Variance of permutation entropy and the influence of ordinal pattern selection

Permutation entropy (PE) is a widely used measure for complexity, often used to distinguish between complex systems (or complex systems in different states). Here, the PE variance for a stationary time series is derived, and the influence of ordinal pattern selection, specifically whether the ordinal patterns are permitted to overlap or not, is examined. It was found that permitting ordinal patterns to overlap reduces the PE variance, improving the ability of this statistic to distinguish between complex system states for both numeric (fractional Gaussian noise) and experimental (semiconductor laser with optical feedback) systems. However, with overlapping ordinal patterns, the precision to which the PE variance can be estimated becomes diminished, which can manifest as increased incidences of false positive and false negative errors when applying PE to statistical inference problems.

Figure 1

Ordinal-pattern selection regimes for $D=3$ where patterns are (a) permitted to overlap and (b) not permitted to overlap. The labels $x_t$ depict the $t\mathrm{th}$ element in the time series.

Figure 2

Comparison of $\gamma$ calculations for white noise and $D=4$. Analytic calculations were performed by calculating the appropriate coincidence matrix and using Eq. (21). Numeric calculations were performed by generating 10 000 white noise sets ($N_{\mathrm{ol}}=99998$) and directly calculating the variance of observed ordinal pattern populations. Note that numeric labels for ordinal pattern motifs are completely interchangeable.

Figure 3

Comparison between PE variance for (a) nonoverlapping ordinal patterns and (b) overlapping ordinal patterns, calculated numerically (blue line) and using Eqs. (12) and (25) for a single noise realization (red line), and averaged over 10 000 realizations (orange line). PE variance is calculated for fractional Gaussian noise as a function of Hurst index (where a Hurst index of 0.5 indicates white noise), and for a fixed time-series length of 100 000 points.

Figure 4

2D histogram of 510 000 noise realizations with uniformly distributed Hurst indices for $H^{\prime }$ calculated using overlapping ordinal patterns. This histogram is approximately proportional to the joint probability $p(I,H^{\prime })$ if a uniform marginal distribution over $I$ is assumed.

Figure 5

Hurst index standard deviation as a function of $H^{\prime }$ calculated using the histogram in Fig. 4 in the case of nonoverlapping and overlapping ordinal patterns. Arrows indicate points where the standard deviation begins to roll off due to the histogram edge.

Figure 6

Chart depicting the categorization of tests based on values of $C_{\text{est}}$ and $C$ for a significance level of 0.05.

Figure 7

(a) Calculated PE as a function of drive current for the semiconductor laser with optical feedback calculated from the experimental output-power time series, and modeled joint probability distributions for (b) nonoverlapping and (c) overlapping ordinal patterns over $H$ and $I$, assuming a uniform marginal distribution over $I$. Distributions over fixed $I$ were modeled assuming Gaussian probability distributions.