Time reversibility from visibility graphs of nonstationary processes

Abstract

Visibility algorithms are a family of methods to map time series into networks, with the aim of describing the structure of time series and their underlying dynamical properties in graph-theoretical terms. Here we explore some properties of both natural and horizontal visibility graphs associated to several nonstationary processes, and we pay particular attention to their capacity to assess time irreversibility. Nonstationary signals are (infinitely) irreversible by definition (independently of whether the process is Markovian or producing entropy at a positive rate), and thus the link between entropy production and time series irreversibility has only been explored in nonequilibrium stationary states. Here we show that the visibility formalism naturally induces a new working definition of time irreversibility, which allows us to quantify several degrees of irreversibility for stationary and nonstationary series, yielding finite values that can be used to efficiently assess the presence of memory and off-equilibrium dynamics in nonstationary processes without the need to differentiate or detrend them. We provide rigorous results complemented by extensive numerical simulations on several classes of stochastic processes.

Detection of time reversibility in time series by ordinal patterns analysis

“…links or paths-based characteristics). Recently, this approach has been extended for the study of non-stationary processes 30,31 .…”

Visibility graph analysis for re-sampled time series from auto-regressive stochastic processes

“…Our detailed comparison demonstrates that network measures often converge to different values between the original VGs and HVGs, when increasing the time delay τ . Note that the delay τ introduced in the re-sampling algorithm is different from the case where time-shifted series are used to test irreversibility of non-stationary processes [16,[33][34][35] .…”

Visibility graphs for fMRI data: multiplex temporal graphs and their modulations across resting state networks

“…Note 74 that the resulting Horizontal visibility graph (HVG) is simply a core subgraph of the Natural visibility graphs 75 (NVG), the former being analytically tractable [26]. As a matter of fact, HVG can be understood as an order 76 statistic [28] and therefore filters out any dependency on the series marginal distributions (that's not true for degrees of non-stationarity in the associated time series [28].…”

“…Also, visibility graphs naturally filter out linear trends so they don't require such detrending [29]. 182 Furthermore, since HVG is an order statistic, it is also invariant under monotonic (order-preserving) rescaling on 183 the data [28]. The NVG is not invariant under this latter transformation though, so nonlinear rescaling to make 184 data more 'peaky' will necessarily modify the associated NVG in a nontrivial way.…”

“…Naive implementation of HVG is already 64 O(N log N ) in most of the cases of practical interest. Finally, these methods are well-suited to handle several 65 degrees of non-stationarity in the associated time series [21].…”