State Space Reconstruction for Multivariate Time Series Prediction

In the nonlinear prediction of scalar time series, the common practice is to reconstruct the state space using time-delay embedding and apply a local model on neighborhoods of the reconstructed space. The method of false nearest neighbors is often used to estimate the embedding dimension. For prediction purposes, the optimal embedding dimension can also be estimated by some prediction error minimization criterion. We investigate the proper state space reconstruction for multivariate time series and modify the two abovementioned criteria to search for optimal embedding in the set of the variables and their delays. We pinpoint the problems that can arise in each case and compare the state space reconstructions (suggested by each of the two methods) on the predictive ability of the local model that uses each of them. Results obtained from Monte Carlo simulations on known chaotic maps revealed the non-uniqueness of optimum reconstruction in the multivariate case and showed that prediction criteria perform better when the task is prediction.