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基于核主成分分析的多输出模型确认方法

胡嘉蕊 吕震宙

胡嘉蕊, 吕震宙. 基于核主成分分析的多输出模型确认方法[J]. 北京航空航天大学学报, 2017, 43(7): 1470-1480. doi: 10.13700/j.bh.1001-5965.2016.0519
引用本文: 胡嘉蕊, 吕震宙. 基于核主成分分析的多输出模型确认方法[J]. 北京航空航天大学学报, 2017, 43(7): 1470-1480. doi: 10.13700/j.bh.1001-5965.2016.0519
HU Jiarui, LYU Zhenzhou. Model validation method with multivariate output based on kernel principal component analysis[J]. Journal of Beijing University of Aeronautics and Astronautics, 2017, 43(7): 1470-1480. doi: 10.13700/j.bh.1001-5965.2016.0519(in Chinese)
Citation: HU Jiarui, LYU Zhenzhou. Model validation method with multivariate output based on kernel principal component analysis[J]. Journal of Beijing University of Aeronautics and Astronautics, 2017, 43(7): 1470-1480. doi: 10.13700/j.bh.1001-5965.2016.0519(in Chinese)

基于核主成分分析的多输出模型确认方法

doi: 10.13700/j.bh.1001-5965.2016.0519
基金项目: 

国家自然科学基金 51475370

中央高校基本科研业务费专项资金 3102015BJ(Ⅱ)CG009

详细信息
    作者简介:

    胡嘉蕊  女, 硕士研究生。主要研究方向:可靠性工程、模型确认

    吕震宙  女, 教授, 博士生导师。主要研究方向:飞行器设计及可靠性工程

    通讯作者:

    吕震宙, E-mail:zhenzhoulu@nwpu.edu.cn

  • 中图分类号: O212.4;TP391.9

Model validation method with multivariate output based on kernel principal component analysis

Funds: 

National Natural Science Foundation of China 51475370

the Fundamental Research Funds for the Central Universities 3102015BJ(Ⅱ)CG009

More Information
  • 摘要:

    目前对于不确定性环境下多个相关的复杂计算模型进行确认的方法存在计算困难及稳定性较差的问题。针对这类复杂计算模型,提出了一种新的基于核主成分分析(KPCA)的多输出模型确认方法。该方法将核主成分分析与面积法的思想相结合,构造了一个新的易于计算且稳定性高的模型确认指标。所提方法通过核主成分分析将相关的输出变量转化为不相关的核主成分,再对每一核主成分进行模型与实验的对比,从而避免了传统多输出模型确认方法中需要求解多个输出的联合累积分布函数的困难。由于核主成分分析(PCA)方法能够有效提取分析对象的非线性成分,因此基于核主成分分析的多输出模型确认方法较基于主成分分析的模型确认方法更为稳定,这表现在相同的实验样本数据下核主成分分析的方法具有更低的出错率。另外核主成分分析通过核主成分提取,可以实现多输出模型的降维,从而降低多输出模型确认的复杂度。所提方法既可以用于一般的多输出模型的确认,也可以用于多确认点的输出模型的确认。最后通过数值算例和工程算例证明了该方法的正确性与有效性。

     

  • 图 1  单个确认点处的面积指标

    Figure 1.  Area metric at single validation site

    图 2  多个确认点处的u-pooling模型确认过程

    Figure 2.  Validation process of u-pooling model at multiple validation sites

    图 3  基于核主成分分析的多输出模型确认指标求解流程

    Figure 3.  Validation metric solving flow of multivariate output model based on KPCA

    图 4  数值算例实验与模型每一核主成分的对比

    Figure 4.  Comparison of each kernel principal component between experiments and models of the numerical example

    图 5  汽车前轴示意图

    Figure 5.  Schematic of automobile front axle

    图 6  工字梁截面

    Figure 6.  Joist steel section

    图 7  工程算例的实验与模型每一核主成分的对比

    Figure 7.  Comparison of each kernel principal component between experiments and models of the engineering example

    表  1  数值算例的3个备选的计算模型

    Table  1.   Three alternative computational models of the numerical example

    模型 公式
    模型1 y1m1=θ1cos(2πx1z)+zsin x2θ1=1.5
    y2m1=sin(0.5πx1+z)+2cosx2θ2=1.5
    模型2 y1m2=θ1cos(2πx1z)+zsinx2θ1=1.7
    y2m2=sin(0.5πx1+z)+2cosx2θ2=1.5
    模型3 y1m3=θ1cos(2πx1z)+zsinx2θ1=1.7
    y2m3=sin(0.5πx1+z)+2cos x2θ2=1.7
    下载: 导出CSV

    表  2  数值算例的模型确认结果

    Table  2.   Model validation results of the numerical example

    模型 模型1 模型2 模型3
    指标值 0.012 0 0.063 6 0.101 2
    下载: 导出CSV

    表  3  数值算例的实验数据分别为10、100和1 000组时与10 000组模型数据确认结果对比

    Table  3.   Validation results of the numerical example of comparing 10, 100, 1 000 experimental observations and 10 000 model responses

    指标类型 10组实验数据 100组实验数据 1 000组实验数据
    标准差 错误率/% 标准差 错误率/% 标准差 错误率/%
    模型1 模型2 模型3 模型1 模型2 模型3 模型1 模型2 模型3
    基于PCA 0.027 9 0.025 1 0.023 4 35 0.012 7 0.014 7 0.013 5 18 0.004 4 0.006 8 0.006 6 2
    基于KPCA 0.012 4 0.012 0 0.011 9 17 0.004 3 0.006 0 0.006 0 3 0.001 9 0.003 0 0.003 3 0
    下载: 导出CSV

    表  4  工程算例的3个备选的计算模型

    Table  4.   Three alternative computational models of the engineering example

    模型 公式
    模型1
    a=12 mm  b=65 mm
    模型2
    a=10 mm  b=65 mm
    模型3
    a=10 mm  b=63 mm
    下载: 导出CSV

    表  5  工程算例的模型确认结果

    Table  5.   Model validation results of the engineering example

    模型 模型1 模型2 模型3
    指标值 0.008 0 0.044 5 0.105 8
    下载: 导出CSV
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出版历程
  • 收稿日期:  2016-06-15
  • 录用日期:  2016-09-30
  • 刊出日期:  2017-07-20

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