# 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
#### 4.3.2 On the Concept of the How from a Functional Perspective  
Recall the definition of variance-based GSI and moment-independent GSI, the thought of fixing an input random variable \(\Theta_{i}\) to a certain value \(\bar{\theta}_{i}\) will lead to a \"collapse\" of the PDF of \(\mathbf{\Theta}\), i.e.,  
\[p_{\mathbf{\Theta}}(\theta)\to p_{\mathbf{\Theta}}^{+}(\theta)\stackrel{{ \mathrm{def}}}{{=}}\prod_{k=1,k\
eq i}^{s}p_{\Theta_{k}}(\theta_{k}) \phi_{\mathrm{D}}(\theta_{i}-\bar{\theta}_{i}), \tag{25}\]  
and the corresponding total variation distance between \(p_{\mathbf{\Theta}}\) and \(p_{\mathbf{\Theta}}^{+}\) is  
\[ \begin{split} d_{\mathrm{TV}}\big{(}p_{\mathbf{\Theta}},p_{ \mathbf{\Theta}}^{+}\big{)}&=\frac{1}{2}\int_{\Omega_{\mathbf{ \Theta}}}\big{|}p_{\mathbf{\Theta}}^{+}-p_{\mathbf{\Theta}}\big{|}\mathrm{d} \theta\\ &=\frac{1}{2}\int_{\Omega_{\Theta_{i}}}\big{|}\delta_{\mathrm{D}} \big{(}\theta_{i}-\bar{\theta}_{i}\big{)}-p_{\Theta_{i}}(\theta_{i})\big{|} \mathrm{d}\theta_{i}=1,\end{split}  \tag{26}\]  
which means that the input space undergoes a unit-volume change when the \(i\)-th random variable collapses to a deterministic value. Let \(p_{X}\) and \(p_{X}^{+}\) be the PDFs of \(X\), \(\mu_{X}\) and \(\mu_{X}^{+}\) be the mean values of \(X\), with respect to \(p_{\mathbf{\Theta}}\) and \(p_{\mathbf{\Theta}}^{+}\). The changes in the output spaces can be defined by \(d_{\mathrm{F}}\big{(}\mu_{X},\mu_{X}^{+}\big{)}\) and \(d_{\mathrm{TV}}\big{(}p_{X},p_{X}^{+}\big{)}\) in terms of the variance-based GSI and the moment-independent GSI [7], respectively, where \(d_{\mathrm{F}}(a,b)\) is the Bregman distance, i.e., the squared Euclidean distance defined by \(d_{\mathrm{F}}(a,b)=(a-b)^{2}\). Then, the variance-based GSI in Equation (13) and the moment-independent GSI in Equation (14) can be rewritten by  
\[\mathcal{S}_{l}^{\mathrm{S}}=\frac{1}{\sigma_{X}^{2}}\mathbb{E}\left[\frac{d_ {\mathrm{F}}\big{(}\mu_{X},\mu_{X}^{+}\big{)}}{d_{\mathrm{TV}}\big{(}p_{ \mathbf{\Theta}},p_{\mathbf{\Theta}}^{+}\big{)}}\right],\;\mathcal{S}_{l}^{ \delta}=\mathbb{E}\left[\frac{d_{\mathrm{TV}}\big{(}p_{X},p_{X}^{+}\big{)}}{d_ {\mathrm{TV}}\big{(}p_{\mathbf{\Theta}},p_{\mathbf{\Theta}}^{+}\big{)}} \right], \tag{27}\]  
where the expectation is calculated by taking the average of values in the dashed regions in Figure 4a,b. Similarly, considering that the distribution parameter has a small perturbation from \(\xi_{i,j}\) to \(\xi_{i,j}^{*}\), the input PDF is changed from \(p_{\mathbf{\Theta}}\) to \(p_{\mathbf{\Theta}}^{*}\). Correspondingly, the failure probability  
Figure 4: A functional perspective to illustrate the essences of four classes of GSIs. and the PDF of \(X\) are changed to \(P_{\rm F}^{*}\) and \(p_{X}^{*}\), respectively. By finite difference methods, the failure-probability-based GSI in Equation (15) and the importance measure of the Fre-GSI in Equation (22) can be rewritten by  
\[\mathcal{S}_{i,j}^{\rm F}=\frac{P_{\rm F}^{*}-P_{\rm F}}{\xi_{i,j}^{*}-\xi_{i,j }},\ \mathcal{S}_{i,j}^{\mathcal{F}}=\frac{d_{\rm TV}(p_{X},p_{X}^{*})}{d_{\rm TV}(p _{\Theta},p_{\Theta}^{*})}. \tag{28}\]  
Now, the physical significance of the four GSIs is abundantly clear. Analogous to the deformation gradient in continuum mechanics, the GSIs in Equations (27) and (28) reflect the average rate and the rate of volume change (for the Fre-GSI, it is actually the absolute rate of volume change) from the input space to the output space, when there is a specific perturbation in the input PDF. More specifically, for the variance-based GSI and the moment-independent GSI, the perturbation is set by fixing an input random variable to a deterministic value, while for the failure-probability-based GSI and the Fre-GSI, it is to make a distribution parameter slightly perturbed. Additionally, assume the distribution parameters of \(\Theta_{i}\) are uniquely determined by the mean value \(\mu_{i}\) and the standard deviation \(\sigma_{i}\) of \(\Theta_{i}\), i.e., \(p_{\Theta_{i}}(\theta_{i};\mu_{i},\sigma_{i})\).