## 4 Experimental Results  
The fPCA model was employed to analyze the initial 100 cycles of data of voltage, current, and temperature to provide fPC scores and eigenfunctions. Based on the BIC value, the cubic spline was adopted to fit the model. We set the number of knots to 35, which provided better results than other options. Figure 2 illustrates examples of the fitted B-spline basis expansion model for the first 25 cycles of voltage curves. The fPCA for voltage provided fPC scores and eigenfunctions, where the first fPC was responsible for 75.4% of the data variation and the second fPC was accountable for 21.1% of the data difference. Similar results were found in the case of current. The first fPC was responsible for about 67.2% of the variation and the second fPC was responsible for 29.2% of the variation in the functional current data. The temperature shows multiple fPCs for the variation, where the majority was covered by both the first and second fPCs. The first fPC was responsible for 42% and the second fPC covered 33% of data variation. Figure 3 demonstrates the corresponding eigenfunctions against fPC scores that we obtained from voltage, current, and temperature. The battery capacity prediction begins with predicting the voltage, current, and temperature of the forthcoming cycles using simple linear regression. We obtained similar shapes of cycles as those of the preceding 100 cycles. Figure 4 illustrates the predicted 20 cycles of current, temperature, and voltage. The fPC scores that we obtained through the model are the input for the generalized regression model, where we used adaptive LASSO regression with five-fold cross-validation. LASSO was used to train the preceding fPC scores with their corresponding capacity and to help predict the capacity based on the fPC scores of predicted cycles. We considered the initial 100 cycles as a training dataset, and the predicted 20 cycles were used for model testing. Figure 5 shows the plot for predicted capacity versus capacity residuals for the training dataset. Figure 2: B-spline fitted for the first 25 cycles of voltage curves. Figure 4: The predicted 20 forthcoming cycles of current, temperature, and voltage. Figure 5: Actual versus predicted capacity for training and testing datasets. Figure 3: Graphical demonstration of the fPCA for monitoring data. The mean functions and eigenfunctions of (**a**) voltage, (**b**) current, and (**c**) temperature are displayed. As shown in Figure 5, the residuals are randomly distributed, which corroborates the model validity and facilitates credible prediction outcomes. The result provides evidence that the capacity prediction appears to be closer to the actual curve and is more accurate with more capacity data if available for further updates. Initially, the study considered 100 cycles and predicted the capacity for the subsequent 20 cycles. We continued the process considering 120 cycles and predicted the following 20 cycles. This process helped us to check the model and verify its accuracy of prediction. Figure 6 shows the outcome of the model capacity from cycle 1 to 120. The fitted capacity curve is color coded, so that fitted capacity and predicted capacity are evident. The errors, such as root mean square error (RMSE) and mean absolute percentage error (MAPE), show that the model works well at the beginning, when both the RMSE and MAPE are minimal. However, both errors increase as the model predicts end cycles. Table 2 illustrates the prediction performances using different training cycles for three separate cases. Note that the small error values from MAPE and RMSE represent better accuracy, as shown in Table 2. The study considered three scenarios to develop the empirical model and test the dataset capable of predicting the capacity at any point of the battery's lifespan. For the first case, we took the first 100 cycles to obtain the mean function and derive eigenfunctions sufficient to explain the overall variation with the help of LASSO regression.