State space reconstruction is the foundation of chaotic system modeling. Selection of reconstructed variables is essential to the analysis and prediction of multivariate chaotic time series. As most existing state space reconstruction theorems deal with univariate time series, we have presented a novel nonuniform state space reconstruction method using information criterion for multivariate chaotic time series. We derived a new criterion based on low dimensional approximation of joint mutual information for time delay selection, which can be solved efficiently through the use of an intelligent optimization algorithm with low computation complexity. The embedding dimension is determined by conditional entropy, after which the reconstructed variables have relatively strong independence and low redundancy. The scheme, which integrates nonuniform embedding and feature selection, results in better reconstructions for multivariate chaotic systems. Moreover, the proposed nonuniform state space reconstruction method shows good performance in forecasting benchmark and actual multivariate chaotic time series.

In multivariate chaotic time series prediction, correlation analysis is important for reducing input dimensions and improving prediction performance. Grey relational analysis (GRA) has proved to be an effective method for data correlation analysis, especially for inexact data and incomplete data. In GRA, points are usually regarded as objects, and the distance between points or the concave and convex degree are mostly used to measure the correlations. However, with discrete variables, correlation analysis results always tend to have some deviations when using prior GRA methods. Furthermore, GRA methods cannot directly use vector datasets. Therefore, in this paper, an improved GRA method is proposed based on vector projections. The input and output variables are expressed as vectors by linking two adjacent points. The vectors, instants of the points, are regarded as the objects, and the projection length of input variables to output variables is used to measure the correlations. The smaller the difference between the projection length and the input variables, the higher the correlation. Then, a hybrid variable selection and prediction model is proposed based on the improved GRA method for multivariate chaotic time series predictions, in order to overcome the negative effects of irrelevant and redundant variables caused by phase-space reconstruction. The experimental results based on the gas furnace dataset and San Francisco river runoff dataset demonstrate that the improved GRA method is effective for data correlation analysis, and the prediction accuracy is better than prior GRA-based methods.

Echo state network (ESN), a type of special recurrent neural network with a large-scale randomly fixed hidden layer (called a reservoir) and an adaptable linear output layer, has been widely employed in the field of time series analysis and modeling. However, when tackling the problem of multidimensional chaotic time series prediction, due to the randomly generated rules for input and reservoir weights, not only the representation of valuable variables is enriched but also redundant and irrelevant information is accumulated inevitably. To remove the redundant components, reduce the approximate collinearity among echo-state information, and improve the generalization and stability, a new method called hierarchical ESN with sparse learning (HESN-SL) is proposed. The HESN-SL mines and captures the latent evolution patterns hidden from the dynamic system by means of layer-by-layer processing in stacked reservoirs, and leverage monotone accelerated proximal gradient algorithm to train a sparse output layer with variable selection capability. Meanwhile, we further prove that the HESN-SL satisfies the echo state property, which guarantees the stability and convergence of the proposed model when applied to time series prediction. Experimental results on two synthetic chaotic systems and a real-world meteorological dataset illustrate the proposed HESN-SL outperforms both original ESN and existing hierarchical ESN-based models for multidimensional chaotic time series prediction.