# Statistical Analysis on Random Matrices of Echo State Network in PEMFC System's Lifetime Prediction
## 3 Experimental Results
### Steady-State Operating Condition  
In the steady-state operating condition, the stack output power is used as the degradation health indicator. 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. During the prediction procedure, 100 individual ESNs have 100 groups of prediction results. Then, the mean value, variance, and 95% CI of the predicted results can be calculated at each time step. When \(\mathbf{W}_{in}\) and \(\mathbf{W}\) follow the uniform distribution, the RUL prediction results are shown in Figure 4. Taking the first 10 groups of ESNs as an example, their hyperparameters and RMSE are shown in Table 1. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Test Group** & **ESN\({}_{1}\)** & **ESN\({}_{2}\)** & **ESN\({}_{3}\)** & **ESN\({}_{4}\)** & **ESN\({}_{5}\)** & **ESN\({}_{6}\)** & **ESN\({}_{7}\)** & **ESN\({}_{8}\)** & **ESN\({}_{9}\)** & **ESN\({}_{10}\)** \\ \hline \multirow{3}{*}{Hyper-para.} & \(\alpha\) & 0.80000 & 0.24477 & 0.80000 & 0.80000 & 0.10000 & 0.61598 & 0.55070 & 0.28000 & 0.10000 & 0.10000 \\  & \(\rho\) & 0.38125 & 0.69227 & 0.79748 & 0.93218 & 0.20756 & 0.57188 & 0.56348 & 0.72581 & 0.18374 & 0.17359 \\  & \(\gamma\) & 0.00100 & 0.00900 & 0.00900 & 0.00900 & 0.00164 & 0.00363 & 0.00100 & 0.00900 & 0.00136 & 0.00340 \\ \hline \multirow{3}{*}{RMSE} & \multicolumn{1}{c}{0.58050} & 0.59487 & 0.59647 & 0.60032 & 0.59554 & 0.57550 & 0.57331 & 0.57092 & 0.60588 & 0.63377 \\ \cline{1-1} \cline{2-11}  & \(\alpha\) & 0.80000 & 0.24477 & 0.80000 & 0.80000 & 0.10000 & 0.61598 & 0.55070 & 0.28000 & 0.10000 & 0.10000 \\ \cline{1-1} \cline{2-11}  & \(\rho\) & 0.38125 & 0.69227 & 0.79748 & 0.93218 & 0.20756 & 0.57188 & 0.56348 & 0.72581 & 0.18374 & 0.17359 \\ \cline{1-1}  & \(\gamma\) & 0.00100 & 0.00900 & 0.00900 & 0.00900 & 0.00164 & 0.00363 & 0.00100 & 0.00900 & 0.00136 & 0.00340 \\ \hline \multirow{3}{*}{RMSE} & \multicolumn{1}{c}{0.58050} & 0.59487 & 0.59647 & 0.60032 & 0.59554 & 0.57550 & 0.57331 & 0.57092 & 0.60588 & 0.63377 \\ \cline{1-1} \cline{2-11}  & \(\alpha\) & 0.80000 & 0.24477 & 0.80000 & 0.80000 & 0.10000 & 0.61598 & 0.55070 & 0.28000 & 0.10000 & 0.10000 \\ \cline{1-1} \cline{2-11}  & \(\rho\) & 0.38125 & 0.69227 & 0.79748 & 0.93218 & 0.20756 & 0.57188 & 0.56348 & 0.72581 & 0.18374 & 0.17359 \\ \cline{1-1} \cline{2-11}  & \(\gamma\) & 0.00100 & 0.09000 & 0.00900 & 0.00900 & 0.00164 & 0.00363 & 0.00100 & 0.00900 & 0.00136 & 0.00340 \\ \hline \multirow{3}{*}{RMSE} & \multicolumn{1}{c}{0.58050} & 0.59487 & 0.59647 & 0.60032 & 0.59554 & 0.57550 & 0.57331 & 0.57092 & 0.60588 & 0.63377 \\ \cline{1-1} \cline{2-11}  & \(\gamma\) & 0.00100 & 0.09000 & 0.00900 & 0.00900 & 0.00164 & 0.00363 & 0.00100 & 0.00900 & 0.00136 & 0.00340 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Steady-state results of \(\text{ESN}_{1}\)-\(\text{ESN}_{10}\) when \(\mathbf{W}_{in}\) and \(\mathbf{W}\) follow the uniform distribution. Figure 4: Steady-state 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. After changing the distribution shapes of \(\mathbf{W}_{in}\) and \(\mathbf{W}\) to Gaussian distribution, the RUL prediction results under the steady-state operating condition are shown in Figure 5. The hyperparameters and RMSE of the first 10 groups of ESNs are shown in Table 2. 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 ESNs is shown in Figure 6.  
The average RMSE under the Gaussian distribution (0.57488) is lower than the average RMSE under the uniform distribution (0.58602) for the steady-state test. The mean prediction results of the ensemble ESN are calculated, i.e., 0.57841 for uniform distribution and 0.56966 for Gaussian distribution.