Employing Fixed-Point Theory for Fuzzy Regression Analysis: Methodology and Empirical Application

Traditional fuzzy regression methods, including the fuzzy minimum method of Tanaka and fuzzy least squares may not have theoretical existence and uniqueness guarantees as well as numerical stability. Of the greatest relevance in this case are these restrictions in the engineering applications where the level of uncertainty is taken into consideration such as estimating the compressive strength of concrete. This paper solves these problems by developing a mathematically precise fuzzy regression model, founded on the fixed-point methodology, on the entire metric space of trapezoidal fuzzy numbers in the metric scale d $\infty$ .

The fuzzy coefficient is defined on the trapezoidal space of vectors and a coefficient of contraction is defined which is shown to exist and to be unique by a fixed-point theory put forward by Banach. The coefficients are estimated through an iterative algorithm that has alpha computation and Lipschitz continuity to calculate the coefficients. The dataset of University of California, Irvine concrete compressive strength data was scaled by ASTM and ACI-equipped uncertainty coefficients and the suggested fixed-point model was compared with Tanaka method and least-squares fuzzy regression. Mean squared error (MSE), coefficient ambiguity, convergence behavior, and toughness at a noise of ±5%.

were used to evaluate performance. Findings The offered approach showed a steady geometric convergence with the average number of 12.3 iterations and zero divergence in all experiments. It also achieved by a factor of 12.5 on the average and a maximum level of 25.1 on best decreased the overall mean squared error as compared to the comparator methods. Coefficient ambiguity; coefficient ambiguity is calculated by the width of ambiguity and was 18.3% lower than Tanaka method and was 13.3% lower than ambiguity squares. In noise perturbation, the model does not show a major growth in the mean error (+6.2) as Tanaka approach (+24.7) and the LS-based approach (+18.3) thus, robustness is substantially enhanced.

The use of fuzzy regression as a part of a fixed-point theoretical model can address the stability issues, existence issues, and singularity issues long dogging classical models of fuzzy regression. This solution constitutes a conceptually and computationally stable, mathematically consistent uncertainty-aware regression solver to be used in engineering problems that require imprecise measurement. This is to be expanded to include in future work the extrapolation of the model to nonlinear fuzzy structures and Gaussian/LR representation of fuzzy and wider uses in data-driven prediction under uncertain conditions.