## 4 Results and Observations
### Further Validation on the Test Bearings under Condition C3  
Next, further validation is introduced to the test bearings operating under condition 3, where the shaft speed is 40 Hz with 10 kN hydraulic loading. For Bearing 5, the ESI exhibits a steady plateau before the inner-race fault is found at the 327 th minute, after which it surges rapidly to the maximum value at the 353 rd minute. This indicates that Bearing 5 undergoes a rapid degradation stage that lasts for merely 26 minutes. Based on the EKF algorithm outlined in Section 3.4.4, the bearing end-of-life time can be estimated using the ESI observations from the 327 th minute to the various prediction times, \(t_{PT}\). With the noise levels of the state process and the measurement set at \(v_{t}=0.1\) and \(w_{t}=0.1\), the results of the state estimation and further extrapolation based on the EKF are shown in Figure 14.  
As for Bearing 6, after the detection of the inner-race fault at the 1418 th minute (\(t_{FDT}=1418\)), the ESI observations since the 1407 th minute (as marked with red circles in Figure 15) are used for degradation state estimation in the EKF. It is seen that the state extrapolation captures the overall increasing trend of the ESI and provides a fairly precise estimate of end-of-life time, \(\overset{\sim}{t}_{\text{EOL}}\). For performance evaluation, the RUL at the prediction time of Bearings 5 and 6 are summarized in Table 7. For Bearing 5, it can be concluded that the absolute residuals of the predicted RUL are less than 5 minutes. As the degradation state represented by the ESI steeply rises to approach the end-of-life threshold, a prompt replacement is suggested for Bearing 5 to avoid critical failures. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **Bearing** \\ **ID** \\ \end{tabular} } & **Baseline** & **Baseline** & **RUL PT \({}^{*}\)\(t_{PT}\)** & **Estimated** & **Baseline** & **Estimated RUL** & **RUL** \\  & **FDT \({}^{*}\)\(t_{FDT}\)** & **EOLT \({}^{*}\)\(t_{FOL}\)** & **/minute** & **Time \(t_{FOL}\)** & **Time \(t_{FOL}\)** & **/minute** & **/minute** & **/minute** \\ \hline \multirow{4}{*}{3} & \multirow{4}{*}{446} & \multirow{4}{*}{490} & 458 & 486 & 32 & 28 (\(-\)4) & [21,34] \\ \cline{3-8}  & & & 459 & 489 & 31 & **30 (\(-\)1)** & [24,38] \\ \cline{3-8}  & & & 460 & 492 & 30 & 32 (\(+\)2) & [26,41] \\ \cline{3-8}  & & & 462 & 485 & 28 & 23 (\(-\)5) & [19,28] \\ \hline \multirow{4}{*}{4} & \multirow{4}{*}{47} & \multirow{4}{*}{160} & 67 & 159 & 93 & **92 (\(-\)1)** & [80,105] \\ \cline{3-8}  & & & 68 & 158 & 92 & 90 (\(-\)2) & [78,102] \\ \cline{3-8}  & & & 69 & 151 & 91 & 82 (\(-\)9) & [73,95] \\ \cline{3-8}  & & & 70 & 150 & 90 & 80 (\(-\)10) & [71,91] \\ \hline \hline \end{tabular}  
* Baseline FDT: first detection time of fault. Baseline EDLT: bearing end-of-life time of ground truth. PT: RUL prediction time.  
\end{table} 
Table 6: RUL Prediction Results of Bearing 3 and Bearing 4 (Condition 2). Figure 14: Bearing 5 degradation state prediction based on the EKF: (**a**) State prediction at the 343 rd minute. The corresponding estimated RUL is \(\dot{r}ul_{343}=\dot{t}_{EOL}-t_{PT}=12\) minutes. (**b**) State prediction at the 345 th minute, with \(\dot{r}ul_{345}=5\) minutes. (**c**) State prediction at the 346 th minute, with \(\dot{r}ul_{346}=3\) minutes. (**d**) State prediction at the 347 th minute, with \(\dot{r}ul_{347}=5\) minutes.