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Proposal for the Application of Fractional Operators in Polynomial Regression Models to Enhance the Determination Coefficient $R^2$ on Unseen Data

Anthony Torres-Hernandez

TL;DR

This paper tackles improving regression generalization by embedding fractional calculus into polynomial regression. It develops a formal framework of sets and groups of fractional operators, anchored by Riemann–Liouville and Caputo definitions, and introduces a Fractional Regression Model that replaces polynomial terms with operatored equivalents $o_x^\alpha x^i$ while preserving the intercept. A multidimensional logistic variant is proposed, pointing toward fractional neural-network activations, and a concrete avocado-price dataset example demonstrates interpolation/extrapolation performance. The work provides both a rigorous mathematical foundation and an empirical case study, highlighting potential gains in predicting unseen data and informing future directions in fractional-feature regression and related architectures.

Abstract

Since polynomial regression models are generally quite reliable for data with a linear trend, it is important to note that, in some cases, they may encounter overfitting issues during the training phase, which could result in negative values of the coefficient of determination $R^2$ for unseen data. For this reason, this work proposes the partial implementation of fractional operators in polynomial regression models to generate a fractional regression model. The goal of this proposal is to attempt to mitigate overfitting, which could improve the value of the coefficient of determination for unseen data, compared to the polynomial model, under the assumption that this would contribute to generating predictive models with better performance. The methodology for constructing these fractional regression models is detailed, and examples applicable to both Riemann-Liouville and Caputo fractional operators are presented.

Proposal for the Application of Fractional Operators in Polynomial Regression Models to Enhance the Determination Coefficient $R^2$ on Unseen Data

TL;DR

This paper tackles improving regression generalization by embedding fractional calculus into polynomial regression. It develops a formal framework of sets and groups of fractional operators, anchored by Riemann–Liouville and Caputo definitions, and introduces a Fractional Regression Model that replaces polynomial terms with operatored equivalents while preserving the intercept. A multidimensional logistic variant is proposed, pointing toward fractional neural-network activations, and a concrete avocado-price dataset example demonstrates interpolation/extrapolation performance. The work provides both a rigorous mathematical foundation and an empirical case study, highlighting potential gains in predicting unseen data and informing future directions in fractional-feature regression and related architectures.

Abstract

Since polynomial regression models are generally quite reliable for data with a linear trend, it is important to note that, in some cases, they may encounter overfitting issues during the training phase, which could result in negative values of the coefficient of determination for unseen data. For this reason, this work proposes the partial implementation of fractional operators in polynomial regression models to generate a fractional regression model. The goal of this proposal is to attempt to mitigate overfitting, which could improve the value of the coefficient of determination for unseen data, compared to the polynomial model, under the assumption that this would contribute to generating predictive models with better performance. The methodology for constructing these fractional regression models is detailed, and examples applicable to both Riemann-Liouville and Caputo fractional operators are presented.

Paper Structure

This paper contains 8 sections, 3 theorems, 57 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Sets of Fractional Operators
  3. Groups of Fractional Operators
  4. Polynomial Regression Model
  5. Seen and Unseen Data in Regression Models
  6. The Coefficient of Determination $R^2$
  7. Fractional Regression Model
  8. Examples Using Fractional Regression Model

Key Result

Corollary 3.2

Let $o_x^\alpha$ be a fractional operator such that $o_x^\alpha \in \mathop{\mathrm{MO}}\nolimits_{x,\alpha}^{\infty,u}(h)$. Let $(\mathbb{Z}, +)$ be the group of integers under addition, and $\mathbb{H}$ a subgroup of $\mathbb{Z}$. The set of fractional matrix operators is given by: This forms a subgroup of the group generated by $A_\alpha(o_x^\alpha)$:

Figures (11)

  • Figure 1: Comparison of metrics and fractional regression models in the Chicago region
  • Figure 2: Comparison of metrics and fractional regression models in the HartfordSpringfield region
  • Figure 3: Comparison of metrics and fractional regression models in the NewYork region
  • Figure 4: Comparison of metrics and fractional regression models in the Northeast region
  • Figure 5: Comparison of metrics and fractional regression models in the RaleighGreensboro region
  • ...and 6 more figures

Theorems & Definitions (7)

  • proof
  • Corollary 3.2
  • Example 1
  • Corollary 3.3
  • Example 2
  • Corollary 3.4
  • Example 3