Model validation method with multivariate output based on kernel principal component analysis
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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。
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关键词:
- 模型确认 /
- 多输出 /
- 相关性 /
- 核主成分分析（KPCA） /
- 面积指标
Abstract:At present, for the multiple correlated complex computational models with uncertainty, the traditional validation methods still have some problems, such as difficult calculation and poor stability.Aimed at such complex computational models, a new multivariate model validation method is proposed based on kernel principal component analysis (KPCA). By combining the KPCA with the idea of area metric, the proposed method constructs a new model validation metric which is easy to be calculated and has high stability. In proposed method, the correlated multivariate output variables are transformed into uncorrelated kernel principal component by the KPCA, and then for each kernel principal component, the computational model is compared with the experiment. Thus this method avoids the difficulties of solving the joint cumulative distribution function of multivariate output in the traditional methods. Because the KPCA can effectively extract the nonlinear characteristic of the analyzed model, the multivariate output model validation method based on the KPCA is more robust than that based on the principal component analysis (PCA). Under the same experiment sample data, the method based on the KPCA has a lower error rate than that based on PCA. Furthermore, by extracting the kernel principal component, dimensionality reduction of the multivariate output can be implemented; thereby the complexity of the multivariate output validation can also be reduced. The proposed method can be applied not only to the general multivariate output model validation, but also to the model validation with multiple validation sites. Finally, the correctness and effectiveness of the proposed method are demonstrated by the numerical and engineering examples.
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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 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 x2 θ1=1.5 y2m1=sin(0.5πx1+z)+zθ2cosx2 θ2=1.5 模型2 y1m2=θ1cos(2πx1z)+zsinx2 θ1=1.7 y2m2=sin(0.5πx1+z)+zθ2cosx2 θ2=1.5 模型3 y1m3=θ1cos(2πx1z)+zsinx2 θ1=1.7 y2m3=sin(0.5πx1+z)+zθ2cos x2 θ2=1.7 
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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
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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
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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
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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
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