# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 4 Results and Discussion
### Sensitivity Analysis of the Turboshaft Engine Performance  
After analyzing the engine performance uncertainty, a next important task is to explain the contributions of the uncertain parameters to the performance uncertainties of the PMs and determine the important uncertain parameters. To this end, a variance-based sensitivity analysis which is described in Section 3.3 is implemented for each PM. In this section, all the uncertain parameters are assumed to be independent identically distributed and follow the truncated Gaussian distribution \(N\big{(}0,0.01^{2}/3^{2}\big{)}\). As demonstrated in Section 3.3, the estimation of the sensitivity index (i.e., \(\bar{S}_{X,N}\)) is mainly affected by the approximation error and sampling error. The former results from the function values predicted by the GPMs  
Figure 5: (a) The variation of the distribution of \(T_{45}\) versus standard deviations of the six uncertain parameters; (b) the variation of the distribution of \(P_{e}\) versus standard deviations of the six uncertain parameters; (c) the variation of the distribution of SFC versus standard deviations of the six uncertain parameters. and the latter is caused by the finite sample size. Thus, it is essential to firstly investigate the effects of these two kinds of errors. To investigate the approximation error, the sample size \(n\) in the formula of \(\widetilde{S}_{X,N}\) is set to 20,000. For each sample, the distributions of three PMs are predicted by using the built GPMs. Moreover, for each predicted distribution, 100 samples are sampled. Note that all the sampling are based on the LHS method. Figure 6 presents the boxplots of the first-order sensitivity indices of six uncertain parameters. The first-order sensitivity indices of six uncertain parameters are denoted as \(S_{1}\), \(S_{2}\), \(S_{3}\), \(S_{4}\), \(S_{5}\), and \(S_{6}\), which corresponds to the order of the parameters in Table 3. This study focuses on the first-order sensitivity index since the first-order sensitivity indices of six uncertain parameters can account for almost 100% of the total variability in each PM, which will be illustrated in the following analysis. As shown in Figure 6, the variation of each sensitivity index is so small that it can be negligible in this study. It also shows the interquartile range for each boxplot, which is displayed above the corresponding boxplot. The small interquartile ranges indicate that the approximation error has a very weak effect on the sensitivity indices. To analyze the sampling error, the mean values and the standard deviations of the sensitivity index \(\widetilde{S}_{X,N}\) under different sample size \(n\) are compared, ignoring the effect of the approximation error. The sample size varies from 2000 to 20,000 with a step of 2000. To estimate the mean value and standard deviation, the calculation of sensitivity indices is repeated 30 times for each sample size. The mean estimates of sensitivity indices with respect to sample size are presented in Figure 7a,c,e. It can be seen that the mean estimates do not change much when the sample size increases from 2000 to 20,000. Thus, mean estimate of sensitivity index is insensitive to the sample size. Since the mean estimate  
Figure 6: (**a**) The boxplot of first-order sensitivity index for \(T_{45}\); (**b**) the boxplot of first-order sensitivity index for \(P_{e}\); (**c**) the boxplot of first-order sensitivity index for SFC (the value above each boxplot represents corresponding the interquartile range). predominantly reduces the influence of sampling error, it can be adopted to perform sensitivity analysis for the three PMs of the turboshaft engine. In Figure 7b,d,f, a significant decrease in standard deviations can be clearly observed as the sample size increases, which means the sample error can be significantly reduced by taking a large sample size. Eventually, the sample size is chosen as 20,000 and the mean value of \(\widehat{S}_{X,N}\) is used to estimate the sensitivity index.