# Statistical Analysis on Random Matrices of Echo State Network in PEMFC System's Lifetime Prediction
## 3 Experimental Results
### Quasi-Dynamic Operating Condition  
In the quasi-dynamic operating condition, the stack output power is used as the degradation-related health indicator. Similar to the steady-state operating condition, half of the data (0-500 h) is used for training the ESN, and the rest of the data (501-1000 h) is used for prediction. When \(\textbf{W}_{in}\) and **W** follow the uniform distribution, the RUL prediction results are shown in Figure 7. Taking the first 10 groups of ESNs as an example, their hyperparameters and RMSE are shown in Table 3.  
After changing the distribution shapes of \(\textbf{W}_{in}\) and **W** to Gaussian distribution, the RUL prediction results under the quasi-dynamic operating condition are shown in Figure 8. The hyperparameters and RMSE of the first 10 groups of ESNs are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Test Group**} & \multicolumn{2}{c}{**ESN\({}_{1}\)**} & \multicolumn{2}{c}{**ESN\({}_{2}\)**} & \multicolumn{2}{c}{**ESN\({}_{3}\)**} & \multicolumn{2}{c}{**ESN\({}_{4}\)**} & \multicolumn{2}{c}{**ESN\({}_{5}\)**} & \multicolumn{2}{c}{**ESN\({}_{6}\)**} & \multicolumn{2}{c}{**ESN\({}_{7}\)**} & \multicolumn{2}{c}{**ESN\({}_{8}\)**} & \multicolumn{2}{c}{**ESN\({}_{9}\)**} & \multicolumn{2}{c}{**ESN\({}_{10}\)**} \\ \hline Hyper- & \(\alpha\) & 0.87000 & 0.35830 & 0.38420 & 0.10110 & 0.87420 & 0.54400 & 0.84620 & 0.82820 & 0.10000 & 0.10900 \\ para. & \(\rho\) & 1.36500 & 1.80570 & 1.58110 & 1.51860 & 1.47940 & 1.90000 & 1.83080 & 1.43060 & 1.76060 & 1.86000 \\  & \(\gamma\) & 0.00790 & 0.00830 & 0.00900 & 0.00900 & 0.00330 & 0.00900 & 0.00880 & 0.00100 & 0.00100 & 0.00580 \\ \hline RMSE & 1.03890 & 0.93920 & 1.28170 & 0.89560 & 0.96380 & 1.37308 & 1.05020 & 0.97690 & 1.13970 & 0.90080 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Quasi-dynamic results of \(\text{ESN}_{1}\)-\(\text{ESN}_{10}\) when \(\textbf{W}_{in}\) and **W** follow the Gaussian distribution.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Test Group**} & \multicolumn{2}{c}{**ESN\({}_{1}\)**} & \multicolumn{2}{c}{**ESN\({}_{2}\)**} & \multicolumn{2}{c}{**ESN\({}_{3}\)**} & \multicolumn{2}{c}{**ESN\({}_{4}\)**} & \multicolumn{2}{c}{**ESN\({}_{5}\)**} & \multicolumn{2}{c}{**ESN\({}_{6}\)**} & \multicolumn{2}{c}{**ESN\({}_{7}\)**} & \multicolumn{2}{c}{**ESN\({}_{8}\)**} & \multicolumn{2}{c}{**ESN\({}_{9}\)**} & \multicolumn{2}{c}{**ESN\({}_{10}\)**} \\ \hline Hyper- & \(\alpha\) & 0.29901 & 0.10970 & 0.10820 & 0.13740 & 0.10000 & 0.34770 & 0.10000 & 0.41590 & 0.10800 & 0.12280 \\ para. & \(\rho\) & 1.90000 & 1.80230 & 1.74110 & 1.90000 & 1.62237 & 1.55670 & 1.90000 & 1.42740 & 1.42460 & 1.61780 \\  & \(\gamma\) & 0.00299 & 0.00710 & 0.00860 & 0.00150 & 0.00900 & 0.00110 & 0.00900 & 0.00890 & 0.00900 & 0.00100 \\ \hline RMSE & 1.07394 & 1.01290 & 1.02710 & 1.10380 & 1.41801 & 1.03080 & 1.88830 & 1.28040 & 1.09540 & 1.49140 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Quasi-dynamic results of \(\text{ESN}_{1}\)-\(\text{ESN}_{10}\) when \(\textbf{W}_{in}\) and **W** follow the uniform distribution.  
Figure 7: Quasi-dynamic RUL results under the uniform distribution: (**a**) Prediction results of 100 individual ESNs (blue lines); (**b**) Mean prediction results (blue line) of ensemble ESN with 95% CI.  
To analyze the effects of the two matrices' distribution shapes of \(\mathbf{W}_{in}\) and \(\mathbf{W}\) on the prediction accuracy, the RMSE of 100 groups of ESNs is shown in Figure 9. The average RMSE under the Gaussian distribution (1.19634) is lower than the average RMSE under the uniform distribution (1.27562) for the quasi-dynamic test. The mean prediction results of the ensemble ESN are calculated, i.e., 0.95657 for uniform distribution and 0.90618 for Gaussian distribution.