Permutation Entropy: Theory and Applications

Dear Colleagues,

Introduced 16 years ago in a seminal paper by C. Bandt and B. Pompe, the concept of permutation entropy (PE) has attracted much attention from researchers from a plethora of fields. Undeniably, part of this success originates from the elegance and simplicity of PE. It is based on comparing neighbouring values in a time series, for then finding the order patterns that result in sorted (ascending) sequences, and finally on studying the corresponding probability distribution. PE thus allows us to synthesise, in a simple metric, the temporal causality of the time series, and to do this in a computationally efficient and almost parameter-free way. If originally introduced to assess the complexity of a time series, it was soon modified to, e.g., discriminate between chaotic and random dynamics, to quantify the irreversibility of a time series, or to test the presence of causality relationships.

The theoretical advantages offered by PE have fostered its application to heterogeneous real-world problems. The best examples can be found in the biomedical field, where it has been used to characterise the dynamics of different entities in healthy and pathological conditions—including individual neurons, brain, heart or gait. However, further examples can also be found in economics, laser physics, mechanical engineering, or climate analysis, to name a few. In synthesis, whenever it is required to characterise the complexity of the mechanisms behind a time series, PE offers an efficient and effective option.

In spite of this, many questions remain open, and the community has only scratched the surface of some theoretical problems. To illustrate, if PE allows characterising the temporal causality in one time series, few options are available to move to multivariate measurement. Additionally, the community still has to reach a consensus on how to handle equal values, or on how to optimise parameters like the embedding delay or the order of permutation.

This Special Issue seeks contributions from researchers working on this active topic, from both theoretical and applied perspectives, in the form of both original research and review papers. The application of PE to real-world problems, and the adaptation of PE to tackle new theoretical problems will especially be welcome.

Dr. Massimiliano Zanin
Guest Editor

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• Permutation entropy
• Forbidden patterns
• Complexity
• Time series analysis
• Causality