## 4 Results and Discussion
### Experiment and Discussion  
Figure 6 shows the results of normal degradation and capacity regeneration decoupling of the first 100 cycle data of each battery. We can see that the starting time of the capacity regeneration in batteries #5, #6, and #7 is the same, because the starting time of each charge-discharge cycle is the same during the three battery tests. However, due to the influence of initial conditions and other uncertainties in the test process, the regeneration amplitude and quantity of those three batteries are different. Figure 5: The SOH degradation with the cycle of batteries #5, #6, #7, and #18. The absolute error of the RUL prediction results and the root mean square error of the SOH prediction results are used to quantify and evaluate the prediction results, which are defined as follows:  
\[AE=\big{|}\beta-\beta^{p}\big{|}, \tag{28}\]  
\[RMSE=\sqrt{\frac{1}{n-\alpha}\sum_{i=\alpha+1}^{n}\big{(}H_{i}-H_{i}^{p}\big{)} ^{2}}, \tag{29}\]  
The prediction comparison for the RVM, the GM, and the hybrid method for batteries #5, #6, #7, and #18 are shown in Table 1 and Figure 7. The prediction start cycle means predicting the RUL at the current cycle, and the length of the training data is equal to the started prediction cycle. It can be seen from Table 1 that the RUL prediction errors of our hybrid are less than the single RVM and the single GM method at a different started predicton cycle. Moreover, the AE and RMSE are also smaller than the single model. The error message illustrates that our method has high accuracy over the prediction period. The performance of the hybrid model is still satisfied with the expansion of the prediction horizon. More specifically, the prognostic AE started prediction at cycle 60 of batteries #5, #6, #7, and #18 among the single RVM methods 54, 15, 74, and 1, respectively. But, the counterparts of our hybrid model are 4, 15, 1, and 4, respectively. Although the AE on battery #18 is slightly larger, we can see that the prognostic of the proposed method is more accurate and stable. Similarly, as compared with the single GM method, the prognostic results on four batteries are the smallest. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Prediction Start**} & \multirow{2}{*}{**Battery**} & \multirow{2}{*}{**Hybrid Model**} & \multirow{2}{*}{**RVM**} & \multirow{2}{*}{**GM**} \\ \cline{3-3} \cline{5-8}  & & & & & & & \\ \cline{3-8} \multirow{4}{*}{**Cycle**} & \multirow{2}{*}{**Number**} & **AE** & **RMSE (\%)** & **AE** & **RMSE (\%)** & **AE** & **RMSE (\%)** \\ \hline \multirow{4}{*}{100} & \multirow{4}{*}{80} & \#5 & 4 & 0.960422 & - & 3.59411 & 4 & 1.025602 \\  & & \#6 & 1 & 1.241741 & 16 & 6.886879 & 2 & 2.576045 \\  & & \#7 & 1 & 0.921971 & - & 4.072111 & 5 & 1.099363 \\  & & \#18 & - & 1.520362 & - & 0.803822 & - & 1.95701 \\ \hline \multirow{4}{*}{80} & \multirow{4}{*}{80} & \#5 & 1 & 0.8253 & 1 & 2.27695 & 1 & 1.132771 \\  & & \#6 & 15 & 3.06826 & 14 & 6.679174 & 16 & 6.349727 \\ \cline{1-1}  & & \#7 & 7 & 1.191789 & - & 2.726045 & 16 & 2.48364 \\ \cline{1-1}  & & \#18 & 1 & 1.015091 & - & 2.278513 & 3 & 2.664143 \\ \hline \multirow{4}{*}{60} & \#5 & 4 & 1.033175 & 54 & 69.82189 & 13 & 2.724726 \\  & & \#6 & 15 & 3.333834 & 15 & 5.40352 & 26 & 8.576316 \\ \cline{1-1}  & & \#7 & 1 & 0.996845 & 74 & 69.76426 & 12 & 2.203527 \\ \cline{1-1}  & & \#18 & 4 & 1.111508 & 1 & 0.708308 & 8 & 3.333486 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of prediction results of the hybrid method with the relevance vector machine (RVM) and the gray model (GM). Figure 6: The extraction of normal degradation and capacity regeneration. To compare more broadly with other types of prediction methods, we investigated many other model-based and data-driven approaches.