# A Review on Local Failure Probability Sensitivity Analysis
## 4 Local RSA in the Standard Normal Space
### Sensitivity through an Approximation of the Failure Domain Indicator Function
#### 4.3.1 Weak Approach: Approximation of the Indicator Function  
The failure domain indicator function makes the direct differentiation of the failure probability integral of Equation (3) not possible. By replacing the failure domain indicator function with a smoother function, the differentiation can be performed and results in a domain integral [42]. Several smooth approximations of the indicator function have been derived in the literature [51], which are typically cdfs of continuous univariate variables [50]. Here we derive the approximation chosen in [42,52], which comes from the following limit  
\[\forall\mathbf{x}\in\mathbb{R}^{d}\ \ \ \ \ \mathbb{I}_{D_{f}}(\mathbf{x})= \lim_{\sigma\to 0}\Psi\left(-\frac{g(\mathbf{s},\mathbf{x})}{\sigma}\right),\]  
where \(\Psi\) is the standard normal univariate cdf, and \(\sigma>0\). An illustration of this approximation of the indicator function is displayed in Figure 1. It should be noted, however, that the uniform distribution defined on \([-\sigma/2,\sigma/2]\), with \(\sigma>0\), is another univariate cdf often considered for the approximation of the indicator function [20,50].  
In the sense of distributions, the derivative of \(P_{f}\) with respect to \(s_{\ell}\) can then be defined as [50]  
\[\frac{\partial P_{f}(\mathbf{s})}{\partial s_{\ell}} =\lim_{\sigma\to 0}\int_{\mathbb{R}^{d}}\frac{\partial\Psi(-g( \mathbf{s},\mathbf{x})/\sigma)}{\partial s_{\ell}}f_{\mathbf{X}}(\mathbf{x}) \mathrm{d}\mathbf{x}\] \[=-\lim_{\sigma\to 0}\int_{\mathbb{R}^{d}}\frac{1}{\sigma} \frac{\partial g(\mathbf{s},\mathbf{x})}{\partial s_{\ell}}\phi\left(-\frac{ g(\mathbf{s},\mathbf{x})}{\sigma}\right)f_{\mathbf{X}}(\mathbf{x}) \mathrm{d}\mathbf{x},\]  
where \(\phi\) is the univariate standard normal pdf. The convergence is proven in [42] under some regularity conditions. Consequently, an approximation of the failure probability sensitivity can be obtained with the following domain integral\[\frac{\partial P_{f}(\mathbf{s})}{\partial s_{\ell}}\approx-\int_{ \mathbb{R}^{d}}\frac{1}{\widetilde{\varphi}}\frac{\partial g(\mathbf{s},\mathbf{ x})}{\partial s_{\ell}}\phi\bigg{(}-\frac{g(\mathbf{s},\mathbf{x})}{\widetilde{ \sigma}}\bigg{)}f_{\mathbf{X}}(\mathbf{x})\mathrm{d}\mathbf{x}=\frac{\partial \widetilde{P_{f}}(\mathbf{s},\widetilde{\sigma})}{\partial s_{\ell}}\] \[\text{where}\quad\quad\widetilde{P_{f}}(\mathbf{s},\widetilde{ \sigma})=\int_{\mathbb{R}^{d}}\Psi\bigg{(}-\frac{g(\mathbf{s},\mathbf{x})}{ \widetilde{\sigma}}\bigg{)}f_{\mathbf{X}}(\mathbf{x})\mathrm{d}\mathbf{x},\]  
for \(\widetilde{\sigma}\) a fixed positive value. One should notice that the following convergence also holds: \(\widetilde{P_{f}}(\mathbf{s},\sigma)\) converges to \(P_{f}(\mathbf{s})\) as \(\sigma\to 0\). The domain integral \(\partial\widetilde{P_{f}}(\mathbf{s},\widetilde{\sigma})/\partial s_{\ell}\) can be estimated with the Monte Carlo method with the following estimate  
\[\frac{\partial\widetilde{P_{f}}(\mathbf{s},\widetilde{\sigma})}{\partial s_{ \ell}}^{\text{MC}}=-\frac{1}{N}\sum_{j=1}^{N}\frac{1}{\widetilde{\varphi}} \frac{\partial g\left(\mathbf{s},\mathbf{X}^{(j)}\right)}{\partial s_{\ell}} \phi\left(-\frac{g\left(\mathbf{s},\mathbf{X}^{(j)}\right)}{\widetilde{\sigma }}\right), \tag{19}\]  
where the \(\mathbf{X}^{(j)}\) are iid from \(f_{\mathbf{X}}\). The same sample can be reused to compute both the failure probability estimate with MC and this sensitivity estimate. However, for each observation \(\mathbf{X}^{(j)}\), the derivative of the lsf with respect to \(s_{\ell}\) has to be evaluated. Therefore, the simulation budget increases. This estimate is biased, since \(\widetilde{\sigma}\
eq 0\) and its statistical properties are detailed in the next section.