Online sequential echo state network with sparse RLS algorithm for time series prediction

Abstract

Recently, the echo state networks (ESNs) have been widely used for time series prediction. To meet the demand of actual applications and avoid the overfitting issue, the online sequential ESN with sparse recursive least squares (OSESN-SRLS) algorithm is proposed. Firstly, the ℓ₀ and ℓ₁ norm sparsity penalty constraints of output weights are separately employed to control the network size. Secondly, the sparse recursive least squares (SRLS) algorithm and the subgradients technique are combined to estimate the output weight matrix. Thirdly, an adaptive selection mechanism for the ℓ₀ or ℓ₁ norm regularization parameter is designed. With the selected regularization parameter, it is proved that the developed SRLS shows comparable or better performance than the regular RLS. Furthermore, the convergence of OSESN-SRLS is theoretically analyzed to guarantee its effectiveness. Simulation results illustrate that the proposed OSESN-SRLS always outperforms other existing ESNs in terms of estimation accuracy and network compactness.

Keywords: Echo state networks; Online sequential learning; Regularization method; Sparse recursive least squares algorithm; Time series prediction.