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Novel Order Patterns Recurrence Plot-Based Quantification Measures to Unveil Deterministic Dynamics from Stochastic Processes

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Theory and Applications of Time Series Analysis (ITISE 2018)

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Abstract

Forbidden ordinal patterns are known to be useful to discriminate between chaotic and stochastic systems. However, while uncorrelated noise can be separated from deterministic signals using forbidden ordinal patterns, correlated noise exhibits apparently forbidden ordinal patterns, which can impede distinguishing noise from chaos. Here, we introduce order patterns recurrence plots to visualise the difference among deterministic chaotic systems, and stochastic systems of uncorrelated and correlated noise. In an order pattern plot of a chaotic system with an optimal embedding dimension, the diagonal lines remain preserved, while uncorrelated noise shows up as thinly isolated dots and correlated noise forms clusters. We propose two measures, the mean and the median of relative frequencies of order patterns that appear in a time series to distinguish those dynamics. The effectiveness of the two measures is analysed using bifurcation diagrams of the logistic map, the tent map, the delayed logistic map and the Hénon map. Our results show, that both, the mean and the median, distinguish chaos from quasiperiodicity in the delayed logistic map. The mean of relative frequencies of order pattern is reciprocal to the number of order patterns that occur in a given time series and thus can be a measure of forbidden structures—which becomes unbounded. While the mean is robust to the change of parameters in the bifurcation diagrams, the median exhibits sensitive changes, which is significant to characterise chaotic signals.

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Notes

  1. 1.

    For example, in our numerical experiments, \(N=\text {20,000},m=6\), for the logistic map with \(r=3.55\), eight order patterns appear, so that the total amount of these eight order patterns is \(N-m+1=\text {19,995}\). However, 19,995 cannot be divided by 8, so the count of some order patterns is 2,500, that of the others is 2,499, resulting in the difference of unity between the mean and the median of \(\{C_{1},\ldots ,C_{n}\}\). This further shows the difference between the \(\mathrm {mean}^{l}\) and the \(\mathrm {median}^{l}\) according to Eqs. 7 and 8.

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Authors and Affiliations

  1. Centre for Audio, Acoustics and Vibration (CAAV), University of Technology Sydney, Ultimo, NSW, 2007, Australia

    Shuixiu Lu, Sebastian Oberst & Guoqiang Zhang

  2. Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, China

    Shuixiu Lu & Zongwei Luo

Authors
  1. Shuixiu Lu
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  2. Sebastian Oberst
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  3. Guoqiang Zhang
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  4. Zongwei Luo
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Correspondence to Shuixiu Lu .

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Editors and Affiliations

  1. Faculty of Sciences, University of Granada, Granada, Spain

    Olga Valenzuela

  2. ETSIIT, CITIC-UGR, University of Granada, Granada, Spain

    Fernando Rojas

  3. ETSIIT, CITIC-UGR, University of Granada, Granada, Spain

    Héctor Pomares

  4. ETSIIT, CITIC-UGR, University of Granada, Granada, Spain

    Ignacio Rojas

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Lu, S., Oberst, S., Zhang, G., Luo, Z. (2019). Novel Order Patterns Recurrence Plot-Based Quantification Measures to Unveil Deterministic Dynamics from Stochastic Processes. In: Valenzuela, O., Rojas, F., Pomares, H., Rojas, I. (eds) Theory and Applications of Time Series Analysis. ITISE 2018. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-26036-1_5

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