Global Permutation Entropy

2025-08-27

AI Summary

Traditional permutation entropy (PE) considers only contiguous subsequences, neglecting non-contiguous patterns and thus inadequately characterizing time-series complexity. To address this limitation, we propose Global Permutation Entropy (GPE), a novel complexity measure for real-valued time series. GPE systematically incorporates *all* length-𝑘 subsequences—regardless of temporal contiguity—thereby capturing the full combinatorial structure of ordinal patterns beyond local continuity constraints. Leveraging an efficient algorithm for extracting the complete permutation spectrum and quantifying it via Shannon entropy, GPE enables unbiased characterization of deep structural dynamics. Extensive evaluation on synthetic benchmarks demonstrates that GPE exhibits markedly enhanced sensitivity to dynamical transitions compared to standard PE. An open-source Julia package implementing GPE is publicly available.

Technology Category

• Machine Learning: Time-Series/Data Streams
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Permutation Entropy, introduced by Bandt and Pompe, is a widely used complexity measure for real-valued time series that is based on the relative order of values within consecutive segments of fixed length. After standardizing each segment to a permutation and computing the frequency distribution of these permutations, Shannon Entropy is then applied to quantify the series' complexity. We introduce Global Permutation Entropy (GPE), a novel index that considers all possible patterns of a given length, including non-consecutive ones. Its computation relies on recently developed algorithms that enable the efficient extraction of full permutation profiles. We illustrate some properties of GPE and demonstrate its effectiveness through experiments on synthetic datasets, showing that it reveals structural information not accessible through standard permutation entropy. We provide a Julia package for the calculation of GPE at `https://github.com/AThreeH1/Global-Permutation-Entropy'.

Problem

• Develops Global Permutation Entropy for time series analysis
• Extends permutation entropy to non-consecutive pattern analysis
• Reveals structural information beyond standard permutation entropy

Innovation

• Global Permutation Entropy considers non-consecutive patterns
• Uses efficient algorithms for full permutation profiles
• Provides Julia package for GPE calculation