Estimating Shannon Entropy from Recurrence Plots

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.

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.

 A second Fortran code is provided. It computes the Shannon entropy using a recurrence plot from a data file. you have to specify the number of data point (Npoint). Your data file is expected to have a single column. The code returns

• the recurrence rate ;
• the "determinism" rate ;
• the Shannon entropy.

Be aware that the so-called "determinism" rate does not provide in fact a determinism rate for the simple reason that, for instance, it is equal to 0.82 when estimated from a time series produced by the Logistic map with
\($\mu=3.99$\), a signal which is 100% deterministic ! So, please, interpret with great care this rate. 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].