  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.
