# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 3 Review of Three Classical Global Sensitivity Indices
### Variance-Based GSI  
The variance-based GSI, also known as Sobol' index [13; 31], is one of the most popular GSIs in many disciplines [2]. The original Sobol' index is defined on the assumption of independence of \(\Theta_{i}\)'s, but it can be extended to dependent variables as well; see work in [14; 32]. For the sake of simplicity, it is mentioned again that hereinafter only the independent case is considered. The variance-based GSI is based on the contribution of basic input variables to the variance of the QoI, of which the first-order definition is given by  
\[\mathcal{S}_{i}^{\rm S}=\frac{\mathbb{D}[\mathbb{E}[X|\Theta_{i}]]}{\mathbb{D }[X]},\ \mathcal{S}_{i}^{\rm TS}=1-\frac{\mathbb{D}[\mathbb{E}[X|\boldsymbol{\Theta}_{ \sim i}]]}{\mathbb{D}[X]},\ i=1,2,\cdots,s, \tag{11}\]  
where \(\boldsymbol{\Theta}_{\sim i}=(\Theta_{1},\cdots,\Theta_{i-1},\Theta_{i+1}, \cdots,\Theta_{s})^{\top}\), \(\mathbb{E}[\cdot]\) and \(\mathbb{D}[\cdot]\) denote the expectation operator and the variance operator, respectively. \(\mathcal{S}_{i}^{\rm S}\) is called the main effect index, measuring the individual contribution of \(\Theta_{i}\) to the total variance \(\mathbb{D}[X]\), while \(\mathcal{S}_{i}^{\rm TS}\) is called the total effect index, which not only shows the individual contribution of \(\Theta_{i}\) to the total variance \(\mathbb{D}[X]\), but also reflects the interaction effect of \(\Theta_{i}\) with other input variables. It is obvious that \(0\leq\mathcal{S}_{i}^{\rm S}\leq\mathcal{S}_{i}^{\rm TS}\leq 1\). Moreover, the difference between \(\mathcal{S}_{i}^{\rm TS}\) and \(\mathcal{S}_{i}^{\rm S}\), i.e., \(\mathcal{S}_{i}^{\rm TS}-\mathcal{S}_{i}^{\rm S}\) stands for the interaction effect of \(\Theta_{i}\), and it equals to zero if no interaction effect exits for \(\Theta_{i}\). The intuition underlying the variance-based GSI is that, for each fixed \(\Theta_{i}=\theta_{i}\), one can obtain a conditional output \(X=g(\boldsymbol{\Theta}_{\sim i},\theta_{i})\), which is still a random variable but follows the conditional PDF \(p_{X|\Theta_{i}}(x|\Theta_{i}=\theta_{i})\). Then, the mean value of this conditional output produces the inner integral in Equation (11), i.e.,  
\[ \begin{split}\mathbb{E}[X|\Theta_{i}=\theta_{i}]&= \int_{\Omega_{\boldsymbol{\Theta}_{\sim i}}}g(\theta_{\sim i},\Theta_{i}= \theta_{i})p_{\boldsymbol{\Theta}_{\sim i}}(\theta_{\sim i})\mathrm{d} \boldsymbol{\theta}_{\sim i}\\ &=\int_{\Omega_{X}}xp_{X|\Theta_{i}}(x|\Theta_{i}=\theta_{i}) \mathrm{d}x\stackrel{{\rm def}}{{=}}g_{i}(\theta_{i})\end{split}  \tag{12}\]and then we have \(\mathbb{E}[X|\Theta_{i}]=g_{i}(\Theta_{i})\), considering now \(\Theta_{i}\) is a random variable (no longer fixed to \(\theta_{i}\)). Let \(\mathbb{E}[X]\stackrel{{\text{def}}}{{=}}\mu_{X}\) and \(\mathbb{D}[X]\stackrel{{\text{def}}}{{=}}\sigma_{X}^{2}\). The ratio between the variance of \(g_{i}(\Theta_{i})\) that gives the outer integral in Equation (11), and the one of \(X\) is calculated by  
\[ \begin{split}\mathcal{S}_{i}^{\text{S}}&=\frac{1}{ \sigma_{X}^{2}}\int_{\Omega_{\Theta_{i}}}(g_{i}(\theta_{i})-\mu_{X})^{2}p_{ \Theta_{i}}(\theta_{i})\text{d}\theta_{i}\\ &=\frac{1}{\sigma_{X}^{2}}\int_{\Omega_{\Theta_{i}}}\left(\int_{ \Omega_{X}}xp_{X|\Theta_{i}}(x|\Theta_{i}=\theta_{i})\text{d}x-\mu_{X}\right) ^{2}p_{\Theta_{i}}(\theta_{i})\text{d}\theta_{i},\end{split}  \tag{13}\]  
which is nothing but the main effect index \(\mathcal{S}_{i}^{\text{S}}\). The idea of fixing an input random variable in calculating the variance-based GSI is based on the pure thought that the variance of output can be reduced if uncertainty in this variable is eliminated [15]. Nevertheless, the variance-based GSI is somewhat questioned for the reason that the variance, i.e., the second-order statistics may not be sufficiently sensitive or informative [10]. Likewise, the variance-based GSI may lead to erroneous inferences or even opposite conclusions when the PDF of the QoI has multiple modes [23; 33].