Complexity-Entropy Causality Plane as a Complexity Measure for Two-dimensional Patterns

Complexity measures are essential to understand complex systems and there are numerous definitions to analyze one-dimensional data. However, extensions of these approaches to two or higher-dimensional data, such as images, are much less common. Here, we reduce this gap by applying the ideas of the permutation entropy combined with a relative entropic index. We build up a numerical procedure that can be easily implemented to evaluate the complexity of two or higher-dimensional patterns. We work out this method in different scenarios where numerical experiments and empirical data were taken into account. Specifically, we have applied the method to i) fractal landscapes generated numerically where we compare our measures with the Hurst exponent; ii) liquid crystal textures where nematic-isotropic-nematic phase transitions were properly identified; iii) 12 characteristic textures of liquid crystals where the different values show that the method can distinguish different phases; iv) and Ising surfaces where our method identified the critical temperature and also proved to be stable.Comment: Accepted for publication in PLoS On

Dependence of the entropic indexes on the temperature of a lyotropic liquid crystal.

<p>We plot versus the temperature in (a) and versus the temperature in (b), where we employ . Figures (c) and (d) present the results for and , and also for and . The different shaded areas represent the different liquid crystal phases. Note that the phase transitions are properly identified in all cases. Due to the asymmetry of the elongated capillary tube where the liquid crystal sample is placed, and present slight differences under the rotation and .</p

Dependence of the entropic indexes on the reduced temperature for Ising surfaces.

<p>(a) The permutation entropy and (b) the complexity measure versus the reduced temperature for and , and also for and . We note invariance of indexes under the rotation and . (c) A 3d visualization of the Ising model phase transition when considering . The gray shadows represent the dependences of on and of on .</p

Schematic representation of the construction of the accessible states.

<p>In this example we have a array (left panel) and we choose the embedding dimensions and . In the right panel we illustrate the construction of the states. We first obtain the sub-matrix corresponding to and that have as elements and, after sorting, this sub-matrix leads to the state “0132”. We thus move to next sub-matrix and which have the elements and that, after sorting, leads to the state “1023”. The last two remaining matrices lead to the states “1230” and “0132”. Finally, we estimate the probabilities , that are, , and which are then used in the <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040689#pone.0040689.e064" target="_blank">equations (1</a>) and (2), leading to and .</p

Examples of fractal surfaces obtained through the random midpoint displacement method.

<p>These are surfaces () for different values of the Hurst exponent . For easier visualization, we have scaled the height of the surfaces in order to stay between and . We note that for small values of the surfaces display an alternation of peaks and valleys (anti-persistent behavior) much more frequent than those one obtained for larger values of . For larger values of , the surfaces are smoother reflecting the persistent behavior induced by the value of .</p

Dependence of the entropic indexes on the number of Monte Carlo steps.

<p>Here, denotes the number of Monte Carlo steps and the reduced temperatures are indicated in the plots. In (a) we show versus and in (b) versus for . We note the stability of both indexes after Monte Carlo steps.</p

Examples of Ising surfaces for three different temperatures.

<p>These surfaces were obtained after Monte Carlo steps for three different temperatures: below , at and above . In these plots, the height values were scaled to stay between and . We note that for temperatures higher or lower than , the surfaces exhibit an almost random pattern. For values of the temperature closer to the surfaces exhibit a more complex pattern, reflecting the long-range correlations that appear among the spin sites during the phase transition.</p

Dependence of the complexity-entropy causality plane on Hurst exponent <i>h</i>.

<p>We have employed fractal surfaces of size (). In (a) we plot and versus for the embedding dimensions and (circles) and also for and (squares). We note the invariance of the index against the rotation and . In (b) we plot the diagram for . We observe changes in the scale of and caused by the increasing number of states. In both cases, as increases the complexity also increases while the permutation entropy decreases. This behavior reflects the differences in the roughness shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0040689#pone-0040689-g002" target="_blank">Fig. 2</a>. For values of the surface is anti-persistent which generates a flatter distribution for the values of leading to values of and closer to the aleatory limit ( and ). For values of there is a persistent behavior in the surfaces heights which generates a more intricate distribution of and, consequently, values of and that are closer to the middle of the causality plane (region of higher complexity).</p