Let $X:=(X_1, ldots, X_p)$ be random objects (the inputs), defined on some probability space $(Omega,{mathcal{F}}, mathbb P)$ and valued in some measurable space $E=E_1timesldots times E_p$. Further, let $Y:=Y = f(X_1, ldots, X_p)$ be the output. Here, $f$ is a measurable function from $E$ to some Hilbert space $mathbb{H}$ ($mathbb{H}$ could be either of finite or infinite dimension). In this work, we give a natural generalization of the Sobol indices (that are classically defined when $Yinmathbb R$ ), when the output belongs to $mathbb{H}$. These indices have very nice properties. First, they are invariant. under isometry and scaling. Further they can be, as in dimension $1$, easily estimated by using the so-called Pick and Freeze method. We investigate the asymptotic behaviour of such estimation scheme.
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