Using dynamics to analyse time series

Abstract

We present a review of recent work to analyze time series in a robust manner using Wasserstein distances which are numerical costs of an optimal transportation problem. Given a time series, the long-term behavior of the dynamical system represented by the time series is reconstructed by Takens delay embedding method. This results in probability distributions over phase space and to each pair we then assign a numerical distance that quantifies the differences in their dynamical properties. From the totality of all these distances a low-dimensional representation in a Euclidean space is derived. This representation shows the functional relationships between the time series under study. For example, it allows to assess synchronization properties and also offers a new way of numerical bifurcation analysis. Several examples are given to illustrate our results. This work is based on ongoing joint work with Michael Muskulus [19, 20].