Confidence bounds of recurrence-based complexity measures
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 (), 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): with to the logistic map with mutual transitions (Fig. 3, middle panel) given as: In difference to the standard logistic map, in the latter the control parameter a is changed with every iteration of the map ().
For control parameter a in the range of () we find an island of stability starting at 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.
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