基于高斯过程模型的多响应稳健参数设计

doi:10.12305/j.issn.1001-506X.2021.12.32

摘要/Abstract

摘要：

针对高维试验数据的稳健参数设计问题, 在高斯过程(Gaussian process, GP)的建模框架下, 采用部分平行的GP(parallel partial GP, PPGP)模型来构建试验因子与多质量特性之间的响应曲面, 在此基础上运用多元质量损失函数作为优化指标来获得可控因子的最佳参数设计值。并且以一个经典仿真算例和两个实际案例验证了所提方法的有效性和优劣性。研究结果表明，与独立建模的单变量GP模型或Kriging模型比较而言, 所提方法不仅能够有效地处理高维试验数据的建模与参数优化问题, 而且能够获得更为稳健的优化结果, 运行效率更高。

关键词: 高斯过程, 高维数据, 多元质量损失函数, 稳健参数设计

Abstract:

To address the problem of robust parameter design for high-dimensional experimental data, the parallel partial Gaussian process (PPGP) model is adopted to construct the response surfaces between the test factors and the multivariate quality characteristics under the modeling framework of Gaussian process (GP). On this basis, the multivariate quality loss function is used as an optimization index to obtain the optimal parameter design values of the controllable factors. A classic simulation examples and two actual cases are used to verify the effectiveness and advantages of the proposed method. The research results show that compared with the independent modeling of the univariate GP model or the Kriging model, the proposed method can not only deal with the modeling and parameter optimization problems of high-dimensional experimental data effectively, but also obtain more robust optimization results and higher operational efficiency.

Key words: Gaussian process (GP), high-dimensional data, multivariate quality loss function, robust parameter design

中图分类号:

F273.2

翟翠红, 汪建均, 冯泽彪. 基于高斯过程模型的多响应稳健参数设计[J]. 系统工程与电子技术, 2021, 43(12): 3683-3693.

Cuihong ZHAI, Jianjun WANG, Zebiao FENG. Robust parameter design of multiple responses based on Gaussian process model[J]. Systems Engineering and Electronics, 2021, 43(12): 3683-3693.

图/表 10

图1

表1

图2

表2

表3

表4

表5

图3

表6

图4

参考文献 53

1	YAN H , PAYNABAR K , SHI J J . Real-time monitoring of high-dimensional functional data streams via spatio-temporal smooth sparse decomposition[J]. Technometrics, 2018, 60 (2): 181- 197. doi: 10.1080/00401706.2017.1346522
2	TANSEL I C Y , YILDIRIM S . MOORA-based Taguchi optimisation for improving product or process quality[J]. International Journal of Production Research, 2013, 51 (11): 3321- 3341. doi: 10.1080/00207543.2013.774471
3	SHAH H K , MONTGOMERY D C , CARLYLE W M . Response surface modeling and optimiza-tion in multiresponse experiments using seemingly unrelated regressions[J]. Quality Engineering, 2004, 16 (3): 387- 397. doi: 10.1081/QEN-120027941
4	WANG J J , MA Y Z , TSUNG F , et al. Economic parameter design for ultra-fast laser micro-drilling process[J]. International Journal of Production Research, 2019, 57 (20): 6292- 6314. doi: 10.1080/00207543.2019.1566660
5	TAGUCHI G . Introduction to quality engineering[M]. New York: Quality Resources, 1986.
6	WU C F J . Post-fisherian experimentation: from physical to virtual[J]. Journal of the American Statistical Association, 2015, 110 (510): 612- 620. doi: 10.1080/01621459.2014.914441
7	WENG T W , MELATI D , MELLONI A , et al. Stochastic simulation and robust design optimization of integrated photonic filters[J]. Nanophotonics, 2017, 6 (1): 299- 308. doi: 10.1515/nanoph-2016-0110
8	RAJAN A, KUANG Y C, OOI M, et al. Moment-based mea-surement uncertainty evaluation for reliability analysis in design optimization[C]//Proc. of the IEEE Instrumentation and Mea-surement Technology Conference, 2016.
9	LIU H W , WANG M R , WANG J K , et al. Monte carlo simulations of gas flow and heat transfer in vacuum packaged mems devices[J]. Applied Thermal Engineering, 2007, 27 (2): 323- 329.
10	WANG G G , SHAN S . Review of metamodeling techniques in support of engineering design optimization[J]. Journal of Mechanical Design, 2007, 129 (4): 370- 380. doi: 10.1115/1.2429697
11	GU L. A comparison of polynomial based regression models in vehicle safety analysis[C]//Proc. of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2001: 509-514.
12	SHAN S Q , WANG G G . Survey of modeling and optimization strategies to solve high-dimen-sional design problems with computationally-expensive black-box functions[J]. Structural and Multidisciplinary Optimization, 2010, 41 (2): 219- 241. doi: 10.1007/s00158-009-0420-2
13	LIANG H Q , ZHU M , WU Z . Using cross-validation to design trend function in Kriging surrogate modeling[J]. American Institute of Aeronautics and Astronautics Journal, 2014, 52 (10): 2313- 2327. doi: 10.2514/1.J052879
14	ZHAO L , CHOI K K , LEE I . Metamodeling method using dynamic kriging for design optimization[J]. American Institute of Aeronautics and Astronautics Journal, 2011, 49 (9): 2034- 2046. doi: 10.2514/1.J051017
15	CHEN R B , WANG W C , WU C F J . Building surrogates with overcomplete bases in computer experiments with applications to bistable laser diodes[J]. ⅡE Transactions, 2010, 43 (1): 39- 53.
16	JIN R C, CHEN W, SUDJIANTO A. On sequential sampling for global metamodeling in engineering design[C]//Proc. of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2002: 539-548.
17	MYERS R H , MONTGOMERY D C . Response surface me-thodology: process and product optimization using designed experiments[M]. Toronto: Wiley, 1995.
18	PAPADRAKAKIS M , LAGAROS N D , TSOM-PANAKIS Y . Structural optimization using evolution strategies and neural networks[J]. Computer Methods in Applied Mechanics and Engineering, 1998, 156 (1-4): 309- 333. doi: 10.1016/S0045-7825(97)00215-6
19	VARADARAJAN S , CHEN W , PELKA C J . Robust concept exploration of propulsion systems with enhanced model approximation capabilities[J]. Engineering Optimization, 2000, 32 (3): 309- 334. doi: 10.1080/03052150008941302
20	FRIEDMAN J H . Multivariate adaptive regressive splines[J]. The Annals of Statistics, 1991, 19 (1): 1- 67.
21	MULLUR A A , MESSAC A . Extended radial basis functions: more flexible and effective metamodeling[J]. American Institute of Aeronautics and Astronautics Journal, 2005, 43 (6): 1306- 1315. doi: 10.2514/1.11292
22	FANG H , HORSTEMEYER M F . Global response approximation with radial basis functions[J]. Journal of Engineering Optimization, 2006, 38 (4): 407- 424. doi: 10.1080/03052150500422294
23	SIMPSON T W , MAUERY T M , KORTE J J , et al. Kriging metamodels for global approximation in simulation-based multidisciplinary design optimization[J]. American Institute of Aeronautics and Astronautics Journal, 2001, 39 (12): 2233- 2241. doi: 10.2514/2.1234
24	MARTIN J D , SIMPSON T W . On the use of Kriging models to approximate deterministic computer models[J]. American Institute of Aeronautics and Astronautics Journal, 2005, 43 (4): 853- 863. doi: 10.2514/1.8650
25	BEERS W V , KLEIJNEN J . Kriging for interpolation in random simulation[J]. Journal of the Operational Research Society, 2003, 54 (3): 255- 262. doi: 10.1057/palgrave.jors.2601492
26	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. doi: 10.1115/1.1897403
27	LEE K , CHO H , LEE I . Variable selection using Gaussian process regression-based metrics for high-dimensional model approximation with limited data[J]. Structural and Multidisciplinary Optimization, 2018, 59 (4): 1439- 1454.
28	LEE I , CHOI K K , ZHAO L . Sampling-based RBDO using the stochastic sensitivity analysis and dynamic Kriging (D-Kriging) method and stochastic sensitivity analysis[J]. Structural and Multidisciplinary Optimization, 2011, 44 (3): 299- 317. doi: 10.1007/s00158-011-0659-2
29	SOBOL I . Sensitivity estimates for nonlinear mathematical models[J]. Mathematical Modelling and Computational Experiments, 1993, 1 (1): 112- 118.
30	汪建均, 屠雅楠, 马义中. 结合SUR与因子效应原则的多响应质量设计[J]. 管理科学学报, 2020, 23 (12): 12- 29.
	WANG J J , TU Y N , MA Y Z . Multi-response quality design intergrating SUR models with factorial effect principles[J]. Journal of Management Sciences in China, 2020, 23 (12): 12- 29.
31	CRAIG P S , GOLDSTEIN M , SEHEULT A H , et al. Bayes linear strategies for matching hydrocarbon reservoir history[J]. Bayesian Statistics, 1996, 5, 69- 95.
32	KENNEDY M C, O'HAGAN A, HIGGINS N. Bayesian analy- sis of computer code outputs[C]//Proc. of the Quantitative Methods for Current Environmental Issues, 2002: 227-243.
33	LIU F , WEST M . A dynamic modelling strategy for Bayesian computer model emulation[J]. Bayesian Analysis, 2009, 4 (2): 393- 411.
34	QIAN P Z G , WU H , WU C F J . Gaussian process models for computer experiments with qualitative and quantitative factors[J]. Technometrics, 2008, 50 (3): 383- 396. doi: 10.1198/004017008000000262
35	MELKUMYAN A, RAMOS F. Multi-kernel Gaussian processes[C]//Proc. of the 22nd International Joint Conference on Artificial Intelligence, 2011: 1408-1413.
36	LI B , GENTON M G , SHERMAN M . Testing the covariance structure of multivariate random fields[J]. Biometrika, 2008, 95 (4): 813- 829. doi: 10.1093/biomet/asn053
37	SUNG C L , WANG W , PLUMLEE M , et al. Multiresolution functional ANOVA for large-scale, many-input computer experiments[J]. Journal of the American Statistical Association, 2020, 115 (530): 908- 919. doi: 10.1080/01621459.2019.1595630
38	CHEN T , HADINOTO K , YAN W J , et al. Efficient meta-modelling of complex process simulations with time-space dependent outputs[J]. Computers and Chemical Engineering, 2011, 35 (3): 502- 509. doi: 10.1016/j.compchemeng.2010.05.013
39	GHOSH M , LI Y X , ZENG L , et al. Modeling multivariate profiles using Gaussian process-controlled B-splines[J]. ⅡSE Transactions, 2020, 53 (7): 787- 798.
40	LI Y X , ZHOU Q . Pairwise meta-modeling of multivariate output computer models using nonseparable covariance function[J]. Technometrics, 2016, 58 (4): 483- 494. doi: 10.1080/00401706.2015.1079244
41	冯泽彪, 汪建均, 马义中. 基于多变量高斯过程模型的贝叶斯建模与稳健参数设计[J]. 系统工程理论与实践, 2020, 40 (3): 703- 713.
	FENG Z B , WANG J J , MA Y Z . Bayesian modeling and robust parameter design based on multivariate Gaussian process model[J]. Systems Engineering-Theory & Practice, 2020, 40 (3): 703- 713.
42	GU M Y , BERGER J O . Parallel partial Gaussian process emulation for computer models with massive output[J]. The Annals of Applied Statistics, 2016, 10 (3): 1317- 1347.
43	RASMUSSEN C E , WILLIAMS C K I . Gaussian processes for machine learning (adaptive computation and machine learning)[M]. Cambridge: MIT Press, 2005.
44	FORRESTER A , SOBESTER A , KEANE A . Engineering design via surrogate modelling: a practical guide[M]. New York: Wiley, 2008.
45	SACKS J , WELCH W J , MITCHELL T J , et al. Design and analysis of computer experiments: rejoinder[J]. Statistical Science, 1989, 4 (4): 433- 435.
46	GAMERMAN D , LOPES H F . Markov chain Monte Carlo: stochastic simulation for Bayesian inference[M]. 2nd. New York: Chapman & Hall/CRC, 2006.
47	FENG Z B , WANG J J , MA Y Z , et al. Robust parameter design based on Gaussian process with model uncertainty[J]. International Journal of Production Research, 2021, 59 (9): 2772- 2788. doi: 10.1080/00207543.2020.1740344
48	ARTILES-LEÓN N . A pragmatic approach to multi-response problems using loss functions[J]. Quality Engineering, 1996, 9 (2): 213- 220. doi: 10.1080/08982119608919037
49	PIGNATIELLO J J . Strategies for robust multi-response quality engineering[J]. ⅡE Transactions, 1993, 25 (3): 5- 15.
50	NAIR V N , ABRAHAM B , MACKAY J , et al. Taguchi's parameter design: a panel discussion[J]. Technometrics, 1992, 34 (2): 127- 161. doi: 10.1080/00401706.1992.10484904
51	NAIR V N , TAAM W , YE K Q . Analysis of functional responses from robust design studies[J]. Journal of Quality Technology, 2002, 34 (4): 355- 370. doi: 10.1080/00224065.2002.11980169
52	ALSHRAIDEH H , CASTILLO E D . Gaussian process modeling and optimization of profile response experiments[J]. Quality and Reliability Engineering International, 2014, 30 (4): 449- 462. doi: 10.1002/qre.1497
53	GOVAERTS B , NOEL J . Analysing the results of a designed experiment when the response is a curve: methodology and application in metal injection moulding[J]. Quality and Reliability Engineering International, 2005, 21 (5): 509- 520. doi: 10.1002/qre.737

函数名称	函数特征	函数详情
Cross-in-Tray	多个局部极小值	$ \begin{array}{c}f\left( {{x_1}, {x_2}} \right) = - 0.0001{\left( {\left\| {\sin \;{x_1}\;\sin {x_2} \cdot \exp \left( {\left\| {100 - \frac{{\sqrt {x_1^2 + x_2^2} }}{\pi }} \right\|} \right)} \right\| + 1} \right)^{0.1}} + {\varepsilon _1}\\{\boldsymbol{x}^} = ( \pm 1.349\;1, \pm 1.349\;1), f\left( {{\boldsymbol{x}^}} \right) = - 2.062\;61\end{array}$
Bohachevsky	碗状	$ \begin{array}{c}f\left( {{x_1}, {x_2}} \right) = x_1^2 + 2x_2^2 - 0.3\cos \left( {3{\rm{ \mathsf{ π} }}{x_1}} \right)\cos \left( {4{\rm{ \mathsf{ π} }}{x_2}} \right) + 0.3 + {\varepsilon _2}\\{\boldsymbol{x}^} = (0, 0), \;\;\;f\left( {{\boldsymbol{x}^}} \right) = 0\end{array}$
McCormick	盘状	$ \begin{array}{c}f\left( {{x_1}, {x_2}} \right) = \sin \left( {{x_1} + {x_2}} \right) + {\left( {{x_1} - {x_2}} \right)^2} - 1.5{x_1} + 2.5{x_2} + 1 + {\varepsilon _3}\\{\boldsymbol{x}^} = ( - 8.919\;01, - 9.657\;51), \;\;\;f\left( {{\boldsymbol{x}^}} \right) = - 8.951\;84\end{array}$
Easom	陡降	$\begin{array}{c}f\left( {{x_1}, {x_2}} \right) = - \cos \left( {{x_1}} \right)\cos \left( {{x_2}} \right)\exp \left( { - {{\left( {{x_1} - {\rm{ \mathsf{ π} }}} \right)}^2} - {{\left( {{x_2} - {\rm{ \mathsf{ π} }}} \right)}^2}} \right) + {\varepsilon _4}\\{\boldsymbol{x}^} = (\pi , \pi ), \;\;\;f\left( {{\boldsymbol{x}^}} \right) = - 1\end{array} $

测试函数	真实极值	本文方法优化结果				单变量GP方法优化结果
测试函数	真实极值	优化极值	极值偏差	均方根误差	95%后验置信区间覆盖率	优化极值	极值偏差	均方根误差	95%后验置信区间覆盖率
Cross-in-Tray	-2.062 6	-1.945 5	0.117 1	0.178 1	0.845 0	-1.657 3	0.405 3	0.221 7	0.955 0
Bohachevsky	0	0.147 1	0.147 1	1.594 9	1.000 0	0.524 9	0.524 9	98.538 3	0.690 0
McCormick	-8.951 8	-8.204 1	0.747 7	2.904 8	0.995 0	-8.223 3	0.728 5	110.529 3	0.990 0
Easom	-1	-0.009 1	0.990 9	0.046 0	0.935 0	-0.000 3	0.999 7	0.001 9	0.975 0
平均值	—	—	0.500 7	1.181 0	0.943 8	—	0.664 6	52.322 8	0.902 5

因子类型	控制因子								噪声因子		信号因子
因子类型	x₁	x₂	x₃	x₄	x₅	x₆	x₇	x₈	x₉	x₁₀	s
描述	每个线圈的匝数	材料厚度	功率	宽度间距比1	宽度间距比2	直径比	直径长度比	外直径	气隙变化	温度	每分钟转数
水平数	2	3	3	3	3	3	3	3	2	3	7

不同转速下电流取值标准	转速/(r/min)
不同转速下电流取值标准	s₁=1 375	s₂=1 500	s₃=1 750	s₄=2 000	s₅=2 500	s₆=3 500	s₇=5 000
上规格限U(s_j)	190	210	215	220	225	230	230
下规格限L(s_j)	120	140	155	170	185	200	200
目标值T(s_j)	155	175	185	195	205	215	215

研究方法	多元质量损失	95%后验置信区间覆盖率	累加偏差	寻优时间/s
GRF方法	111.845 55	1.000 0	47.344 5	NA
单变量GP方法	52.810 6	0.463 0	37.818 2	505.060 0
本文方法	2.679 6	0.996 0	10.150 4	21.440 0