# A Review on Local Failure Probability Sensitivity Analysis
## 4 Local RSA in the Standard Normal Space
### Sensitivity with Direct Simulation Method
#### 4.2.1 Sensitivity Estimation with Directional Sampling  
Directional Sampling (DS) is a sampling technique first introduced in [38] and derived for structural reliability problems in [39; 40]. We focus here on the case where the limit state function is star-shaped. It is recalled here that the random normal inputs \(\mathbf{X}\) can be written \(\mathbf{X}=R\mathbf{T}\)[41] where \(R\) is a positive random variable and \(\mathbf{T}\) is a \(d\)-variate random vector which is independent of \(R\) and uniformly distributed on the unit sphere \(\mathcal{T}^{d}:\{(t_{1},\ldots,t_{d})\in\mathbb{R}^{d}:t_{1}^{2}+\ldots+t_{d}^ {2}=1\}\). The probability of failure, with the dependency on \(\mathbf{s}\) included, is written  
\[P_{f}(\mathbf{s})=\mathbb{P}(g(\mathbf{s},R\mathbf{T})\leq 0)=\int_{\mathcal{T}^{d} }\mathbb{P}(g(\mathbf{s},R\mathbf{t})\leq 0)f_{\mathbf{T}}(\mathbf{t}) \mathrm{d}t=\int_{\mathcal{T}^{d}}(1-F_{R}(r(\mathbf{s},\mathbf{t}))f_{ \mathbf{T}}(\mathbf{t})\mathrm{d}t,\]  
where \(\mathbf{t}\mapsto r(\mathbf{s},\mathbf{t})\) is the function which assigns to every direction \(\mathbf{t}\) its root (where the dependency of \(\mathbf{s}\) is underlined), \(f_{\mathbf{T}}\) is the density of a random vector \(\mathbf{T}\) and \(F_{R}\) is the cdf of the radial component \(R\) of the standard normal inputs \(\mathbf{X}\). In the DS framework, the root \(r(\mathbf{s},\mathbf{t})\) is the distance between the limit state surface \(\{g(\mathbf{s},\mathbf{x})=0\}\) and the origin, in the direction \(\mathbf{t}\). The sensitivity of \(P_{f}\) can be simply derived with the equation  
\[\frac{\partial P_{f}(\mathbf{s})}{\partial s_{\ell}}=\int_{\mathcal{T}^{d}} \biggl{(}-\frac{\partial r(\mathbf{s},\mathbf{t})}{\partial s_{\ell}}f_{R}(r( \mathbf{s},\mathbf{t}))\biggr{)}f_{\mathbf{T}}(\mathbf{t})\mathrm{d}t,\]  
where \(f_{R}\) is the pdf of \(R\). The derivatives of \(r\) for \(\ell\in[1,\ldots,p]\) are equal to [42]  
\[\forall\mathbf{t}\in\mathcal{T}^{d}\quad\frac{\partial r(\mathbf{s},\mathbf{t })}{\partial s_{\ell}}=-\frac{1}{\
abla_{\mathbf{x}}g(\mathbf{s},r(\mathbf{s}, \mathbf{t})\mathbf{t})^{\top}}\frac{\partial g(\mathbf{s},r(\mathbf{s}, \mathbf{t})\mathbf{t})}{\partial s_{\ell}}.\]Therefore, the failure probability sensitivity can be estimated with  
\[\frac{\widehat{\partial P_{f}(\mathbf{s})}^{\text{DS}}}{\partial s_{\ell}}=\frac{ 1}{N}\sum_{j=1}^{N}\frac{f_{R}\Big{(}r\Big{(}\mathbf{s},\mathbf{T}^{(j)}\Big{)} \Big{)}}{\
abla_{\mathbf{x}}g\Big{(}\mathbf{s},r\Big{(}\mathbf{s},\mathbf{T}^{ (j)}\Big{)}\mathbf{T}^{(j)}\Big{)}^{\top}\mathbf{T}^{(j)}}\frac{\partial g\Big{(} \mathbf{s},r\Big{(}\mathbf{s},\mathbf{T}^{(j)}\Big{)}\mathbf{T}^{(j)}\Big{)}}{ \partial s_{\ell}}, \tag{15}\]  
where the observations \(\mathbf{T}^{(j)}\) are iid from \(f_{\mathbf{T}}\). Introducing the IS framework, the sensitivity estimate in the Directional IS (DIS) context is written  
\[\frac{\widehat{\partial P_{f}(\mathbf{s})}^{\text{DIS}}}{\partial s_{\ell}}= \frac{1}{N}\sum_{j=1}^{N}\frac{f_{R}\Big{(}r\Big{(}\mathbf{s},\mathbf{T}^{(j)} \Big{)}\Big{)}}{\
abla_{\mathbf{x}}g\Big{(}\mathbf{s},r\Big{(}\mathbf{s}, \mathbf{T}^{(j)}\Big{)}\mathbf{T}^{(j)}\Big{)}^{\top}\mathbf{T}^{(j)}}\frac{ \partial g\Big{(}\mathbf{s},r\Big{(}\mathbf{s},\mathbf{T}^{(j)}\Big{)} \mathbf{T}^{(j)}\Big{)}}{\partial s_{\ell}}\frac{f_{\mathbf{T}}\Big{(} \mathbf{T}^{(j)}\Big{)}}{h_{\mathbf{T}}\Big{(}\mathbf{T}^{(j)}\Big{)}}, \tag{16}\]  
where the observations \(\mathbf{T}^{(j)}\) are iid from \(h_{\mathbf{T}}\). The estimates Equations (15) and (16) are unbiased. Since all the observations are iid, the variance and mean square error are derived with the classical formulas. The observations are the same as the ones used to compute the failure probability estimate. However, it should be noted that the denominator of the first term of both estimates is the cosine between the sampling direction and the gradient of the lsf evaluated at the intersection between the sampling direction and the limit state surface.