This work is motivated by goal-oriented sensitivity analysis of inputs/outputs of complex simulators. More precisely we are interested in the ranking of the uncertain input variables that impact the most a feasible design domain. Most sensitivity analysis methods deal with scalar outputs. In this paper, we propose a way to perform sensitivity analysis when dealing with set-valued outputs. Our new methodology is driven by sensitivity analysis on excursion sets but can also be applied to setvalued simulators as in viability field, or when dealing with maps such as pollutant concentration maps or flooding zone maps. We propose a method based on the Hilbert Schmidt Independence Criterion (HSIC) with a kernel tailored to sets as outputs. A first contribution is the proof that this kernel is characteristic (i.e injectivity of the embedding in the associated Reproducing Kernel Hilbert Space), a required property for the HSIC interpretation in a sensitivity analysis context. We propose then to compute the HSIC-ANOVA indices which allow a decomposition of the input contributions. Using these indices, we can identify which inputs should be neglected (screening) and we can rank the others by influence (ranking). The estimation of these indices is also adapted to the set-valued outputs. Finally we test the proposed method on two test cases of excursion sets.

翻译：本文源于对复杂模拟器输入/输出进行目标导向灵敏度分析的动机。更准确地说，我们关注的是对可行设计域影响最大的不确定输入变量的排序。大多数灵敏度分析方法处理的是标量输出。本文针对集合值输出的场景，提出了一种灵敏度分析方法。新方法最初针对超出集灵敏度分析设计，但同样适用于生存力场等集合值模拟器，以及污染物浓度图、洪水区域图等地图数据。我们提出基于希尔伯特-施密特独立性准则（HSIC）的方法，并特别针对集合输出设计了专用核函数。首个贡献是证明了该核函数具有特征性（即嵌入到再生核希尔伯特空间中的单射性），这是HSIC在灵敏度分析中成立的必要性质。随后提出计算HSIC-ANOVA指标，该指标可实现输入贡献的分解。通过该指标，既能识别可忽略的输入（筛选），又能对其他输入按影响程度排序（排序）。指标的估计方法同样适应集合值输出。最后，我们在两个超出集测试案例上验证了所提方法。