# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 4 Results and Discussion
### Gaussian Process Modeling for the Turboshaft Engine Performance  
Since it is time-consuming to perform uncertainty analysis based on the simulation model, the functional relationships between three PMs and the uncertain parameters are fitted by three GPMs using the training data from a carefully designed experiment. Firstly, the maximin Latin hypercube design (LHD) is employed to obtain the experimental design. The reason for using the maximin LHD is that it can uniformly scatter the points across the experimental region. Thus, the constructed GPM can make predictions with a small prediction error. In most instances, the maximin LHD is often chosen as the standard design for fitting the GPM [19]. Secondly, the run size of the experimental design is another key factor that has a large influence on the prediction accuracy of the GPM. A recommendation shows that the run size of an effective LHD should be at least 10 times the dimension of input parameters [39]. In this study, there are six input parameters involved in the GPMs. Thus, a suitable design requires at least 60 points. In terms of the GPMs of \(P_{e}\), SFC and \(T_{85}\), the prior processes of these PMs are assumed to be independent, which implies the GPMs are constructed respectively. For the GPM of each PM, a constant prior mean function and the Matern (\(v=5/2\)) correlation function are used. This is because the training data obtained from the simulation experiments do not show any strong trend over the experimental region and the Matern correlation function can give better fitting and prediction performance. Finally, the posterior GPMs are estimated with the training data. To evaluate the influence of the run size, several maximin LHDs of different sizes \(\Re\), where \(\Re\in\{60,90,120,150,180\}\), are compared using two performance criteria: normalized root mean squared error (NRMSE) and predictivity coefficient (\(Q^{2}\)). These two criteria are defined as follows:  
\[\text{NRMSE}=\frac{1}{Y_{base}}\sqrt{\frac{1}{n}\left(\sum_{i=1}^{n}\left(Y \big{(}\mathbf{x}^{(i)}\big{)}-\hat{Y}\big{(}\mathbf{x}^{(i)}\big{)}\right)^ {2}\right)}, \tag{37}\]  
\[Q^{2}=1-\frac{\sum_{i=1}^{n}\left(Y\big{(}\mathbf{x}^{(i)}\big{)}-\hat{Y} \big{(}\mathbf{x}^{(i)}\big{)}\right)^{2}}{\sum_{i=1}^{n}\left(Y\big{(}\mathbf{ x}^{(i)}\big{)}-\overline{Y}\right)^{2}},\,\overline{Y}=\frac{1}{n}\sum_{i=1}^{n}Y \Big{(}\mathbf{x}^{(i)}\Big{)}, \tag{38}\]  
where \(\hat{Y}\big{(}\mathbf{x}^{(i)}\big{)}\) denotes the predicted mean value provided by the GPM, \(Y\big{(}\mathbf{x}^{(i)}\big{)}\) denotes the simulated value given by the simulation model, and \(Y_{base}\) denotes the baseline value of the PM. Generally, the closer \(Q^{2}\) is to 1 and the smaller NRMSE is, the more accurate the GPM is. To assess the accuracy of the GPMs with different LHDs, a maximin LHD of size 50 is generated as the validation data. Moreover, for each size in \(\Re\), the design is repeated 30 times to investigate the effect of sampling variability in the LHD. To illustrate the effects of run size and sampling variability, six boxplots regarding NRMSE and \(Q^{2}\) for the GPMs of three PMs are presented in Figure 3. It is seen that all the results show a good convergence. More specifically, from Figure 3a,c,e, it can be observed that the NRMSEs decrease obviously as the run size varies from 60 to 180. Meanwhile, the \(Q^{2}\)s approach to 1 as the size varies from 60 to 180, as shown in Figure 3b,d,f. In addition, the variations of these two criteria resulting from sampling variability also decrease as the run size increases