Quantifying the irregularity of a time series

TL;DR

Circulance introduces a topology-based scalar, defined as $ ext{Φ}= ext{min}_{ au}oldsymbol{ au}_ au$ with $oldsymbol{ au}_ au=| ext{K}^+ \,\cap\, ext{K}^-|$, to quantify irregularity in time series by mapping them to ordinal pattern transition networks (OPTNs). By embedding the series into dimension $d$ with delay $ au$ and representing transitions between unique ordinal patterns as a directed graph, circulance captures deviations from a circular (periodic) OPTN structure, distinguishing periodic, quasiperiodic, chaotic, and stochastic regimes. The approach is validated on canonical model systems and applied to real data from EEG brain activity and sunspot numbers, revealing meaningful temporal variations and regime shifts; Embedding length and parameter choices critically affect separability, with $N oughly d!^2$ recommended for robust differentiation. Overall, circulance offers a computationally efficient, interpretable metric that bridges nonlinear time-series analysis and network topology, supporting real-time tracking and potential control of complex dynamical systems.

Abstract

We introduce circulance, a scalar measure for classifying time series of dynamical systems. Circulance captures the extent of temporal regularity or irregularity that is encoded in the topology of a directed ordinal pattern transition network derived from a time series. We demonstrate numerically that circulance sensitively and robustly positions time series of canonical model systems, representative of preset dynamical regimes, along a continuous spectrum from regularity to randomness. Analyzing empirical data from long-term observations of high-dimensional, complex systems -- human brain and the Sun -- reveals that circulance aids in elucidating different dynamical regimes.

Figures (11)

• Figure 1: Different types of dynamics and the corresponding ordinal pattern transition networks (OPTNs). Within each subfigure, vertices of the same color have the same number of adjacent vertices. (a) Schematic of the construction of an OPTN from a strictly periodic dynamics (with $d=4$ and $\tau=1$). The resulting OPTN consists of one closed trail or circuit on the OPTN. (b) A quasiperiodic dynamics results in a broader degree distribution (more distinct colors) of the OPTN. (c) Chaotic and (d) stochastic dynamics result in even more vertices and a higher amount of vertices with a distinct number of adjacent vertices in the respective OPTN. For a stochastic dynamics, the deviation from a simple circuit is maximum. Time series shown in (b)-(d) are of same length $N$.
• Figure 2: Circulance $\text{}$ in dependence of time series length $N$ for a single realization of a stochastic process and different embedding dimensions $d$. For a given time series length $N$, the embedding dimension $d$ is chosen such that circulance is maximum. Blue vertical lines mark $N=d!^2$ (cf. Eq. \ref{['eq:upper_bound']}).
• Figure 3: Time series classification in the $\text{}-N$-plane with dynamical regimes ranging from periodic via quasiperiodic and chaotic to stochastic (marker color resp. red, green, pink, and blue). Background colors schematically depict the continuum inferred empirically. Overlapping regimes between chaos and stochasticity point to limited distinguishability of high-dimensional chaos and white noise. Model systems are associated with different markers as follows: white noise - circle; generalized Hénon map - downward triangle; Zaslavskii map - plus; Hénon map - diamond; Lotka-Volterra system - square; Lorenz96 system - rotated Y. Circulance values of time series from model systems with $N=10000$ from Table \ref{['tab:dynamics_circulance']} accurately fit into the $\text{}-N$-plane.
• Figure 4: Bifurcation diagram of the standard Hénon map (a). Mean largest Lyapunov exponent $\lambda_{\rm max}$parlitz1992 (b) and mean circulance $\overline{ \text{}}$ (c) of time series ($x$-component; $N=10000$) for different values of the bifurcation parameter $a$. Each datapoint is the mean of $10$ realizations of time series with the same bifurcation parameter $a$ but with different initial conditions. The gray line and the red-shaded area in (c) indicate mean and standard deviation of circulance ($\overline{ \text{}}=16.98\pm0.65$) of white noise time series of the same length and averaged over $1000$ realizations. (d): Scatterplot of $\lambda_{\rm max}$ (from (b)) and $\overline{ \text{}}$ (from (c)).
• Figure 5: Time-dependent changes of circulance (mean over all electrodes) from a multi-day EEG recording. The black line (grey shaded area) indicates the moving average (standard deviation) over 250 datapoints and serves as visual guidance. The inset schematically visualizes the potential association of ultradian rhythms during nighttimes with a sleep cycle and different sleep stages. The blue line indicates mean circulance of 100 stochastic time series with a length that equals the window size used for the sliding-window approach.