Laplacian Echo State Network for Multivariate Time Series Prediction.

Echo state network is a novel kind of recurrent neural networks, with a trainable linear readout layer and a large fixed recurrent connected hidden layer, which can be used to map the rich dynamics of complex real-world data sets. It has been extensively studied in time series prediction. However, there may be an ill-posed problem caused by the number of real-world training samples less than the size of the hidden layer. In this brief, a Laplacian echo state network (LAESN), is proposed to overcome the ill-posed problem and obtain low-dimensional output weights. First, an echo state network is used to map the multivariate time series into a large reservoir. Then, assuming that an unknown underlying manifold is inside the reservoir, we employ the Laplacian eigenmaps to estimate the manifold by constructing an adjacency graph associated with the reservoir states. Finally, the output weights are calculated by the low-dimensional manifold. In addition, some criteria of transient stability, local controllability, and local observability are given. Experimental results based on two real-world data sets substantiate the effectiveness and characteristics of the proposed LAESN model.