[Paper Review] Generating correlated random vector by polynomial normal transformation

Qing Xiao|arXiv (Cornell University)|Aug 26, 2015

Advanced Statistical Methods and Models10 references3 citations

TL;DR

This paper proposes a high-degree polynomial normal transformation (PNT) model to generate correlated non-normal random vectors by transforming standard normal variables. It introduces probability weighted moment (PWM) matching and percentile matching to efficiently estimate polynomial coefficients, enabling accurate simulation of complex distributions and precise correlation structure preservation through derived polynomial equations for equivalent normal-space correlation coefficients.

ABSTRACT

This paper develops a polynomial normal transformation model, whereby various non-normal probability distributions can be simulated by the standard normal distribution. Two methods are presented to determine the coefficients of polynomial model: (1) probability weighted moment (PWM) matching (2) percentile matching. Compared to the existing raw moment or L-moment matching, the proposed methods are more computationally convenient, and can be used to estimate the coefficients of polynomial model with a higher degree. Furthermore, for two correlated random variables, a polynomial equation is derived to estimate the equivalent correlation coefficient in standard normal space, and random vector with non-normal marginal distributions and prescribed correlation matrix can be generated. Finally, numerical examples are worked to demonstrate the proposed method.

Motivation & Objective

• To develop a flexible polynomial normal transformation model capable of simulating a wide range of non-normal distributions beyond existing third- or fifth-order methods.
• To address the computational complexity of existing moment-matching methods by introducing probability weighted moment (PWM) matching and percentile matching for higher-degree polynomial models.
• To enable accurate generation of correlated random vectors with arbitrary non-normal marginal distributions and a prescribed correlation matrix.
• To derive analytical polynomial equations that relate the desired correlation in the original space to the equivalent correlation in the standard normal space.

Proposed method

• The polynomial normal transformation model expresses a non-normal random variable X as a polynomial function of a standard normal variable Z: X ≈ Σ aₖZᵏ for k = 0 to n.
• Coefficients aₖ are determined using two novel methods: (1) PWM matching, which equates the PWMs of the polynomial model to those of the target distribution, forming a linear system; and (2) percentile matching, which uses a least-squares approach to match empirical percentiles.
• For correlated variables, the method derives a polynomial equation linking the desired correlation ρₓ in the original space to the equivalent ρ_z in the standard normal space using product moment formulae.
• The method leverages the Weierstrass approximation theorem, ensuring that the monotonic CDF transformation F⁻¹[Φ(Z)] can be well-approximated by a polynomial over a closed interval.
• The proposed approach avoids solving complex nonlinear systems of equations, unlike traditional raw moment or L-moment matching, especially for high-degree polynomials.
• The correlation structure is preserved by first computing the required ρ_z via the derived polynomial equation and then generating standard normal deviates with that correlation before applying the inverse CDF transformation.

Experimental results

Research questions

• RQ1 Can probability weighted moments (PWMs) be used to efficiently estimate coefficients for high-degree polynomial normal transformations?
• RQ2 How does percentile matching compare to PWM matching in terms of accuracy and computational feasibility for simulating non-normal distributions?
• RQ3 Can a closed-form polynomial equation be derived to estimate the equivalent normal-space correlation coefficient ρ_z for a given target correlation ρ_x between two non-normal variables?
• RQ4 To what extent can the proposed method simulate complex, non-normal distributions with higher statistical fidelity than existing TPNT or FPNT techniques?

Key findings

• The PWM matching method enables efficient, linear system-based estimation of polynomial coefficients, making it computationally superior to nonlinear raw moment or L-moment matching, especially for high-degree polynomials.
• The percentile matching method provides a robust alternative with good accuracy, validated through numerical investigation, and is particularly useful when analytical moments are difficult to compute.
• The derived polynomial equation for ρ_z allows precise determination of the required normal-space correlation, ensuring that the generated random vectors match the target correlation matrix.
• Numerical examples show that the simulated correlation matrix of generated samples (Rₓ*) closely matches the desired matrix (Rₓ), with Rₓ* = [1.000, 0.900, 0.499; 0.900, 1.000, 0.300; 0.499, 0.300, 1.000] for a target Rₓ = [1.000, 0.900, 0.500; 0.900, 1.000, 0.300; 0.500, 0.300, 1.000].
• The method successfully generates multivariate random vectors with both prescribed non-normal marginal distributions and a given correlation matrix, demonstrating high accuracy and generality.