## 1 Introduction: Problem Setting
### [PERSON]' Approach  
In variance-based sensitivity analyses, the primary indicator associated with a given normalized input parameter \(x_{i},\ i=1,\ldots,d\) (normalized) is typically defined as  
\[\frac{\mathbf{V}_{x_{i}}[\mathbf{E}(\mathbf{u}|x_{i})]}{\mathbf{V}\mathbf{u}}, \tag{4}\]  
where \(\mathbf{V}_{x_{i}}[\mathbf{E}(\mathbf{u}|x_{i})]\) represents the variance of the conditional expectation of \(\mathbf{u}\) given \(x_{i}\) and \(\mathbf{V}\mathbf{u}\) denotes the total variance of \(\mathbf{u}\). This indicator is termed the _first-order sensitivity index_ by [PERSON]' [36] or _correlation ratio_ by [PERSON] [37]. The _total sensitivity index_[38] quantifies the overall effect of a given parameter, encompassing all possible coupling terms between that parameter and all others. The total sensitivity index (TSI) of an input parameter \(x_{i},i\in\{1,\ldots,d\}\) is defined in the following way [36; 38; 39]:  
\[S_{i}^{tot}=S_{i}+\sum_{l_{1}\
eq i}S_{il_{1}}+\sum_{l_{1},l_{2}\
eq i,l_{1}< l_{2}}S_{il_{1}l_{2}}+\ldots+S_{il_{1}\ldots l_{d-1}}, \tag{5}\]  
where \(S_{i}\) is called _the main effect (first-order sensitivity index)_ of \(x_{i}\) and \(S_{il_{1}\ldots l_{j-1}}\) is the \(j^{-}\) th order sensitivity index. Higher-order terms capture the interaction effects between the unknown input parameters \(x_{i_{1}},\ldots,x_{i_{v}}\), where \(\
u\in\{2,\ldots,d\}\) on the output variance. Later on, we show how sensitivity indices \(S_{l_{1}\ldots l_{v}}\) are defined via the variances of conditional expectations \(\mathbf{V}_{l_{1}},\mathbf{V}_{l_{1}\ldots l_{v}},\ \ 2\leq\
u\leq d\). The global SA method employed in this work is grounded in decomposing an integrable model function \(f\) in the \(d\)-dimensional factor space into terms of escalating dimensionality [36]:  
\[f(\mathbf{x})=f_{0}+\sum_{\
u=1}^{d}\sum_{l_{1}<\ldots<l_{v}}f_{l_{1}\ldots l_ {v}}(x_{l_{1}},x_{l_{2}},\ldots,x_{l_{v}}), \tag{6}\]where \(f_{0}\) is a constant. The total number of summands in Equation (6) is \(2^{d}\). The representation Equation (6) is referred to as the ANOVA representation of the model function \(f(\mathrm{x})\) if each term is chosen to satisfy the following condition [33]:  
\[\int_{0}^{1}f_{l_{1}\ldots l_{v}}(x_{l_{1}},x_{l_{2}},\ldots,x_{l_{v}})\mathrm{d }x_{l_{k}}=0,\quad 1\leq k\leq\
u,\quad\
u=1,\ldots,d.\]  
An important comment here is that, if the entire presentation Equation (6) of the right-hand side is used, then it does not simplify the initial problem. The main expectation is that a truncated sequence  
\[\sum_{\
u=1}^{d_{tr}}\sum_{l_{1}<\ldots<l_{v}}f_{l_{1}\ldots l_{v}}(x_{l_{1}},x _{l_{2}},\ldots,x_{l_{v}}),\]  
where \(d_{tr}<d\), is considered as a sufficiently good approximation to the model function \(f\). Then,  
\[\mathbf{V}=\int_{U^{d}}f^{2}(\mathrm{x})\mathrm{d}\mathrm{x}-f_{0}^{2},\quad \mathbf{V}_{l_{1}\ldots l_{v}}=\int f_{l_{1}\ldots l_{v}}^{2}\mathrm{d}x_{l_{1 }}\ldots\mathrm{d}x_{l_{v}} \tag{7}\]  
The quantities represent the total and partial variances, respectively. They are obtained by squaring and integrating over \(U^{d}\) Equation (6), assuming that \(f(\mathrm{x})\) is a square-integrable function (ensuring that all terms in Equation (6) are also square-integrable functions). Consequently, the total variance of the model output is partitioned into partial variances in a manner analogous to the model function, resulting in a unique ANOVA decomposition: \(\mathbf{V}=\sum_{\
u=1}^{d}\sum_{l_{1}<\ldots<l_{v}}\mathbf{V}_{l_{1}\ldots l _{v}}\). The use of probability theory concepts is based on the assumption that the input parameters are random variables distributed in \(U^{d}\), which defines \(f_{l_{1}\ \ldots\ l_{v}}(x_{l_{1}},x_{l_{2}},\ldots,x_{l_{v}})\), and also as random variables with variances regarding Equation (7).