## 1 Introduction: Problem Setting
### [PERSON]' Approach
For example, \(f_{l_{1}}\) is presented by a conditional expectation:  
\[f_{l_{1}}(x_{l_{1}})=\mathbf{E}(\mathbf{u}|x_{l_{1}})-f_{0}\quad\text{and respectively}\quad\mathbf{V}_{l_{1}}=\mathbf{V}[f_{l_{1}}(x_{l_{1}})]= \mathbf{V}[\mathbf{E}(\mathbf{u}|x_{l_{1}})].\]  
Based on the above assumptions about the model function and the output variance, the following quantities  
\[S_{l_{1}\ \ldots\ l_{v}}=\frac{\mathbf{V}_{l_{1}\ \ldots\ l_{v}}}{\mathbf{V}}, \quad\
u\in\{1,\ldots,d\} \tag{8}\]  
are referred to as the global sensitivity indices [33]. For \(\
u=1\), this formula coincides with Equation (4), and the so-defined measures correspond to the main effect of input parameters as well as the effect of the cross-interactions. Based on Formulas (7) and (8), it becomes apparent that addressing the problem of global sensitivity analysis mathematically involves calculating total sensitivity indices (Equation (5)) of the relevant order. This process requires computing multidimensional integrals of the form in Equation (2). In general, obtaining \(S_{i}^{tot}\) entails computing \(2^{d}\) integrals of type in Equation (7). Earlier discussions highlighted that the fundamental assumption behind representation Equation (6) is that the essential features of model functions (1), which describe typical real-world scenarios, can be captured using low-order subsets of input variables. These subsets include terms up to order \(d_{tr}\), where \(d_{tr}<d\). Leveraging this assumption allows us to reduce the dimensionality of the original problem. Following [PERSON]' [33], we consider any set of \(m\) variables (\(1\leq m\leq d-1\)): \(\mathrm{y}=(x_{k_{1}},\ldots,x_{k_{m}})\), \(1\leq k_{1}<\ldots<k_{m}\leq d\), and let \(\mathrm{z}\) be the set of \(d-m\) complementary variables. Thus, \(\mathrm{x}=(\mathrm{y},\mathrm{z})\). Let \(K=(k_{1},\ldots,k_{m})\). The variances corresponding to the subsets y and z can be defined as  
\[\mathbf{V}_{\text{y}}=\sum_{n=1}^{m}\sum_{(i_{1}<\ldots<i_{n})\in K}\mathbf{V}_{i _{1}\ \ldots\ i_{n}},\qquad\mathbf{V}_{\text{z}}=\sum_{n=1}^{d-m}\sum_{(j_{1}<\ldots< j_{n})\in\bar{K}}\mathbf{V}_{j_{1}\ \ldots\ j_{n}}, \tag{9}\]  
where the complement of the subset \(K\) in the set of all parameter indices is denoted by \(\bar{K}\). The left sum in Equation (9) is extended over all subsets \((i_{1},\ldots,i_{n})\), where all indices \(i_{1},\ldots,i_{n}\) belong to \(K\). Then, the total variance, corresponding to the subset y, is \(\mathbf{V}_{\text{y}}^{tot}=\mathbf{V}-\mathbf{V}_{\text{z}}\) and is extended over all subsets \((i_{1},\ldots,i_{v}),\ 1\leq v\leq d\), where at least one \(i_{l}\in K,\ 1\leq l\leq\
u\). The procedure for computing global sensitivity indices is based on the following representation of the variance:  
\[\mathbf{V}_{\text{y}}:\mathbf{V}_{\text{y}}=\int\,f(\text{x})\;f(\text{y}, \text{z}^{\prime})\text{d}\text{x}\text{d}\text{z}^{\prime}-f_{0}^{2} \tag{10}\]  
(see [33]). The equality (10) enables the construction of a Monte Carlo algorithm for evaluating \(f_{0},\mathbf{V}\) and \(\mathbf{V}_{\text{y}}\):  
\[S_{i}^{tot}=S_{i}+\sum_{l_{1}\
eq i}S_{il_{1}}+\sum_{l_{1},l_{2}\
eq i,l_{1}< l_{2}}S_{il_{1}l_{2}}+\ldots+S_{il_{1}\ldots l_{d-1}}, \tag{5}\]0 \[S_{i}^{tot}=S_{i}+\sum_{l_{1}\
eq i}S_{il_{1}}+\sum_{l_{1},l_{2}\
eq i,l_{1}< l_{2}}S_{il_{1}l_{2}}+\ldots+S_{il_{1}\ldots l_{d-1}}, \tag{5}\]1  
where \(\xi=(\eta,\zeta)\) is a random sample and \(\eta\) corresponds to the input subset denoted by y. For example, for \(m=1,\text{y}=\{x_{l_{1}}\},l_{1}\in\{1,\ldots,d\}\) and \(\text{z}=\{1,\ldots,d\}\backslash l_{1}\): \(S_{l_{1}}=S_{(l_{1})}=\mathbf{V}_{(l_{1})}/\mathbf{V}\), \(S_{l_{1}}^{tot}=\mathbf{V}_{l_{1}}^{tot}/\mathbf{V}=1-S_{\text{z}}\). Calculating the impact of higher-order interactions can be achieved through an iterative process. For instance, consider the following equation: \(S_{(l_{1}l_{2})}=\mathbf{V}_{(l_{1}l_{2})}/\mathbf{V}=S_{l_{1}}+S_{l_{2}}+S_{l_ {1}l_{2}}\).