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Journal Articles Electronic Journal of Statistics Year : 2014

## Sensitivity analysis for multidimensional and functional outputs

Fabrice Gamboa
• Function : Author
• PersonId : 935425
Alexandre Janon
• Function : Author
• PersonId : 947965
Thierry Klein
Agnès Lagnoux

#### 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 $Y\in\R$ ), 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.

### Dates and versions

hal-00881112 , version 1 (07-11-2013)

### Identifiers

• HAL Id : hal-00881112 , version 1
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### Cite

Fabrice Gamboa, Alexandre Janon, Thierry Klein, Agnès Lagnoux. Sensitivity analysis for multidimensional and functional outputs. Electronic Journal of Statistics , 2014, 8 (1), pp.575-603. ⟨hal-00881112⟩

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