# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## Appendix A Proof of Some Properties of the Fre-GSI
### Link of the Fre-GSI with the Moment-Independent GSI  
Denote that  
\[s_{i}^{\delta}(\theta_{i};\xi_{i})=\frac{1}{2}\int_{\Omega_{X}}\Bigl{|}p_{X| \Theta_{i}}(x|\Theta_{i}=\theta_{i})-p_{X}(x;\xi_{i})\Bigr{|}\text{d}x, \tag{14}\]  
where \(\xi_{i}=\{\xi_{i,j}\}_{j=1}^{s_{i}}\) stands for the distribution parameter(s) of \(\Theta_{i}\). Then, Equation (14) can be rewritten by  
\[\mathcal{S}_{i}^{\delta}=\mathbb{E}\Bigl{[}s_{i}^{\delta}(\Theta_{i};\xi_{i}) \Bigr{]}. \tag{15}\]  
Define \(f(x;\xi_{i})=p_{X|\Theta_{i}}(x|\Theta_{i}=\theta_{i})-p_{X}(x;\xi_{i})\). Taking the derivative of \(s_{i}^{\delta}\) with respect to the distribution parameter \(\xi_{i,j}\) leads to  
\[\frac{\partial s_{i}^{\delta}(\theta_{i};\xi_{i})}{\partial\xi_{i,j}} =\frac{1}{2}\frac{\partial}{\partial\xi_{i,j}}\int_{\Omega_{X}}|f(x; \xi_{i})|\text{d}x \tag{16}\] \[=\frac{1}{2}\int_{\Omega_{X}}\frac{\partial|f(x;\xi_{i})}{|f(x; \xi_{i})|}\frac{\partial f(x;\xi_{i})}{\partial\xi_{i,j}}\text{d}x\] \[=-\frac{1}{2}\int_{\Omega_{X}}\frac{f(x;\xi_{i})}{|f(x;\xi_{i})|} \frac{\partial p_{X}(x;\xi_{i})}{\partial\xi_{i,j}}\text{d}x.\]Hence,  
\[ \begin{split}\left|\frac{\partial s_{i}^{\delta}(\theta_{i};\mathbf{\xi}_{ i})}{\partial\xi_{i,ij}}\right|&=\left|\frac{1}{2}\int_{\Omega_{X}}\frac{f(x;\mathbf{\xi}_{ i})}{|f(x;\mathbf{\xi}_{i})|}\frac{\partial p_{X}(x;\mathbf{\xi}_{i})}{\partial\xi_{i,j}} \mathrm{d}x\right|\\ &\leq\frac{1}{2}\int_{\Omega_{X}}\left|\frac{\partial p_{X}(x; \mathbf{\xi}_{i})}{\partial\xi_{i,ij}}\right|\mathrm{d}x=\left\|\frac{\partial p_{ \mathbf{\Theta}}(\mathbf{\theta};\mathbf{\xi})}{\partial\xi_{i,ij}}\right\|\cdot\mathcal{ S}_{i,j}^{F}.\end{split}  \tag{103}\]  
Therefore, by Equations (102) and (103) there is  
\[ \begin{split}\left|\frac{\partial\mathcal{S}_{i}^{\delta}}{ \partial\xi_{i,ij}}\right|&=\left|\frac{\partial\mathbb{E}\left[ s_{i}^{\delta}(\Theta_{i};\mathbf{\xi}_{i})\right]}{\partial\xi_{i,ij}}\right|\\ &=\left|\mathbb{E}\left[\frac{\partial s_{i}^{\delta}(\Theta_{i}; \mathbf{\xi}_{i})}{\partial\xi_{i,ij}}\right]+\int_{\Omega_{\Theta_{i}}}s_{i}^{ \delta}\frac{\partial p_{\Theta_{i}}}{\partial\xi_{i,j}}\mathrm{d}\theta_{i} \right|\\ &\leq\mathbb{E}\left[\left|\frac{\partial s_{i}^{\delta}(\Theta_{i };\mathbf{\xi}_{i})}{\partial\xi_{i,ij}}\right|\right]+\left|\int_{\Omega_{\Theta_ {i}}}s_{i}^{\delta}\frac{\partial p_{\Theta_{i}}}{\partial\xi_{i,j}}\mathrm{d} \theta_{i}\right|\\ &\leq\left\|\frac{\partial p_{\mathbf{\Theta}}(\mathbf{\theta};\mathbf{ \xi})}{\partial\xi_{i,j}}\right\|\cdot\mathcal{S}_{i,j}^{F}+\left|\int_{\Omega _{\Theta_{i}}}s_{i}^{\delta}\frac{\partial p_{\Theta_{i}}}{\partial\xi_{i,j}} \mathrm{d}\theta_{i}\right|.\end{split}  \tag{104}\]  
Now consider the second term on the right side of the inequality. Note that \(\int_{\Omega_{\Theta_{i}}}\partial p_{\Theta_{i}}/\partial\xi_{i,j}\mathrm{d }\theta_{i}=0\), denote \(\Omega_{\Theta_{i}}=\{\theta_{i}:\partial p_{\Theta_{i}}(\theta_{i})/ \partial\xi_{i,j}\leq 0\}\) and \(\Omega_{\Theta_{i}}^{+}=\Omega_{\Theta_{i}}\backslash\Omega_{\Theta_{i}}^{+}\). Since \(0\leq s_{i}^{\delta}\leq 1\), there is  
\[ \begin{split}\left|\int_{\Omega_{\Theta_{i}}}s_{i}^{\delta}\frac{ \partial p_{\Theta_{i}}}{\partial\xi_{i,j}}\mathrm{d}\theta_{i}\right|& =\left|\int_{\Omega_{\Theta_{i}}^{-}}s_{i}^{\delta}\frac{\partial p _{\Theta_{i}}}{\partial\xi_{i,j}}\mathrm{d}\theta_{i}+\int_{\Omega_{\Theta_{i }}^{+}}s_{i}^{\delta}\frac{\partial p_{\Theta_{i}}}{\partial\xi_{i,j}}\mathrm{ d}\theta_{i}\right|\\ &\leq\left|\int_{\Omega_{\Theta_{i}}^{+}}s_{i}^{\delta}\frac{ \partial p_{\Theta_{i}}}{\partial\xi_{i,j}}\mathrm{d}\theta_{i}\right|\leq \int_{\Omega_{\Theta_{i}}^{+}}\frac{\partial p_{\Theta}}{\partial\xi_{i,j}} \mathrm{d}\theta_{i}.\end{split}