# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 4 Frechet-Derivative-Based GSI
### Links of the Frechet-Derivative-Based GSI with the Three Classical GSIs
#### 4.3.3 Links Between the Fre-GSI to the Three Classical GSIs  
The derivative of the variance-based GSI with respect to the distribution parameter can be derived from the Fre-GSI (the proof is given in Appendix A.3), i.e.,  
\[\frac{\partial\mathcal{S}_{i}^{\rm S}}{\partial\bar{\xi}_{i,j}}=-\left\|\frac {\partial p_{\Theta_{i}}(\theta_{i};\bar{\mathbf{\xi}}_{i})}{\partial\bar{\xi}_{ i,j}}\right\|\mathcal{S}_{i}^{\rm S}\cdot\int_{\Omega_{X}}\frac{(x-\mu_{X})^{2}}{ \sigma_{X}^{2}}\mathcal{F}_{i,j}\mathrm{d}x+\int_{\Omega_{\Theta_{i}}}\hat{s }_{i}^{\rm S}(\theta_{i};\bar{\mathbf{\xi}}_{i})\frac{\partial p_{\Theta_{i}}}{ \partial\bar{\xi}_{i,j}}\mathrm{d}\theta_{i}\stackrel{{\rm def}}{{= }}\text{EQ1}, \tag{29}\]  
where  
\[\hat{s}_{i}^{\rm S}(\theta_{i};\bar{\mathbf{\xi}}_{i})=\frac{1}{\sigma_{X}^{2}( \bar{\mathbf{\xi}}_{i})}\left(\int_{\Omega_{X}}xp_{X|\Theta_{i}}(x|\Theta_{i}= \theta_{i})\mathrm{d}x-\mu_{X}(\bar{\mathbf{\xi}}_{i})\right)^{2} \tag{30}\]  
is assumed to be known. Actually, \(\hat{s}_{i}^{\rm S}(\theta_{i};\bar{\mathbf{\xi}}_{i})\) is a one-dimensional function of \(\theta_{i}\), which can be computed by regression methods as suggested in [41].  
Similarly, the upper bound of the absolute value of the derivative of the moment-independent GSI with respect to the distribution parameter can be given from the Fre-GSI by  
\[\left|\frac{\partial\mathcal{S}_{i}^{\rm S}}{\partial\bar{\xi}_{i,j}}\right| \leq\left\|\frac{\partial p_{\Theta}(\theta;\bar{\mathbf{\xi}})}{\partial\bar{ \xi}_{i,j}}\right\|\cdot\mathcal{S}_{i,j}^{\mathcal{F}}+\left\|\frac{\partial p _{\Theta_{i}}(\theta_{i};\bar{\mathbf{\xi}}_{i})}{\partial\bar{\xi}_{i,j}}\right\| \stackrel{{\rm def}}{{=}}\text{UB2}, \tag{31}\]  
which provides convenience to learn the robustness of \(\mathcal{S}_{i}^{\rm S}\) with respect to the basic distribution parameters without estimating \(\mathcal{S}_{i}^{\rm S}\). The proof is given in Appendix A.4.  
A practical conversion from the Fre-GSI to the failure-probability-based GSI reads  
\[\mathcal{S}_{i,j}^{\rm F}=\left\|\frac{\partial p_{\Theta}(\theta;\bar{\mathbf{ \xi}})}{\partial\bar{\xi}_{i,j}}\right\|\cdot\int_{\Omega_{\rm F}}\mathcal{F} _{i,j}\mathrm{d}x\stackrel{{\rm def}}{{=}}\text{EQ3}, \tag{32}\]which means the failure-probability-based GSI is proportional to the integral of the Fre-GSI in the failure domain, where the proportional coefficient is exactly the norm term in the Fre-GSI. Additionally, an inequality between \(\left|\mathcal{S}_{i,j}^{\text{F}}\right|\) and \(\mathcal{S}_{i,j}^{\mathcal{F}}\) can be derived by  
\[\left|\mathcal{S}_{i,j}^{\text{F}}\right|\leq\left\|\frac{\partial p_{\Theta}( \theta;\xi)}{\partial\xi_{i,j}}\right\|\cdot\mathcal{S}_{i,j}^{\text{ \emph{ref} def}}\text{ UB3}, \tag{33}\]  
where the proof of Equations (32) and (33) is available in Appendix A.5. From Equation (33), it is interesting to find that the upper bound of the absolute value of \(\mathcal{S}_{i,j}^{\text{F}}\) is independent of the failure criterion. Hence, by Equation (33), the decision-maker can flexibly know in advance which parameters may be influential on the failure probability, without calculating the failure probability in terms of a certain criterion of failure.