We investigate a generalised version of the recently proposed ordinal partition time series to network transformation algorithm. Firstly we introduce a fixed time lag for the elements of each partition that is selected using techniques from traditional time delay embedding.The resulting partitions define regions in the embedding phase space that are mapped to nodes in the network space. Edges are allocated between nodes based on temporal succession thus creating a Markov chain representation of the time series. We then apply this new transformation algorithm to time series generated by the Rössler system and find that periodic dynamics translate to ring structures whereas chaotic time series translate to band or tube-like structures -thereby indicating that our algorithm generates networks whose structure is sensitive to system dynamics. Furthermore we demonstrate that simple network measures including the mean out degree and variance of out degrees can track changes in the dynamical behaviour in a manner comparable to the largest Lyapunov exponent. We also apply the same analysis to experimental time series generated by a diode resonator circuit and show that the network size, mean shortest path length and network diameter are highly sensitive to the interior crisis captured in this particular data set. 1 arXiv:1501.06656v1 [nlin.CD] 27 Jan 2015Within the last ten years a novel approach to time series analysis has emerged whereby data is transformed into a complex network and then analysed using various measures from network science. The choice of transformation algorithm is critical in this process as different methods are inherently more effective at capturing certain aspects of dynamics and less effective at capturing others. In this paper we investigate a recently proposed algorithm known as the method of ordinal partitions. This computationally simple algorithm explicitly embeds temporal information in the network structure by partitioning the time series into a set of symbolic states which become network nodes, and then connecting these nodes based on the transition sequence present in the data. New in this work, we generalise the algorithm by introducing a time lag parameter for the elements in each partition, as is done in traditional methods of time delay embedding. Our results demonstrate that this new approach generates networks which are measurably sensitive to the dynamics present in the source time series, and has the potential to be useful as a tool for change point detection in continuous chaotic systems.

In this study, we propose a new information theoretic measure to quantify the complexity of biological systems based on time-series data. We demonstrate the potential of our method using two distinct applications to human cardiac dynamics. Firstly, we show that the method clearly discriminates between segments of electrocardiogram records characterized by normal sinus rhythm, ventricular tachycardia and ventricular fibrillation. Secondly, we investigate the multiscale complexity of cardiac dynamics with respect to age in healthy individuals using interbeat interval time series and compare our findings with a previous study which established a link between age and fractal-like long-range correlations. The method we use is an extension of the symbolic mapping procedure originally proposed for permutation entropy. We build a Markov chain of the dynamics based on order patterns in the time series which we call an ordinal network, and from this model compute an intuitive entropic measure of transitional complexity. A discussion of the model parameter space in terms of traditional time delay embedding provides a theoretical basis for our multiscale approach. As an ancillary discussion, we address the practical issue of node aliasing and how this effects ordinal network models of continuous systems from discrete time sampled data, such as interbeat interval time series.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

It has been established that the count of ordinal patterns, which do not occur in a time series, called forbidden patterns, is an effective measure for the detection of determinism in noisy data. A very recent study has shown that this measure is also partially robust against the effects of irregular sampling. In this paper, we extend said research with an emphasis on exploring the parameter space for the method's sole parameter-the length of the ordinal patterns-and find that the measure is more robust to under-sampling and irregular sampling than previously reported. Using numerically generated data from the Lorenz system and the hyper-chaotic Rössler system, we investigate the reliability of the relative proportion of ordinal patterns in periodic and chaotic time series for various degrees of under-sampling, random depletion of data, and timing jitter. Discussion and interpretation of results focus on determining the limitations of the measure with respect to optimal parameter selection, the quantity of data available, the sampling period, and the Lyapunov and de-correlation times of the system.

We are motivated by real-world data that exhibit severe sampling irregularities such as geological or paleoclimate measurements. Counting forbidden patterns has been shown to be a powerful tool towards the detection of determinism in noisy time series. They constitute a set of ordinal symbolic patterns that cannot be realised in time series generated by deterministic systems. The reliability of the estimator of the relative count of forbidden patterns from irregularly sampled data has been explored in two recent studies. In this paper, we explore highly irregular sampling frequency schemes. Using numerically generated data, we examine the reliability of the estimator when the sampling period has been drawn from exponential, Pareto and Gamma distributions of varying skewness. Our investigations demonstrate that some statistical properties of the sampling distribution are useful heuristics for assessing the estimator's reliability. We find that sampling in the presence of large chronological gaps can still yield relatively accurate estimates as long as the time series contains sufficiently many densely sampled areas. Furthermore, we show that the reliability of the estimator of forbidden patterns is poor when there is a high number of sampling intervals, which are larger than a typical correlation time of the underlying system.

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.