# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 5 Test Examples
### Example 3: [PERSON]'s Growing Size Model  
Consider [PERSON]'s growing size model [16] as follows:  
\[X=\sum_{i=1}^{s}\Theta_{i}, \tag{39}\]  
where the \(\Theta_{i}\)'s are i.i.d. normal random variables with mean value \(\mu_{i}\) and standard deviation \(\sigma_{i}\), and \(s\) stands for the size of the model. Obviously, \(X\) follows the normal distribution with mean value \(\mu_{X}=\sum_{i=1}^{s}\mu_{i}\) and variance \(\sigma_{X}^{2}=\sum_{i=1}^{s}\sigma_{i}^{2}\). The derivatives of PDF of \(X\) with respect to \(\mu_{i}\) and \(\sigma_{i}\) for \(i=1,2,\cdots,s\) can be read as follows:  
\[ \begin{cases}\dfrac{\partial p_{X}}{\partial\mu_{i}}=\dfrac{x-\mu_{X}}{ \sigma_{X}^{2}}p_{X}(x;\mu_{X},\sigma_{X}),\\ \dfrac{\partial p_{X}}{\partial\sigma_{i}}=\dfrac{(x-\mu_{X})^{2}-\sigma_{X}^ {2}}{\sigma_{X}^{3}}\dfrac{\sigma_{i}}{\sigma_{X}}p_{X}(x;\mu_{X},\sigma_{X}). \end{cases}  \tag{40}\]  
Note that for the normal distribution with mean value \(\mu_{i}\) and standard deviation \(\sigma_{i}\) there are [10]  
\[ \begin{cases}\left\|\dfrac{\partial p_{\Theta_{i}}(\theta_{i};\mu_{i},\sigma_ {i})}{\partial\mu_{i}}\right\|=\dfrac{1}{\sqrt{2\pi}\sigma_{i}},\\ \left\|\dfrac{\partial p_{\Theta_{i}}(\theta_{i};\mu_{i},\sigma_{i})}{\partial \sigma_{i}}\right\|=\dfrac{2}{\sqrt{2\pi}\sigma_{i}}e^{-1/2}.\end{cases}  \tag{41}\]  
Then, the analytical IMs from the Fre-GSIs in terms of \(\mu_{i}\) and \(\sigma_{i}\) are \(\sigma_{i}/\sigma_{X}\) and \(\sigma_{i}^{2}/\sigma_{X}^{2}\), respectively. Let \(\mu_{i}=0\) and \(\sigma_{i}=1\) for \(i=1,2,\cdots,s\). Obviously, the variance-based GSI for \(\Theta_{i}\) is \(1/s\). The conditional PDF \(X|\Theta_{i}\) admits \(\mathcal{N}(\Theta_{i},\sqrt{s-1})\); therefore, the moment-independent GSI can be numerically computed by  
\[\mathcal{S}_{i}^{s}=\frac{1}{2}\mathbb{E}_{\Theta_{i}}\bigg{[}\int_{\Omega_{ X}}\Big{|}\mathcal{N}(0,\sqrt{s})-\mathcal{N}(\Theta_{i},\sqrt{s-1})\Big{|} \mathrm{d}x\bigg{]}, \tag{42}\]  
and the IMs computed from the Fre-GSIs in terms of \(\mu_{i}\) and \(\sigma_{i}\) are \(1/\sqrt{s}\) and \(1/s\), respectively. The failure-probability-based GSIs can be obtained by integrating Equation (40) in the failure domain, whose formulas are omitted for the sake of brevity. Let the model size \(s\) grow from 3 to 50. The values of the four GSIs with increasing \(s\) are summarized in Figure 8. Since the failure-probability-based GSI is dependent on the failure domain, we set \(\Omega_{\text{F}}:x>0\), which ensures that the same failure probability equals 0.5 as the model size increases. The four GSIs decrease expectedly as \(s\) increases, which is in accordance with our intuition [16]: one out of 3 summed i.i.d. inputs plays a more important role than one out of 50 summed i.i.d. inputs on one's degree of belief. This result can also be observed in Figure 9, where the amplitude of Fre-CSI decreases as the model size increases. The analytical values of \(\partial S_{i}^{\text{S}}/\partial\mu_{i}\) and \(\partial S_{i}^{\text{S}}/\partial\sigma_{i}\) are 0 and \(2(s-1)/s^{2}\), respectively, while \(\partial S_{i}^{\delta}/\partial\mu_{i}\) and \(\partial  
Moreover, in Figure 8, one can see that \(\mathcal{S}_{i}^{\mathcal{S}}\) decreases faster than \(\mathcal{S}_{i}^{\mathcal{S}}\), \(\mathcal{S}_{i}^{\mathcal{F}}(\mu)\) and \(\mathcal{S}_{i}^{\mathcal{F}}(\mu)\). Note that \(\mathcal{S}_{i}^{\mathcal{F}}(\mu)\) has the largest value among the GSIs, while \(\mathcal{S}_{i}^{\mathcal{F}}(\sigma)\) is zero, thus cannot provide informative results for global sensitivity analysis. Another feature of the variance-based GSI can be observed when considering [PERSON]'s growing size model with a multiplicative structure [16], i.e.,  
\[X=\prod_{i=1}^{s}\Theta_{i}, \tag{43}\]  
where \(\Theta_{i}\) is independently distributed with the lognormal distribution \(\mathcal{LN}(\theta_{i};\mu_{i},\sigma_{i})\), in which \(\mu_{i}\) and \(\sigma_{i}\) are the mean value and standard deviation of \(\ln\Theta_{i}\), respectively.