Time series prediction with input noise based on the ESN and the EM and its industrial applications

Industrial time series data usually have a high noise. In this paper, an echo state network (ESN) model with input noise is proposed to address the problem of predicting time series with noise. In the ESN, the introduction of the input noise makes it difficult to accurately estimate the non-linear states of the dynamical reservoir, therefore, in this study, the states are approximated by linearizing it through an extended Kalman filter (EKF). For the learning of the model parameters, the expectation maximization algorithm (EM) is used to iteratively update all the uncertain parameters to construct the prediction intervals, where the state estimation is performed using a forward back algorithm. To verify the effectiveness of the proposed method, two benchmark data sets and three real gas data sets from steel enterprises are used in this paper. Experimental results show that the prediction accuracy of the proposed method is better than that of the existing methods.