# Frechet-Derivative-Based Global Sensitivity Analysis and Its Physical Meanings in Structural Design
## 4 Frechet-Derivative-Based GSI  
### Definition of the Frechet-Derivative-Based GSI and Its Parametric Expression  
As mentioned above, an ideal GSI is expected to not only measure the importance of each model input but also characterize the direction of this importance. These two features will undoubtedly help to reflect the basic physical mechanisms of stochastic parametric models. In probability theory, it is well known that the PDF of the QoI can be uniquely defined if the studied stochastic model is well-posed and the PDF of model inputs is also precisely determined. This fact is also known as the rule of change of random variables (CRV) [37], which can be compactly expressed in an operator form; see the formula in Equation (9). In this sense, a natural GSI in terms of stochastic models can be heuristically defined as an extension of the \"partial derivative\" of a function in the context of the operator of uncertainty propagation (UP), as shown in Figure 2. From the perspective of functional analysis, the Frechet derivative can perform this task in theory. In summary, based on the conceptual idea of the Frechet derivative, the Frechet-derivative-based GSI (Fre-GSI) can serve as a natural GSI that may be more compatible with stochastic models. The Fre-GSI \(\mathcal{F}_{i}\) for the \(i\)-th parameter \(\Theta_{i}\) is defined by [10]  
\[\lim_{\left\|\phi_{\mathcal{P}_{\Theta_{i}}}\right\|_{V}\to 0}\frac{\left\|\psi\left[p_{ \Theta}+\delta p_{\Theta_{i}}\right]-\psi[p_{\Theta}]-\mathcal{F}_{i}\delta p _{\Theta_{i}}\right\|_{V}}{\left\|\delta p_{\Theta_{i}}\right\|_{V}}=0, \tag{18}\]  
Figure 2: The functional perspective on the stochastic model and the Fréchet-derivative-based GSI. where \(\delta p_{\Theta_{i}}\) is a small perturbation of \(p_{\Theta_{i}}\) satisfying \(\int\delta p_{\Theta_{i}}=0\) and \(\delta p_{\Theta_{i}}>-p_{\Theta}\). \(\|\cdot\|_{W}\) and \(\|\cdot\|_{V}\) are two appropriately equipped norms (e.g., half of the \(\ell_{1}\)-norm) on the Banach spaces \(W\) and \(V\) of \(p_{X}\) and \(p_{\Theta_{i}}\), respectively. It should be noted that, here, the Fre-GSI is defined by the Frechet derivative of \(\psi\), and since \(\psi\) is a linear operator, the Frechet derivative of \(\psi\) is exactly itself [10]. If \(p_{\Theta}(\mathbf{\theta})\) is a parametric distribution with certain distribution parameters \(\mathbf{\xi}\), i.e., \(p_{\Theta}(\mathbf{\theta};\mathbf{\xi})\), then the parametric form of the Fre-GSI can be derived as [10]  
\[\mathcal{F}_{i,j}(x;\mathbf{\xi})=\frac{\partial p_{X}(x;\mathbf{\xi})}{\partial \xi_{i,j}}/\left\|\frac{\partial p_{\Theta}(\mathbf{\theta};\mathbf{\xi})}{\partial\xi _{i,j}}\right\|, \tag{19}\]  
where \(\|\cdot\|\) is defined by \(\frac{1}{2}\int|p|\). Based on the assumption of independence of \(\mathbf{\Theta}\), the denominator in Equation (19) (the norm term) can be further simplified to \(\left\|\partial p_{\Theta_{i}}(\theta_{i};\mathbf{\xi}_{i})/\partial\xi_{i,j}\right\|\). In addition, it is obvious that \(\mathcal{F}_{i,j}(x;\mathbf{\xi})\) satisfies  
\[\int_{\Omega_{X}}\mathcal{F}_{i,j}(x;\mathbf{\xi})\mathrm{d}x=0 \tag{20}\]  
due to the fact that \(\int_{\Omega_{X}}p_{X}(x;\mathbf{\xi})\mathrm{d}x=1\). In the following, \(x\) and \(\mathbf{\xi}\) in \(\mathcal{F}_{i,j}(x;\mathbf{\xi})\) are omitted for simplicity. Compared with the variance-based GSI in Equation (13), the moment-independent GSI in Equation (14), and the failure-probability-based GSI in Equation (15), the Fre-GSI in Equation (19) is not a scalar but a function of \(x\). In this sense, the Fre-GSI reflects the sensitivity of each realization of \(X\) to inputs. Computing the Fre-GSI in Equation (19) mainly includes two steps: (1) Estimate the PDF of \(X\) in terms of \(\mathbf{\xi}\) and the perturbed PDF of \(X\) in terms of \(\mathbf{\xi}^{*}\). This can be carried out using, for example, moment-based methods [38], kernel density estimation [22], probability density evolution method [39], etc.