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In: Mathematical Methods in Time Series Analysis and Digital Image Processing, Eds.: R. Dahlhaus and J. Kurths and P. Maas and J. Timmer, Springer, Berlin, Heidelberg, ISBN: 978-3-540-75631-6, 153–182 (2008) DOI:10.1007/978-3-540-75632-3_5
Analysis of Bivariate Coupling by Means of Recurrence
C. Bandt, A. Groth, N. Marwan, M. C. Romano, M. Thiel, M. Rosenblum, J. KurthsIn the analysis of coupled systems, various techniques have been developed to model and detect dependencies from observed bivariate time series. Most well-founded methods, like Granger-causality and partial coherence, are based on the theory of linear systems: on correlation functions, spectra and vector autoregressive processes. In this paper we discuss a nonlinear approach using recurrence.
Recurrence, which intuitively means the repeated occurrence of a very similar situation, is a basic notion in dynamical systems. The classical theorem of Poincar?e says that for every dynamical system with an invariant probability measure P, almost every point in a set B will eventually return to B. Moreover, for ergodic systems the mean recurrence time is 1/P(B). Details of recurrence patterns were studied when chaotic systems came into the focus of research, and it turned out that they are linked to Lyapunov exponents, generalized entropies, the correlation sum, and generalized dimensions.
Our goal here is to develop methods for time series which typically contain a few hundreds or thousands of values and which need not come from a stationary source. While Poincaré's theorem holds for stationary stochastic processes, and linear methods require stationarity at least for suficiently large windows, recurrence methods need less stationarity. We outline different concepts of recurrence by specifying different classes of sets B. Then we visualize recurrence and define recurrence parameters similar to autocorrelation.
We are going to apply recurrence to the analysis of bivariate data. The basic idea is that coupled systems show similar recurrence patterns. We can study joint recurrences as well as cross-recurrence. We shall see that bothapproaches have their benefits and drawbacks.
Model systems of coupled oscillators form a test bed for analysis of bivariate time series since the corresponding differential equations involve a parameter which precisely defines the degree of coupling. Changing the parameter we can switch to phase synchronization and generalized synchronization. The approaches of cross- and joint recurrence are compared for several models. In view of possible experimental requirements, recurrence is studied on ordinal scale as well as on metric scale. Several quantities for the description of synchronization are derived and illustrated. Finally, two different applications to EEG data will be presented.