Characterizing pseudoperiodic time series through the complex network approach

Abstract

Recently a new framework has been proposed to explore the dynamics of pseudoperiodic time series by constructing a complex network [Phys. Rev. Lett. 96, 238701 (2006)]. Essentially, this is a transformation from the time domain to the network domain, which allows for the dynamics of the time series to be studied via the organization of the network. In this paper, we focus on the deterministic chaotic Rössler time series and stochastic noisy periodic data that yield substantially different structures of the networks. In particular, we test an extensive range of network topology statistics, which have not be discussed in previous works, but which are capable of providing a comprehensive statistical characterization of the dynamics from different angles. Our goal is to find out how they reflect and quantify different aspects of specific dynamics, and how they can be used to distinguish different dynamical regimes. For example, we find that the joint degree distribution appears to fundamentally characterize the spatial organizations of the cycles in phase space, and this is quantified via assortativity coefficient. We applied the network statistics to the electrocardiograms of a healthy individual and an arrythmia patient. Such time series are typically pseudoperiodic, but are noisy and nonstationary and degrade traditional phase-space based methods. These time series are, however, better differentiated by our network-based statistics.

Dynamical Systems Induced on Networks Constructed from Time Series

“…Cycle networks [7,19,20] were first proposed to study the pseudo-periodic time series, where nodes represent the individual cycles, and edges are constructed based on the similarity between cycles. Those researchers have demonstrated that cycle networks can be used to distinguish different dynamical systems, such as periodic and chaotic systems.…”

Change-point detection with recurrence networks

“…Recently, a considerable number of studies have been conducted on complex network analyses of time series. Cycle networks [10][11][12] are network representations of pseudo periodic time series, where each node corresponds to a cycle and each link represents a correlation between the cycles. Networks based on correlations between short segments were also proposed [13].…”

Dynamical Disorder and Self-Correlation in the Characterization of Nonlinear Systems: Application to Deterministic Chaos

“…Although the classical Nonlinear Dynamics Theory was set up some decades ago, new approaches have been proposed recently: the study of the topology of complex networks derived from time series for the characterization of the underlying dynamics [Zhang & Small, 2006;Zhang et al, 2008;Xu et al, 2008]; the use of the combination of a complexity measure and the Shannon entropy for distinguishing noise from chaos [Rosso et al, 2007]; the search for forbidden patterns in time series for detecting determinism [Amigo et al, 2008;Amigo et al, 2006;Carpi et al, 2010;Zanin, 2008]; the use of the Modified Sample Entropy as a regularity measure in time series [Xie et al, 2008;Xie et al, 2010]; the application of the 0-1 test for finding chaos in deterministic systems [Gottwald & Melbourne, 2009] and the extraction of the qualitative dynamical states in a system using Fuzzy c-Means Clustering [Shao et al, 2007].…”