Criterion for determining the optimal delay of attractor reconstruction using persistent homology

Abstract

Persistent homology computes the number and the widths of holes of a shape. The delay-coordinate embedding reconstructs an attractor from time series data. Recently the combination of persistent homology and attractor reconstruction is proposed for clustering time series data. We propose a criterion to determine the delay of attractor reconstruction for periodic and chaotically periodic signals. Our criterion chooses the delay that maximizes the hole width of reconstructed attractors. We compare it to the criterion with mutual information and that by Perea and Harer. It is found that the combination of our criterion with Perea and Harer's criterion works well.