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：本研究旨在解决复杂模拟器输入/输出的目标导向型灵敏度分析问题。具体而言，我们关注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。本文提出了一种处理集值输出的灵敏度分析方法。新方法受游走集合灵敏度分析驱动，但同样适用于生存性领域的集值模拟器，以及处理污染物浓度图或洪水淹没区图等地图数据。我们提出基于希尔伯特-施密特独立性准则的方法，采用专为集合输出定制的核函数。首要贡献是证明该核函数具有特征性（即嵌入到相关再生核希尔伯特空间中的单射性），这是HSIC在灵敏度分析背景下解释的必备性质。随后我们提出计算HSIC-ANOVA指标，实现输入贡献的分解。利用这些指标，我们既可以识别应忽略的输入变量（筛选），又可以按影响程度对剩余变量排序（排序）。这些指标的估计方法也针对集值输出进行了适应性改进。最后在两个游走集合测试案例上验证了所提方法。