Kernel-based Sensitivity Analysis for (Excursion) Sets

Abstract

In this article, we aim to perform sensitivity analysis of set-valued models and, in particular, to quantify the impact of uncertain inputs on feasible sets, which are key elements in solving a robust optimization problem under constraints. While most sensitivity analysis methods deal with scalar outputs, this article introduces a novel approach to perform sensitivity analysis with set-valued outputs. Our innovative methodology is designed for excursion sets, but is versatile enough to be applied to set-valued simulators, including those found in viability fields, or when working with maps like pollutant concentration maps or flood zone maps. We propose to use the Hilbert-Schmidt Independence Criterion (HSIC) with a kernel designed for set-valued outputs. After setting a probabilistic framework for random sets, a first contribution is the proof that this kernel is characteristic, an essential property in a kernel-based sensitivity analysis context. To measure the contribution of each input, we then propose to use HSIC-ANOVA indices. With these indices, we can identify which inputs should be neglected (screening) and we can rank the others according to their influence (ranking). The estimation of these indices is also adapted to the set-valued outputs. Finally, we test the proposed method on three test cases of excursion sets.

KEYWORDS:

• HSIC-ANOVA indices
• Random sets
• Screening
• Ranking

Supplementary Materials

The supplementary materials contain details of the proofs of Lemma 3.1 and of Proposition 3.3–Proposition 3.6, additional numerical results, and codes to reproduce Figure 3.

Acknowledgments

The authors thank Gabriel Sarazin for his numerous fruitful discussions and comments. We are also grateful to the reviewers and the associate editor for their relevant and helpful comments and to Adan Reyes Reyes for providing the model of Section 4.3. This research was conducted with the support of the consortium in Applied Mathematics CIROQUO, gathering partners in technological research and academia in the development of advanced methods for Computer Experiments.

Disclosure Statement

The authors report there are no competing interests to declare.