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Special Issue "Entropy Measures for Data Analysis: Theory, Algorithms and Applications"

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: 14 December 2018

Special Issue Editor

Guest Editor
Prof. Dr. Karsten Keller

Institut für Mathematik, Universität zu Lübeck, D-23562 Lübeck, Germany
E-Mail
Phone: +49 451 500 3165
Interests: data analysis; time series analysis; computational statistics; information theory; ergodic theory; automatic learning

Special Issue Information

Dear Colleagues,

Entropies and entropy-like quantities are playing an increasing role in modern non-linear data analysis. Fields of their application reach from diagnostics in physiology, for instance, electroencephalography (EEG), magnetoencephalography (MEG) and electrocardiography (ECG), to econophysics and engineering. During the last few years, classical concepts as the Approximate entropy and the Sample entropy have been supplemented by new entropy measures, like the Permutation entropy and various variants of it. Recent developments are focused on multidimensional generalizations of the concepts with a special emphasize on the quantification of coupling between time series and system components behind them. Some of the main future challenges in the field are a better understanding of the nature of the various entropy measures and their relationships, with the aim of their adequate application including good parameter choices. The utilization of entropy measures as features in automatic learning and their application to large and complex data for tasks as classification, discrimination and finding structural changes requires fast and well-founded algorithms.

Prof. Dr. Karsten Keller
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1500 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • data analysis
  • complexity measures
  • entropy
  • approximate entropy
  • sample entropy
  • permutation entropy
  • classification
  • discrimination
  • automatic learning

Published Papers (8 papers)

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Research

Open AccessArticle Improving Entropy Estimates of Complex Network Topology for the Characterization of Coupling in Dynamical Systems
Entropy 2018, 20(11), 891; https://doi.org/10.3390/e20110891 (registering DOI)
Received: 24 October 2018 / Revised: 13 November 2018 / Accepted: 19 November 2018 / Published: 20 November 2018
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Abstract
A new measure for the characterization of interconnected dynamical systems coupling is proposed. The method is based on the representation of time series as weighted cross-visibility networks. The weights are introduced as the metric distance between connected nodes. The structure of the networks,
[...] Read more.
A new measure for the characterization of interconnected dynamical systems coupling is proposed. The method is based on the representation of time series as weighted cross-visibility networks. The weights are introduced as the metric distance between connected nodes. The structure of the networks, depending on the coupling strength, is quantified via the entropy of the weighted adjacency matrix. The method has been tested on several coupled model systems with different individual properties. The results show that the proposed measure is able to distinguish the degree of coupling of the studied dynamical systems. The original use of the geodesic distance on Gaussian manifolds as a metric distance, which is able to take into account the noise inherently superimposed on the experimental data, provides significantly better results in the calculation of the entropy, improving the reliability of the coupling estimates. The application to the interaction between the El Niño Southern Oscillation (ENSO) and the Indian Ocean Dipole and to the influence of ENSO on influenza pandemic occurrence illustrates the potential of the method for real-life problems. Full article
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Open AccessArticle Cross-Sectoral Information Transfer in the Chinese Stock Market around Its Crash in 2015
Entropy 2018, 20(9), 663; https://doi.org/10.3390/e20090663
Received: 3 August 2018 / Revised: 23 August 2018 / Accepted: 24 August 2018 / Published: 3 September 2018
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Abstract
This paper applies effective transfer entropy to research the information transfer in the Chinese stock market around its crash in 2015. According to the market states, the entire period is divided into four sub-phases: the tranquil, bull, crash, and post-crash periods. Kernel density
[...] Read more.
This paper applies effective transfer entropy to research the information transfer in the Chinese stock market around its crash in 2015. According to the market states, the entire period is divided into four sub-phases: the tranquil, bull, crash, and post-crash periods. Kernel density estimation is used to calculate the effective transfer entropy. Then, the information transfer network is constructed. Nodes’ centralities and the directed maximum spanning trees of the networks are analyzed. The results show that, in the tranquil period, the information transfer is weak in the market. In the bull period, the strength and scope of the information transfer increases. The utility sector outputs a great deal of information and is the hub node for the information flow. In the crash period, the information transfer grows further. The market efficiency in this period is worse than that in the other three sub-periods. The information technology sector is the biggest information source, while the consumer staples sector receives the most information. The interactions of the sectors become more direct. In the post-crash period, information transfer declines but is still stronger than the tranquil time. The financial sector receives the largest amount of information and is the pivot node. Full article
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Open AccessArticle Analog Circuit Fault Diagnosis via Joint Cross-Wavelet Singular Entropy and Parametric t-SNE
Entropy 2018, 20(8), 604; https://doi.org/10.3390/e20080604
Received: 31 May 2018 / Revised: 24 July 2018 / Accepted: 25 July 2018 / Published: 14 August 2018
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Abstract
In this paper, a novel method with cross-wavelet singular entropy (XWSE)-based feature extractor and support vector machine (SVM) is proposed for analog circuit fault diagnosis. Primarily, cross-wavelet transform (XWT), which possesses a good capability to restrain the environment noise, is applied to transform
[...] Read more.
In this paper, a novel method with cross-wavelet singular entropy (XWSE)-based feature extractor and support vector machine (SVM) is proposed for analog circuit fault diagnosis. Primarily, cross-wavelet transform (XWT), which possesses a good capability to restrain the environment noise, is applied to transform the fault signal into time-frequency spectra (TFS). Then, a simple segmentation method is utilized to decompose the TFS into several blocks. We employ the singular value decomposition (SVD) to analysis the blocks, then Tsallis entropy of each block is obtained to construct the original features. Subsequently, the features are imported into parametric t-distributed stochastic neighbor embedding (t-SNE) for dimension reduction to yield the discriminative and concise fault characteristics. Finally, the fault characteristics are entered into SVM classifier to locate circuits’ defects that the free parameters of SVM are determined by quantum-behaved particle swarm optimization (QPSO). Simulation results show the proposed approach is with superior diagnostic performance than other existing methods. Full article
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Open AccessArticle Sample Entropy of Human Gait Center of Pressure Displacement: A Systematic Methodological Analysis
Entropy 2018, 20(8), 579; https://doi.org/10.3390/e20080579
Received: 18 June 2018 / Revised: 27 July 2018 / Accepted: 2 August 2018 / Published: 6 August 2018
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Abstract
Sample entropy (SampEn) has been used to quantify the regularity or predictability of human gait signals. There are studies on the appropriate use of this measure for inter-stride spatio-temporal gait variables. However, the sensitivity of this measure to preprocessing of the signal and
[...] Read more.
Sample entropy (SampEn) has been used to quantify the regularity or predictability of human gait signals. There are studies on the appropriate use of this measure for inter-stride spatio-temporal gait variables. However, the sensitivity of this measure to preprocessing of the signal and to variant values of template size (m), tolerance size (r), and sampling rate has not been studied when applied to “whole” gait signals. Whole gait signals are the entire time series data obtained from force or inertial sensors. This study systematically investigates the sensitivity of SampEn of the center of pressure displacement in the mediolateral direction (ML COP-D) to variant parameter values and two pre-processing methods. These two methods are filtering the high-frequency components and resampling the signals to have the same average number of data points per stride. The discriminatory ability of SampEn is studied by comparing treadmill walk only (WO) to dual-task (DT) condition. The results suggest that SampEn maintains the directional difference between two walking conditions across variant parameter values, showing a significant increase from WO to DT condition, especially when signals are low-pass filtered. Moreover, when gait speed is different between test conditions, signals should be low-pass filtered and resampled to have the same average number of data points per stride. Full article
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Open AccessArticle Assessing Information Transmission in Data Transformations with the Channel Multivariate Entropy Triangle
Entropy 2018, 20(7), 498; https://doi.org/10.3390/e20070498
Received: 3 May 2018 / Revised: 11 June 2018 / Accepted: 20 June 2018 / Published: 27 June 2018
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Abstract
Data transformation, e.g., feature transformation and selection, is an integral part of any machine learning procedure. In this paper, we introduce an information-theoretic model and tools to assess the quality of data transformations in machine learning tasks. In an unsupervised fashion, we analyze
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Data transformation, e.g., feature transformation and selection, is an integral part of any machine learning procedure. In this paper, we introduce an information-theoretic model and tools to assess the quality of data transformations in machine learning tasks. In an unsupervised fashion, we analyze the transformation of a discrete, multivariate source of information X¯ into a discrete, multivariate sink of information Y¯ related by a distribution PX¯Y¯. The first contribution is a decomposition of the maximal potential entropy of (X¯,Y¯), which we call a balance equation, into its (a) non-transferable, (b) transferable, but not transferred, and (c) transferred parts. Such balance equations can be represented in (de Finetti) entropy diagrams, our second set of contributions. The most important of these, the aggregate channel multivariate entropy triangle, is a visual exploratory tool to assess the effectiveness of multivariate data transformations in transferring information from input to output variables. We also show how these decomposition and balance equations also apply to the entropies of X¯ and Y¯, respectively, and generate entropy triangles for them. As an example, we present the application of these tools to the assessment of information transfer efficiency for Principal Component Analysis and Independent Component Analysis as unsupervised feature transformation and selection procedures in supervised classification tasks. Full article
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Open AccessArticle Vague Entropy Measure for Complex Vague Soft Sets
Entropy 2018, 20(6), 403; https://doi.org/10.3390/e20060403
Received: 12 March 2018 / Revised: 16 May 2018 / Accepted: 17 May 2018 / Published: 24 May 2018
Cited by 3 | PDF Full-text (5725 KB) | HTML Full-text | XML Full-text
Abstract
The complex vague soft set (CVSS) model is a hybrid of complex fuzzy sets and soft sets that have the ability to accurately represent and model two-dimensional information for real-life phenomena that are periodic in nature. In the existing studies of fuzzy and
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The complex vague soft set (CVSS) model is a hybrid of complex fuzzy sets and soft sets that have the ability to accurately represent and model two-dimensional information for real-life phenomena that are periodic in nature. In the existing studies of fuzzy and its extensions, the uncertainties which are present in the data are handled with the help of membership degree which is the subset of real numbers. However, in the present work, this condition has been relaxed with the degrees whose ranges are a subset of the complex subset with unit disc and hence handle the information in a better way. Under this environment, we developed some entropy measures of the CVSS model induced by the axiomatic definition of distance measure. Some desirable relations between them are also investigated. A numerical example related to detection of an image by the robot is given to illustrate the proposed entropy measure. Full article
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Open AccessArticle Identification of Auditory Object-Specific Attention from Single-Trial Electroencephalogram Signals via Entropy Measures and Machine Learning
Entropy 2018, 20(5), 386; https://doi.org/10.3390/e20050386
Received: 17 April 2018 / Revised: 16 May 2018 / Accepted: 16 May 2018 / Published: 21 May 2018
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Abstract
Existing research has revealed that auditory attention can be tracked from ongoing electroencephalography (EEG) signals. The aim of this novel study was to investigate the identification of peoples’ attention to a specific auditory object from single-trial EEG signals via entropy measures and machine
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Existing research has revealed that auditory attention can be tracked from ongoing electroencephalography (EEG) signals. The aim of this novel study was to investigate the identification of peoples’ attention to a specific auditory object from single-trial EEG signals via entropy measures and machine learning. Approximate entropy (ApEn), sample entropy (SampEn), composite multiscale entropy (CmpMSE) and fuzzy entropy (FuzzyEn) were used to extract the informative features of EEG signals under three kinds of auditory object-specific attention (Rest, Auditory Object1 Attention (AOA1) and Auditory Object2 Attention (AOA2)). The linear discriminant analysis and support vector machine (SVM), were used to construct two auditory attention classifiers. The statistical results of entropy measures indicated that there were significant differences in the values of ApEn, SampEn, CmpMSE and FuzzyEn between Rest, AOA1 and AOA2. For the SVM-based auditory attention classifier, the auditory object-specific attention of Rest, AOA1 and AOA2 could be identified from EEG signals using ApEn, SampEn, CmpMSE and FuzzyEn as features and the identification rates were significantly different from chance level. The optimal identification was achieved by the SVM-based auditory attention classifier using CmpMSE with the scale factor τ = 10. This study demonstrated a novel solution to identify the auditory object-specific attention from single-trial EEG signals without the need to access the auditory stimulus. Full article
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Open AccessArticle The Relationship between Postural Stability and Lower-Limb Muscle Activity Using an Entropy-Based Similarity Index
Entropy 2018, 20(5), 320; https://doi.org/10.3390/e20050320
Received: 28 March 2018 / Revised: 14 April 2018 / Accepted: 21 April 2018 / Published: 26 April 2018
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Abstract
The aim of this study is to see if the centre of pressure (COP) measurements on the postural stability can be used to represent the electromyography (EMG) measurement on the activity data of lower limb muscles. If so, the cost-effective COP data measurements
[...] Read more.
The aim of this study is to see if the centre of pressure (COP) measurements on the postural stability can be used to represent the electromyography (EMG) measurement on the activity data of lower limb muscles. If so, the cost-effective COP data measurements can be used to indicate the level of postural stability and lower limb muscle activity. The Hilbert–Huang Transform method was used to analyse the data from the experimental designed to examine the correlation between lower-limb muscles and postural stability. We randomly selected 24 university students to participate in eight scenarios and simultaneously measured their COP and EMG signals during the experiments. The Empirical Mode Decomposition was used to identify the intrinsic-mode functions (IMF) that can distinguish between the COP and EMG at different states. Subsequently, similarity indices and synchronization analyses were used to calculate the correlation between the lower-limb muscle strength and the postural stability. The IMF5 of the COP signals and the IMF6 of the EMG signals were not significantly different and the average frequency was 0.8 Hz, with a range of 0–2 Hz. When the postural stability was poor, the COP and EMG had a high synchronization with index values within the range of 0.010–0.015. With good postural stability, the synchronization indices were between 0.006 and 0.080 and both exhibited low synchronization. The COP signals and the low frequency EMG signals were highly correlated. In conclusion, we demonstrated that the COP may provide enough information on postural stability without the EMG data. Full article
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