# Statistical Analysis on Random Matrices of Echo State Network in PEMFC System's Lifetime Prediction
## 1 Introduction  
Energy and environment are the focal points attracting the attention of the international community due to the depletion of fossil fuels and the increasing emissions of carbon dioxide [1, 2]. Hydrogen (H\({}_{2}\)) is considered to be a promising energy carrier of sustainable clean energy in the 21 st century. The proton exchange membrane fuel cells (PEMFC) system can transform the chemical energy of H\({}_{2}\) into electricity directly, and the by-products during the electrochemical process are heat and water [3, 4]. It is a promising power conversion device because of its satisfactory characteristics except for the high cost and limited lifetime. Predicting the lifetime of the PEMFC system can help the users to take action in advance to extend its working life. Model-based and data-driven methods are two typical classifications of existing lifetime prediction methods [5, 6]. Data-driven methods, which are mainly based on machine learning (ML) theory, construct the nonlinear degradation relations [7, 8]. One of the patterns of ML, the recurrent neural network (RNN), has shown great power in nonlinear time series prediction. Nevertheless, the RNN has some weaknesses such as slow convergence rate and the existence of bifurcation. To overcome the weaknesses of RNN, the echo state network (ESN) was proposed by Prof. [PERSON] et al. in [9]. There are two characteristics of ESN: one is a large, randomly generated reservoir that is used to replace the hidden layers of RNN; the other is the weight matrices (input and internal) that are generated randomly. The ESN has a faster convergence rate and lower computational training cost than thetraditional RNN. Nevertheless, the hyperparameters of ESN (leaking rate, spectral radius, regularization coefficient, reservoir neuron number, reservoir connectivity, etc.) need to be carefully tuned from human expertise [10]. For the PEMFC system's lifetime prediction application, the ESN has first been applied in [11] to estimate the steady-state cell voltage by the single-step ahead prediction pattern. Four kinds of reservoirs are compared in [12], and the delay line reservoir (DLR) and the simple cycle reservoir (SCR) have the best precision values. A cycle reservoir with a jump (CRJ) model has been proposed in [13], and this structure is validated on the steady-state datasets. The CRJ model can speed up the linear fitting process and improve prediction accuracy. In ref. [14], the steady-state cell voltages are divided into the approximation part and detail part by wavelet transform, and then double ESNs with different dynamic characteristics are utilized to deal with the voltage signal separately. This double ESN structure could improve prediction accuracy. The hyperparameters of ESN in [11; 12; 13; 14] are set by the trial-and-error method. Inspired by the separation concept in [14], the wavelet transform and multiple ESNs are proposed in [15] to deal with the multi-timescale degradation features under the dynamic operating conditions in the long-term prediction horizon. In ref. [15], the hyperparameters of the spectral radius and leaking rate are optimized by the grid-search method, and the prediction accuracy has been improved. To improve the prediction efficiency, the original degradation data prediction is replaced by the shortened coefficients in the discrete wavelet transform process [16]. The prediction data points can be shortened by a factor of 8, and the hyperparameters of ESN (spectral radius, leaking rate, and regularization coefficient) are optimized by the genetic algorithms (GA) method. In ref. [17], the analysis of variance (ANOVA) method is applied to analyze the four hyperparameters of the reservoir neuron number, reservoir connectivity, spectral radius, and leaking rate under the steady-state dataset. Results show that the spectral radius has the biggest effects on the prediction performance.