Optimal Markov approximations and generalized embeddings

Based on information theory, we present a method to determine an optimal Markov approximation for modeling and prediction from time series data. The method finds a balance between minimal modeling errors by taking as much as possible memory into account and minimal statistical errors by working in embedding spaces of rather small dimension. A key ingredient is an estimate of the statistical error of entropy estimates. The method is illustrated with several examples, and the consequences for prediction are evaluated by means of the root-mean-squared prediction error for point prediction.