# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 4 Results and Discussion
### Uncertainty Quantification of the Turboshaft Engine Performance  
#### 4.3.1 Comparison of the GPM-Based LHS with the Simulation-Based MCS  
In this study, the propagation of the component performance uncertainty is implemented by LHS. Given each set of uncertain parameters, the engine performance is predicted using the built GPMs. Thus, it is essential to validate the GPM-based LHS. To this end, the uncertainty analysis results under different sample sizes are firstly compared with those obtained by applying MCS into the simulation model. To this end, all the uncertain parameters are assumed to be independent identically distributed. As mentioned in Section 4.1, each uncertain parameter follows a truncated Gaussian distribution. The mean value is set to 0 and three times the standard deviation is set to 0.01. After that, a detailed comparison between the proposed GPM-based LHS and the classical simulation-based MCS is given in Figure 4 by varying the sample size from 1000 to 20,000. With the simulation-based MCS, the mean and variance of each PM are obtained for each given sample size, whereas, the distributions of these two statistics of each PM are estimated through the GPM-based LHS. As shown in Figure 4, the blue dashed lines represent the MCS results with different sample sizes, including the mean and variance of each PM, and the green solid lines represent the mean values of the two statistics estimated by the GPM-based LHS. Moreover, Figure 4 also provides the 0.005-quantiles and 0.995-quantiles of the distributions of the statistics, as represented by the red dotted lines. It can be observed that the proposed method shows a much better convergence in mean estimates than the MCS, as indicated by Figure 4a,c,e. The maximum relative difference of the mean estimates between the MCS and the GPM-based LHS is only about 0.028% when the sample size is larger than 10,000. This validates the accuracy of the GPM-based LHS in estimating the mean values of the PMs. When the sample size is less than 10,000, the proposed method also outperforms the MCS in the robustness of mean estimates and maintains the equal accuracy. In addition, the narrow prediction intervals (between 0.005-quantile and 0.995-quantile) also indicate the mean estimates are credible. In the meantime, it can be observed in Figure 4b,d,f that the sample size seems to have a significant influence on the variance estimates of the PMs for both methods. The convergences of the variance estimates do not perform well like those of the mean estimates in the GPM-based LHS method. However, the proposed method still gives a better convergence than the MCS. For the GPM-based LHS, as the sample size varies from 10,000 to 20,000, the maximum variations of the variance estimate for \(T_{45}\), \(P_{e}\), and SFC are 9.83 \(\times\) 10\({}^{-7}\), 1.87 \(\times\) 10\({}^{-6}\), and 4.08 \(\times\) 10\({}^{-7}\), respectively. As a comparison, the corresponding results for the MCS are 1.31 \(\times\) 10\({}^{-6}\), 3.11 \(\times\) 10\({}^{-6}\), and 5.43 \(\times\) 10\({}^{-7}\). Recall that the above results are based on the normalized sample data. Moreover, taking the results of the MCS as a baseline, the maximum relative errors of the variance (or standard deviation) estimated by the GPM-based LHS are 3.58% (or 1.77%), 3.73% (or 1.85%), and 3.04% (or 1.51%) for \(T_{45}\), \(P_{e}\), and SFC, respectively. The differences between these two methods are very small in terms of the variance estimates, which demonstrates the high prediction accuracy of the proposed method. Additionally, the estimation accuracy of the variance is desired due to the narrow prediction interval. Therefore, the proposed GPM-based LHS method for uncertainty analysis is validated. The method is more robust than the MCS method since the variations of mean estimates and variance estimates are all