Structural and Multidisciplinary Optimization

, Volume 60, Issue 5, pp 1841–1866 | Cite as

A novel approach to uncertainty analysis using methods of hybrid dimension reduction and improved maximum entropy

  • Zhiying Chen
  • Ping ZhouEmail author
  • Yong Liu
  • Pengfei Ji
Research Paper


Methods of uncertainty analysis based on statistical moments are more convenient than methods that use a Taylor series expansion because the moments methods require neither an iteration process to locate the most probable point nor the computation of derivatives of the performance function. However, existing moments estimation methods are either computationally expensive (e.g., the full factorial numerical integration method) or produce large errors (e.g., the univariate dimension-reduction method). In this paper, a hybrid dimension-reduction method taking account of interactions among variables is presented for estimating the probability moments of the system performance function. In this method, a contribution-degree analysis with finite changes is implemented to identify the relative importance of the input variables on the output. Then, an approximate performance function is generated with the hybrid dimension-reduction method that is based on the results of contribution-degree analysis. Finally, the statistical moments of the performance function can be calculated from the approximate performance function. Once the probability moments are obtained, an improved maximum entropy method is used to generate the probability density function of the performance function. The uncertainty analysis can be implemented by using the approximation probability density function. Five illustrative numerical examples are presented, and different methods are compared in those examples. The statistical moments estimation results reveal that the proposed moments estimation method can dramatically improve efficiency and also guarantee accuracy. Compared with the other probability density function approximation methods, our improved maximum entropy method, using more statistical moments, is more accurate and robust.


Hybrid dimension-reduction method Contribution-degree analysis Uncertainty analysis Maximum entropy method 



We acknowledge TopEdit LLC for the linguistic editing and proofreading during the preparation of this manuscript.

Funding information

This study is supported by the National Natural Science Foundation of China (Grant No. 51275024).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Energy and Power EngineeringBeihang UniversityBeijingChina

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