[Paper Review] Derivative-based global sensitivity measures: general links with Sobol' indices and numerical tests

Matieyendou Lamboni, Bertrand Iooss+2 more|arXiv (Cornell University)|Feb 5, 2012

Probabilistic and Robust Engineering Design26 references4 citations

TL;DR

This paper establishes a general inequality linking derivative-based global sensitivity measures (DGSM) to total Sobol’ indices for Boltzmann and log-concave probability measures, demonstrating that DGSM provides a computationally efficient upper bound for total Sobol’ indices. The proposed bound, denoted Υⱼ, enables effective screening of influential inputs with significantly fewer model evaluations, especially useful for high-dimensional models where variance-based methods are too costly.

ABSTRACT

The estimation of variance-based importance measures (called Sobol' indices) of the input variables of a numerical model can require a large number of model evaluations. It turns to be unacceptable for high-dimensional model involving a large number of input variables (typically more than ten). Recently, Sobol and Kucherenko have proposed the Derivative-based Global Sensitivity Measures (DGSM), defined as the integral of the squared derivatives of the model output, showing that it can help to solve the problem of dimensionality in some cases. We provide a general inequality link between DGSM and total Sobol' indices for input variables belonging to the class of Boltzmann probability measures, thus extending the previous results of Sobol and Kucherenko for uniform and normal measures. The special case of log-concave measures is also described. This link provides a DGSM-based maximal bound for the total Sobol indices. Numerical tests show the performance of the bound and its usefulness in practice.

Motivation & Objective

• To address the computational burden of variance-based sensitivity analysis in high-dimensional models with many input variables.
• To extend the theoretical link between derivative-based sensitivity measures (DGSM) and Sobol’ indices beyond uniform and Gaussian distributions to a broader class of probability measures.
• To develop a computationally efficient screening method using DGSM that provides a reliable upper bound for total Sobol’ indices.
• To validate the practical utility of the bound Υⱼ in identifying influential inputs when full variance-based analysis is infeasible due to high computational cost.

Proposed method

• Derive a general inequality linking DGSM and total Sobol’ indices under the assumption of Boltzmann probability measures, using Poincaré-type inequalities.
• Extend the theoretical link to the class of log-concave measures, which includes uniform, normal, and Laplace distributions.
• Define a new sensitivity index Υⱼ as a constant multiple of the crude DGSM, serving as a maximal upper bound for the total Sobol’ index.
• Use numerical simulations on two test models (a quadratic function and a flood model) to compare DGSM-based indices with classical Sobol’ indices.
• Apply renormalization and weighting techniques to improve the reliability of derivative-based indices when input variables have different units.
• Combine DGSM with first-order Sobol’ indices in a two-stage sensitivity analysis to detect non-influential variables and assess interaction effects.

Experimental results

Research questions

• RQ1 Can a theoretical inequality be established between DGSM and total Sobol’ indices for a broader class of probability distributions beyond uniform and Gaussian?
• RQ2 How does the DGSM-based upper bound Υⱼ compare to actual total Sobol’ indices in terms of accuracy and computational efficiency?
• RQ3 Can Υⱼ reliably identify the most influential input variables in high-dimensional models where full variance-based analysis is computationally prohibitive?
• RQ4 What is the impact of variable units and distributional assumptions on the performance of derivative-based sensitivity indices?
• RQ5 How can DGSM be effectively combined with first-order Sobol’ indices to improve global sensitivity analysis in practice?

Key findings

• A general inequality is established linking DGSM and total Sobol’ indices for Boltzmann probability measures, extending prior results limited to uniform and normal distributions.
• For log-concave measures, the same theoretical link holds, broadening the applicability of DGSM as a screening tool.
• The proposed index Υⱼ, defined as a constant multiple of the crude DGSM, provides a valid and computable upper bound for the total Sobol’ index, enabling efficient screening.
• Numerical tests on a flood model show that Υⱼ correctly identifies the most influential variables (Q, Zᵥ, Hₐ, Kₛ) for both overflow and cost outputs, outperforming unnormalized derivative indices νⱼ and τⱼ.
• The bound Υⱼ successfully prioritizes inputs even when full Sobol’ index estimation is infeasible due to high computational cost, confirming its utility in practice.
• The combination of DGSM and first-order Sobol’ indices enables efficient detection of non-influential inputs and assessment of interaction effects, supporting a two-stage sensitivity analysis framework.