两种基于方差的全局灵敏度分析W指标改进算法
收稿日期: 2015-04-10
修回日期: 2016-03-22
网络出版日期: 2016-03-25
基金资助
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)
Two improved methods for variance-based global sensitivity analysis' W-indices
Received date: 2015-04-10
Revised date: 2016-03-22
Online published: 2016-03-25
Supported by
National Natural Science Foundation of China (51475370);Fundamental Research Funds for the Central University (3102015BJ(II)CG009)
在全局灵敏度分析(SA)中,基于方差的灵敏度分析指标(包括Sobol指标和W指标)应用广泛。其中Sobol指标是将输入随机变量固定于特定点时,求得其对输出响应量的平均影响;而W指标求解当输入随机变量在各自分布区间上缩减变化时,输入变量对输出响应量的影响程度。相比Sobol指标,W指标所反映的信息更加全面。但目前对W指标的求解方法还比较欠缺,双层重复抽样蒙特卡罗(DLRS MC)方法和双层一次抽样蒙特卡罗(DLSS MC)方法是两种传统的求解方法。针对W指标的求解问题,提出了两种新算法:改进的蒙特卡罗模拟(AMCS)和基于稀疏网格积分(SGI)的方法。AMCS只需抽取一组样本便可计算出所有变量的各阶W指标,由于该方法是通过筛选策略来计算条件区间上的方差,避免了DLSS MC法中由于小数取整带来的计数误差,从而提高了计算W指标的精度。基于SGI的方法则利用稀疏网格积分来计算三重矩进而得到W指标,由于该方法继承了稀疏网格积分的高效性,因而进一步提高了W指标的计算效率。最后,给出了两个数值算例和一个工程算例,用于验证所提方法求解W指标的准确性和高效性。
关键词： 全局灵敏度分析; Sobol指标; W指标; 改进的蒙特卡罗模拟法; 稀疏网格积分法
巩祥瑞 , 吕震宙 , 左健巍 . 两种基于方差的全局灵敏度分析W指标改进算法[J]. 航空学报, 2016 , 37(6) : 1888 -1898 . DOI: 10.7527/S1000-6893.2016.0090
In the global sensitivity analysis (SA), the variance-based sensitivity indices, such as Sobol's indices and W-indices, are used widely. Sobol's indices estimate the average variation of model output when input variables are fixed in their points. W-indices measure the average reduction of model output if input variables are reduced in their distributions. Compared with Sobol's indices, W-indices include more information. The double loop repeated set Monte Carlo (DLRS MC) and double loop single set Monte Carlo (DLSS MC) are two traditional methods, but these available methods for solving W-indices are still defective. In order to calculate W-indices efficiently, two new methods are presented. They are advanced Monte Carlo simulation (AMCS) and sparse grid integration (SGI)-based method. The AMCS only needs one set of samples to estimate all W-indices. Since screening method is used to estimate the variance in the conditional interval, and the count error induced by taking an integer in DLSS MC can be avoided, the accuracy of AMCS is higher than that of DLSS MC. The SGI-based method estimates W-indices by evaluating threefold statistic moment by SGI, in which the efficiency of the SGI is inherited. Finally, two numerical examples and an engineering example are employed to demonstrate the reasonability and efficiency of the presented methods.
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