Parallel computing for efficient calculation of multidimensional Bernstein copulas in modeling nonlinear dependence between random variables

Copulas are a versatile tool for modeling the dependence structure between random variables. By defining marginal distributions, copulas can capture complex joint distributions that are often beyond the scope of traditional statistical methods, which typically rely on linearity and normality assumptions. Unlike these methods, copulas are marginal-free and can effectively model nonlinear dependencies. A Bernstein copula is an empirical, data-driven model capable of reproducing intricate relationships between variables. While highly effective for real-world data, computing Bernstein copulas becomes computationally demanding in higher dimensions. In an 𝑚 − 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑎𝑙 case with a sample size of (𝑛), the computation requires evaluating an 𝑛 𝑚 grid of points, which leads to significant resource demands in terms of processing time and memory as (𝑚) and (𝑛) increase. In this paper, we propose an efficient method for implementing multidimensional Bernstein copulas. We introduce both an optimized algorithm for calculating a multidimensional empirical copula and a parallelized approach for computing the Bernstein copula.