## 4 Results and Comparison
### Comparison with Previous Studies
#### 4.2.2 Performance Comparison  
The performances of the proposed approach and the eight previous solutions [32, 33, 34, 35, 36, 37, 38, 39] are compared in Tables 9 and 10, which compare the relative error and score, respectively. The overall scores are computed by averaging five test bearings under the operating condition of 1800 rpm/4000 N.  
From the comparison, we can observe that the proposed IBL approach can achieve a comparable performance with the eight representative solutions to the challenging RUL estimation problem. The overall score of the proposed approach (i.e., \(Score=0.3951\)) and the other eight solutions are all in the range from 0.0000 to 0.5000. However, none of the nine approaches can consistently provide accurate RUL estimations for all five test bearings. For example, the proposed approach has excellent performance for Bearing1-5 with almost full score, while it is unable to estimate as accurately for Bearing1-6. On the other hand, the estimation results from [PERSON] et al. show high errors for Bearing1-5 (i.e., \(Score=0.0313\)), even though this solution for Bearing1-3 is ranked at the top on the score table (i.e., \(Score=0.9526\)). The latest study of [PERSON] et al. provided very actuate RUL prediction for Bearing1-6 (i.e., \(Score=0.9683\)), but performed extremely poorly with zero scores for Bearing1-3 and Bearing1-4. Even though there is the limited number of test instances in this study, we tentatively apply the Mann-Withney test [24] to see whether the proposed approach is significantly different from the existing solutions statistically. Table 11 provides the \(p\)-values of the test between the proposed approach with the eight existing solutions. It can be found that the approach is significant different with the [PERSON]'s solution but not considered differently with the others by setting a significance level \(p\)-value = 0.05. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Test Bearing**} & \multicolumn{3}{c}{**[PERSON]**} & \multirow{2}{*}{**[PERSON] [36]**} & **[PERSON] et al.** & **[PERSON]** & **[PERSON] et al.** [39] & **[PERSON] et al.** & **[PERSON] et al.** & **[PERSON] et al.** & **Proposed** \\  & & & [37] & **et al.** [38] & **et al.** [38] & **[PERSON] et al.** [39] & **[40]** & **[41]** & **[42]** & **Approach** \\ \hline
1-3 & 0.2774 & 0.0420 & 0.8657 & 0.8706 & 0.9526 & 0.2231 & 0.7679 & 0.0772 & 0.1877 \\
1-4 & 0.0625 & 0.0346 & 0.0549 & 1.0000 & 0.8236 & 0.0962 & 0.0000 & 0.1044 & 0.3299 \\
1-5 & 0.7320 & 0.0897 & 0.0000 & 0.1539 & 0.0313 & 0.0413 & 0.0000 & 0.6543 & 0.9688 \\
1-6 & 0.5000 & 0.1000 & 0.5144 & 0.2031 & 0.3779 & 0.4791 & 0.9683 & 0.8922 & 0.0909 \\
1-7 & 0.7579 & 0.0391 & 0.3722 & 0.1250 & 0.0665 & 0.5391 & 0.0508 & 0.6892 & 0.3032 \\ \hline Overall & 0.4660 & 0.0611 & 0.3614 & 0.4705 & 0.4504 & 0.2758 & 0.3574 & 0.4835 & 0.3747 \\ \hline \end{tabular}
\end{table} 
Table 10: Comparison of performance scores of the proposed approach with the previous solutions. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Test Bearing**} & \multicolumn{6}{c}{**Relative Error of RUL, (\%+\(Er\))**} \\ \cline{2-10}  & **[PERSON]** & **et al.** [35] & **[PERSON] [36] & **[PERSON] et al.** & **[PERSON]** & **[PERSON] et al.** [39] & **[PERSON] et al.** & **[PERSON] et al.** & **[PERSON] et al.** & **Proposed** \\  & & & [37] & **et al.** [38] & **et al.** [38] & **et al.** [40] & **[41]** & **[42]** & **Approach** \\ \hline
1-3 & 37 & 91.4 & \(-1.04\) & \(4\) & \(-0.35\) & \(43.28\) & \(7.62\) & \(73.89\) & \(-12.1\) \\
1-4 & 80 & 97.1 & \(-20.94\) & \(0\) & \(5.60\) & \(67.55\) & \(-157.71\) & \(65.19\) & \(32.4\) \\
1-5 & 9 & 69.6 & \(-278.26\) & \(54\) & \(100\) & \(-22.98\) & \(-72.52\) & \(12.24\) & \(0.92\) \\
1-6 & \(-6\) & 66.4 & \(19.18\) & \(46\) & \(28.08\) & \(21.23\) & \(0.93\) & \(3.29\) & \(69.2\) \\
1-7 & \(-2\) & 93.5 & \(-7.13\) & \(60\) & \(-19.55\) & \(17.83\) & \(85.99\) & \(10.74\) & \(-8.61\) \\ \hline \end{tabular}
\end{table}