RESEARCH PAPER
Nonparametric uncertainty representation method with different
insufficient data from two sources
Xiang Peng
1,2
&
Zhenyu Liu
2
&
Xiaoqing Xu
1
&
Jiquan Li
1
&
Chan Qiu
2
&
Shaofei Jiang
1
Received: 8 December 2017 /Revised: 30 April 2018 /Accepted: 6 May 2018
#
Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract
The uncertainty information of design variables is included in the available representation data, and there are differences among
representation data from different sources. Therefore, the paper proposes a nonparametric uncertainty representation method of
design variables with different insufficient data from two sources. The Gaussian interpolation model for sparse sampling points
and/or sparse sampling intervals from a single source is constructed through maximizing the logarithmic likelihood estimation
function of insufficient data. The weight ratios of probability density values at sampling points are optimized through minimizing
the total deviation of the fusion model, and the fusion Gaussian model is constructed based on the weight sum of the optimum
probability density values of sampling points for Source 1 and Source 2. The methodology is extended to five different fusion
conditions, which contain the fusion of uncertain distribution parameters, the fusion of insufficient data and interval data, etc.
Five application examples are illustrated to verify the effectiveness of the proposed methodology.
Keywords Uncertainty representation
.
Insufficient data
.
Data fusion
.
Maximum likelihood estimation
1 Introduction
In traditional engineering design problems, design variables
are assumed to be determinate values. However, many vari-
ables are uncertain in practical conditions due to varying op-
eration conditions, manufacturing tolerances, uncertainties of
measurement data, differences in material properties, etc. To
quantify the influence of these uncertainties and enhance
product performance in practical conditions, there is a strong
demand to quantify the uncertainties of design variables and
incorporate the uncertainty information into the optimization
design of engineering products, and many uncertainty optimi-
zation design methods have been widely applied to engineer-
ing design, such as uncertain topology optimization design
(Chen et al. 2009, Luo et al. 2017,Wangetal.2017a,
2017b, 2017c), uncertain structural optimization design
(Echard et al. 2011,Wangetal.2017a, 2017b, 2017c), reli-
ability design (Jiang et al. 2013a, 2013b; Cheng et al. 2016;
Keshtegar and Meng 2017), and robust design (Shah et al.
2015; Chakraborty et al. 2017).
According to the available amount of uncertainty represen-
tation data, the uncertain variables are decomposed into strong
statistical variables, sparse variables and interval variables
(Oberkampf et al. 2004). The strong statistical variables are
aleatory uncertain variables, which have a large amount of
uncertainty representation data. Their distribution types and
distribution parameters are determinate. Sparse variables and
interval variables are epistemic uncertain variables. The ex-
perimental data of interval variables cannot be acquired, only
sub-intervals and corresponding basic probability assignments
can be provided according to expert knowledge. The sampling
* Shaofei Jiang
jsf75@zjut.edu.cn
Xiang Peng
pengxiang@zjut.edu.cn
Zhenyu Liu
liuzy@zju.edu.cn
Xiaoqing Xu
837300473@qq.com
Jiquan Li
lijq@zjut.edu.cn
Chan Qiu
qc@zju.edu.cn
1
Key Laboratory of E&M, Zhejiang University of Technology,
Hangzhou 310014, China
2
State Key of CAD&CG, Zhejiang University, Hangzhou 310027,
China
Structural and Multidisciplinary Optimization
https://doi.org/10.1007/s00158-018-2003-6
@Phil_Robichaud
@deepthiw
@JoseServera