Characterization of time series via Rényi complexity–entropy curves

Abstract

One of the most useful tools for distinguishing between chaotic and stochastic time series is the so-called complexity-entropy causality plane. This diagram involves two complexity measures: the Shannon entropy and the statistical complexity. Recently, this idea has been generalized by considering the Tsallis monoparametric generalization of the Shannon entropy, yielding complexity-entropy curves. These curves have proven to enhance the discrimination among different time series related to stochastic and chaotic processes of numerical and experimental nature. Here we further explore these complexity-entropy curves in the context of the Rényi entropy, which is another monoparametric generalization of the Shannon entropy. By combining the Rényi entropy with the proper generalization of the statistical complexity, we associate a parametric curve (the Rényi complexity-entropy curve) with a given time series. We explore this approach in a series of numerical and experimental applications, demonstrating the usefulness of this new technique for time series analysis. We show that the Rényi complexity-entropy curves enable the differentiation among time series of chaotic, stochastic, and periodic nature. In particular, time series of stochastic nature are associated with curves displaying positive curvature in a neighborhood of their initial points, whereas curves related to chaotic phenomena have a negative curvature; finally, periodic time series are represented by vertical straight lines.

Spillover effects of Great Recession on Hong-Kong’s Real Estate Market: An analysis based on Causality Plane and Tsallis Curves of Complexity–Entropy

“…For further analysis of complexity-entropy curves in the context of the Rényi entropy please see M. Jauregui et al[44].…”

ordpy: A Python package for data analysis with permutation entropy and ordinal network methods

“…In a similar direction, there are also the developments of complexity-entropy curves proposed by Ribeiro et al [22] and Jauregui et al [23]. These works have further extended the complexity-entropy plane concept by considering the Tsallis and Rényi entropies combined with proper generalizations of statistical complexity [62].…”

“…Permutation entropy's success is not limited to its practical usage as this approach has inspired numerous time series analysis tools. Some of these related methods consider different quantifiers for the ordinal probability distribution [14][15][16][17][18][19][20][21][22][23], generalize the Bandt-Pompe symbolization algorithm to evaluate ordinal structures on multiple temporal scales [24][25][26][27], include signal amplitude information [28][29][30][31], and account for equal values in time series [32,33]. Other works have generalized permutation entropy and its ordinal approach to twodimensional data such as images [34,35].…”

Feature Extraction of Ship-Radiated Noise Based on Intrinsic Time-Scale Decomposition and a Statistical Complexity Measure

“…It is worth noting that this plane is a valuable tool for differentiating time series. As a matter of fact, it has achieved great success in distinguishing between chaotic time series and stochastic ones [ 33 , 34 , 35 ].…”