Sensitivity analysis for multidimensional and functional outputs

Abstract

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 $E=E_{1}\times\ldots\times E_{p}$. 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 $Y\in\mathbb{R}$), 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 an estimation scheme.