﻿ 一种平均矩独立重要性指标及其拒绝抽样方法<sup>*</sup>
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An average moment-independent importance index and its rejection sampling method
CHENG Lei, ZHANG Leigang, LEI Bao, LIANG Zudian, LIU Peng
China Academy of Launch Vehicle Technology, Beijing 100076, China
Received: 2018-05-08; Accepted: 2018-07-27; Published online: 2018-09-10 16:36
Foundation item: Equipment Development Department the "13th Five-year" Equipment Research Field Foundation of China Central Military Commission (6140244010216HT15001)
Corresponding author. ZHANG Leigang, E-mail: leigang_zhang@163.com
Abstract: In the process of moment-independent importance analysis, importance index are always used to quantify the inverse allocation of structural system output uncertainty to input uncertainty.An assumption is given that the variance of a given factor can be reduced by future research, which leads to the development the moment-independent index function. The moment-independent importance index function provides an index for a given factor as a function of the amount of variance of that factor can be reduced. Meanwhile, by assuming the reduction amount of a particular factor variance as a random variable, the average moment-independent importance index is defined by taking average of the moment-independent importance index function. Estimating the average moment-independent importance index involves a large amount of computation using Sobol's method, and thus rejection sampling (RS) method is introduced here with the generated samples used in Sobol's method. Consequently, RS can use the samples generated during Sobol's method to accurately estimate the moment-independent importance index function and the average moment-independent importance index without any further model evaluation, which greatly reduces the computational cost. Numerical and engineering examples are demonstrated to show the effectiveness of the proposed measures and the accuracy and availability of the RS method.
Keywords: importance analysis     moment-independent importance index function     average moment-independent importance index     Sobol's method     rejection sampling (RS)

1 平均矩独立重要性指标 1.1 矩独立重要性指标

 (1)

ηI反映了输入变量XI在其整个取值区间内的不确定性对失效概率的影响程度。由于式(1)中定义的指标不便求解，Li和Lu[17]提出了一种新的重要性指标δIP，即在原有指标的基础上将绝对值形式转化为方差形式，其定义如下：

 (2)

 (3)

Wei等[18]提出了一种可以高效求解条件方差的方法，那么矩独立重要性指标δIP可以改为

 (4)

 (5)

 (6)

1.2 平均矩独立重要性测度

 (7)

 (8)

 (9)

2 拒绝抽样方法

RS方法可以从某给定分布产生理想分布样本，该技术对于伪随机样本[20-21]和随机样本已经得到较好应用，本文将该方法用于失效概率函数的求解中。

2) 低偏差序列：产生低偏差序列的点(ξi, νi)∈[0, 1]2, i=1, 2, …, M。将ξi映射到xi，使用累积分布函数(CDF)的反变换xi=Fxi-1(ξi)来得到输入变量xi的样本(其中Fxixi的CDF)。

2) 低偏差序列：ui=νi

RS方法可以用来求解式(8)中转化后的基于方差的重要性指标，从而得到式(9)中的平均矩独立重要性指标，步骤如下：

1) 对于均匀分布U(a, b)，在区间[a+λi1/2·(b-a), b]上产生b′的均匀分布样本，并使a′=b′-λi1/2(b-a)。

2) 对于正态分布N(μ, σ2)，令μ′=μσ′=λ1/2σ

3 算例 3.1 数值算例——Ishigami函数

Ishigami函数被广泛用于可靠性分析中，首先被Ishigami和Homma引入，然后被用于测试重要性和不确定性分析技术中[2, 22]，可表示为

 (10)

 图 1 Sobol方法求解的Ishigami测试函数重要性指标结果 Fig. 1 Importance index results solved by Sobol's method for Ishigami test function
 图 2 RS方法求解的Ishigami测试函数重要性指标结果 Fig. 2 Importance index results solved by RS method for Ishigami test function

3.2 工程算例——屋架结构

 图 3 屋架结构模型的示意图[17] Fig. 3 Schematic diagram of a roof truss structure model[17]

 分布参数 均值 变异系数 q 20 000 N/m 0.07 l 12 m 0.01 AS 9.82×10-4 m2 0.06 AC 0.04 m2 0.12 ES 1×1011 N/m2 0.06 EC 2×1010 N/m2 0.06

 图 4 Sobol方法求解的屋架结构模型重要性指标结果 Fig. 4 Importance index results solved by Sobol's method for roof truss structure model
 图 5 RS方法求解的屋架结构模型重要性指标结果 Fig. 5 Importance index results solved by RS method for roof truss structure model
3.3 工程算例——十杆桁架结构

 图 6 平面十杆桁架结构模型示意图 Fig. 6 Schematic diagram of a planar ten-bar truss structure model

 分布参数 均值 变异系数 L 1 m 0.05 E 100 GPa 0.05 P1 80 kN 0.05 P2 10 kN 0.05 P3 10 kN 0.05 Ai 0.001 m2 0.15

 图 7 Sobol方法求解的十杆桁架结构模型重要性指标结果 Fig. 7 Importance index results solved by Sobol's method for ten-bar truss structure model
 图 8 RS方法求解的十杆桁架结构模型重要性指标结果 Fig. 8 Importance index results solved by RS method for ten-bar truss structure model
4 结论

1) 输入变量分布参数发生变化时，各个变量对应的全局矩独立重要性指标也发生变化，而本文提出的平均矩独立重要性指标可以衡量输入变量分布参数变化时对输出响应的平均影响。

2) RS方法可以使用原始全局重要性分析中的样本，额外获取输入变量分布参数发生变化时的矩独立重要性指标，从而计算得到平均矩独立灵敏度指标。这样在不增加额外计算成本的情况下获取更多的信息，为研究人员进一步进行工程设计和优化提供了丰富的指导信息。

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#### 文章信息

CHENG Lei, ZHANG Leigang, LEI Bao, LIANG Zudian, LIU Peng

An average moment-independent importance index and its rejection sampling method

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(1): 66-73
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0266