基于Mehler公式的等效相关系数求解技术

A technique for solution of equivalent correlation coefficients based on the mehler’s formula

摘要:

首先基于等效相关系数的传统二维积分方程，引入二维相关标准正态密度函数的Mehler级数展开公式，然后导出了等效相关系数的无穷次代数方程及其收敛特性，实现了积分方程向代数方程的转变，进一步完善了Nataf变换理论.同时，通过方程截断近似的方式给出了求解等效相关系数的迭代方法.由于避免了二维相关标准正态密度函数的积分和利用了代数方程系数的可重复性及一维积分特性，本文方法具有广泛的适用范围，且兼顾了计算的精度和效率.最后，通过算例验证了方法的有效性和精确性.

关键词:等效相关系数; 相关随机向量; Nataf变换; Mehler公式; 代数方程

Abstract:

Firstly, the Mehler’s formula, an equivalent series expansion of the bivariate normal probability density function (PDF), is introduced into the original equation with respect to the equivalent correlation coefficients, which is defined in the two dimensional dependent normal space. Then the equivalent algebraic equation with infinite terms is deduced straightforwardly, together with its convergence property. Theoretically, this work can be treated as the improvement of the Nataf transformation. Meanwhile, an iterative method for approximate solutions of equivalent correlation coefficients is proposed based on the truncated algebraic equation. Without the integral of bivariate normal PDF and with the merits of the reusability of coefficients of algebraic equations, the proposed method is of wide applicability, high efficiency and high precision. Lastly, the accuracy and rationality of this method are verified by examples.

Key words:equivalent correlation coefficients; dependent random vector; Nataf transformation; Mehler’s formula; algebraic equation