Purpose

Normal transformation is often required in structural reliability analysis to convert the non-normal random variables into independent standard normal variables. The existing normal transformation techniques, for example, Rosenblatt transformation and Nataf transformation, usually require the joint probability density function (PDF) and/or marginal PDFs of non-normal random variables. In practical problems, however, the joint PDF and marginal PDFs are often unknown due to the lack of data while the statistical information is much easier to be expressed in terms of statistical moments and correlation coefficients. This study aims to address this issue, by presenting an alternative normal transformation method that does not require PDFs of the input random variables.

Design/methodology/approach

The new approach, namely, the Hermite polynomial normal transformation, expresses the normal transformation function in terms of Hermite polynomials and it works with both uncorrelated and correlated random variables. Its application in structural reliability analysis using different methods is thoroughly investigated via a number of carefully designed comparison studies.

Findings

Comprehensive comparisons are conducted to examine the performance of the proposed Hermite polynomial normal transformation scheme. The results show that the presented approach has comparable accuracy to previous methods and can be obtained in closed-form. Moreover, the new scheme only requires the first four statistical moments and/or the correlation coefficients between random variables, which greatly widen the applicability of normal transformations in practical problems.

Originality/value

This study interprets the classical polynomial normal transformation method in terms of Hermite polynomials, namely, Hermite polynomial normal transformation, to convert uncorrelated/correlated random variables into standard normal random variables. The new scheme only requires the first four statistical moments to operate, making it particularly suitable for problems that are constraint by limited data. Besides, the extension to correlated cases can easily be achieved with the introducing of the Hermite polynomials. Compared to existing methods, the new scheme is cheap to compute and delivers comparable accuracy.

中文翻译：

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用于结构可靠性分析的 Hermite 多项式正态变换

目的

在结构可靠性分析中经常需要进行正态变换，将非正态随机变量转换为独立的标准正态变量。现有的正态变换技术，例如 Rosenblatt 变换和 Nataf 变换，通常需要非正态随机变量的联合概率密度函数 (PDF) 和/或边际 PDF。然而，在实际问题中，联合PDF和边缘PDF由于缺乏数据往往是未知的，而统计信息更容易用统计矩和相关系数来表示。本研究旨在通过提出一种不需要输入随机变量的 PDF 的替代正态变换方法来解决这个问题。

设计/方法/方法

新方法，即 Hermite 多项式正态变换，用 Hermite 多项式表示正态变换函数，它适用于不相关和相关的随机变量。通过许多精心设计的比较研究，对它在使用不同方法的结构可靠性分析中的应用进行了彻底的研究。

发现

进行综合比较以检查所提出的 Hermite 多项式正态变换方案的性能。结果表明，所提出的方法与以前的方法具有相当的准确性，并且可以以封闭形式获得。而且，新方案只需要前四个统计矩和/或随机变量之间的相关系数，大大拓宽了正态变换在实际问题中的适用性。

原创性/价值

本研究从 Hermite 多项式的角度解释了经典的多项式正态变换方法，即 Hermite 多项式正态变换，将不相关/相关的随机变量转换为标准正态随机变量。新方案只需要前四个统计矩即可运算，特别适用于受有限数据约束的问题。此外，通过引入 Hermite 多项式可以很容易地扩展到相关案例。与现有方法相比，新方案的计算成本低，并且精度相当。