A high dimensional delay selection for the reconstruction of proper Phase Space with Cross auto-correlation

TL;DR

This paper introduces cross autocorrelation as a nonlinear measure for delay selection in phase space reconstruction, demonstrating it produces less distorted reconstructions than mutual information in various dynamical models.

Contribution

The paper proposes cross autocorrelation as an alternative to mutual information for better delay selection in high-dimensional phase space reconstruction.

Findings

Cross autocorrelation yields less distorted phase space reconstructions.

Multidimensional MI sometimes fails to produce satisfactory reconstructions.

Cross autocorrelation outperforms MI in the tested dynamical models.

Abstract

For the purpose of phase space reconstruction from nonlinear time series, delay selection is one of the most vital criteria. This is normally done by using a general measure viz., mutual information (MI). However, in that case, the delay selection is limited to the estimation of a single delay using MI between two variables only. The corresponding reconstructed phase space is also not satisfactory. To overcome the situation, a high-dimensional estimator of the MI is used; it selects more than one delay between more than two variables. The quality of the reconstructed phase space is tested by shape distortion parameter (SD), it is found that even this multidimensional MI sometimes fails to produce a less distorted phase space. In this paper, an alternative nonlinear measure cross autocorrelation (CAC) is introduced. A comparative study is made between the reconstructed phase spaces of a known three dimensional Neuro dynamical model, Lorenz dynamical model and a three dimensional food web model under MI for two and higher dimensions and also under cross auto-correlation separately. It is found that the least distorted phase space is obtained only under the notion of cross autocorrelation.