Hurst entropy: A method to determine predictability in a binary series based on a fractal-related process

Shannon's concept of information is related to predictability. In a binary series, the value of information relies on the frequency of 0's and 1's, or how it is expected to occur. However, information entropy does not consider the bias in randomness related to autocorrelation. In fact, it is possible for a binary temporal series to carry both short- and long-term memories related to the sequential distribution of 0's and 1's. Although the Hurst exponent measures the range of autocorrelation, there is a lack of mathematical connection between information entropy and autocorrelation present in the series. To fill this important gap, we combined numerical simulations and an analytical approach to determine how information entropy changes according to the frequency of 0's and 1's and the Hurst exponent. Indeed, we were able to determine how predictability depends on both parameters. Our findings are certainly useful to several fields when binary times series are applied, such as neuroscience to econophysics.

Figure 1

(a) Examples of a binary series with $p_1=0.1$ and 0.5 and $H=0.5$ (black lines) and 0.8 (gray lines). (b) Entropy rate $S\left(L\right)/L$ by $1/L$ and $p_1$ for two series with $H=0.5$ (black lines) and $H=0.8$ (gray lines). Dashed lines represent the tendency of the entropy rate for $p_1=0.5$. (c) Coefficients of the linear fit for the dashed lines. The continuous line represents the linear coefficients, and the x lines are the angular coefficients. The inset shows the relative drop in the entropy rate due to the temporal correlations, associated with the gain in predictability.

Figure 2

The analytical solution of the entropy rate considering the Hurst exponent $S^H$. (a) View in two dimensions (2D) of $S^H$ versus $H$. (b) View in 3D of $S^H$ versus $p_1$ and $H$. Notice that each line represents distinct values of $p_1$, and the lines show a symmetrical shape around $p_1=0.5$. These lines are the same as those represented in panel (a).

Figure 3

The analytical solution for the relative drop of the entropy rate considering the Hurst exponent $\mathrm{\Delta }{S}^{\%}$. (a) View in 2D of $\mathrm{\Delta }{S}^{\%}$ versus $H$. (b) View in 3D of $\mathrm{\Delta }{S}^{\%}$ versus $p_1$ and $H$.

Figure 4

Graphical representation of the Hurst entropy rate $S^H$ obtained from simulations (black dots) for $H=(0,1)$ with intervals of 0.1 and $p_1=[0,1]$ with intervals of 0.01. The simulation results overlie the analytical solution (gray lines).