OFFPRINT Testing time series irreversibility using complex network methods

The absence of time-reversal symmetry is a fundamental property of many nonlinear time series. Here, we propose a new set of statistical tests for time series irreversibility based on standard and horizontal visibility graphs. Specifically, we statistically compare the distributions of time-directed variants of the common complex network measures degree and local clustering coefficient. Our approach does not involve surrogate data and is applicable to relatively short time series. We demonstrate its performance for paradigmatic model systems with known time-reversal properties as well as for picking up signatures of nonlinearity in neuro-physiological data. editor’s choice Copyright c © EPLA, 2013 Introduction. – Nonlinear processes govern the dynamics of many real-world systems. Therefore, a sophisticated diagnostics and identification of such processes from observational data is a common problem in time series analysis important for model development. Consequently, in the last decades, testing for nonlinearity of time series has been of great interest. Various approaches have been developed for identifying signatures of different types of nonlinearity as a necessary precondition for the possible emergence of chaos ([1], § 5.3). Since linearity of Gaussian processes directly implies time reversibility [2–4] (see [5], § 4 for further details), nonlinearity results (among other features) in an asymmetry of certain statistical properties under time reversal [6]. Therefore, studying reversibility properties of time series is an important alternative to the direct quantitative assessment of nonlinearity [7]. In contrast to classical higher-order statistics requiring surrogate data techniques [6], most recently developed approaches for testing irreversibility have been based on symbolic dynamics [8–10] or statistical-mechanics concepts [11–13]. Motivated by the enormous success of complex network theory in many fields of science [14], in the last years several techniques for network-based time series analysis have been proposed [15–21]. As a particularly successful example, visibility graphs (VGs) and related methods [16,17] (a)E-mail: [email protected] (see “Methods” section) are based on the mutual visibility relationships between points in a one-dimensional landscape representing a univariate (scalar-valued) time series. The degree distributions of the thus constructed VGs allow classifying time series according to the type of recorded dynamics and obey characteristic scaling in case of fractal or multifractal behaviour of the data under study [22,23]. These relationships make VGs promising candidates for studying observational time series from various fields of research such as turbulence [24], finance [23,25,26], physiology [22,27], or geosciences [28–32]. In [33], Lacasa et al. demonstrated that horizontal visibility graphs (HVGs) [17], an algorithmic variant of VGs (see “Methods” section), allow discriminating between reversible and irreversible time series. Based on a time-directed version of HVGs, they could show that irreversible dynamics results in an asymmetry between the probability distributions of the numbers of incoming and outgoing edges of all network vertices, which can be detected by means of the associated Kullback-Leibler divergence. In this work, we thoroughly extend this idea and provide a set of rigorous statistical tests for time series irreversibiliby, which can be formulated based on both standard and horizontal VGs and utilise different network properties. Specifically, we demonstrate that for VGs and HVGs, degrees as well as local clustering coefficients can be decomposed into contributions from past and