Analyzing the Role of Nonlinear Regression Models in Capturing Complex Functional Relationships Between Variables

Posted: Oct 23, 2017

• Liam Gonzalez
• Joseph Garcia
• Jacob Scott

Abstract

The exploration of relationships between variables constitutes a fundamental pursuit across scientific disciplines, with regression analysis serving as a cornerstone methodology for quantifying these associations. While linear regression models have dominated statistical practice due to their interpretability and computational simplicity, their inherent limitations in capturing complex, nonlinear dependencies have become increasingly apparent in contemporary data-rich environments. This research addresses the critical challenge of effectively modeling intricate functional relationships that characterize many natural and engineered systems, where linear approximations often prove inadequate for both explanatory and predictive purposes. Nonlinear regression models offer a theoretically appealing alternative, capable of representing a vast array of functional forms through various mathematical structures, including polynomial expansions, spline functions, neural networks, and kernel methods. However, the practical implementation of these models presents significant challenges related to parameter estimation, model selection, computational complexity, and interpretability. The conventional wisdom that increasing model flexibility necessarily improves performance has been repeatedly challenged by the curse of dimensionality and the risk of overfitting, particularly in high-dimensional settings with limited observations. Our investigation builds upon existing nonlinear modeling frameworks while introducing several novel methodological innovations. We propose an integrated approach that combines fractal-based feature engineering with adaptive regularization techniques, specifically designed to enhance the capacity of nonlinear