Transition-network-based Wasserstein metrics: powerful techniques for estimating time-delays from chaotic time series

This article proposes a novel methodology for efficiently and accurately unveiling potential time-delay characteristics concealed within chaotic time series. The implementation of this approach leverages a newly developed quantifier which is a specialized Wasserstein metric defined within the framework of time series analysis incorporating transition network strategies. We refer to this quantifier as the transition-network-based Wasserstein metric. Throughout the paper, we use the multi-scale transition network paradigm as an illustrative framework. Firstly, we provide a rigorous mathematical definition of the proposed metric. Secondly, we demonstrate the effectiveness of the proposed method through numerical experiments using time series consisting of state variable observations derived from the well-known Mackey-Glass system. Thirdly, we discuss the influence of the intrinsic parameters of the transition network paradigm on the proposed method, offering guidelines for their optimal selection. Finally, for time series respectively impacted by dynamical and observational Gaussian white noise, we discover the upper bound of the possible range of applicable noise intensities (for dynamical noise) and noise levels (for observational noise). We also corroborate that the identification performance of this new approach surpasses that of four prevalent techniques proven to work well for these two types of noise.