Elsevier

Physics Letters A

Volume 373, Issue 26, 15 June 2009, Pages 2245-2250
Physics Letters A

Confidence bounds of recurrence-based complexity measures

https://doi.org/10.1016/j.physleta.2009.04.045Get rights and content

Abstract

In the recent past, recurrence quantification analysis (RQA) has gained an increasing interest in various research areas. The complexity measures the RQA provides have been useful in describing and analysing a broad range of data. It is known to be rather robust to noise and nonstationarities. Yet, one key question in empirical research concerns the confidence bounds of measured data. In the present Letter we suggest a method for estimating the confidence bounds of recurrence-based complexity measures. We study the applicability of the suggested method with model and real-life data.

Introduction

Recurrence Plots (RP) and their quantification (recurrence quantification analysis, RQA) [11] have become rather popular in various fields of science. The complexity measures based on RPs have helped to gain a deeper insight into diverse kinds of phenomena and experimental data. In this Letter we propose a straightforward extension to the existing RQA framework which allows us to not only compute these complexity measures, but also to estimate their confidence bounds. We do this by using a well-known resampling paradigm – the bootstrap. We show that the confidence bounds of RQA measures come with the regular analysis at virtually no extra costs and that the method can be useful for comparing univariate time series in a statistically sound fashion.

Section snippets

Recurrence Plots and their quantification

Recurrence is a fundamental property of dynamical systems. On this basis the data analysis tool called Recurrence Plot (RP) has been devised by Eckmann et al. [1] which visualises recurrences in the phase space of an n-dimensional state vector xi (i=1,,N),Ri,j=Θ(εxixj), where Θ is the Heaviside function, is a norm and ε is the recurrence threshold. The threshold ε can be defined as an absolute value or in dependence on other criteria. For the examples in Fig. 1 we chose ε so that

Confidence bounds of RQA measures

The RQA measures have been quite useful for the analysis of a variety of data. Yet, in order to not only detect qualitative changes in a system's dynamics but to be able to judge their significance or to compare two univariate time series, it is necessary to derive a quantitative judgement such as a confidence interval. For recurrence-based complexity measures those intervals can be estimated using a resampling paradigm.

Prototypical example

As a prototypical example we compare the logistic map in the chaotic regime (Fig. 3, upper panel):xi+1=axi(1xi) with a=3.92 to the logistic map with mutual transitions (Fig. 3, middle panel) given as:xi+1=aixi(1xi). In difference to the standard logistic map, in the latter the control parameter a is changed with every iteration of the map (ai+1=ai+Δa).

For control parameter a in the range of [3.8;3.88] (Δa=0.00001) we find an island of stability starting at a=1+83.828 with a period-3 window.

Application

We apply the suggested procedure to measurements of electric brain signal (electroencephalogram, EEG) in a study on event-related potentials (ERPs).

Discussion

The extension to the existing RQA framework presented here is straightforward and easily applicable. The RQA is known to be a useful tool for analysing noisy and nonstationary data, as is the case with EEG recordings. By providing confidence bounds to a number of recurrence-based complexity measures, we cannot only detect properties of the investigated data, but we can also extend the analysis to statistical comparisons, which is often called for in empirical and experimental research. As shown

Acknowledgements

This work supported by grants of the German Research Foundation (DFG) in the Research Group FOR 868 and the SFB 555 Komplexe nichtlineare Systeme and by the EU in the COST Action BM0601 and the BioSim Network of Excellence. The software used for this Letter, the CRPtoolbox & extensions, is available for download at http://tocsy.agnld.uni-potsdam.de.

References (20)

  • N. Marwan et al.

    Phys. Rep.

    (2007)
  • J.-P. Eckmann et al.

    Europhys. Lett.

    (1987)
  • B. Efron et al.

    An Introduction to the Bootstrap

    (1993)
  • R.A. Fisher

    The Design of Experiments

    (1935)
  • W.S. Gosset

    Biometrika

    (1908)
  • A. Groth

    Phys. Rev. E

    (2005)
  • R. Hubbard et al.

    Theory Psychol.

    (2008)
  • H. Kantz et al.

    Nonlinear Time Series Analysis

    (1997)
  • N. Marwan et al.

    Int. J. Bifur. Chaos Cognition Complex Brain Dyn.

    (February 2004)
  • N. Marwan et al.

    Phys. Rev. E

    (2002)
There are more references available in the full text version of this article.

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