Abstract
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 fractal landscapes generated numerically where we compare our measures with the Hurst exponent; liquid crystal textures where nematic-isotropic-nematic phase transitions were properly identified; 12 characteristic textures of liquid crystals where the different values show that the method can distinguish different phases; and Ising surfaces where our method identified the critical temperature and also proved to be stable.
Highlights
Investigations related to the so called complex systems are widely spread among different scientific communities, ranging from physics and biology to economy and psychology
We show that the complexity-entropy causality plane can be extended for higher-dimensional patterns. We apply this new approach in different scenarios related to two-dimensional structures and the results indicate that the method is very promising for distinguishing between two-dimensional patterns
We have proposed a generalization of the complexity-entropy causality plane to higher-dimensional patterns
Summary
Investigations related to the so called complex systems are widely spread among different scientific communities, ranging from physics and biology to economy and psychology. A considerable part of these works deals with empirical data aiming to extract patterns, regularities or laws that rule the dynamics of the system In this direction, the concept of complexity measures often emerges. Complexity measures can compare empirical data such as time series and classify them in somewhere between regular, chaotic or random [1], while other complexity measures can differentiate between degrees of correlations [2]. Examples of these measures include algorithmic complexity [3], entropies [4], relative entropies [5], fractal dimensions [6], and Lyapunov exponents [7]. These seminal works are still motivating new definitions, and today there are numerous definitions of complexity, which have been successful applied to different areas such as medicine [8,9], ecology [10,11,12,13], astrophysics [14,15,16], and music [17,18]
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