Soft Nonlinear Quantile-Based AI Regression Under Uncertainty

Abstract

Regression analysis in real-world applications is often challenged by nonlinear relationships, uncertainty, and imprecise observations. Traditional regression models typically assume crisp inputs and outputs, limiting their ability to represent ambiguity inherent in many domains such as risk analysis, finance, and healthcare. To address these limitations, this paper proposes a novel soft nonlinear quantile-based artificial intelligence (AI) regression framework that integrates fuzzy logic with gradient boosting quantile regression. The proposed approach models the conditional distribution of the response variable by estimating multiple quantiles (e.g., 0.25, 0.50, and 0.75) using gradient boosting with quantile loss functions, and reconstructs fuzzy outputs in the form of triangular fuzzy numbers. This design enables simultaneous handling of nonlinear dependencies, asymmetric uncertainty, and imprecision in the data within a unified framework. Unlike existing fuzzy regression models, which are predominantly linear or semi-parametric, and AI-based quantile models, which assume crisp outputs, the proposed method bridges this gap by producing interpretable fuzzy predictions using modern ensemble learning techniques. The effectiveness of the model is evaluated on a synthetic dataset with nonlinear structure and artificially generated fuzzy responses. Experimental results demonstrate that the proposed model significantly outperforms classical linear quantile regression, achieving lower prediction errors (RMSE = 0.15 vs. 0.425; MAE = 0.112 vs. 0.367), improved coverage probability (96.7%), and more accurate estimation of uncertainty intervals. Statistical validation using a paired t-test confirms that these improvements are highly significant (p < 10⁻¹⁸). These findings highlight the potential of combining quantile-based estimation with fuzzy representation and ensemble learning for robust regression under uncertainty. The proposed framework is particularly suitable for applications involving imprecise data and complex nonlinear relationships, and provides a promising direction for future research in interpretable and uncertainty-aware machine learning.