On Rényi Permutation Entropy

Among various modifications of the permutation entropy defined as the Shannon entropy of the ordinal pattern distribution underlying a system, a variant based on Rényi entropies was considered in a few papers. This paper discusses the relatively new concept of Rényi permutation entropies in dependence of non-negative real number q parameterizing the family of Rényi entropies and providing the Shannon entropy for q=1. Its relationship to Kolmogorov–Sinai entropy and, for q=2, to the recently introduced symbolic correlation integral are touched.

Ordinal Pattern Dependence in the Context of Long-Range Dependence

Ordinal pattern dependence is a multivariate dependence measure based on the co-movement of two time series. In strong connection to ordinal time series analysis, the ordinal information is taken into account to derive robust results on the dependence between the two processes. This article deals with ordinal pattern dependence for a long-range dependent time series including mixed cases of short- and long-range dependence. We investigate the limit distributions for estimators of ordinal pattern dependence. In doing so, we point out the differences that arise for the underlying time series having different dependence structures. Depending on these assumptions, central and non-central limit theorems are proven. The limit distributions for the latter ones can be included in the class of multivariate Rosenblatt processes. Finally, a simulation study is provided to illustrate our theoretical findings.

Extracting Robust Biomarkers From Multichannel EEG Time Series Using Nonlinear Dimensionality Reduction Applied to Ordinal Pattern Statistics and Spectral Quantities

In this study, ordinal pattern analysis and classical frequency-based EEG analysis methods are used to differentiate between EEGs of different age groups as well as individuals. As characteristic features, functional connectivity as well as single-channel measures in both the time and frequency domain are considered. We compare the separation power of each feature set after nonlinear dimensionality reduction using t-distributed stochastic neighbor embedding and demonstrate that ordinal pattern-based measures yield results comparable to frequency-based measures applied to preprocessed data, and outperform them if applied to raw data. Our analysis yields no significant differences in performance between single-channel features and functional connectivity features regarding the question of age group separation.

CHARACTERIZATION OF SAR IMAGES WITH WEIGHTED AMPLITUDE TRANSITION GRAPHS

We propose a new technique for SAR image texture characterization based on ordinal pattern transition graphs. The proposal consists in (i) transforming a 2-D patch of data into a time series using a Hilbert Space Filling Curve, (ii) building an Ordinal Pattern Transition Graph with weighted edges; (iii) obtaining a probability distribution function from this graph; (iv) computing the Entropy and Statistical Complexity of this distribution. The weight of the edges is related to the absolute difference of observations. This modification takes into account the scattering properties of the target, and leads to a good characterization of several types of textures. Experiments with data from Munich urban areas, Guatemala forest regions, and Cape Canaveral ocean samples demonstrate the effectiveness of our technique, which achieves satisfactory levels of separability.

A Multivariate Weighted Ordinal Pattern Transition Network for Characterizing Driver Fatigue Behavior from EEG Signals

Driver fatigue has caused numerous vehicle crashes and traffic injuries. Exploring the fatigue mechanism and detecting fatigue state are of great significance for preventing traffic accidents, and further lessening economic and societal loss. Due to the objectivity of EEG signals and the availability of EEG acquisition equipment, EEG-based fatigue detection task has raised great attention in recent years. Although there exist various methods for this task, the study of fatigue mechanism and detection of fatigue state still remain much to be explored. To investigate these problems, a multivariate weighted ordinal pattern transition (MWOPT) network is proposed in this paper. To be specific, a simulated driving experiment was first conducted to obtain the EEG signals of subjects in alert state and fatigue state respectively. Then the MWOPT network is constructed based on a novel Shannon entropy. To probe into the mechanism underlying fatigue behavior, the small-worldness index is extracted from the generated MWOPT network. Furthermore, the nodal degree index is input into a classifier to distinguish the fatigue state from alert state. The obtained high accuracy indicates the effectiveness of the proposed network for EEG-based fatigue detection. Besides, four nodes are found to play an important role in identifying fatigue state. These results suggest that the proposed method enables to analyze nonlinear multivariate time series and investigate the driving fatigue behavior.