Proposal for the application of fractional operators in polynomial regression models to enhance the determination coefficient R2 on unseen data

Abstract

Since polynomial regression models are generally quite reliable for data that can be handled using a linear system, it is important to note that in some cases, they may suffer from overfitting during the training phase. This can lead to negative values of the coefficient of determination 𝑅2 when applied to unseen data. To address this issue, this work proposes the partial implementation of fractional operators in polynomial regression models to construct a fractional regression model. The aim of this approach is to mitigate overfitting, which could potentially improve the 𝑅2 value for unseen data compared to the conventional polynomial model, under the assumption that this could lead to predictive models with better performance. The methodology for constructing these fractional regression models is presented along with examples applicable to both Riemann–Liouville and Caputo fractional operators, where some results show that regions with initially negative or near-zero 𝑅2 values exhibit remarkable improvements after the application of the fractional operator, with absolute relative increases exceeding 800% on unseen data. Finally, the importance of employing sets in the construction of the fractional regression model within this methodological framework is emphasized, since from a theoretical standpoint, one could construct an uncountable family of fractional operators derived from the Riemann–Liouville and Caputo definitions that, although differing in their formulation, would yield the same regression results as those shown in the examples presented in this work.