Abstract: M. McCullough, K. Sakellariou, T. Stemler, M. Small (2017)

Recently proposed ordinal networks not only afford novel methods of nonlinear time series analysis but also constitute stochastic approximations of the deterministic flow time series from which the network models are constructed. In this paper, we construct ordinal networks from discrete sampled continuous chaotic time series and then regenerate new time series by taking random walks on the ordinal network. We then investigate the extent to which the dynamics of the original time series are encoded in the ordinal networks and retained through the process of regenerating new time series by using several distinct quantitative approaches. First, we use recurrence quantification analysis on traditional recurrence plots and order recurrence plots to compare the temporal structure of the original time series with random walk surrogate time series. Second, we estimate the largest Lyapunov exponent from the original time series and investigate the extent to which this invariant measure can be estimated from the surrogate time series. Finally, estimates of correlation dimension are computed to compare the topological properties of the original and surrogate time series dynamics. Our findings show that ordinal networks constructed from univariate time series data constitute stochastic models which approximate important dynamical properties of the original systems.

It can be useful to build models based solely on observed time series data when attempting to analyse complex nonlinear systems for which the driving mechanism is not fully understood. For example, an ordinal network is such a model produced by mapping time series to a set of symbolic states based on order patterns in short segments of data and then constructing an abstract network representation of the time series based on the temporal succession of these states. Thus far, this class of models has primarily been investigated as a tool for the computation of measures that are sensitive to the properties of the dynamics. However, if ordinal networks are actually encoding the information necessary to capture the essence of complex dynamics, then it should be possible to generate new time series from these models which have similar properties to the original data from which they are built. Herein, we investigate this hypothesis.

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Regenerating time series from ordinal networks