Estimating Shannon Entropy from Recurrence Plots

Christophe LETELLIER

Recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics [1]. A few years later, the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis [2]. Among the different measures introduced, a Shannon entropy was found to be correlated with the inverse of the largest Lyapunov exponent. The discrepancy between this and the usual interpretation of a Shannon entropy has been solved by replacing the probabilities P_n’s used in the entropy formula
  S = -\sum_{n=1}^H{ P_n \log (P_n)} \, .
The key point is to replace P_n with the number of diagonal segments of non-recurrent points (made of white dots) divided by the number of recurrent points [3]. Indeed, a white dot representing a non-recurrent point is nothing more than a signature of complexity within the data. With this definition, S_{\rm RP} increases as the bifurcation parameter increases (as shown for the Logistic map in Fig. 1b). There is a one-to-one correspondence between the new definition of S_{\rm RP} and the positive largest Lyapunov exponent.

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Fig. 1 : Comparison between the Largest Lyapunov Exponent and the Shannon Entropy.
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C++ code for computing Shannon entropy for the Logistic map

The algorithm provided here computed the Shannon Entropy from Recurrence Plots for the Logistic map versus parameter \mu. It produced Fig. 1. It is also possible to add noise.

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C++ code for computing the Shannon entropy from a data file

The code here provided computes the Shannon entropy from your data file (please, do not forget to set the number of points in your data file and to replace the name "datafile" by the name of your file).

Note also that, for flows, it is definitely better to compute recurrence plots from a "discrete" time series recorded in the Poincaré section of the attractor than from a "continuous" time series. This is discussed in Ref. [3].

[1] J.-P. Eckmann, S. Oliffson Kamphorst & D. Ruelle, Recurrence Plots of Dynamical Systems, Europhysics Letters, 4, 973-977, 1987.

[2] L. L. Trulla, A. Giuliani, J. P. Zbilut & C. L. Webber Jr., Recurrence quantification analysis of the logistic equation with transients, Physics Letters A, 223 (4), 255-260, 1996.

[3] This latter detail was missing in the definition provided in C. Letellier, Estimating the Shannon entropy : recurrence plots versus symbolic dynamics, Physical Review Letters, 96, 254102, 2006.


C++ code for computing the Shannon entropy from a data file
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5770 - 25/07/24

C++ code for computing Shannon entropy for the Logistic map
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5204 - 25/07/24

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