## 3 Experimental Study
### Experimental Results  
There are 100 engines in the data set. Here, we choose engine No. 21, No. 35, No. 51, and No. 100 as examples. The RUL prediction results are shown in Figure 9.  
Figure 8: Information proportion of variance of principal components. The figures show that there is great uncertainty in the initial stage. Then, the CI gradually narrows as the aero-engine degrades, and, finally, the fault interval is accurately predicted. From the partially enlarged image, it can be seen that even if the point prediction does not completely overlap with the real-life degradation curve, the CI can still provide a reliable prediction range. The same conclusion can be made from the results of other engines in the data set, which are not displayed due to the space limitation. To illustrate the effectiveness of the BiGRU-IVBI model in the proposed method more comprehensively, under the same network configuration conditions, we compared our model with the Bayesian multi-layer perception (BMILP), Bayesian long short-term memory (BLSTM), Bayesian gate recurrent unit (BGRU), and Bayesian bidirectional long short-term memory (BBiLSTM) models. Table 2 shows the RUL prediction results of the comparison with 95% CI. It can be clearly seen from the data in Table 2 that our BiGRU-IVBI method has the best performance among the four models with the lowest _RMSE_, _MAE_, _Score_, and _SMAPE_. To make the results more intuitive, a line chart is also provided in Figure 10.  
Simultaneously, we conducted an experiment using unprocessed data without the PCA feather-extraction process. It is evident from Table 2 and Figure 10 that the experimental results are inferior to the results obtained using PCA, which explains the effectiveness of PCA in improving the prediction results. All of the above results indicate that our model is feasible and is the most cost-effective one. Figure 9: RUL prediction results of four engines: (**a**) engine No. 21; (**b**) engine No. 35; (**c**) engine No. 51; (**d**) engine No. 100. As seen in Table 3, we compare the proposed method with some advanced methods on the CMAPSS data sets mentioned in the literature, and the results show that our method has a significant improvement in the \(RMSE\) on data set FD001. This means that the RUL predicted by our proposed method is closer to the real RUL and achieves more satisfactory accuracy. We also conducted comparative experiments for different window sizes and finally set the window size to \(T=30\). The comparative experimental results of the different window sizes are shown in Table 4 and Figure 11. \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \(RMSE\) & \(MAE\) & \(Score\) & \(SMAPE\) (\%) \\ \hline BMLP & 37.90 & 35.09 & 22.34 & 38.80 \\ BLSTM & 25.13 & 20.08 & 9.05 & 23.16 \\ BGRU & 24.58 & 19.49 & 8.33 & 22.07 \\ BBiLSTM & 11.04 & 10.30 & 3.02 & 13.23 \\ BiGRU-IVBI & 9.91 & 7.60 & 1.84 & 11.96 \\ BiGRU-IVBI (non-PCA) & 11.47 & 9.43 & 2.75 & 13.86 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Prediction performance comparison with other models. Figure 10: Evaluation indexes based on different models. In order to illustrate the effect of the parameter \(\rho\) in the Gaussian mixture model on the prediction performance, we also compare the experimental results of several Gaussian mixture models with different mixing ratios \(\rho\), and the comparison results are shown in Table 5 and Figure 12. In order to ensure the fairness of the experiment, the average value of five experiments is taken for all of the above test data. We regularly take \(\rho\) within the range from 0 to 1 with a step size of 0.25. It can be found from the table and the figure that the RUL prediction error of the network is at its minimum when \(\rho=0.25\). It is worth noting that the prior of the Gaussian mixture distribution used in this paper will degenerate to a single Gaussian distribution when \(\rho=0\) and \(\rho=1\).