In this paper, 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 paper introduces a novel approach for performing 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 proposing 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.

翻译：本文旨在对集合值模型进行敏感性分析，特别关注不确定输入对可行集的影响，这是求解约束条件下鲁棒优化问题的关键要素。尽管多数敏感性分析方法处理标量输出，本文提出了一种针对集合值输出的新型敏感性分析方法。我们创新的方法论专为超限集合设计，但具有足够通用性，可应用于集合值模拟器（包括生存域领域的模拟器），或处理污染物浓度图、洪水区域图等地图数据。我们提出使用希尔伯特-施密特独立性准则（HSIC），并设计了适用于集合值输出的核函数。在建立随机集的概率框架后，首个贡献是证明该核具有特征性——这是基于核的敏感性分析中不可或缺的性质。为衡量每个输入的贡献，我们进一步提出使用HSIC-ANOVA指标。通过这些指标可识别应忽略的输入（筛选），并根据影响程度对其他输入排序（排序）。该指标的估计方法也针对集合值输出进行了适配。最终，我们在三个超限集合测试案例中验证了所提方法的有效性。