Optimal embedding parameters: A modelling paradigm

Reconstruction of a dynamical system from a time series requires the selection of two parameters, the embedding dimension $d_e$ and the embedding lag $\tau$. Many competing criteria to select these parameters exist, and all are heuristic. Within the context of modeling the evolution operator of the underlying dynamical system, we show that one only need be concerned with the product $d_e\tau$. We introduce an information theoretic criteria for the optimal selection of the embedding window $d_w=d_e\tau$. 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 superior to estimation of embedding parameters with the standard techniques.