Genetic Algorithm with Calibration Variables for Pareto Front Approximation in Prediction Intervals

Abstract

In regression tasks, point estimates are insufficient—interval uncertainty must be quantified. The two main criteria for evaluating prediction intervals—Prediction Interval Coverage Probability (PICP) and Normalized Average Width (PINAW)—are conflicting, forming a Pareto front. This paper presents GA-PC, a novel genetic algorithm that modifies NSGA-II by introducing auxiliary calibration variables with a quadratic penalty. Unlike heuristic approaches, the introduction of these variables is theoretically justified via Noether’s second theorem and Bianchi identities: they correspond to gauge degrees of freedom, and the penalty acts as gauge fixing, improving convergence without altering the Pareto set. On ZDT2, GA-PC yields results that match the best reported values. On real financial data, the method provides full PICP coverage (0.000–1.000) and a wider PINAW range than NSGA-II and MOEA/D. The algorithm is scalable, exhibits transferability of hyperparameters, and is applicable to any number of criteria. Code available in the Supplementary File.