Improving Discrepancy Measures for Global Sensitivity Analysis

Abstract

Sensitivity analysis methods based on Sobol'total-order indices ($T_i$) are well-founded but computationally demanding. A recently proposed ersatz discrepancy measure offers a cheaper alternative by quantifying deviations from uniformity in input--output scatterplots, yet lacks theoretical grounding and has not been benchmarked against other data-given estimators. We introduce an adjusted ersatz discrepancy that rank-transforms the output before gridding and imputes isolated empty cells via a Moore-neighbourhood rule, substantially improving agreement with $T_i$. We prove, via a copula-theoretic argument, that the adjustment is a consistent screening statistic with a zero condition, an explicit full-support ceiling bounding its use as a magnitude estimator, and a documented failure mode for purely interaction-mediated dependencies. We benchmark the adjusted ersatz against three zero-extra-cost comparators -- polynomial chaos expansion (PCE), PCE-derived Shapley effects, and a PAWN-type maximum Kolmogorov--Smirnov index -- across seven benchmark functions and a real-world hydrological model. The adjusted ersatz is the only estimator achieving perfect rank agreement on a non-smooth hydrological output where PCE is misspecified. A joint sensitivity analysis of five algorithmic parameters shows grid resolution, not the imputation threshold or sampling method, drives performance variability.