# Statistical Analysis on Random Matrices of Echo State Network in PEMFC System's Lifetime Prediction
## 1 Introduction
Additionally, based on the ANOVA method, the effects of three hyperparameters (i.e., leaking rate, spectral radius, and regularization coefficient) have been analyzed under the dynamic operating conditions [18]. Results show that the interaction of the spectral radius and regularization parameter has the biggest effect, and the effects of the spectral radius rank second. Based on the ANOVA analysis results of [17], a multi-reservoir ESN structure is proposed in [19], and each reservoir has a different spectral radius. This multi-reservoir ESN can avoid the optimization of the spectral radius, and its effectiveness is validated under the steady-state cell voltage prediction condition. As an extension of the work in [19], the ESN and Markov chains are combined to predict the cell voltage under the dynamic micro-combined heat and power (\(\mu\)-CHP) load profile [20]. The RUL prediction by the combined structure can be realized with a mean relative accuracy below 17%. An ensemble ESN network has recently been proposed to avoid hyperparameter optimization and improve the adaptiveness of lifetime prediction [21]. A dynamic health indicator of \"virtual steady-state voltage (VSV)\" is proposed based on linear parameter varying (LPV) models. Then, a number of ESNs with different combinations of hyperparameters (leaking rate and spectral radius) are used for the VSV prediction. Finally, the long-term RUL prediction and a confidence interval of 95% confidence level are given in detail. In the past two years, the multi-input ESN structure is proposed to improve the prediction accuracy of the RUL [18; 22; 23; 24]. The effects of the operating parameters (i.e., current, temperature, and pressure) are fully analyzed under the steady-state and dynamic operating conditions in [18; 22] separately. The multi-input ESN structure could improve the prediction accuracy regarding the health indicator and the scheduled current as the double inputs are more meaningful in practice. In ref. [23], the ambient temperature and stack voltage are used as the double inputs of ESN to improve the RUL prediction accuracy. The normalized root mean square error is equal to 0.098 with a prediction over 2000 h. In ref. [24], the input parameters of ESN are evaluated and selected by the least absolute shrinkage and selection operator (LASSO), and then the degradation-related parameters are used for the RUL prediction. Results show that the ESN with eight input parameters is the optimal choice. Both [23] and [24] are validated by the steady-state datasets. In the current state-of-art, the developments of ESN for the PEMFC system's lifetime prediction are focused on the ESN structure (reservoir forms, ensemble ESN, multi-input ESN, etc.) and hyperparameter optimization. In the practical implementation process of ESN, the input and internal weight matrices (\(\mathbf{W}_{in}\) and \(\mathbf{W}\)) remain unchanged after they have been drawn from a random distribution. Nevertheless, the effects of the distribution shape of \(\mathbf{W}_{in}\) and \(\mathbf{W}\) on the lifetime prediction accuracy have not been reported yet. There is abundant evidence that the ensemble structure can improve the prediction robustness and stability such as the ensemble neural network model [25], particle filtering [26], and extreme learning machine [27]. The ensemble ESN is proposed in [21], and the effects of the leaking rate and spectral radius have been fully analyzed. Nevertheless, the statistical property of the ESN, e.g., the uncertainty caused by the random matrices, has not been analyzed yet. Based on these two motivations, the contributions of this paper are summarized as follows:  
1. The effects of two distribution shapes (uniform and Gaussian distribution) of \(\mathbf{W}_{in}\) and \(\mathbf{W}\) on the prediction accuracy are explored;
2. The metaheuristic technique of particle swarm optimization (PSO) is utilized to optimize the hyperparameters of the leaking rate, spectral radius, and regularization coefficient;
3.