# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 3 Methodologies  
### Gaussian Process Model  
The GPM generally treats the response \(Y(\mathbf{x})\) as a realization of a stationary GP superimposed on a regression model. The prior for the functional relationship between inputs and output can be defined by [17; 18; 19]:  
\[Y(\mathbf{x})=\sum_{j=1}^{p}\beta_{j}f_{j}(\mathbf{x})+Z(\mathbf{x})=\mathbf{F}(\mathbf{x})^{T} \mathbf{\beta}+Z(\mathbf{x}), \tag{9}\]  
where \(\mathbf{x}\in\left[0,1\right]^{d}\) is a vector of normalized inputs, \(d\) is the number of scalar inputs, \(\mathbf{F}(\mathbf{x})=\left(f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),\ldots,f_{p}(\mathbf{x})\right)^ {T}\) is a \(p\times 1\) vector of known regression functions, \(\mathbf{\beta}=\left(\beta_{1},\beta_{2},\ldots,\beta_{p}\right)^{T}\) is a \(p\times 1\) vector of unknown regression coefficients, and \(Z(\mathbf{x})\) is a zero mean stationary GP with covariance function \(Cov(\cdot,\cdot)\). Given any two points \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\), the covariance function of \(Y(\mathbf{x}_{i})\) and \(Y\big{(}\mathbf{x}_{j}\big{)}\) is given by \(Cov\big{[}Y(\mathbf{x}_{i}),Y(\mathbf{x}_{j})\big{]}=Cov\big{[}Z(\mathbf{x}_{i}),Z\big{(} \mathbf{x}_{j}\big{)}\big{]}=\sigma_{z}^{2}R\big{(}\mathbf{x}_{i},\mathbf{x}_{j}\big{)}\), where \(\sigma_{z}^{2}\) is the process variance, \(R(\cdot,\cdot)\) is the correlation function. For a stationary GP, the correlation between \(Z(\mathbf{x}_{i})\) and \(Z(\mathbf{x}_{j})\) only depends on the difference between \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\), which means the correlation function \(R(\cdot,\cdot)\) is a function of \(\mathbf{x}_{i}-\mathbf{x}_{j}\), i.e., \(R\big{(}\mathbf{x}_{i},\mathbf{x}_{j}\big{)}=R\big{(}\mathbf{x}_{i}-\mathbf{x}_{j}\big{)}\). It is noticed that a valid correlation function must be nonnegative for all inputs and satisfy \(R(0)=1\)[19; 28]. This is due to the fact that \(Y(\mathbf{x}_{i})\) and \(Y(\mathbf{x}_{j})\) should be similar if \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\) are sufficiently close. The correlation function is the most important term in the GPM since it gives the GPM the properties of interpolation and spatial correlation [29]. Moreover, it has a direct impact on the smoothness of \(Y(\mathbf{x})\). In general, the power exponential correlation function and the Matern correlation function are two commonly used correlation functions. These two types of correlation functions with \(d\)-dimensional inputs \(\mathbf{x}\) are given by:  
\[R\big{(}\mathbf{x}_{i},\mathbf{x}_{j}\left|\mathbf{\theta}\right.\big{)}=\exp\Bigg{\{}- \sum_{k=1}^{d}\theta_{k}h_{k}^{p_{k}}\Bigg{\}}, \tag{10}\]  
\[R\big{(}\mathbf{x}_{i},\mathbf{x}_{j}|\mathbf{\
u},\mathbf{\theta}\big{)}=\prod_{k=1}^{d} \frac{1}{\Gamma(v)2^{v-1}}\left(\frac{2\sqrt{v}h_{k}}{\theta_{k}}\right)^{v}K_ {v}\left(\frac{2\sqrt{v}h_{k}}{\theta_{k}}\right), \tag{11}\]  
where \(\mathbf{\theta}=(\theta_{1},\theta_{2},\ldots,\theta_{d})^{T}\) is a vector of scale parameters with \(\theta_{k}\geq 0\), \(\mathbf{p}=(p_{1},p_{2},\ldots,p_{d})^{T}\) is a vector of power parameters with \(0<p_{k}\leq 2\), \(h_{k}=\left|x_{ik}-x_{jk}\right|\), \(x_{ik}\) is the \(k\)th element of \(\mathbf{x}_{i}\), \(v\) is the smoothness parameter, \(K_{v}(\cdot)\) is the modified Bessel function of order \(v\) and \(\Gamma(\cdot)\) is the Euler Gamma function. For the power exponential correlation function, Equation (10) provides the exponential correlation function with \(p_{k}=1\) and the Gaussian correlation function with \(p_{k}=2\).