Permutation Entropy and Statistical Complexity Analysis of earthquake sequences

Earthquake sequences exhibit intricate space–time–magnitude patterns that have motivated the use of statistical methods to uncover properties and relationships that standard time-series analysis techniques are unable to capture. Although these methods have made it possible to highlight properties such as clustering, scaling, long-range dependencies and related features, appropriate analyses of the complexity of the seismic phenomenon have not yet been developed or applied.
Since Bandt and Pompe’s seminal work, the permutation entropy and statistical complexity form the basis for constructing the so-called complexity–entropy causality plane (CECP). Permutation entropy and statistical complexity provide insight into two different aspects of a dataset. Permutation entropy measures the level of intrinsic randomness: data that are more predictable and tend to repeat a limited number of ordinal patterns exhibit lower permutation entropy, whereas data with a greater variety of patterns and less predictability show higher values. For a fixed value of permutation entropy, statistical complexity indicates the extent to which certain ordinal patterns are favored over others. In other words, higher complexity—at a given entropy level—reflects a greater deviation from a uniform distribution, suggesting that some ordinal patterns occur more frequently than others. By computing both measures for a time series, one can simultaneously assess the randomness of the data and the degree of structural or correlational organization within its fluctuations. 
While the CECP has been widely used to investigate the complex patterns of continuous time series, it has yet to be applied to analyze point processes, particularly in the context of seismic events. Thus, the present paper aims at analyzing the dynamics of seismic point processes in the CECP, offering new insights into their underlying patterns and behaviors. 
We first analyzed in the CECP the magnitude series generated by the physics-based numerical model developed by Olami, Feder, and Christensen (OFC) in 1992. Although introduced several decades ago, the OFC model remains a robust framework, successfully reproducing key qualitative features of real-world seismicity, such as the Gutenberg-Richter law, the Omori law, and the Ruff–Kanamori diagram.
We further investigated magnitude sequences from Italian seismic regions affected by the strongest earthquakes since 1985. Our results indicate that these magnitude sequences display in the CECP a pattern that aligns very well with that observed in the OFC model and apparently correlated with the magnitude of the strongest events. 
Although preliminary, these results underscore the potential of CECP analysis for seismicity studies, providing new and diverse ways to describe, interpret, and explore earthquake dynamics.