Kernel-based sensitivity analysis for (excursion) sets

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.