## 4 Results
### Capacity Fade Forecasting Results  
This section analyzes the evolution of the capacity fade of the four analyzed cells with the number of charge-discharge cycles. Figure 7 shows three ARIMA predictions at different cycles for each analyzed cell. The training range varies between 30% (left), 50% (middle), and 80% (right) of the cell lifetime. Each plot shows the behavior of the measured capacity fade \(\Delta C\) (black line, which corresponds to the objective time series) and the forecasting results using the proposed health indicator \(\Delta V_{RMS}\) to estimate \(\Delta C\) (blue line) and the forecast directly using \(\Delta C\) (red line). Note that the solid lines of \(\Delta V_{RMS}\) and \(\Delta C\) refer to known values while the dashed lines refer to predicted values. From Figure 7, it can be seen that both forecasting methods improve their accuracy as more data points are available. It can also be seen that both methods give similar results for the same initial conditions. To measure the performance of the methods, Table 3 shows the root-mean-square error (RMSE) of the blue (\(\Delta V_{RMS}\) prediction) and red lines (\(\Delta C\) prediction) of Figure 7, which is calculated against the black line (objective data series) at different cycles. Figure 5: DWT filter applied to the computed values of \(\Delta V_{RMS}\). The figure shows the original signal, its approximation or filtered signal, and the three details removed from the original signal, corresponding to successive high-pass filters. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Cell**} & \multicolumn{2}{c}{**B0005**} & \multicolumn{2}{c}{**B0006**} & \multicolumn{2}{c}{**B0007**} & \multicolumn{2}{c}{**B0018**} \\ \hline \multicolumn{2}{c}{**Forecast Start**} & & & & & & & & \\
**Point (\% of Total** & **Method** & **\(\Delta V_{\text{RMS}}\)** & **\(\Delta C\)** & **\(\Delta V_{\text{RMS}}\)** & **\(\Delta C\)** & **\(\Delta V_{\text{RMS}}\)** & **\(\Delta C\)** & **\(\Delta V_{\text{RMS}}\)** & **\(\Delta C\)** \\
**Useful Life** & & **[Ah]** & **[Ah]** & **[Ah]** & **[Ah]** & **[Ah]** & **[Ah]** & **[Ah]** & **[Ah]** & **[Ah]** \\ \hline \multirow{3}{*}{30\%} & LF & 0.375 & 0.220 & 0.275 & 0.067 & 0.267 & 0.156 & 0.226 & 0.149 \\  & ARIMA & 0.235 & 0.164 & 0.230 & 0.054 & 0.168 & 0.067 & 0.172 & 0.049 \\  & LF & 0.303 & 0.136 & 0.134 & 0.128 & 0.181 & 0.071 & 0.219 & 0.050 \\  & ARIMA & 0.068 & 0.026 & 0.083 & 0.191 & 0.161 & 0.124 & 0.263 & 0.056 \\  & LF & 0.204 & 0.052 & 0.070 & 0.182 & 0.091 & 0.024 & 0.167 & 0.046 \\  & ARIMA & 0.059 & 0.024 & 0.080 & 0.089 & 0.060 & 0.026 & 0.077 & 0.073 \\  & LF & 0.154 & 0.027 & 0.065 & 0.137 & 0.066 & 0.029 & 0.089 & 0.064 \\  & ARIMA & 0.054 & 0.030 & 0.033 & 0.064 & 0.036 & 0.023 & 0.044 & 0.056 \\  & LF & 0.110 & 0.029 & 0.070 & 0.113 & 0.045 & 0.036 & 0.046 & 0.078 \\  & ARIMA & 0.060 & 0.025 & 0.020 & 0.033 & 0.040 & 0.015 & 0.040 & 0.087 \\  & LF & 0.075 & 0.037 & 0.064 & 0.081 & 0.030 & 0.036 & 0.056 & 0.092 \\  & ARIMA & 0.038 & 0.034 & 0.024 & 0.033 & 0.021 & 0.022 & 0.023 & 0.037 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparative RMSE results between the proposed health indicator \(\Delta V_{\text{RMS}}\) and the capacity fade \(\Delta C\) with respect to the measured capacity fade \(\Delta C\) (objective time series) for each studied cell with a discharge current of 2 A (C-rate of 1). Figure 6: Raw and filtered values of the \(\Delta V_{\text{RMS}}\) health indicator for the analyzed cells when discharged at a C-rate of 1. Figure 7: ARIMA predictions at different cycles for each analyzed cell with a discharge current of 2 A (C-rate of 1). Behavior of the measured capacity fade \(\Delta C\) (black line, which corresponds to the objective time series), the prediction results with the proposed \(\Delta V_{\text{RMS}}\) health indicator in the scale of \(\Delta C\) (blue line), and the prediction of \(\Delta C\) (red line).