# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 3 Methodologies
### Gaussian Process Model
By conditioning Equation (14) on the sample data, a conditional normal distribution can be derived to be:  
\[P\Big{(}Y(\mathbf{x}^{*})|\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2},\mathbf{\theta}\Big{)} \sim N\Big{(}\mu\Big{(}Y(\mathbf{x}^{*})|\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2},\bm {\theta}\Big{)},var\Big{(}Y(\mathbf{x}^{*})|\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2}, \mathbf{\theta}\Big{)}\Big{)}, \tag{15}\]  
where,  
\[\mu\Big{(}Y(\mathbf{x}^{*})|\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2},\mathbf{\theta}\Big{)} =\mathbf{F}(\mathbf{x}^{*})^{T}\mathbf{\beta}-\mathbf{r}(\mathbf{x}^{*})^{T}\mathbf{R}_{D} {}^{-1}\Big{(}\mathbf{Y}^{n}-\mathbf{F}_{D}(\mathbf{x})^{T}\mathbf{\beta}\Big{)}, \tag{16}\]  
\[var\Big{(}Y(\mathbf{x}^{*})|\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2},\mathbf{\theta}\Big{)} =\sigma_{z}^{2}\Big{(}1-\mathbf{r}(\mathbf{x}^{*})^{T}\mathbf{R}_{D}{}^{-1}\mathbf{r}(\mathbf{x}^{ *})\Big{)}, \tag{17}\]  
When considering multiple new sites, a posterior GP can be derived to be [30]:  
\[Y(\cdot)\Big{|}\Big{(}\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2},\mathbf{\theta}\Big{)} \sim GP\Big{(}\mu\Big{(}\cdot|\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2},\mathbf{\theta} \Big{)},cov\Big{(}\cdot,\cdot|\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2},\mathbf{\theta} \Big{)}\Big{)}, \tag{18}\]  
where \(\mu\big{(}\cdot|\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2},\mathbf{\theta}\big{)}\) has the same expression as Equation (16) and \(cov\big{(}\cdot,\cdot|\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2},\mathbf{\theta}\big{)}\) is given by:  
\[cov\Big{(}\mathbf{x}_{i},\mathbf{x}_{j}|\mathbf{Y}^{n},\mathbf{\beta},\sigma_{z}^{2},\mathbf{ \theta}\Big{)}=\sigma_{z}^{2}\Big{(}R\big{(}\mathbf{x}_{i},\mathbf{x}_{j}\big{)}-\mathbf{r }(\mathbf{x}_{i})^{T}\mathbf{R}_{D}{}^{-1}\mathbf{r}(\mathbf{x}_{j})\Big{)}. \tag{19}\]  
It can be observed that the conditional mean and covariance functions given by Equations (16), (17), and (19) depend on the unknown parameters \(\mathbf{\beta}\), \(\sigma_{z}^{2}\), and \(\mathbf{\theta}\). Based on the collected sample data, a classical maximum likelihood estimation (MLE) method is employed to estimate the unknown parameters. Given the multivariate normal assumption for \(\mathbf{Y}^{n}\), the likelihood function can be written as:  
\[L\Big{(}\mathbf{Y}^{n}\Big{|}\mathbf{\beta},\sigma_{z}^{2},\mathbf{\theta}\Big{)}=\frac{ \big{(}2\pi\sigma_{z}^{2}\big{)}^{-n/2}}{\sqrt{|\mathbf{R}|}}\exp\Bigg{(}-\frac{ \Big{(}\mathbf{Y}^{n}-\mathbf{F}(\mathbf{x})^{T}\mathbf{\beta}\Big{)}^{T}\mathbf{R}^{-1}\Big{(} \mathbf{Y}^{n}-\mathbf{F}(\mathbf{x})^{T}\mathbf{\beta}\Big{)}  
Plugging \(\hat{\mathbf{\beta}}\) and \(\hat{\sigma}_{z}^{2}\) into Equation (21) and omitting the constant term, the log likelihood can be derived to be:  
\[L\Big{(}\mathbf{Y}^{n}\Big{|}\hat{\mathbf{\beta}},\hat{\sigma}_{z}^{2},\mathbf{\theta}\Big{)} =-\frac{1}{2}\Big{[}ln\Big{(}\hat{\sigma}_{z}^{2}(\mathbf{\theta})\Big{)}+ln(|\mathbf{ R}(\mathbf{\theta})|)\Big{]}. \tag{24}\]  
Thus the likelihood function explicitly depends on \(\mathbf{\theta}\) since \(\hat{\mathbf{\beta}}\) and \(\hat{\sigma}_{z}^{2}\) are only functions of \(\mathbf{\theta}\). It turns out that the MLE \(\hat{\mathbf{\theta}}\) of \(\mathbf{\theta}\) is obtained by maximizing \(L\Big{(}\mathbf{Y}^{n}\Big{|}\hat{\mathbf{\beta}},\hat{\sigma}_{z}^{2},\mathbf{\theta} \Big{)}\), which is equivalent to minimizing:  
\[g(\mathbf{\theta})=ln\Big{(}\hat{\sigma}_{z}^{2}(\mathbf{\theta})\Big{)}+ln(|\mathbf{R}( \mathbf{\theta})|). \tag{25}\]  
A suitable numerical optimization method can be used to obtain \(\hat{\mathbf{\theta}}\) and then a model prediction at new site \(\mathbf{x}^{*}\) can be performed by replacing \(\mathbf{\beta},\sigma_{z}^{2}\), and \(\mathbf{\theta}\) with \(\hat{\mathbf{\beta}},\hat{\sigma}_{z}^{2}\), and \(\hat{\mathbf{\theta}}\), respectively.