Optimal embedding parameters: a modelling paradigm

The reconstruction of a dynamical system from a time series requires the selection of two parameters: the embedding dimension de and the embedding lag τ. Many competing criteria to select these parameters exist, and all are heuristic. Within the context of modelling the evolution operator of the underlying dynamical system, we show that one only need be concerned with the product deτ. We introduce an information theoretic criterion for the optimal selection of the embedding window dw=deτ. For infinitely long time series, this method is equivalent to selecting the embedding lag that minimises the nonlinear model prediction error. For short and noisy time series, we find that the results of this new algorithm are data-dependent and are superior to estimation of embedding parameters with the standard techniques.