An orthogonal normal transformation of correlated non-normal random variables for structural reliability
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 with the probability of structural failure expressed as the multifold integral in terms of X (). 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.
Section snippets
Basic idea
Essentially, QPNT [25], [26] attempts to define a new distribution to fit well the statistical data of an arbitrary random variable only using its first three moments, with the normal transformation based on the defined distribution. In the transformation technique of QPNT for correlated non-normal random variables, the equivalent correlation coefficients in non-orthogonal standard normal space are expressed as an explicit formula of the skewnesses and original correlation coefficients.
Accuracy of the proposed equivalent correlation coefficients
In this section, the accuracy of the equivalent correlation coefficients obtained from the proposed explicit formula is investigated through a set of numerical examples. There are 13 different pairs of marginal distributions (i.e., Cases 1–13) selected to calculate their respective F values (defined as F ) using various normal transformation, including Nataf transformation, QPNT and TLT, with the results presented in Table 2 and in Fig. 2. It is worth noting that the values obtained
Application of the proposed transformation in structural reliability analysis involving correlated random variables
With the aid of proposed transformation, current reliability analysis methods (e.g. FORM) that are established in orthogonal standard normal space can be applied for estimating the failure probability. The computational procedure of TLT-based FORM is identical to that of general FORM, except for the proposed normal transformation and the computation of Jacobian matrix elements with respect to U using the following equation:
In the follows,
Conclusions
This paper proposes an efficient and explicit normal transformation technique, i.e., TLT, to map correlated non-normal random variables onto independent standard normal variables based on the 3P lognormal distribution. Herein, an explicit expression of the equivalent correlation coefficient in non-orthogonal standard normal space is proposed, while the applicable range of original correlation coefficient is also determined based on the three fundamental properties of the preceding formula of
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The study is partially supported by the National Natural Science Foundation of China (Grant Nos.: 51738001, 51820105014 and U1934217). The support is gratefully acknowledged.
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