## 4 Cases Study
### RUL Prediction of PCCS  
#### 4.2.1 The RUL Prediction of Sample 1  
In light of the PCCS maintenance requirement, the failure threshold of Sample 1 is \(w\) = 14 mm and the actual failure LM \(L_{\text{fail}}\) = 43 thousand kilometers. Fourteen mileage points are recorded. The inspection of mileage intervals are unequal. Firstly, the unequal interval wear series are converted to equal interval series. Figure 9 shows the result of the conversion process of unequal interval wear series to equal interval ones.  
In order to calculate the prediction of the wear rate coefficient and the PDF distribution of the RUL prediction for the PCCS at each mileage point, firstly, the initial model parameters are taken as \(\Theta\) = [0.3,0.2,0.01,0.1]T, the equation \(\hat{\mathcal{X}}_{k_{+1}}=X_{k}+\hat{\theta}_{k_{+1}}(T_{k+1}-T_{k})\) is used to predict the wear path by one-step, and then, the predicted results are restored by Equation (29). The predicted wear trend of the proposed method and the actual wear pa  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Category**} & **Original** & **Thickness of** & **Original** & **Mileage of the** & **LM** \\  & **Thickness** & **the Last Point** & **Mileage** & **Last Point** & **1000 km** \\  & **mm** & **mm** & **1000 km** & **1000 km** & \\ \hline Sample 1 & 18.5 & 4.6 & 850.786 & 893.473 & 42.687 \\ Sample 2 & 19.10 & 4.7 & 1394.295 & 1504.763 & 110.468 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The selected wear data for PCCS.  
Figure 8: The wear path of H type for PCCS.  
It can be seen from Figure 10 that the predicted wear loss is very close to the actual wear loss. The mean squared error (MSE) of the predicted wear and the actual wear at each mileage point is used to assess the prediction results. The MSE index at each monitoring mileage point \(\tau_{k}\) is defined by:  
\[MSE(k)=\sum_{i=1}^{k}{[Z(\tau_{k})-Z_{pre}(\tau_{k})]^{2}/k} \tag{30}\]  
where \(Z_{pre}(\tau_{k})\) is the restored wear loss at \(\tau_{k}\) after one-step prediction.  
The MSE of Sample 1 at each mileage point is illustrated in Figure 11.  
Figure 11: MSE of sample 1.  
Figure 9: The conversion process.  
The average MSE between the actual wear and predicted wear is 0.0047 which is small. For example, at the last mileage point MSE(14) = 0.0041, the prediction result is very good, which demonstrates that the proposed model can predict the wear of the PCCS very well.  
In order to demonstrate the prediction ability of the proposed method, the PDF of RUL at life quantiles of 30%, 50%, 70%, and 90% are applied to compare the results, \(\tau_{k}\)\(\in\)[0.30\(\tau_{a}\), 0.50\(\tau_{a}\), 0.70\(\tau_{a}\), 0.90\(\tau_{a}\)], where \(\tau_{a}\) is the actual LM of the PCCS. The prediction results are shown in Figure 12.  
Figure 12 shows that the PDF of the RUL becomes narrower with the accumulation of wear data. Hence it indicates that the prediction result is better and better.  
The relative error (RE) between the real and predicted RUL is adopted as a performance measure of prediction ability. The RE at each monitoring mileage point \(\tau_{k}\) is formulated by:  
\[RE(k)=|L_{real}(\tau_{k})-E_{pre}(\tau_{k})|/L_{real}(\tau_{k}) \tag{31}\]  
where \(L_{\text{real}}(\tau_{k})\) is the real RUL, here \(L_{\text{real}}(\tau_{k})=L_{\text{fail}}\)-\(\tau_{k}\), \(E_{\text{pre}}(\tau_{k})\) is the expectation of RUL prediction at \(\tau_{k}\).  
The RE of Sample 1 is calculated by the Equation (31). The illustration of the results is given in Figure 13.  
In Figure 13, it can observe that the RE at the last three mileage point exceeds 0.1, but the RE of the other mileage points are smaller than 0.1. At the last mileage point, the RE(14) = 0.0083. It indicates that the proposed method in this paper is feasible.  
Figure 12: The PDF of the RUL prediction at life quantiles.  
Figure 13: RE of sample 1.