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An efficient method for estimating various variance-based sensitivity indices
YUN Wanying, LYU Zhenzhou , MU Shanshan
School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China
Received: 2015-05-14; Accepted: 2015-07-04; Published online: 2015-09-08 16:52
Foundation items: Natural Science Foundation of China (51475370); the Fundamental Research Funds for the Central Universities (3102015 BJ (II) CG009)
Corresponding author. Tel.: 029-88460480 E-mail: zhenzhoulu@nwpu.edu.cn
Abstract: In order to simultaneously estimate various variance-based sensitivity indices, a method is proposed by combining space-partition idea and unscented transformation (UT) method, which can estimate variance-based global sensitivity index (Sobol index), variance-based regional sensitivity index and variance-based W index by repeatedly using a set of UT samples. Besides, a modified variance-based W index is proposed, which can analyze the sensitivity of the model input variables comprehensively and reasonably. What's more, the modified variance-based W index includes both the original one and the variability of effect on the variance of model output when the model input variable is fixed in different intervals. Thus, the modified one more reasonably reflects the average impact on the variance of the model output when model input variable is fixed in different intervals. The results of numerical and engineering examples illustrate the accuracy and efficiency of the proposed method and the reasonability of the modified variance-based W index.
Key words: space-partition     unscented transformation(UT)     Sobol index     variance-based regional sensitivity index     W index     modified W index

1 各类基于方差的灵敏度指标 1.1 基于方差的全局灵敏度指标

Y=g(X),X=(X1,X2,…,Xn)为n维随机变量,XiY方差贡献的1阶指标定义如下[4]:

1阶方差贡献总指标以及r阶方差贡献总指标定义如下:

1.2 基于方差的区域灵敏度指标

1.3 基于方差的W指标

VX(Y|XiUi)为Xi属于Ui区间而X-i变量在其整个分布范围内变化时所对应的输出方差。Qi包括了Ui的所有可能,因此,基于方差的W指标反映的是输入变量固定在其分布范围的任意子区间内对输出方差的平均影响。对于W总指标,文献[10]也给出了相应的定义:

2 改进的基于方差的W指标

3 各类基于方差灵敏度指标的近似计算方法

3.1 输出响应均值和方差的计算

3.2 基于方差的全局灵敏度指标的计算

wlsl则是由如下密度函数产生的X的第l个权值及相应的n维Sigma点。

E(Y)(Xi∈Aji)采用SP-UT估算的表达式为

3.3 均值比函数、方差比函数、基于方差的W指标及改进的基于方差的W指标的计算

1) 在计算无条件均值和方差中,每一维输入变量区间被划分为Ni个子区间,则Xi变量分布区间相应分位数向量B为:

2) 满足q1,q2Bq2>q1的区间有Ni(Ni+1)/2个,计算每一个可能区间的HMi和HVi,如下:

3) 根据步骤2)计算的结果画出HMi和HVi与分位数q1q2的关系图。

3.4 SP-UT方法的计算量

4 算例验证

 方法 S1 S2 S3 S12 S23 S13 SP-UT 0.6427 0.1607 0.0714 0.8548 0.2378 0.7370 MCS 0.6305 0.1563 0.0719 0.9205 0.2396 0.7839 解析解[14] 0.6694 0.1674 0.0744 0.8926 0.2479 0.7686 注:MCS的计算量为1.001×106。
 图 1 Sobol’s G函数均值比函数三维图 Fig. 1 3D plots of mean ratio functions of Sobol’s G function
 图 2 Sobol’s G函数方差比函数三维图 Fig. 2 3D plots of variance ratio functions of Sobol’s G function

 方法 W1 W2 W3 WT1 WT2 WT3 SP-UT 0.5240 0.2020 0.1056 0.6923 0.3933 0.3252 (0.0062) (0.0066) (0.0052) (0.0084) (0.0084) (0.0059) MCS 0.5291 0.1615 0.0830 0.7716 0.3966 0.3137 注:小括号中的数表示抽取30组2000个插值区间计算出的基于方差的W指标的标准差,MCS的计算量为6×106。
 图 3 Sobol’s G函数的改进的基于方差的W指标的计算结果 Fig. 3 Calculation results of variance-based modified W indices of Sobol’s G function

 图 4 钢筋混凝土梁结构的均值比函数三维图 Fig. 4 3D plots of mean ratio functions of reinforced concrete beam

 输入变量 均值 变异系数 分布类型 FY/(kN·cm-2) 44 0.105 正态 As/cm2 4.08 0.02 正态 Fc/(kN·cm-2) 3.12 0.14 正态

 方法 SFY SAs SFc SFYAs SFYFc SAsFc SP-UT 0.8424 0.0304 0.0805 0.8767 0.9231 0.1169 MCS 0.8686 0.0312 0.0871 0.8988 0.9575 0.1221 注:MCS的计算量为1.001×106。

 方法 WFY WAs WFc WTFY WTAs WTFc SP-UT 0.7509 0.0187 0.0372 0.9417 0.1698 0.2245 (0.0027) (0.00023) (0.016) (0.002) (0.0036) (0.0036) MCS 0.7784 0.0155 0.0361 0.9457 0.1268 0.1966 注:小括号中的数表示抽取30组2000个插值区间计算出的基于方差的W指标的标准差,MCS的计算量为6×106。
 图 5 钢筋混凝土梁结构的方差比函数三维图 Fig. 5 3D plots of variance ratio functions of reinforced concrete beam
 图 6 钢筋混凝土梁结构的改进的基于方差的W指标的计算结果 Fig. 6 Calculation results of variance-based modified W indices of reinforced concrete beam

5 结 论

1) 本文对原始的基于方差的W指标进行了改进,使其在包含原始指标所提供的信息外,还包含了Xi在不同分布子区间内对输出方差影响的变异性,更全面合理地反映了Xi取不同实现区间时对输出方差的平均影响。

2) 鉴于各类基于方差的灵敏度指标从不同的角度衡量了输入变量的重要性,提出了高效的同时计算这些指标的基于空间分割、UT和函数插值技术的方法,该方法利用一组UT样本点就能计算出所有指标,从而高效地从多角度为设计人员提供更有价值的输入变量的重要性信息。

3) 本文方法可以适用于复杂的非线性响应函数,通过算例进行了验证。

4) 由于本文方法的计算量会随输入变量维数的增加而呈指数增长,因此,其仅适用于输入变量维数较低的问题。

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#### 文章信息

YUN Wanying, LYU Zhenzhou, MU Shanshan

An efficient method for estimating various variance-based sensitivity indices

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(4): 796-805.
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0309