为了清晰地掌握相关输入变量情况下响应量方差的来源,非常有必要将基于方差的重要性测度(VBIM)分离为相关部分和独立部分。为此,在二阶非线性回归的基础上,提出了一种适用于非线性响应量的相关变量重要性分析的新方法。一个输入变量对响应量方差的相关贡献由该变量与每一个剩余变量两两相关的贡献分量组成,所以又进一步提出了一种概念简单有效的求解相关贡献分量的方法,并在此基础上定义了重要性矩阵,以便清晰地表达相关变量对响应量方差贡献的各个分量。数值算例和工程应用算例中,各相关输入变量的重要性测度分析的结果表明:本文方法可以在二次非线性响应量情况下合理分解相关变量的相关贡献和独立贡献;对于更复杂的非线性响应量情况,通过将所提方法中的二次非线性回归扩展为更合理的近似模型,可以得到重要性分析的高精度结果。
To explore the origin of the variance of the output response in cases where the correlated input variables are involved, it is necessary to divide the variance based importance measure (VBIM) into correlated and uncorrelated contributions. For this purpose, a novel method adaptable for the nonlinear output responses is proposed based on second order nonlinear regression. The correlated contribution is composed of the components of the individual input variable correlated with each of the other input variables. An effective method simple in concept is further proposed to decompose the correlated contribution into its components, based on which an importance matrix is defined for explicitly exposing the contribution components of the correlated input variable to the variance of the output response. The VBIMs of the numerical and engineering examples show that the proposed novel method can accurately decompose the contribution of the correlated input variables to the variance of the second order nonlinear output response. For output models more complicated than the second order nonlinear output response, the VBIM decomposition with high precision can be obtained by extending the second order nonlinear regression to a more reasonable approximation of the response.
[1] Xu C G, George Z G. Uncertainty and sensitivity analysis for models with correlated parameters[J]. Reliability Engineering and System Safety, 2008, 93(6): 1563-1573.
[2] Guo J, Du X P. Sensitivity analysis with mixture of epistemic and aleatory uncertainties[J]. AIAA Journal, 2007, 45(9): 2337-2349.
[3] Aven T, Nokland T E. On the use of uncertainty importance measures in reliability and risk analysis[J]. Reliabi-lity Engineering and System Safety, 2010, 95(2): 127-133.
[4] Helton J C, Davis F J. Latin hypercube sampling and the propagation of uncertainty in analysis of complex systems[J]. Reliability Engineering and System Safety, 2003, 81(1): 23-69.
[5] Doksum K, Samarov A. Nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression[J]. Annals of Statistics, 1995, 23(5): 1443-1473.
[6] Saltelli A. The role of sensitivity analysis in the corroboration of models and its link to model structural and parametric uncertainty[J]. Reliability Engineering and System Safety, 1997, 57(1): 1-4.
[7] Sobolá I M. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates[J]. Mathe-matics and Computers in Simulation, 2001, 55(1-3): 271-280.
[8] Iman R L, Hora S C. A robust measure of uncertainty importance for use in fault tree system analysis[J]. Risk Analysis, 1990, 10(3): 401-406.
[9] Saltelli A. Sensitivity analysis for importance assessment[J]. Risk Analysis, 2002, 22(3): 579-590.
[10] Borgonovo E. A new uncertainty importance measure [J]. Reliability Engineering and System Safety, 2007, 92(6): 771-784.
[11] Cui L J, Lu Z Z, Zhao X P. Moment-independent importance measure of basic random variable and its probability density evolution solution[J]. Science China Technological Sciences, 2010, 53(4): 1138-1145.
[12] Sobol I M. Theorem and examples on high-dimensional model representation[J]. Reliability Engineering and System Safety, 2003, 79(2): 187-193.
[13] Campolongo F, Saltelli A. Sensitivity analysis of an environmental model: an application of different analysis methods[J]. Reliability Engineering and System Safety, 1997, 57(1): 49-69.
[14] 宋笔锋. 结构体系失效概率计算方法研究. 西安:西北工业大学航空学院, 1994. Song Bifeng. A study on method for computing structural system failure probability. Xi'an: School of Aeronautics, Northwestern Polytechnical University, 1994. (in Chinese)
[15] 吕震宙, 宋述芳, 李洪双, 等. 结构机构可靠性及可靠性灵敏度分析[M]. 北京:科学出版社, 2009. Lu Zhenzhou, Song Shufang, Li Hongshuang, et al. The reliability and reliability sensitivity analysis for structure and machine system[M]. Beijing: Science Press, 2009. (in Chinese)
CJA