Recurrence Plot for binary variable?

[holistic]

I was wondering, if it is possible to compute the RP also for a binary variable which alternates between 0 and 1? I'm really not sure, if the phase space reconstruction makes sense in such a case

[Norbert]

Yes, this is possible. This is also called "recurrence on a nominal scale" or "symbolic recurrences". There are several studies already using this approach. The difference to the standard recurrence approach is that instead of a neighbourhood of some size ε a recurrence can now be defined by the exact match of x(i) and x(j). Embedding of such binary or symbolic series mean that we consider words of length m instead of reconstructing an m-dimension phase space.

Here is some literature related to this variant of recurrence:

• C. Bandt, A. Groth, N. Marwan, M. C. Romano, M. Thiel, M. Rosenblum, J. Kurths: Analysis of Bivariate Coupling by Means of Recurrence, 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
• R. V. Donner, U. Hinrichs, B. Scholz-Reiter: Symbolic recurrence plots: A new quantitative, European Physical Journal – Special Topics, 164(1), 85–104 (2008). DOI:10.1140/epjst/e2008-00836-2
• P. Faure, A. Lesne: Recurrence plots for symbolic sequences, International Journal of Bifurcation and Chaos, 20(6), 1731–1749 (2010). DOI:10.1142/S0218127410026794
• P. beim Graben, A. Hutt: Detecting Recurrence Domains of Dynamical Systems by Symbolic Dynamics, Physical Review Letters, 110(15), 154101 (2013). DOI:10.1103/PhysRevLett.110.154101

[holistic]

Thank you!

Is this approach implemented in the matlab software (crqa), i.e. when I just compute the RR,LAM,DET asf. for a binary variable as normally done with my continuous variable?

[Norbert]

There is nothing to be implemented further. If you have only a binary time series, you just need to ensure that your threshold is smaller than 1. This will give the same results as using the identity relationship (i.e., threshold = 0).

[holistic]

Ok..thanks..just to be sure. I attached a screenshot of an analysis with a binary timeseries. Here, the treshold is 0.2 and I used euclidian norm. Is this correct? I assume I can interpret the values of RR, DET asf. in the same way as with a continuous variable?

[Norbert]

I would suggest to use the option "nonorm" to avoid the centering and standardisation of the data.

When using embedding dimension of 1 the results should be the same for Euclidean and Maximum norm.

The interpretation is similar but be aware that there is no phase space trajectory. We could also consider a phase space that is highly discretised (due to the binary time series). The interpretation depends, of course, strongly on the kind of data and the considered system.

[holistic]

I see . Thanks a lot!