An Optimal Metric for Predicting Chaotic Time Series

Abstract

The optimal metric for predicting chaotic time series was derived. In the derivation, the effects on the metric of the modeling of dynamics and the state space reconstruction by embedding were taken into account. The obtained metric minimizes the error in prediction with the nearest neighbor approximation, when the metric is used to calculate the distance in the state space. It is shown that the Euclidean metric is not the best choice, in either the reconstructed space or the true state space. The validity of the obtained metric was shown using numerical examples.