﻿ 基于可能性矩的混合不确定性全局灵敏度分析<sup>*</sup>
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Global sensitivity analysis under mixed uncertainty based on possibilistic moments
CHENG Kai, LYU Zhenzhou, SHI Yan
School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China
Received: 2016-07-27; Accepted: 2016-09-02; Published online: 2016-11-17 18:11
Foundation item: Fundamental Research Funds for the Central Universities (3102015BJ(Ⅱ) CG009)
Corresponding author. LYU Zhenzhou, E-mail: zhenzhoulu@nwpu.edu.cn
Abstract: For the structures with fuzzy uncertainty and random uncertainty simultaneously, to measure the influence of fuzzy and random input variables on the statistical characteristic of output response, a new global sensitivity index is proposed. Based on the definition of possibilistic moments of the fuzzy variable, the characteristic of the output response under mixed uncertainty is analyzed. With respect to the possibilistic moments of the output response, the possibilistic expectation of output response is taken as an example, and the average difference between the unconditional probability density function (PDF) and the conditional PDF of the model output possibilistic expectation is used to establish the global sensitivity indices for both the fuzzy input and the random input. The properties of the proposed global sensitivity indices are discussed, and the Kriging surrogate model is applied to solving the proposed index efficiently. Finally, some examples are used to verify the rationality and effectiveness of the proposed method.
Key words: fuzzy variable     random variable     sensitivity analysis     possibilistic moments     Kriging surrogate model

1 模糊数可能性矩的定义

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2 混合输入对输出影响的定性分析

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3 混合不确定性下的灵敏度指标 3.1 随机输入变量灵敏度指标

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δXRi进一步可写为Copula函数的形式，来表达输出响应Y的可能性期望与XRi的相关性大小。

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3.2 模糊输入变量灵敏度指标

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3.3 灵敏度指标的数学性质

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4 灵敏度指标的求解方法 4.1 灵敏度指标一般求解方法

4.1.1 随机输入变量灵敏度指标的求解方法

1) 将随机输入变量在其整个分布范围内抽样，通过不确定性传递和优化算法得到输出响应量隶属函数，求得输出响应的可能性期望，进而估计响应量可能性期望的无条件概率密度函数fM(m)。

2) 根据XRi的概率密度函数抽取其样本，将XRi固定在其中一个样本点(记为xRi*)处，其余的输入变量仍在整个分布范围内变化抽样，通过不确定性传递和优化算法得到此时响应量可能性期望的条件概率密度函数fM|XRi(m)。此时，便可以得到fM|XRi(m)和fM(m)在XRi=xRi*处的差异面积SXRi*

3) 重复这个过程，最终得到SXRi的均值，进而得到δXRi的估计值。

4.1.2 模糊输入变量灵敏度指标的求解方法

1) 将随机输入变量在其整个分布范围内抽样，通过不确定性传递和优化算法得到输出响应量的可能性期望，进而估计响应量可能性期望的无条件概率密度函数fM(m)。

2) 同样将随机输入变量在其整个分布范围内变化抽样。固定某个模糊输入变量XFi的隶属水平γ，根据XFi的隶属函数可得此时XFi∈[XFi(γ), XFi(γ)]，XFi在区间[XFi(γ), XFi(γ)]内任取一实现值xFi*，可通过不确定性传递和优化算法求得输出响应的可能性期望的有条件概率密度函数fM|xFi*(m)。此时，便可以得到fM|XFi(m)和fM(m)在XFi=xFi*处的面积差异SXFi*，当XFi遍历区间[XFi(γ), XFi(γ)]时，便可得到SXFi的一个区间[SXFi(γ), SXFi(γ)]。

3) 遍历XFi的隶属水平，即可相应地得到SXFi的隶属函数，进而得到SXFi的可能性期望值和δXFi的估计值。

4.2 Kriging代理模型法

Kriging代理模型法是一种具有很强全局近似能力的逼近技术，它借助某一点周围的已知信息的加权线性组合来估计该点的未知信息[16]。Kriging代理模型法可以用来逼近模型。

4.2.1 Kriging法中的抽样阶段

Kriging法对试验点的预测需要知道一系列观测点X(k)={X1(k)X2(k), …, Xn(k)}(k=1, 2, …, NtNt为训练点的个数)(称为训练点(TP))的信息，其预测能力取决于训练点所携带的信息量。若所携带的信息量足够反映整个参数空间，那么预测结果将会较为准确。但达到足够的信息量需要大量的样本，这导致计算成本的大幅度增加。因此，训练点的选取对Kriging法的预测能力有至关重要的影响。

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4.2.2 Kriging法的基本理论

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5 算例

 随机变量 灵敏度指标值 一般方法 Kriging法 x1 0.211 3 0.211 3 x2 0.408 7 0.408 7 x3 0.048 2 0.048 2 x4 0.094 7 0.094 8

 图 1 流体管道系统示意图 Fig. 1 Schematic diagram of a sewer pipe system

 随机变量 模糊变量 c σ x1 C1 0.825 0.070 x2 C2 0.825 0.070 x3 C3 0.900 0.050

 参数 灵敏度指标值 一般方法 Kriging法 C1 0.004 9 0.004 9 C2 0.004 9 0.004 9 C3 0.004 3 0.004 3 Y 0.764 2 0.764 2 W 0.053 7 0.053 7

 图 2 屋架结构的简单示意图 Fig. 2 Schematic diagram of a roof truss structure

AcAsql为随机变量，分布参数参见表 4

 随机变量 分布类型 均值 标准差 q/(N·m-1) 正态 20 000 1 400 l/m 正态 12 0.12 As/m2 正态 9.82×10-4 5.982×10-5 Ac/ m2 正态 0.04 0.004 8

 参数 灵敏度指标值 一般方法 Kriging法 q 0.339 4 0.339 1 l 0.673 9 0.673 6 Ac 0.144 4 0.144 3 Ec 0.012 0 0.011 9 As 0.158 5 0.158 4 Es 0.029 1 0.029 0

6 结论

1) 通过分析混合不确定性下输出响应的特征，指出混合不确定性作用下响应输出的概率矩为一个模糊变量，而可能性矩为一个随机变量。

2) 基于模糊变量可能性矩的定义，以输出响应的一阶可能性中心距为例，比较输出响应有条件和无条件可能性期望的概率密度函数的平均差异，分别建立了随机输入变量和模糊输入变量关于输出响应的可能性期望的灵敏度指标。类似地可以将所提指标扩展来衡量输入变量对输出响应的高阶可能性矩的影响。

3) 为了高效求解所提指标，本文采用了Kriging代理模型法，该方法只需对原始模型进行一次代理即可较精确求解所提指标，因此大大较少了计算量。

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#### 文章信息

CHENG Kai, LYU Zhenzhou, SHI Yan

Global sensitivity analysis under mixed uncertainty based on possibilistic moments

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(8): 1705-1712
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0626