## 4 Results
### Regression Algorithms
The initial results indicate that by implementing Linear Regression, the performance of the model is slightly higher compared to the regularization methods; however, as previously mentioned, overfitting may occur in the testing process, so that shrinkage techniques are recommended to obtain a more stable and reliable model. Like binary classifiers, cross-validation is used to validate the model, specifically to note the difference in performance using regularization in the regression algorithms. Nested Cross-Validation has been used to optimize the selection of hyperparameters for Lasso and Ridge Regression; however, in the case of Linear Regression, Non-Nested Cross-Validation uses the same data to fit model parameters and assess model performance. The cross-validation score process is illustrated in Figure 9, which compares the performance with and without regularization. A constant value of the Linear Regression score in the cross-validation step indicates that hyperparameter tuning is not applicable; on the other hand, regularization methods show a different performance due to optimized hyperparameter tuning, which is explained by the shrinkage parameters, providing a performance in the range of 97%. It is crucial to mention that, as indicated in the previous subsection, the final performance of the regression algorithms must not differ substantially from the cross-validation process. Finally, the performance of each regression algorithm is evaluated in the testing. The numerical results of the MSE, MAE, and \(R^{2}\) are shown in Table 2.  
All the regression algorithms show a similar performance in the testing step, corroborating the cross-validation process; however, compared to Linear Regression, the regularization methods provide a more optimal performance in the case of multicollinearity between the independent variables in the dataset, which in this specific problem prevents not only overfitting, but also a possible bias due to model simplicity or erroneous assumptions. Figure 10 illustrates the implementation of the regression algorithms, comparing the testing and predicted results of the SOH during the BESS' operation. It is fundamental to point out that points graphically close to the straight line indicate a high-quality level in the implementation of the regression algorithms. In addition, the predictions of SOH values after the EOL criteria are illustrated to show the robustness of the algorithms across the entire dataset, all because considering a significant sample of data points prevents easy predictions, which may result in overfitting due to high variance and low bias. Regarding the RUL results, Table 3 provides the SOH and the corresponding cycles. As shown in Figure 10, the battery reaches 70% of SOH in cycle 446. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Regression** & **MSE** & **MAE** & \(R^{2}\) \\ \hline Linear & 0.0155 & 0.0119 & 0.9779 \\ \hline Ridge & 0.0153 & 0.0110 & 0.9768 \\ \hline Lasso & 0.0188 & 0.0153 & 0.9650 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance of different regression algorithms. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Cycle** & **RUL** & **SOH** \\ \hline
50 th & 396 cycles & 0.9695 \\ \hline
200 th & 246 cycles & 0.8650 \\ \hline
300 th & 146 cycles & 0.7780 \\ \hline
400 th & 46 cycles & 0.7283 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RUL results considering cycles and SOH. Figure 9: Graphical results of the cross-validation for the regression algorithms, with and without regularization. Five folds were selected in 30 trials.