# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 3 Methodologies
### Variance-Based Sensitivity Analysis  
The variance-based sensitivity analysis (also called Sobol indices or the Sobol method) is a kind of global sensitivity analysis approach [33; 34]. Based on the basic principle of variance decomposition, the total variance of the model output can be decomposed into fractions which are induced by the uncertain inputs and sets of inputs. Let a model output \(Y(\mathbf{x})\) be a function of uncertain inputs \(\mathbf{x}=\{x_{1},x_{2},\ldots,x_{d}\}\), where \(\mathbf{x}\in[0,1]^{d}\). It should be noted that \(Y\) is a scalar output. Assuming that the elements of \(\mathbf{x}\) are mutually independent, a decomposition of the total variance of \(Y\) can be expressed as [35]:  
\[Var(Y)=\sum_{i=1}^{d}V_{i}+\sum_{i<j}^{d}V_{i,j}+\sum_{i<j<k}^{d}V_{i,j,k}+ \ldots+V_{1,2,\ldots,d}, \tag{30}\]  
where \(V_{i}=Var(E(Y|x_{i}))\), \(V_{i,j}=Var\big{(}E(Y|x_{i},x_{j})\big{)}-V_{i}-V_{j}\), and so on. The term \(Var(E(Y|x_{i}))\) implies the expected reduction in the total variance of \(Y\) when the value of \(x_{i}\) is fixed. Similarly, \(Var\big{(}E\big{(}Y|x_{i},x_{j}\big{)}\big{)}\) is the expected reduction in the total variance of \(Y\) given both \(x_{i}\) and \(x_{j}\). In this case, Equation (30) allows us to partition the total variance of \(Y\) into terms that are associated with inputs and interaction effects between them. Consequently, the quantities \(V_{i}\) and \(V_{i,j}\) can be used to measure the sensitivities of \(x_{i}\) and the interaction between \(x_{i}\) and \(x_{j}\), respectively. In general, both measures \(V_{i}\) and \(V_{i,j}\) are often converted into scale-invariant versions, as follows:  
\[S_{i}=\frac{V_{i}}{Var(Y)}=\frac{Var(E(Y|x_{i}))}{Var(Y)}, \tag{31}\]  
\[S_{i,j}=\frac{V_{i,j}}{Var(Y)}=\frac{Var\big{(}E\big{(}Y|x_{i},x_{j}\big{)} \big{)}-Var(E(Y|x_{i}))-Var\big{(}E\big{(}Y|x_{j}\big{)}\big{)}}{Var(Y)}, \tag{32}\]  
where \(S_{i}\) is called the first-order sensitivity index (or main effect) and \(S_{i,j}\) is referred to as the second-order sensitivity index (or two factor interaction). The index \(S_{i}\) describes the fractional contribution of \(x_{i}\) to the total variance while \(x_{i,j}\) exhibits the fractional contribution of the interaction between \(x_{i}\) and \(x_{j}\) to the total variance. It is noticed that higher-order sensitivity indices are usually not considered since the high-order effects are usually not as predominant as the first-order and the second-order effects. To evaluate the sensitivity indices, an effective index estimator based on pick-freeze method is proposed to estimate those indices. The estimator is given by [36]:  
\[S_{X,N}=\frac{\frac{1}{N}\sum_{i=1}^{N}Y_{i}Y_{i}^{X}-\left(\frac{1}{N}\sum_{i=1 }^{N}Y_{i}\right)\left(\frac{1}{N}\sum_{i=1}^{N}Y_{i}^{X}\right)}{\frac{1}{N} \sum_{i=1}^{N}Y_{i}^{2}-\left(\frac{1}{N}\sum_{i=1}^{N}Y_{i}\right)^{2}}, \tag{33}\]  
where \(S_{X,N}\) is the sensitivity index of \(X\), \(X\in\{x_{1},x_{2},\ldots,x_{d}\}\), \(Y_{i}\) (for \(i=1,2,\ldots,N\)) is a deterministic function value with all sampled inputs \(x\), and \(Y_{i}^{X}\) (for \(i=1,2,\ldots,N\)) is also a deterministic function value with all resampled inputs except \(X\). In practice, the computation of \(S_{X,N}\) usually involves hundreds or thousands of evaluations of the model output. If the model is implemented by a complicated computer code, the estimation of \(S_{X,N}\) will be time-consuming. To reduce the computational cost, the simulation model is approximated by the GPM in this study.