Independence test based on total-order HSIC-ANOVA indices ; Test d'indépendance basé sur les indices HSIC-ANOVA d'ordre total

National audience ; Building a surrogate model for an industrial computationally-expensive simulation code is made difficult by the combined effect of the curse of dimensionality andthe lack of input-output data. A preliminary sensitivity analysis may help discard non-influential inputs and rank the remaining inputs according to their impact on the output distribution. In order to perform sensitivity analysis, the historical approach proposed by Sobol' provides a convenient conceptual framework where the output variance is apportioned between input variables. However, the accurate estimation of the corresponding indices requires thousands of model evaluations, which is often unaffordable in an industrial context. To circumvent this pitfall, it has become quite common to resort to a sensitivity measure based on Hilbert-Schmidt independence criterion (denoted by HSIC). This measure is applied to all input-output pairs of variables and allows to define the so-called "HSIC indices". Their interpretation is much less intuitive than the one related to Sobol' indices since their foundations come from the theory of reproducing kernel Hilbert spaces. To ease interpretation, the HSIC-ANOVA indices have been recently introduced to allow for a strict separation of main effects and interactions, akin to what is proposed in Sobol' formalism with Hoeffding decomposition. This breakthrough was obtained after assuming mutual independence between inputs and provided that specific kernels, like Sobolev kernels, are used to compute HSIC-ANOVA indices. In this work, a first contribution consists in demonstrating that Sobolev kernels are characteristic. Because of this property, independence within input-output pairs of variables can be detected from the observed values of HSIC-ANOVA indices. Then, it is shown that a test of independence can be constructed for the total-order HSIC-ANOVA index after adapting existing methodologies in the HSIC-related literature. Finally, an extensive simulation study proves empirically that the ...