# Drone Ground Impact Footprints with Importance Sampling: Estimation and Sensitivity Analysis +
## 5 Reliability-Oriented Sensitivity Analysis  
### Definition of ROSA Sensitivity Indices  
Reliability-oriented sensitivity analysis (ROSA) differs from classical sensitivity analysis in the nature of the output quantity of interest under study. Indeed, sensitivity analysis focuses on the model output, whereas ROSA analyses the impact of the input uncertainty on a reliability measure. Two kinds of ROSA indices can be computed in practice [21]:  
* first, target sensitivity analysis evaluates the impact of inputs over a function of the output, typically the indicator function of a critical domain. In the drone impact application, it answers the question: which uncertain inputs lead to extreme drone impact?
* second, conditional sensitivity analysis, which aims at studying the impact of inputs exclusively within the critical domain, namely, conditionally to the failure event. In the drone application, this indice determines, conditionally to an extreme impact, which uncertain inputs are the most influential.  
In this article, we consider two recent target and conditional ROSA moment-independent indices \(\bar{\eta}_{i}\) and \(\delta^{f}_{i}\)[22] to analyse the influence of the \(i^{th}\) component of \(U\), \(U^{(i)}\), on the scalar output quantity \(\hat{Z}=||Z||_{2}\) for a given failure event. We propose to define here the failure event as \(Z\
ot\in\mathbf{L}(t)\), that is, the ground impact is outside a given volume set and is thus an extreme impact. The two ROSA indices are defined by the following equations for the proposed drone fallout test case with \(i=1,\dots,5\) as there are 5 random inputs:  
\[\bar{\eta}_{i}=\frac{1}{2}\Big{\|}f_{\mathbf{U}^{(i)}}-f_{\mathbf{U}^{(i)}|Z \
ot\in\mathbf{L}(t)}\Big{\|}_{L^{1}(\mathbb{R})}. \tag{12}\]  
Figure 5: (**a**) 1000 Samples generated with the density \(f\) of MIS (**b**) 1000 Samples generated with the density \(h\) of MIS (**c**) Minimum density volume set estimation for different probability values \(\alpha\) (0.5; 0.8; 0.9; 0.95; 0.99; 0.999) with MIS (wind of 5 m.s \({}^{-1}\) with angle \(-90^{\circ}\)).  
and  
\[\delta_{i}^{f}=\frac{1}{2}\Big{\|}f_{(U^{(i)},\hat{Z})|Z\
ot\in\mathbf{L}(t)}-f_{U^ {(i)}|Z\
ot\in\mathbf{L}(t)}f_{\hat{Z}|Z\
ot\in\mathbf{L}(t)}\Big{\|}_{L^{1}( \mathbb{R}^{2})}=\frac{1}{2}\|c_{i}-1\|_{L^{1}(\mathbb{R}^{2})}. \tag{13}\]  
where \(f_{U^{(i)}}\) is the density of \(U^{(i)},f_{U^{(i)}|Z\
ot\in\mathbf{L}(t)}\) is the density of \(U^{(i)}\) conditionally to \(Z\
ot\in\mathbf{L}(t)\), \(f_{(U^{(i)},\hat{Z})|Z\
ot\in\mathbf{L}(t)}\) is the density of the couple \((U^{(i)},\hat{Z})\) conditionally to \(Z\
ot\in\mathbf{L}(t)\) and finally \(c_{i}\) the copula density \((U^{(i)},\hat{Z})\) conditionally to \(Z\
ot\in\mathbf{L}(t)\). The ROSA indices \(\hat{\eta}_{i}\) and \(\delta_{i}^{f}\) take values in \([0,1]\) where the low values of these indices mean this \(i^{th}\) component of \(U\) is not influential on the failure event analysis and conversely. The computation of these indices can be done with the samples generated for density level set estimation (see Section 4) and thus requires no additional calls to \(\mathcal{M}\). Moreover, this methodology can be applied even if the random inputs \(U\) are dependent contrary to ROSA variance based indices [23].