exhibit on Multidisciplinary Analysis and Optimization conference[C]. Atlanta, GA:AIAA, 2002
[139]Alexander I J, Forrester A J K. Recent Advances in Surrogate-Based Optimization[J]. Progress in Aerospace Sciences,2009,45:50-79
[140]Barthelemy J, Hall L E. Automatic Differentiation as a Tool in Engineering Design[A].4th AIAA/NASA/USAF/ISSMO Symposium on Multidisciplinary Analysis and Optimization conference[C]. Cleveland, Ohio:AIAA,1992
[141]Giles M B, Pierce N A. An Introduction to the Adjoint Approach to Design[J]. Flow, Turbulence and Combustion,2000,65(3-4):393-415
[142]Sobieszczanski-Sobieski J, Haftka. Multidisciplinary Aerospace Design Optimization:Survey of Recent Developments Industrial Experiments [J]. Structural and Multidisciplinary Optimization,1997,14(1):1-23
[143]Giunta A A, Watson L T. A Comparison of Approximation Modeling Techniques: Polynomial Versus Interpolating Models[A]. Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis & Optimization, AIAA-98-4758[C]. St. Louis, MO:AIAA,1998
[144]Simpson T W, Peplinski J D, Koch P N, Allen J K. Meta-Models for Computer Based Engineering Design:Survey and Recommendations [J]. Engineering with computers,2001,17:128-150
[145]Li W, Padula S. Approximation Methods for Conceptual Design of Complex Systems[M]. Brentwood, TN:Nashboro Press,2005
[146]Queipo N V, Haftka R T, Shyy W, Goel T, Vaidyanathan R, Tucker P K. Surrogate-Based Analysis and Optimization [J]. Progress in Aerospace Sciences, 2005,41:1-28
[147]Papila N, Shyy W, Griffin L W, Dorney D J. Shape Optimization of Supersonic Turbines Using Response Surface and Neural Network Methods [J]. Journal of Propulsion and Power,2001,18:509-518
[148]Shyy W, Papila N, Vaidyanathan R, Tucker P K. Global Design Optimization for Aerodynamics and Rocket Propulsion Components[J]. Progress in Aerospace Sciences 2001,37:59-118
[149]Vaidyanathan R, Tucker P K, Papila N, Shyy W. Cfd Based Design Optimization for a Single Element Rocket Injector[J]. Journal of Propulsion and Power,2004,20(4):705-717
[150]Stander N, Roux W, Giger M, Redhe M, Fedorova N, Haarhoff J. A Comparison of Meta-Modeling Techniques for Crashworthiness Optimization[A]. Proceedings of the 10th AIAA/ISSMO multidisciplinary analysis and optimization conference[C]. Albany, NY:AIAA,2004
[151]Fang H, Rais-Rohani M, Liu Z, Horstemeyer M F. A Comparative Study of Metamodeling Methods for Multi-Objective Crashworthiness Optimization [J]. Computers & Structures,2005,83:2121-2136
[152]Viana F A C, Haftka R T. Using Multiple Surrogates for Metamodeling[A]. ASMO-UK/ISSMO International Conference on Engineering Design Optimization[C]. Bath, UK:ISSMO,2008
[153]Yang Y. Regression with Multiple Candidate Models:Selecting or Mixing[J]. Statistic Sinica,2003,13(5):783-809
[154]Shaibu A B, Cho B R. Another View of Dual Response Surface Modeling and Optimization in Robust Parameter Design[J]. The International Journal of Advanced Manufacturing Technology 2009,41:631-641
[155]Owen A B. Orthogonal Arrays for Computer Experiments, Integration, and Visualization[J]. Statistica Sinica,1992,2:439-452
[156]Fang K T, Lin D K J, Winker P, Zhang Y. Uniform Design:Theory and Application[J]. Technometrics,2000,39(3):237-248
[157]McKay M D, Bechman R J, Conover W J. A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code[J]. Technometrics,1979,21(2):239-245
[158]Zhou X J, Ma Y Z, Li X F. Ensemble of Surrogates with Recursive Arithmetic Average[J]. Structrual and Multidiscplinary Optimization,2011,44(5):651-671
[159]Box G E P, Draper N R. Expirical Model Building and Response Surface[M]. New York:Wiley,1987
[160]Copeland K A F, Nelson P R. Dual Response Optimization Via Direct Function Minimization[J]. Journal of Quality Technology,1996,28:331-336
[161]Koch P N, Yang R J, Gu L. Design for Six Sigma through Robust Optimization[J]. Structural and Multidisciplinary Optimization,2004,26(3-4):235-248
[162]Wang G G, Simpson T W. Fuzzy Clustering Based Hierarchical Metamodeling for Space Reduction and Design Optimization[J]. Journal of Engineering Optimization,2004,36(3):313-335
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