# Statistical Analysis on Random Matrices of Echo State Network in PEMFC System's Lifetime Prediction
## 4 Conclusions  
The data-driven method of echo state network (ESN) is further explored in the long-term lifetime prediction of the PEMFC system. During the implementation process of the ESN, the input and the internal weight matrices (\(\textbf{W}_{in}\) and \(\textbf{W}\)) are randomly generated. The traditional method keeps these two matrices unchanged once they are generated. Nevertheless, the random characteristics of these matrices will affect the performance of the ESN. To statistically analyze the effects of random characteristics, an ensemble ESN is proposed and tested under three different operating conditions. The conclusions are as follows:  
1. The random characteristics of \(\textbf{W}_{in}\) and \(\textbf{W}\) affect the lifetime prediction results. The prediction results are presented in the statistical form by the ensemble computing technique, and this helps the user to analyze the uncertainties of the randomness. After the data analysis, a 95% confidence interval (CI) is given, which better qualifies the reliability of the result.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Test Group**} & \multicolumn{1}{c}{**ESN\({}_{1}\)**} & \multicolumn{1}{c}{**ESN\({}_{2}\)**} & \multicolumn{1}{c}{**ESN\({}_{3}\)**} & \multicolumn{1}{c}{**ESN\({}_{4}\)**} & \multicolumn{1}{c}{**ESN\({}_{5}\)**} & \multicolumn{1}{c}{**ESN\({}_{6}\)**} & \multicolumn{1}{c}{**ESN\({}_{7}\)**} & \multicolumn{1}{c}{**ESN\({}_{8}\)**} & \multicolumn{1}{c}{**ESN\({}_{9}\)**} & \multicolumn{1}{c}{**ESN\({}_{10}\)**} \\ \hline Hyper- & \(\alpha\) & 0.30376 & 0.47659 & 0.54372 & 0.50078 & 0.13709 & 0.57584 & 0.32100 & 0.28035 & 0.11564 & 0.39271 \\ para. & \(\rho\) & 1.64620 & 1.29902 & 1.55139 & 1.71862 & 1.38232 & 1.44123 & 1.43160 & 1.59497 & 1.50729 & 1.90000 \\  & \(\gamma\) & 0.00634 & 0.00900 & 0.00586 & 0.00900 & 0.00900 & 0.00638 & 0.00157 & 0.00625 & 0.00900 & 0.00900 \\ \hline RMSE & 0.00511 & 0.00527 & 0.00525 & 0.00481 & 0.00413 & 0.00512 & 0.00509 & 0.00473 & 0.00569 & 0.00485 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Dynamic results of ESN\({}_{1}\)-ESN\({}_{10}\) when \(\textbf{W}_{in}\) and \(\textbf{W}\) follow the Gaussian distribution.  
Figure 12: RMSE of 100 groups of ESNs under the dynamic operating condition.  
Based on the uniform and Gaussian distribution shapes of \(\mathbf{W}_{in}\) and \(\mathbf{W}\), the prediction performances are fully compared. To analyze the effects of random matrices, the PSO method is used to optimize the hyperparameters of the ESN. Based on the comparison results, the Gaussian distribution of \(\mathbf{W}_{in}\) and \(\mathbf{W}\) can decrease the prediction error slightly when compared to the uniform distribution. However, the effects of two different distribution shapes on the prediction results are rather insignificant.
* Combining the ESN with other methods to improve the prediction accuracy and exploring the prognostic methods to realize the online lifetime prediction will be the focuses of our future work.  
Conceptualization, Z.H. and Z.Z.; Formal analysis, Z.H. and Z.Z.; Funding acquisition, M.-C.P. and F.G.; Methodology, Z.H. and Z.Z.; Project administration, M.-C.P. and F.G.; Software, Z.H. and Z.Z.; Supervision, Z.Z., M.-C.P. and F.G.; Validation, Z.H. and Z.Z.; Writing--original draft, Z.H.; Writing--review and editing, Z.H., Z.Z., M.-C.P. and F.G. All authors have read and agreed to the published version of the manuscript.  
This work has been supported by the EIPHI Graduate school (contract \"ANR-17-EURE-0002\").  
Not applicable.  
The authors declare no conflict of interest.