# 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 Variance-Based GSI  
Write  
\[s_{i}^{\mathrm{S}}(\theta_{i};\boldsymbol{\xi}_{i})=\frac{1}{\sigma_{X}^{2}( \boldsymbol{\xi}_{i})}\left(\int_{\Omega_{X}}xp_{X|\Theta_{i}}(x|\Theta_{i}= \theta_{i})\mathrm{d}x-\mu_{X}(\boldsymbol{\xi}_{i})\right)^{2}, \tag{11}\]  
in which \(\boldsymbol{\xi}_{i}=\{\boldsymbol{\xi}_{i,j}\}_{j=1}^{s_{i}}\) is for the distribution parameter(s) of \(\Theta_{i}\). Based on this definition, the variance-based GSI in Equation (13) can be equally written by  
\[\mathcal{S}_{i}^{\mathrm{S}}=\mathbb{E}\Big{[}s_{i}^{\mathrm{S}}(\Theta_{i}; \boldsymbol{\xi}_{i})\Big{]}. \tag{12}\]  
Considering the derivative of \(s_{i}^{\mathrm{S}}\) with respect to the distribution parameter \(\xi_{i,j}\), we have  
\[\begin{split}\frac{\partial s_{i}^{\mathrm{S}}(\theta_{i}; \boldsymbol{\xi}_{i,j})}{\partial\xi_{i,j}}&=\underbrace{\frac{- 2}{\sigma_{X}^{2}}\left(\int_{\Omega_{X}}xp_{X|\Theta_{i}}\mathrm{d}x-\mu_{X} \right)\frac{\partial\mu_{X}}{\partial\xi_{i,j}}}_{\mathrm{Part\,I}}\\ &+\underbrace{\frac{-2}{\sigma_{X}^{3}}\left(\int_{\Omega_{X}}xp _{X|\Theta_{i}}\mathrm{d}x-\mu_{X}\right)^{2}\frac{\partial\sigma_{X}}{\partial \xi_{i,j}}}_{\mathrm{Part\,II}}.\end{split} \tag{13}\]  
Note that in Equation (13), the expectation of Part I is zero due to the fact that \(\mathbb{E}[\mathbb{E}(X|\Theta_{i})]=\mathbb{E}[X]=\mu_{X}\), and the expectation of Part II can be simplified by using Equations (11) and (12), so that  
\[\mathbb{E}\Bigg{[}\frac{\partial s_{i}^{\mathrm{S}}(\Theta_{i};\boldsymbol{ \xi}_{i})}{\partial\xi_{i,j}}\Bigg{]}=-\frac{2}{\sigma_{X}}s_{i}^{\mathrm{S}} \cdot\frac{\partial\sigma_{X}}{\partial\xi_{i,j}}. \tag{14}\]  
Further, we adopt the idea in [41] that since \(s_{i}^{\mathrm{S}}(\theta_{i};\boldsymbol{\xi}_{i})\) is an one-dimensional function of \(\theta_{i}\), it can be obtained with less computational effort by one-dimensional semi-parametric or non-parametric regression methods. Hence, assume \(s_{i}^{\mathrm{S}}(\theta_{i};\boldsymbol{\xi}_{i})\) is known. Then, we have \[\frac{\partial S_{i}^{\text{S}}}{\partial\xi_{i,j}} =\frac{\partial\mathbb{E}\big{[}s_{i}^{\text{S}}(\Theta_{i};\xi_{i} )\big{]}}{\partial\xi_{i,j}} \tag{10}\] \[=\mathbb{E}\bigg{[}\frac{\partial s_{i}^{\text{S}}(\Theta_{i};\xi_ {i})}{\partial\xi_{i,j}}\bigg{]}+\int_{\Omega_{\Theta_{i}}}s_{i}^{\text{S}}( \theta_{i};\xi_{i})\frac{\partial p_{\Theta_{i}}}{\partial\xi_{i,j}}\text{d} \theta_{i}\] \[=-\frac{2}{\sigma_{X}}S_{i}^{\text{S}}\cdot\frac{\partial\sigma_{ X}}{\partial\xi_{i,j}}+\int_{\Omega_{\Theta_{i}}}s_{i}^{\text{S}}(\theta_{i};\xi_{i}) \frac{\partial p_{\Theta_{i}}}{\partial\xi_{i,j}}\text{d}\theta_{i}.\]  
Moreover, since  
\[\sigma_{X}^{2}=\int_{\Omega_{X}}(x-\mu_{X}(\xi_{i}))^{2}p_{X}(x;\xi_{i})\text{ d}x, \tag{11}\]  
taking the derivative of \(\sigma_{X}^{2}\) with respect to \(\xi_{i,j}\) can be read as follows  
\[2\sigma_{X}\frac{\partial\sigma_{X}}{\partial\xi_{i,j}} =\int_{\Omega_{X}}-2(x-\mu_{X})\frac{\partial\mu_{X}}{\partial\xi_ {i,j}^{\text{S}}}p_{X}(x;\xi_{i})\text{d}x \tag{12}\] \[+\int_{\Omega_{X}}(x-\mu_{X})^{2}\frac{\partial p_{X}(x;\xi_{i}) }{\partial\xi_{i,j}}\text{d}x\] \[=\left\|\frac{\partial p_{\Theta_{i}}(\theta_{i};\xi_{i})}{ \partial\xi_{i,j}}\right\|\int_{\Omega_{X}}(x-\mu_{X})^{2}\mathcal{F}_{i,j} \text{d}x.\]  
Combining Equations (10) and (12), we have  
\[\frac{\partial S_{i}^{\text{S}}}{\partial\xi_{i,j}}=-\left\|\frac{\partial p_ {\Theta_{i}}(\theta_{i};\xi_{i})}{\partial\xi_{i,j}}\right\|\mathcal{S}_{i}^ {\text{S}}\cdot\int_{\Omega_{X}}\frac{(x-\mu_{X})^{2}}{\sigma_{X}^{2}} \mathcal{F}_{i,j}\text{d}x+\int_{\Omega_{\Theta_{i}}}s_{i}^{\text{S}}(\theta_ {i};\xi_{i})\frac{\partial p_{\Theta_{i}}}{\partial\xi_{i,j}}\text{d}\theta_{i}. \tag{13}\]