Estimating the active dimension of the dynamics in a time series based on an information criterion

We describe a method for estimating the active dimension of the dynamics generating a time series from the observed time series. The active dimension is the minimum number of variables needed to describe the essential part of the dynamics behind the time series, and is equal to the minimal embedding dimension at which the delay-coordinate reconstruction becomes a local embedding. The achievement of embedding was examined based on the results of time series prediction with the optimal metric. To bring further objectivity to the estimation, we introduced an information criterion, the minimum description length criterion. We demonstrated the effectiveness of the method by applying it to chaotic time series obtained from numerical calculations and from an electrochemical experiment. The advantages of the proposed method are the transparency of its parameter settings and its applicability to relatively short time series.