## 3 Methodology  
The data used in this study were obtained from a publicly available data repository of the NASA Ames Prognostic Center of Excellence (PCoE) [32]. The study was conducted utilizing the extracted data from the NASA experimental battery No. 7, which was run through three different operational profiles, namely, charge, discharge, and impedance at room temperature. However, the capacity degradation study of Li-ion batteries requires voltage, current, and temperature to be measured during the discharge phase. Discharge was carried out at a constant current level of 2 A until the voltage dropped to 2.2 V. Repeated discharge cycles result in accelerated aging, and the process was halted once the battery reached its end of life (i.e., 30% rated capacity). Suppose that a battery has been used for \(N\) charge-discharge cycles. The study defined the training data as \(D_{train}=(X_{train},~{}Y_{train})\) where \(X_{train}=\{v_{ij},~{}c_{ij},~{}t_{ij},~{}i=1,~{}\dots,~{}N,~{}j=1,~{}\dots,~{} m_{i}\}\), which includes the monitoring measurements of voltage, current, and temperature for the first \(N\) cycles at \(m_{i}\) time points, and \(Y_{train}=\{y_{i},~{}i=1,~{}\dots,~{}N\}\), the battery capacity data for the corresponding cycles, as the target. Based on the training data, we aim to \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Authors**} & \multicolumn{4}{c}{**Data-Driven Models**} \\ \cline{2-13}  & **Bayesian** & **Gaussian** & **Forecast** & **Kalman** & **Particle** & **Particle** & **Autoergressive** & **Neural** & **Support** & **Relevance** & **Vector** & **Vector** & **Precision** & **LASSO** \\  & **Regression** & **Process** & **Filler** & **Filter** & **Siarman Optimization** & **Based** & **Network** & **Machine** & **Machine** & **PCA** & **Regression** \\ \hline [PERSON] et al., 2016 [20] & ✓ & & & & & & & & & & & \\ \hline [PERSON] et al., 2015 [21] & ✓ & & & & & & & & & & & \\ \hline [PERSON] et al., & & & & & & & & & & & & \\ \hline [PERSON] et al., & & & & & & & & & & & & \\ \hline
2006 [23] & ✓ & & & & & & & & & & & \\ \hline [PERSON] et al., & & & & & ✓ & & & & & & & \\ \hline
2013 [24] & ✓ & & & & & & & ✓ & & & & \\ \hline [PERSON], 2014 [25] & ✓ & & & & & & ✓ & & & & \\ \hline [PERSON] et al., & & & ✓ & & & & & & ✓ & & \\ \hline
2013 [26] & & & & ✓ & & & & & ✓ & & \\ \hline [PERSON] et al., 2015 [27] & & & & & ✓ & & & ✓ & & & \\ \hline [PERSON] et al., & & & & & & & & ✓ & & \\ \hline
2018 [28] & & & & & & & & & & & \\ \hline [PERSON] et al., & & & & & & & & & & & \\ \hline [PERSON] et al., 2014 [30] & & & & ✓ & & & & & ✓ & \\ \hline [PERSON] et al., & & & & & & & & & & ✓ & \\ \hline
2015 [6] & ✓ & & & & & & & & & ✓ & \\ \hline [PERSON] et al., 2017 [21] & & & & & & & & & ✓ & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Literature of data-driven models’ classification. The check mark indicates corresponding models used in each reference item. predict \(D_{test}=(Y_{test})=\{y_{i},\;i=N+1,\;\ldots,\;N+P\}\) the battery capacity for forthcoming \(P\) cycles. Note that the testing data does not include \(X_{test}\) as the monitoring observations are not available when the prediction is made in real applications. We instead attempt to predict \(X_{test}=\{v_{ij},\;c_{ij},\;t_{ij},\;i=N+1,\;\ldots,\;N+P,\;j=1,\;\ldots,\;m _{i}\}\) based on \(X_{train}\), and use the predicted monitoring measurements as the input of the prediction model. Figure 1 shows the graphical representation of the prediction model building and testing. For the model training, the functional principal component (fPC) scores are extracted from the first \(N\) cycles of voltage, current, and temperature measurements. These fPC scores are utilized to predict the battery capacity via the least absolute shrinkage and selection operator (LASSO), which is a penalized regression procedure used to fit the model by shrinking the coefficient to zero; this results in a biased prediction outcome with low prediction variance and enhanced prediction accuracy.