## 4 Cases Study
### Comparative Study  
A comparative study is set up to validate the effectiveness and advantages of the proposed method in applications of unequal interval data. The method in [8] (it is called the Si method in the following) is used to compare the RUL prediction with the proposed method. The equation \(\hat{X}_{k+1}=X_{k}+\hat{\theta}_{k+1}(T_{k+1}-T_{k})\) is used to predict the wear path by one-step in the Si method. The predicted wear paths of the PCCS are obtained by the proposed method in this article. The actual wear path and predicted wear paths of two methods are shown in Figure 18.  
It should be noted that the smaller the MSE, the higher the accuracy of the prediction of wear loss. The MSE of two methods at each mileage point is illustrated in Figure 19.  
It can be seen that the method in this article has a higher accuracy of the prediction of wear loss than the one by Si method. At the last mileage point, the MSE(28) of the two methods is 0.0062 and 0.0272, respectively. It indicates that the algorithm proposed in this paper improves the accuracy of the prediction for the PCCS wear loss effectively.  
It is essential to point out that the smaller the RE, the higher the accuracy of the prediction. The RE of RUL prediction at the life quantiles of 35%, 55%, 75%, and 95% are calculated to compare the RUL prediction performance of the two methods. The prediction results are shown in Table 5.  
Figure 19: MSE of two methods.  
Figure 18: The actual wear path and predicted wear paths of two methods.  
As can be seen that in Table 5 the RE at the four quantiles of the method in this paper are smaller than the Si method. To intuitively describe the change of RE, the RE of the two methods at each mileage point is given in Figure 20.  
Figure 20 shows that the RE of the method in this paper is smaller than the RE of the Si method. The average RE of the method in this paper and Si method are 0.0702 and 0.1514, respectively. It demonstrates that the method in this paper has a higher accuracy of the RUL prediction than the Si method. The performance of the RUL estimation of the two methods is illustrated in Figure 21.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **35\% Quantile** & **55\% Quantile** & **75\% Quantile** & **95\% Quantile** \\ \hline Si method & 57.1711 (21.66\%) & 40.6560 (17.02\%) & 28.2535 (12.16\%) & 1.3780 (10.05\%) \\ \hline Method of this paper & 64.5765 (11.51\%) & 45.9183 (6.28\%) & 31.9183 (0.77\%) & 1.5580 (1.70\%) \\ \hline \hline \end{tabular}
\end{table}
Table 5: RUL Prediction and RE of the two methods.  
Figure 21: Prediction results of the RUL in the two methods.  
Figure 20: RE of the two methods.  
It demonstrates that the proposed method of this article has distinct advantages compared with the Si method in Figure 21. For example, the real RUL is 1.5320 at the last mileage point, the predicted RUL of this paper and the Si method are 1.5580 and 1.3780, respectively. The corresponding absolute error is 0.0260 and 0.1540, respectively. At \(\tau_{28}\) = 110.4680, the RE of the two methods is 0.0170 and 0.1005, respectively. It shows that the method in this paper can effectively improve the accuracy of RUL prediction.