A most probable point method for probability distribution construction
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RESEARCH PAPER
A most probable point method for probability distribution construction Yongyong Xiang 1 & Baisong Pan 1
&
Luping Luo 1
Received: 3 October 2019 / Revised: 5 February 2020 / Accepted: 3 May 2020 # Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract With nonlinearity and uncertainty existing in engineering problems, it is important to predict the probability distribution of a response of an engineering model. The probability distribution is often constructed without sufficient accuracy due to a high computational cost. In this paper, a most probable point (MPP) method for the probability distribution construction is proposed. First, predictive models of the MPP components are established based on the Gaussian mixture distribution (GMD) and the inverse first-order reliability method. A mixture of first- and second-order reliability methods is then used to calculate discrete points of the cumulative distribution function (CDF). Finally, the CDF of the response is constructed by the GMD. A mathematical example and three engineering examples are used to verify the effectiveness of the proposed method. Keywords Uncertainty propagation . Probability distribution construction . Most probable point . Gaussian mixture distribution . Mixture of first- and second-order reliability methods
1 Introduction In practical engineering applications, statistical moments, probabilities of extreme events, and probability distributions are often used to predict system behaviors. Due to more information included in the probability distributions, they are frequently used in applications, for example, in model prediction (Helton et al. 2004; Ghasemi et al. 2019; Zhu et al. 2016), stress-strength interference model for reliability analysis (Xue and Yang 1997; Gorjian et al. 2010; Zhang et al. 2017), and model validation (Jung et al. 2015; Roy and Oberkampf 2011; Li et al. 2014; Moon et al. 2019). Denote an engineering model by Y = H(X), where Y is a response and X ¼ ½X 1 ; …; X nX T is a vector of random variables. The cumulative distribution function (CDF) of Y is given by F Y ðyÞ ¼ PrðH ðXÞ≤ yÞ ¼ ∫H ðX Þ ≤ y f X ðXÞdX Editorial Responsibility: Nam Ho Kim * Baisong Pan [email protected] 1
College of Mechanical Engineering, Zhejiang University of Technology, Hangzhou Zhejiang 310023, People’s Republic of China
ð1Þ
where fX(X) is the joint probability density function (PDF) of X. Given CDF, the PDF can be readily available. Since analytical solutions rarely exist for the probability integration, it is difficult to obtain the CDF of Y when the number of random variables is large and the function is nonlinear. Many methods have been proposed to construct the probability distributions of responses, which can be mainly classified into three categories: (1) sampling-based methods, (2) moment-based methods, and (3) MPP-based methods. Sampling-based methods contain direct Monte Carlo simulation (MCS) (Metropolis and Ulam 1949; Rubinstein and Kroese 2016), subset simulation (Au and Beck 2001), importance sampli
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