# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 5 Test Examples
### Example 1: A Similar Toy Model  
We first study a toy model that is very similar to the one in Equation (10), i.e.,  
\[X=\Theta_{1}-\Theta_{2}, \tag{34}\]  
where \(\Theta_{1}\) and \(\Theta_{2}\) are i.i.d. standard normal random variables. The unconditional \(X\) follows \(\mathcal{N}(0,\sqrt{2})\) in the same way, and conditional \(X|\Theta_{1}\) and \(X|\Theta_{2}\) follow \(\mathcal{N}(\theta_{1},1)\) and \(\mathcal{N}(-\theta_{2},1)\), respectively. Thus, theoretical solutions of the four GSIs can be calculated by definition and are listed in Table 1, where EQ1 in Equation (29), UB2 in Equation (31), EQ3 in Equation (32) and UB3 in Equation (33) are analytically verified. The analytical expressions of Fre-GSIs (see details in [10]) are shown in Figure 5b.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \(\Theta_{i}\) & \(\xi_{i,j}\) & \(\mathcal{S}_{i}^{\text{G}}\) & \(\partial\mathcal{S}_{i}^{\text{S}}/\partial\xi_{i,j}\) & EQ1 & \(\mathcal{S}_{i}^{\text{\emph{ref}}}\) & \(\partial\mathcal{S}_{i}^{\text{\emph{ref}}}/\partial\xi_{i,j}\) & UB2 & \(\mathcal{S}_{i,j}^{\text{F}}\) & EQ3 & UB3 & \(\mathcal{S}_{i,j}^{\mathcal{F}}\) \\ \hline \multirow{4}{*}{\(\Theta_{1}\)} & \(\mu_{1}\) & & 0.000 & 0.000 & & 0.000 & 0.681 & 0.220 \({}^{\ddagger}\) & 0.220 \({}^{\ddagger}\) & 0.282 & 0.707 \\  & \(\sigma_{1}\) & 0.500 & 0.500 & 0.306 & & & 0.005 \({}^{\ddagger}\) & 0.005 \({}^{\ddagger}\) & & & \\  & \(\sigma_{1}\) & & 0.500 & 0.500 & 0.261 & 0.726 & 0.110 \({}^{\ddagger}\) & 0.110 \({}^{\ddagger}\) & 0.242 & 0.500 \\  & \(\mu_{2}\) & & 0.000 & 0.000 & & 0.000 & 0.681 & \(-\)0.220 \({}^{\ddagger}\) & \(-\)0.220 \({}^{\ddagger}\) & 0.282 & 0.707 \\ \(\Theta_{2}\) & \(\sigma_{2}\) & 0.500 & 0.500 & 0.306 & & & \(-\)0.005 \({}^{\ddagger}\) & \(-\)0.005 \({}^{\ddagger}\) & & & \\  & & & & & & 0.261 & 0.726 & 0.110 \({}^{\ddagger}\) & 0.110 \({}^{\ddagger}\) & 0.242 & 0.500 \\  & & & & & & & 0.010 \({}^{\ddagger}\) & 0.010 \({}^{\ddagger}\) & & & & \\ \hline \end{tabular} Note: The superscripts \({}^{\ddagger}\) and \({}^{\ddagger}\) stand for the failure domains defined as \(x>1\) and \(x>4\), respectively.  
\end{table}
Table 1: The theoretical solutions of the four GSIs (Example 1).  
Now we discuss how the four GSIs work for the toy models in Equations (10) and (34). It is seen that the variance-based and the moment-independent GSIs, and even their derivatives with respect to the distribution parameters, fail to distinguish the difference between \(\Theta_{1}\) and \(\Theta_{2}\) for the two toy models. In other words, if there is no information about the model (e.g., as a \"black-box\"), the decision-maker may think that \(\Theta_{1}\) and \(\Theta_{2}\) in Equation (34) are identical in terms of the importance measure as well as the sensitivity to direction, which is somewhat misleading. In this regard, the failure-probability-based GSI can discriminate between the two toy models based on the difference in signs of \(\mathcal{S}_{i,j}^{\text{F}}\). Nonetheless, this sign is not efficient in helping us understand the global properties of the two toy models--it only provides the property of the system on the failure domain of \(X\), i.e., a very narrow domain compared to the whole region of \(X\). This can be improved via the Fre-GSI in Figure 5, where it is intuitive that for the toy model in Equation (10) the Fre-GSIs are identical with respect to \(\mu_{1}\) and \(\mu_{2}\) (Figure 4(a)), but are origin-symmetrical (Figure 4(b)) for the toy model in Equation (34).