An orthogonal normal transformation of correlated non-normal random variables for structural reliability

Highlights

• Propose a normal transformation technique based on the 3P lognormal distribution.

• Derive the complete explicit expressions of equivalent correlation coefficients.

• Identify the applicable range of original correlation coefficients.

• Verify the user-friendliness, efficiency and accuracy of the proposed approach.

Abstract

In this paper, an efficient and explicit technique is proposed for transforming correlated non-normal random variables into independent standard normal variables based on the three-parameter (3P) lognormal distribution. In contrast with the classic Nataf transformation, the derived equivalent correlation coefficient in non-orthogonal standard normal space of the proposed transformation is expressed as an explicit formula, thereby avoiding tedious iteration algorithm or multifarious empirical formulas. Meanwhile, the applicable range of the original correlation coefficient is determined based on fundamental properties of the proposed expression of correlation distortion and the definition of correlation coefficient. The proposed transformation requires only the first three moments (i.e., mean, standard deviation, and skewness) of basic random variables, as well as their correlation matrix. Therefore, the proposed transformation can also be applied even when the joint distribution or marginal distributions of the basic random variables are unknown. Several numerical examples are presented to demonstrate the user-friendliness, efficiency, and accuracy of the proposed transformation applied in structural reliability analysis involving correlated non-normal random variables.

Introduction

In structural reliability analysis, structural system’s input parameters, e.g., material properties, geometric dimensions, loads, and modeling and prediction errors, are treated as random variables ${\mathit{X}}_i,i=1,2,\dots ,n,$ with the probability of structural failure expressed as the multifold integral in terms of X $=$ (${\mathit{X}}_1,{\mathit{X}}_2,\dots ,{\mathit{X}}_n$)${}^T$. These random variables are usually non-normal and often correlated. However, most existing approximation techniques estimating the probability of structural failure are established in orthogonal standard normal space, for example, the first- and second-order reliability methods [1], [2], [3], [4], [5], [6], [7] (FORM and SORM) and the methods of moment (MM) [8]. Therefore, a procedure to transform correlated non-normal random variables into independent standard normal ones (called normal transformation for brevity) is indispensable in structural reliability analysis.

Rosenblatt transformation [7], [9], [10] is available and has been popular for realizing the aforementioned normal transformation when the joint probability density function (PDF) or joint cumulative distribution function (CDF) of basic random variables is known. However, due to often limited experimental data, this precondition is hardly ever fulfilled in engineering practice [11]. In addition, Rosenblatt transformation includes an inherent defect: the conditioning order of input random variables affects the result. This method may lead to n! different transformations for an n-dimensional random vector X, significantly influencing the final results obtained from FORM/SORM [12].

Nataf transformation [13], [14] is a worthwhile alternative for normalizing correlated random variables, requiring only marginal CDFs of basic random variables and correlation matrix, with conditioning order of input random variables having no effect. For each pair of marginal distributions with known Pearson product-moment correlation coefficient (called correlation coefficient for brevity), the equivalent correlation coefficient in non-orthogonal standard normal space can be determined by a complex nonlinear integral equation. To avoid solving this nonlinear equation, Liu and Der Kiureghain [14] have used numerical methods to develop 54 empirical formulas for 10 types of commonly used distributions. Although the empirical formulas provide sufficiently accurate results (maximum error only about 4.5%), a considerable bundle of formulas are inconvenient for engineering application and are inflexible for covering an arbitrary distribution [15], such as a truncated distribution. Fortunately, Li et al. [16] have utilized the bivariate Gaussian–Hermite quadrature and Newton–Raphson method to solve this nonlinear integral equation for the general cases. However, iteration algorithm and eliminating the correlation (equivalent to non-orthogonality) among random variables to enable the Gaussian–Hermite quadrature undoubtedly increase computational burden.

A recent theory states that any multivariate joint distribution can be expressed in terms of marginal distributions through the so-called “copula” function for describing the dependence structure among the variables [17]. Several research works suggest that the classic Nataf transformation essentially adopts a Gaussian copula to reconstruct the joint probability distribution [18] and that more non-Gaussian copulas are further applied in structural reliability analysis involving correlated random variables [19]. However, it remains an outstanding practical challenge to select a best-fit copula model and calculate the copula parameters in a model for the multivariable cases [20], [21]. Furthermore, the marginal CDFs of basic random variables sometimes are unknown due to lack of statistical data, and the probabilistic characteristics of these variables are expressed as the statistical moments and correlation matrix.

To realize the normal transformation under incomplete probability information for independent random variables, polynomial normal transformation provides an alternative approach [22], [23], [24]. Based on the condition that the first three moments of a random variable are known, Zhao and Ono [25] have proposed an isoprobabilistic transformation technique using a quadratic polynomial function in terms of a standard normal variable, with the polynomial coefficients determined by the first three moments (i.e., mean, standard deviation, and skewness) of the variable. This transformation technique is referred to as quadratic polynomial normal transformation (QPNT). Using QPNT, a method has been further developed for orthogonal normal transformation of correlated random variables by determining the equivalent correlation coefficients in non-orthogonal standard normal space as solutions of quadratic polynomials [26]. However, the applicable range of this method has to be limited into a relatively narrow district of the skewnesses of random variables to ensure the sufficient accuracy.

The objectives of the present paper are to overcome the problems of QPNT and to propose a more efficient transformation technique for equivalent correlation coefficients in non-orthogonal standard normal space. To achieve this goal, the rest of the paper is organized as follows. Section 2 proposes complete explicit formulas for mapping correlated non-normal random variables onto independent standard normal variables or the opposite based on the three-parameter (3P) lognormal distribution. Herein, the equivalent correlation coefficients are derived and expressed as explicit formulas, with clarifying the applicable range of the original correlation coefficients. Section 3 investigates the accuracy of the equivalent correlation coefficients obtained from the proposed explicit formulas. Section 4 introduces the application of proposed transformation in current reliability methods and provides three numerical examples to demonstrate the user-friendliness, efficiency, and accuracy of the proposed method. Finally, Section 5 summarizes the findings of the present paper.