## 4 Experimental Results and Discussion
### Result Analysis  
We validate the effectiveness of our experimental method by using four battery sets from the NASA dataset as examples and comparing it with four other methods (DELM, PSO-DELM, WOA-DELM, IWOA-DELM, and WD-IWOA-DELM) under the same conditions. The comparison is denoted as A1, A2, A3, A4, and A5, respectively (as shown in Table 4). We conduct comparative experiments with a prediction starting point set to 80 (except for B0018, which is set to 60). The prediction results and errors are presented in Figure 10 and Table 5.  
Figure 10: SOH prediction results and forecast errors of NASA batteries. (**a**) B0005; (**c**) B0006; (**e**) B0007; (**g**) B0018; (**b**,**d**,**f**,**h**) represent the corresponding forecast errors. To verify the generalization and robustness of this method, we take four battery sets from the CALCE dataset as examples and conduct comparative experiments with M1, M2, M3, and M4 under the same conditions. The prediction starting point was set to 400, and the time length was set to 20. The prediction results are presented in Figure 11 and Table 6. Figure 12 presents the probability density function and linear fitting curve for CS2_38. Figure 13 compares the _RMSE_ values among different models. From Figure 11, it can be observed that the capacity recovery phenomenon in the four groups of batteries from CALCE is more pronounced. Taking CS2_37 as an example, the PSO-DELM and WOA-DELM models show a gradual increase in fitting mid-term errors, and although the IWOA-DELM model improves the fitting effect, the prediction accuracy still needs improvement. The proposed method in this paper has an _RMSE_ mean of 0.72% and an _MAPE_ mean of 0.89%, which is an improvement compared to the _RMSE_ mean of 1.37% and _MAPE_ mean of 1.81% of the IWOA-DELM, with an increased accuracy of 0.65% and 0.92%, respectively. The prediction curve can better fit the real values at local fluctuations, indicating good fitting performance. Figure 12a shows the probability density function for CS2_38, and the battery probability density curve obtained by our method is the highest, narrowest, and follows a normal distribution trend, indicating that the predicted values by our method are very close to the real values. Figure 12b displays  
Figure 11: SOH prediction results and forecast errors of CALCE batteries. (**a**) CS\(-\)35; (**b**) CS\(-\)36; (**c**) CS\(-\)37; (**d**) CS\(-\)38. Additionally, to validate the performance of the proposed model in this study, we compared it with other commonly used SOH estimation models, namely EMD-GRU-ARIMA [33], EMD-TCN [1]. All algorithms utilize identical training and testing sets. Due to varying choices of performance metrics in different literature, this study employs _RMSE_ and _MAPE_ as evaluation criteria. The specific comparative results are presented in Table 7.  
Taking Table 7 of the NASA dataset as an example, in the EMD-TCN method, the capacity signal was first divided into the overall degradation trend and local degradation trend. Then, the dilated techniques in TCN were used to effectively capture the local capacity regeneration signal. However, this method does not consider the mode mixing phenomenon existing in EMD, and a single TCN model lacks sufficient generalization. The EMD-GRU-ARIMA method uses the CMSE to search for the boundary between high and low frequencies, where the high-frequency part is predicted using the GRU model, and the low-frequency part is predicted using the ARIMA model, resulting in improved accuracy. Nevertheless, this method may suffer from overfitting when dealing with a small amount of data or in the presence of high noise. Building upon this, our proposed method reduces the _RMSE_ by approximately 0.003, indicating not only higher prediction accuracy but also stronger generalization capabilities, leading to stable prediction results for different batteries. In addition, [PERSON] et al.