# A Review on Local Failure Probability Sensitivity Analysis
## 2 Local RSA with Respect to Deterministic Inputs
### Expression of the Local RSA with Respect to Distribution Parameters  
We describe here the framework of the local RSA with respect to \(\boldsymbol{\theta}\), the distribution parameters of the original inputs \(\mathbf{Z}\). As mentioned before, the derivatives of \(P_{f}\) with regard to \(\boldsymbol{\theta}\) give valuable information about the influence of the probabilistic model selected for the random inputs on the system's failure occurrence. This sensitivity also allows us to identify which random inputs should be focused [11; 12]. In order to compute the derivatives of \(P_{f}(\boldsymbol{\theta})\), two options are possible. The first option is to evaluate both the failure probability \(P_{f}\) and its derivatives in the original input space. The dependence of \(P_{f}\) on \(\boldsymbol{\theta}\) is then uniquely contained in the joint distribution \(f_{\mathbf{Z}}\) of the original inputs \(\mathbf{Z}\). It is assumed here that \(f_{\mathbf{Z}}\) is continuously differentiable with respect to \(\boldsymbol{\theta}\). Furthermore, we suppose that the integration domain of \(\mathbf{Z}\) does not depend on \(\boldsymbol{\theta}\). The failure probability is thus written  
\[P_{f}(\boldsymbol{\theta})=\int_{D_{\mathbf{Z}}}f_{\mathbf{Z}}(\mathbf{z}; \boldsymbol{\theta})\,\mathrm{d}\mathbf{z}=\int_{\mathbb{R}^{d}}\mathbb{I}_{D_ {\mathbf{Z}}}(\mathbf{z})f_{\mathbf{Z}}(\mathbf{z};\boldsymbol{\theta})\, \mathrm{d}\mathbf{z}=\mathbb{E}_{f_{\mathbf{Z}}}\left(\mathbb{I}_{D_{\mathbf{ Z}}}(\mathbf{Z})\right), \tag{1}\]where \(D_{f_{\mathbf{Z}}}=\left\{\mathbf{z}\in\mathbb{R}^{d}\mid\delta_{\mathbf{Z}}( \mathbf{z})\leq 0\right\}\) is the failure domain in the original space. The derivatives of \(P_{f}\) with respect to \(\theta_{\ell}\) for \(\ell\in[1,\ldots,m]\) are straightforward  
\[\frac{\partial P_{f}(\boldsymbol{\theta})}{\partial\theta_{\ell}}=\int_{D_{f_{ \mathbf{Z}}}}\frac{\partial f_{\mathbf{Z}}(\mathbf{z};\boldsymbol{\theta})}{ \partial\theta_{\ell}}\,\mathrm{d}\mathbf{z}=\int_{\mathbb{R}^{d}}\mathbb{I}_ {D_{f_{\mathbf{Z}}}}(\mathbf{z})\frac{\partial f_{\mathbf{Z}}(\mathbf{z}; \boldsymbol{\theta})}{\partial\theta_{\ell}}\,\mathrm{d}\mathbf{z}. \tag{2}\]  
The resulting expression of the derivative of \(P_{f}\) in Equation (2) is thus a domain integral, which depends on the derivatives of \(f_{\mathbf{Z}}\) with respect to \(\theta_{\ell}\), and is defined on \(D_{f_{\mathbf{Z}}}\). The second option is to evaluate both the failure probability \(P_{f}\) and its derivatives in the standard space. The dependence of \(P_{f}\) on \(\boldsymbol{\theta}\) is then contained in the transformed limit state function \(g\) through the isoprobabilistic transformation while the standardized joint distribution \(f_{\mathbf{X}}\) is parameter-free. Therefore, the failure probability is expressed as  
\[P_{f}(\boldsymbol{\theta})=\int_{D_{f}(\boldsymbol{\theta})}f_{\mathbf{X}}( \mathbf{x})\,\mathrm{d}\mathbf{x}=\int_{\mathbb{R}^{d}}\mathbb{I}_{D_{f}( \boldsymbol{\theta})}(\mathbf{x})f_{\mathbf{X}}(\mathbf{x})\,\mathrm{d} \mathbf{x}, \tag{3}\]  
where \(D_{f}(\boldsymbol{\theta})=\left\{\mathbf{x}\in\mathbb{R}^{d}\mid g( \boldsymbol{\theta},\mathbf{x})\leq 0\right\}\) is the failure domain in the standard normal space.