Activated permutation entropy for graph signals

Abstract

Nonlinear dynamics play an important role in the analysis of signals. A popular, readily interpretable nonlinear measure is Permutation Entropy (PE). PE has recently been extended for the analysis of graph signals, thus providing a framework for non-linear analysis of data sampled on irregular domains. Integrating ideas from Graph Signal Processing (GSP) and Machine Learning, we introduce Activated Permutation Entropy (APEG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$APE_G$$\end{document}) by combining continuous information present in graph signals with traditional ordinal analysis and introducing an ordinal activation function (OAF) akin to the one of neural networks. We also formally extend ordinal contrasts to the graph domain. Activated versions of ordinal contrasts of length 3 are introduced and their advantage is shown in experiments from various domains. Building on recent work in ordinal analysis, we further demonstrate how we can find the continuous ordinal pattern that maximizes the APEG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$APE_G$$\end{document} of various dynamical systems. Simulations with popular non-linear maps and analysis of real-life MRI data show the validity of APEG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$APE_G$$\end{document} and potential benefits over the state of the art. By extending very recent concepts related to PE to the graph domain, we expect to accelerate the development of improved graph-based entropy methods that enable nonlinear analysis of broader data types and establishing relationships with emerging ideas in data science.