Sensitivity analysis for multidimensional and functional outputs

Let X:=(X1, ldots, Xp) be random objects (the inputs), defined on some probability space (Omega,F, mathbb P) and valued in some measurable space E=E1×ldots × Ep. Further, let Y:=Y = f(X1, ldots, Xp) be the output. Here, f is a measurable function from E to some Hilbert space mathbbH (mathbbH 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 Y∈mathbb R ), when the output belongs to mathbbH. 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.