Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems

Abstract

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.

Topological Analysis of Differential Effects of Ketamine and Propofol Anesthesia on Brain Dynamics

“…Given the rarity of this kind of dataset, a small sample size currently is unavoidable, however future work to generalize these findings may yield greater statistical power. Finally, for the OPN analysis, there is a fundamental question of how to select the optimal values for parameters d and τ [38,41,69]. While we have made an effort to make choices that are theoretically grounded and data-driven, further mathematical work may provide better insights and further unify the field.…”

“…representations of the data [57,38], following the procedure discussed in [41]. Due to the difficulties associated with multi-dimensional OPNs [67], we applied this method to a uni-variate time series, constructed from the first temporal principle component of the data.…”

“…There is no agreed-upon method for choosing the optimal d and τ . Different researchers have suggested different criteria, including values that maximize the variance of the degree distribution [38], false-nearest-neighbours criteria [28], or the first local maxima in the permutation entropy [41]. One significant consideration is that the embedding dimension must be large enough that the state-space is sufficiently large to fully capture the range of patterns present in the data (otherwise the resulting OPN fails to capture all the dynamics), but not so large that every moment is unique (in which case, we reconstruct the time series as a path graph).…”

Introduction to Focus Issue: Complex network perspectives on flow systems

“…[49][50][51] Such transition networks belong to the increasing number of approaches for utilizing complex network methods for the analysis of general uni-or multivariate time series. 52 Other notable examples of concepts for generating such time series networks include recurrence networks and related methods.…”

Construction of complex networks from time series based on the cross correlation interval

“…They used the DWCN to characterize the chaotic dynamic behavior of gasliquid slug flow [10] and further proposed a multivariate recurrence network [11][12][13][14]. McCullough et al have recently proposed the ordinal pattern partition networks that are formed from time series by symbolizing the data into ordinal patterns [15]. The above mentioned methods have been used in different fields including medicine [16][17][18], astronomy [19], financial analysis [20] and geophysics [21].…”