基于半参数分层贝叶斯建模与优化

doi:10.12305/j.issn.1001-506X.2023.05.35

摘要/Abstract

摘要：

针对响应服从非正态分布和模型不确定性的稳健参数设计问题, 在Polya树混合建模的框架下, 构建了一种半参数分层贝叶斯响应曲面模型, 并在此基础上实现了稳健参数设计。首先, 建立贝叶斯半参数模型, 并获得模型各参数的后验分布; 其次, 运用马尔可夫链蒙特卡罗(Markov chain Monte Carlo, MCMC)算法获得各参数的估计值; 然后, 基于此构建期望质量损失函数, 并利用混合遗传算法全局寻优, 获得可控因子的最优设置; 最后, 通过数值模拟研究和实际案例验证了所提方法的有效性。所提方法能有效解决小样本数据以及模型不确定性对优化结果影响的问题, 从而能够获得更稳健可靠的可控因子最优设置。

关键词: 非正态分布, 稳健参数设计, Polya树混合, 马尔可夫链蒙特卡罗算法, 质量损失函数

Abstract:

To solve the robust parameter design problem with non-normal distribution and model uncertainty, a semi-parametric hierarchical Bayesian response surface model is constructed under the framework of Polya tree hybrid modeling, and the robust parameter design is realized on this basis.Firstly, the Bayesian semi-parametric model is established and the posterior distributions of the parameters of the model are obtained. Secondly, the Markov chain Monte Carlo (MCMC) algorithm is used to obtain the estimated value of each parameter. Then, the expected quality loss function is constructed based on this, and the hybrid genetic algorithm is used for global optimization to obtain the optimal setting of controllable factors. Finally, a numerical simulation study and a real case are given to verify the effectiveness of the proposed method. The proposed method can effectively solve the problem of the influence of small sample data and model uncertainty on optimization results, so as to obtain a more robust and reliable optimal setting of controllable factors.

Key words: non-normal distribution, robust parameter design, Polya tree mixed, Markov chain Monte Carlo (MCMC) algorithm, quality loss function

中图分类号:

F273.2

陈晓英, 汪建均, 杨世娟. 基于半参数分层贝叶斯建模与优化[J]. 系统工程与电子技术, 2023, 45(5): 1580-1588.

Xiaoying CHEN, Jianjun WANG, Shijuan YANG. Bayesian modeling and optimization based on semi-parametric hierarchy[J]. Systems Engineering and Electronics, 2023, 45(5): 1580-1588.

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方法	可控因子优化结果			EQL	RMSE
方法	x₁	x₂	x₃	EQL	RMSE
GLM方法	-0.514 1	-0.628 6	0.146 5	3.750 0	0.397 6
Bayes方法	0.341 7	1.418 4	-0.298 3	0.473 2	0.286 7
TBays方法	0.177 3	1.471 7	-0.416 0	0.508 0	0.292 3
所提方法	0.711 6	1.318 2	1.372 5	0.164 9	0.286 6

可控因子	水平
可控因子	-1.682	-1	0	1	1.682
x₁	20	24.05	30	35.95	40
x₂	0.8	1.04	1.4	1.76	2.0
x₃	0.8	1.08	1.5	1.92	2.2

序号	编码值			响应y
序号	z₁	z₂	z₃	响应y
1	1	1	1	12.938 1
2	1	1	-1	13.397 7
3	1	-1	1	13.036 8
4	1	-1	-1	13.443 3
5	-1	1	1	13.745 0
6	-1	1	-1	14.520 9
7	-1	-1	1	14.423 2
8	-1	-1	-1	15.146 0
9	1.682	0	0	11.811 2
10	-1.682	0	0	13.920 1
11	0	1.682	0	13.762 1
12	0	-1.682	0	14.365 2
13	0	0	1.682	14.070 5
14	0	0	-1.682	15.055 8
15	0	0	0	13.902 0
16	0	0	0	13.901 7
17	0	0	0	13.902 4
18	0	0	0	13.901 2
19	0	0	0	13.899 7
20	0	0	0	13.902 4

方法	可控因子优化结果			EQL	RMSE
方法	z₁	z₂	z₃	EQL	RMSE
GLM方法	0.562 4	0.266 2	0.841 2	13.600 0	0.022 4
Bayes方法	-0.764 5	0.706 4	0.751 9	0.555 4	0.013 0
TBayes方法	-0.388 7	0.604 6	0.795 2	0.560 8	0.010 5
所提方法	-0.381 4	0.629 8	0.863 9	0.001 3	0.002 4

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[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.
[3]	冯泽彪, 汪建均. 考虑响应共变特性的多响应稳健参数设计[J]. 系统工程与电子技术, 2019, 41(9): 2048-2057.
[4]	杨世娟, 汪建均. 基于分层贝叶斯模型的稳健参数设计[J]. 系统工程与电子技术, 2019, 41(10): 2293-2303.
[5]	顾晓光, 马义中, 刘健, 汪建均. 基于单值数据过程能力指数的多元质量特性稳健参数设计[J]. 系统工程与电子技术, 2017, 39(2): 362-368.
[6]	张流洋, 马义中, 汪建均, 吴锋. 考虑偏度特征的动态多响应稳健参数设计与优化[J]. 系统工程与电子技术, 2016, 38(2): 339-346.