A comprehensive framework for fitting coefficients uncertainty: The RUM methodology

Accurate uncertainty quantification (UQ) is critical for reliable regression analysis. Traditional methods, such as the covariance-based approach, often combine global fitting uncertainty with sample-specific variability and typically assume linearity, which limits their applicability in nonlinear contexts. Furthermore, techniques capable of addressing nonlinear models are often computationally demanding and tend to overestimate uncertainties. To address these limitations, this work presents a comprehensive review of existing UQ strategies and introduces the Regression Uncertainty Model (RUM), a generalized, computationally efficient, and interpretable framework for both linear and nonlinear regressions. RUM captures both small and large uncertainties and provides upper and lower bounds for each coefficient uncertainty as well as for the norm of the global uncertainty vector. The model simplifies the uncertainty representation using a single directional vector, whose optimal selection is explored to improve accuracy. Unlike conventional methods, RUM ensures that the root mean square error of the uncertainty curves matches the residual variance of the original fit, preventing systematic overestimation. We validate the methodology on synthetic and experimental datasets and benchmark it against standard techniques, including the covariance-based method, Monte Carlo simulations, and Bootstrapping. Evaluation using a merit function shows that the RUM is able to outperform traditional approaches, offering more accurate and robust uncertainty estimates across a wide range of different regression scenarios.