Improving Discrepancy Measures for Global Sensitivity Analysis

2026-07-30

AI Summary

This study addresses the high computational cost of traditional global sensitivity analysis based on Sobol’ total-order indices and the lack of theoretical guarantees and thorough validation in existing low-cost alternatives. The authors propose an improved pseudo-variogram-based metric that incorporates output rank transformation, Moore neighborhood-based interpolation to fill grid gaps, and a novel adjusted statistic satisfying consistency, a zero-condition property, and a well-defined upper bound. For the first time, the method’s failure mode under pure interaction effects is identified. Systematic validation is conducted using copula theory and multiple zero-cost estimators. Experiments on seven benchmark functions and a hydrological model demonstrate that the proposed approach achieves perfect ranking agreement with Sobol’ indices for non-smooth outputs and significantly outperforms existing techniques such as polynomial chaos expansion.

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.

Problem

• global sensitivity analysis
• discrepancy measure
• Sobol' indices
• data-driven estimators
• theoretical grounding

Innovation

• adjusted ersatz discrepancy
• global sensitivity analysis
• Sobol' indices
• copula theory
• Moore-neighbourhood imputation