# A Review on Local Failure Probability Sensitivity Analysis
## 4 Local RSA in the Standard Normal Space
### Sensitivity through the Design Point of the System  
#### 4.1.1 Expression in the FORM Context  
In the FORM and SORM context, the limit state function \(g\) is approximated with Taylor series expansions in the neighborhood of the design points in the standard normal space [14]. The design points of the system are the closest points to the origin of each failure region of the failure domain \(D_{f}\). The reliability analysis is then performed using the built approximations of \(g\). We derive here the sensitivities of \(P_{f}\) with the FORM approximation, first in a context where there is a unique design point \(P^{*}\), of norm \(\beta\): \(\beta=\|P^{*}\|\). The failure probability is estimated with the equation  
\[P_{f}^{\text{FORM}}=1-F_{X_{1}}(\beta(\mathbf{s})),\]  
where \(F_{X_{1}}\) is the cumulative distribution function (cdf) of the univariate standard normal variable \(X_{1}\). The quantity \(\beta\) is equal to  
\[\beta(\mathbf{s})=\mathbf{\alpha}^{\top}P^{*}\quad\text{ where }\quad\mathbf{\alpha}=- \frac{\
abla_{\mathbf{x}}g(\mathbf{s},P^{*})}{\|\
abla_{\mathbf{x}}g(\mathbf{s },P^{*})\|}.\]  
Therefore, the value of \(\beta\) is a function of \(\mathbf{s}\). The derivative of \(P_{f}\) according to \(s_{\ell}\) in the FORM context is then simply  
\[\frac{\widehat{\partial P_{f}(\mathbf{s})}}{\partial s_{\ell}}=-\frac{\partial \beta(\mathbf{s})}{\partial s_{\ell}}f_{X_{1}}(\beta(\mathbf{s})),\]  
where \(f_{X_{1}}\) is the pdf of the random variable \(X_{1}\) and \(\partial\beta(\mathbf{s})/\partial s_{\ell}\) is the derivatives of \(\beta\) with respect to \(\mathbf{s}\). After some calculus, see [31], the derivative is equal to  
\[\frac{\partial\beta(\mathbf{s})}{\partial s_{\ell}}=\frac{1}{\|\
abla_{ \mathbf{x}}g(\mathbf{s},P^{*})\|}\frac{\partial g(\mathbf{s},P^{*})}{\partial s _{\ell}}. \tag{13}\]  
Consequently the sensitivity of \(P_{f}\) with respect to \(s_{\ell}\), with \(\ell\in[1,\dots,p]\) is equal to  
\[\frac{\widehat{\partial P_{f}(\mathbf{s})}}{\partial s_{\ell}}^{\text{FORM}}= -\frac{1}{\|\
abla_{\mathbf{x}}g(\mathbf{s},P^{*})\|}\frac{\partial g(\mathbf{ s},P^{*})}{\partial s_{\ell}}f_{X_{1}}(\beta(\mathbf{s})), \tag{14}\]with the FORM approximation context. Therefore, one needs the value of the gradient \(\
abla_{\mathbf{x}\mathbf{g}}\) evaluated in \([\mathbf{s},P^{*}]\), as well as the derivative of \(g\) with respect to \(s_{\ell}\) evaluated in \([\mathbf{s},P^{*}]\). Apart from these two quantities, no additional evaluation of the costly lsf is required to estimate the sensitivities. Depending on the nature of the deterministic inputs \(\mathbf{s}\), this derivative can be furthered detailed [31]. If the vector \(\mathbf{s}\) represents distribution parameters only, \(\mathbf{s}=\boldsymbol{\theta}\), then  
\[\frac{\partial\beta(\boldsymbol{\theta})}{\partial\theta_{\ell}}=\boldsymbol {\alpha}^{\top}\frac{\partial T(\boldsymbol{\theta},\mathbf{z}^{*})}{ \partial\theta_{\ell}},\]  
where \(\mathbf{z}^{*}=T_{\boldsymbol{\theta}}^{-1}(P^{*})\). Therefore, the derivative of \(P_{f}\) is approximated with  
\[\frac{\widehat{\partial P_{f}(\boldsymbol{\theta})}^{\text{ FORM}}}{\partial\theta_{\ell}}=-\boldsymbol{\alpha}^{\top}\frac{\partial T( \boldsymbol{\theta},\mathbf{z}^{*})}{\partial\theta_{\ell}}f_{\mathbf{x}_{1} }(\beta(\boldsymbol{\theta})).\]  
If the vector \(\mathbf{s}\) represents design parameters only, \(\mathbf{s}=\boldsymbol{\delta}\), it follows that \(\partial g(\delta,P^{*})/\partial\delta_{\ell}=\partial g_{\mathbf{Z}}( \boldsymbol{\delta},\mathbf{z}^{*})/\partial\delta_{\ell}\), where \(\mathbf{z}^{*}=T^{-1}(P^{*})\). It should be noted that the study of the bias and the error of the sensitivity estimate can be particularly challenging in a black-box model context. As the failure probability estimation in the FORM framework, the method requires very small simulation budget but the statistical properties of the resulting estimates are arduous to obtain. This remark is also valid when there are multiple design points.