Entropy of weighted recurrence plots

The Shannon entropy of a time series is a standard measure to assess the complexity of a dynamical process and can be used to quantify transitions between different dynamical regimes. An alternative way of quantifying complexity is based on state recurrences, such as those available in recurrence quantification analysis. Although varying definitions for recurrence-based entropies have been suggested so far, for some cases they reveal inconsistent results. Here we suggest a method based on weighted recurrence plots and show that the associated Shannon entropy is positively correlated with the largest Lyapunov exponent. We demonstrate the potential on a prototypical example as well as on experimental data of a chemical experiment.

Figure 1

Recurrence plot of logistic map for (a) $r=3.5$ (periodic regime) and (b) $r=4.0$ (chaotic regime) and (c) weighted recurrence plot of logistic map for $r=4.0$ (chaotic regime).

Figure 2

Comparison between (a) the Lyapunov exponent ${\lambda }_{max}$, (b) $S_{\mathrm{RP}}\left(\ell \right)$ [Eq. (3)], (c) $S_{\mathrm{RP}}^{\mathrm{white}}$ [Eq. (6)], and (d) $S_{\mathrm{WRP}}$ [Eq. (9)] for the logistic map.

Figure 3

Comparison between (a) the Lyapunov exponent ${\lambda }_{max}$, (b) $S_{\mathrm{RP}}\left(\ell \right)$ [Eq. (3)], (c) $S_{\mathrm{RP}}^{\mathrm{white}}$ [Eq. (6)], and (d) $S_{\mathrm{WRP}}$ [Eq. (9)] for the Rössler oscillator.

Figure 4

Complexity measures for 30 electrochemical oscillators as a function of circuit voltage. Color maps shows the entropies associated with the time series of each electrode for different values of voltage. The top shows the entropy $S_{\mathrm{RP}}$ of black dots [Eq. (3)], the middle the entropy of white dots $S_{\mathrm{RP}}^{\mathrm{white}}$ [Eq. (6)], and the bottom the entropy of weighted RPs $S_{\mathrm{WRP}}$ [Eq. (9)].

Figure 5

Experimental results showing the comparison between the mean of the three normalized entropies over all electrodes for each value of the voltage. The top shows the entropy $S_{\mathrm{RP}}$ of black dots [Eq. (3)], the middle the entropy of white dots $S_{\mathrm{RP}}^{\mathrm{white}}$ [Eq. (6)], and the bottom the entropy of weighted RPs $S_{\mathrm{WRP}}$ [Eq. (9)]. Again, 50 bins were considered to obtain the probability density function of strengths $p\left(s\right)$.