Fast procedure to compute empirical and Bernstein copulas.

In this work, a novel technique for efficient computation of bivariate empirical copulas and, by extension, non-parametrical copulas. The algorithm addresses discrete and finite equations, integrating mathematical-statistical components. It introduces two novel concepts: Propagation and Overlapping, to enhance computations and their comprehension during empirical copula construction. The algorithm is presented in pseudo-code for its implementation in any programming language. Comparative performance assessments demonstrate computing speeds ranging from 60 to 250 times faster than the standard algorithm across multiple case studies. Recent research highlights the utility of copulas in Artificial Intelligence (AI) techniques for enhanced predictions [8]. Existing studies center on parametric copulas, underscoring the significance of introducing a methodology for non-parametric copula implementation because this approach facilitates precise modeling of non-linear relationships among random variables, offering substantial improvements over conventional techniques, and boosting its integration, within the realm of artificial intelligence. • This study harnesses the superior capacity of non-parametrical copulas for modeling non-linear relationships compared to parametrical alternatives. However, owing to their substantial computational demands, this analysis primarily centers on the challenges associated with memory use and computation time. • In this work, in subsection 1.1, is addressed an engaging discussion concerning the advantages and disadvantages of employing non-parametrical copulas. In the same subsection are considered expert opinions, it is concluded that non-parametric copulas offer a unique advantage by excellent replicating nonlinear joint distributions, making them suitable candidates for application in artificial intelligence techniques. • In section 2, an algorithm was proposed to compute the empirical copula, exhibiting a speed improvement, of at least, 60-fold compared to the standard method, whose results are shown in Table 3. Its pseudocode, presented in subsection 2.2, is adaptable to any computing language. • In this work, it is proposed a new method based on concepts of propagation and overlapping, they are explained in depth in subsection 2.1. In order to clarify them, extensive graphical depictions of data placement within the empirical copula matrix are shown. These illustrations aid in making a reflection on the reduction in calculations quantity and, consequently, the reduction in computational time. • In section 3, an analysis was conducted to evaluate the precision of outcomes generated by our proposed methodology in comparison to those yielded by the standard method. It is imperative to note that while the precision of the standard method's results is assumed, our proposal's results adhere to the designated tolerance levels assigned to them. This robust evaluation underscores the reliability and accuracy of our proposed approach, ensuring compliance with predefined error tolerances. [ABSTRACT FROM AUTHOR]