# Statistical Analysis on Random Matrices of Echo State Network in PEMFC System's Lifetime Prediction
## 3 Experimental Results
### Dynamic Operating Condition  
In the dynamic operating condition, the RPLR is used as the degradation health indicator, which has been described in detail in our previous work [18]. First, the polarization curve of the PEMFC stack at the beginning of life (BoL) is measured. The BoL power at different currents can be calculated to form a table. Secondly, the power at time step \(t\) can be calculated by the measured current (\(l_{t}\)) and voltage (\(U_{t}\)). Finally, the RPLR at time step \(t\) is calculated by  
\[\text{RPLR}=(P_{t}-P_{0})/P_{t} \tag{11}\]  
where \(P_{t}\) is the power at time step \(t\) and \(P_{0}\) is the BoL power, which is determined by the look-up table method. The \(P_{t}\) and \(P_{0}\) are at the same current level. Half of the data (0-191 h) is used for training the ESN, and the rest of the data (192-382 h) is used for prediction. When \(\mathbf{W}_{in}\) and \(\mathbf{W}\) follow the uniform distribution, the  
Figure 8: Quasi-dynamic RUL results under the Gaussian distribution: (**a**) Prediction results of 100 individual ESNs (blue lines); (**b**) Mean prediction results (blue line) of ensemble ESN with 95% CI. Figure 9: RMSE of 100 groups of ESNs under the quasi-dynamic operating condition. RUL prediction results are shown in Figure 10. Taking the first 10 groups of ESNs as an example, their hyperparameters and RMSE are shown in Table 5.  
After changing the distribution shapes of \(\mathbf{W}_{in}\) and \(\mathbf{W}\) to Gaussian distribution, the RUL prediction results under the dynamic operating condition are shown in Figure 11. The hyperparameters and RMSE of the first 10 groups of ESNs are shown in Table 6.  
Figure 11: Dynamic RUL results under the Gaussian distribution: (**a**) Prediction results of 100 individual ESNs (blue lines); (**b**) Mean prediction results (blue line) of ensemble ESN with 95% CI. Figure 10: 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. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**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- & \(\alpha\) & 0.12483 & 0.25671 & 0.15844 & 0.11213 & 0.34245 & 0.23291 & 0.23923 & 0.14834 & 0.26963 & 0.36064 \\ para. & \(\rho\) & 1.43205 & 1.74959 & 1.35257 & 1.53465 & 1.18192 & 1.33707 & 1.90000 & 1.23153 & 1.37328 & 1.25512 \\  & \(\gamma\) & 0.00218 & 0.00524 & 0.00900 & 0.00900 & 0.00129 & 0.00405 & 0.00100 & 0.00900 & 0.00165 & 0.00253 \\ \hline RMSE & 0.00476 & 0.00527 & 0.00483 & 0.00482 & 0.00578 & 0.00523 & 0.00508 & 0.00485 & 0.00479 & 0.00501 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Dynamic results of ESN\({}_{1}\)-ESN\({}_{10}\) when \(\mathbf{W}_{in}\) and \(\mathbf{W}\) follow the uniform distribution. The RMSE of 100 groups of ESNs is shown in Figure 12. The average RMSE under the Gaussian distribution (0.00489) is lower than the average RMSE under the uniform distribution (0.00502) for the dynamic test. The mean prediction results of the ensemble ESN are calculated, i.e., 0.00473 for uniform distribution and 0.00464 for Gaussian distribution. Among these three different operating conditions, each random generation of \(\textbf{W}_{in}\) and \(\textbf{W}\) produces a different prediction result. The essential reason is that the randomness of the matrices changes the weight values of the connected neurons, and the random weight leads to different neuron behaviors, which are important for the nonlinear relationship construction ability of the reservoir. The numerous repeated tests also illustrate the uncertainty properties of random matrices.