Special Issue "Wavelet Entropy: Computation and Applications"

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

Deadline for manuscript submissions: closed (30 May 2015).

Special Issue Editor

Dr. Raúl Alcaraz
E-Mail Website
Guest Editor
Research Group in Electronic, Biomedical and Telecommunication Engineering, Universidad de Castilla-La Mancha, Campus Universitario s/n, 16071, Cuenca, Spain
Tel. +34969179100
Interests: entropy; complexity; information theory; information geometry; nonlinear dynamics; computational mathematics and statistics in medicine; biomedical time series analysis; cardiac signal processing
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Special Issue Information

Dear Colleagues,

Wavelet Entropy (WE) is a novel tool with the ability to analyze transient features of non-stationary signals. This metric combines wavelet decomposition and entropy to estimate the degree of order/disorder of a signal with a high time-frequency resolution. Initially, Shannon entropy was proposed to quantify the energy distribution in wavelet sub-bands, the metric defined in this way being applied to a wide variety of different scenarios. Special interest has shown WE’s application to physiological signals, such as electrocardiograms, electroencephalograms, intracranial pressure recordings or evoked related potentials, in which it is able to reveal clinically useful information, e.g., in the prevention of cardiac diseases or the detection of driving fatigue. A similar attention has also initiated WE’s use in forecasting faults and dangers in modern power systems and detecting machinery vibration. Moreover, several studies have also shown the superiority of WE in analyzing the variability and complexity of climate processes compared with traditional methods. Finally, it is worth noting that WE-based analysis of electromechanical noise has recently gained an increasing interest regarding remote corrosion monitoring in industrial applications.

However, Shannon entropy can present difficulties, such as wavelet aliasing and energy leakage, in the processing of some non-stationary signals. Thus, recent studies have proposed various alternatives to compute this metric. Novel indices such as Relative WE, Wavelet Singular Entropy, Wavelet Tsallis Entropy and Wavelet Sample Entropy, amongst others, can be found in the literature. Within this context, this special issue aims to provide a forum for contributions on these new ways of computing WE and the most recent advances in its application to every scenario in which useful information can be obtained.

Prof. Dr. Raúl Alcaraz Martínez
Guest Editor

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Keywords

  • Wavelet Transform
  • Wavelet Entropy and Relative Wavelet Entropy
  • Wavelet Singular Entropy
  • Multiscale Wavelet Entropy
  • Wavelet Tsallis Entropy
  • Wavelet Sample Entropy
  • Physiological signal processing (ECG, EEG, ICP, ERP, etc.)
  • Biomedical time series processing (HRV, RR, PP, etc.)
  • Defaults in electric power systems
  • Transmission line fault detection and identification
  • Image processing and compression
  • Hydrologic series data analysis
  • Electromechanical noise

Published Papers (13 papers)

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Research

Open AccessArticle
Wavelet Entropy Based Analysis and Forecasting of Crude Oil Price Dynamics
Entropy 2015, 17(10), 7167-7184; https://doi.org/10.3390/e17107167 - 22 Oct 2015
Cited by 5
Abstract
For the modeling of complex and nonlinear crude oil price dynamics and movement, wavelet analysis can decompose the time series and produce multiple economically meaningful decomposition structures based on different assumptions of wavelet families and decomposition scale. However, the determination of the optimal [...] Read more.
For the modeling of complex and nonlinear crude oil price dynamics and movement, wavelet analysis can decompose the time series and produce multiple economically meaningful decomposition structures based on different assumptions of wavelet families and decomposition scale. However, the determination of the optimal model specification will critically affect the forecasting accuracy. In this paper, we propose a new wavelet entropy based approach to identify the optimal model specification and construct the effective wavelet entropy based forecasting models. The wavelet entropy algorithm is introduced to determine the optimal wavelet families and decomposition scale, that will produce the improved forecasting performance. Empirical studies conducted in the crude oil markets show that the proposed algorithm outperforms the benchmark model, in terms of conventional performance evaluation criteria for the model forecasting accuracy. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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Open AccessArticle
A Vibration Analysis Based on Wavelet Entropy Method of a Scroll Compressor
Entropy 2015, 17(10), 7076-7086; https://doi.org/10.3390/e17107076 - 19 Oct 2015
Cited by 4
Abstract
Vibration-based condition monitoring and fault diagnosis is an effective approach to maintain the reliable operation of a scroll compressor. Unfortunately, the vibration signal from the scroll compressor always has characteristics of being non-linear and non-stationary, which makes vibration signal analysis and fault feature [...] Read more.
Vibration-based condition monitoring and fault diagnosis is an effective approach to maintain the reliable operation of a scroll compressor. Unfortunately, the vibration signal from the scroll compressor always has characteristics of being non-linear and non-stationary, which makes vibration signal analysis and fault feature extraction very difficult. To extract the significant fault features, a vibration analysis method based on Wavelet entropy is proposed in this paper. Two forms of the wavelet entropy, namely the wavelet space feature spectrum entropy (WSFSE) and the wavelet energy spectrum entropy (WESE), are defined to depict instantaneous characteristics of the local variation of the vibration signal. Four types of mechanical faulty vibration signal, namely unbalanced rotor, malfunctioning scroll, loosened mechanical assembly, and loosened bearing, are analyzed by using the proposed approach. The experimental results show that feature components and energy distribution of each fault signal is accurately identified and revealed, which proves that the combined application of WSFSE and WESE approach is a successful scheme for the vibration analysis of scroll compressors. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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Open AccessArticle
Shannon Entropy-Based Wavelet Transform Method for Autonomous Coherent Structure Identification in Fluid Flow Field Data
Entropy 2015, 17(10), 6617-6642; https://doi.org/10.3390/e17106617 - 25 Sep 2015
Cited by 11
Abstract
The coherent secondary flow structures (i.e., swirling motions) in a curved artery model possess a variety of spatio-temporal morphologies and can be encoded over an infinitely-wide range of wavelet scales. Wavelet analysis was applied to the following vorticity fields: (i) a numerically-generated system [...] Read more.
The coherent secondary flow structures (i.e., swirling motions) in a curved artery model possess a variety of spatio-temporal morphologies and can be encoded over an infinitely-wide range of wavelet scales. Wavelet analysis was applied to the following vorticity fields: (i) a numerically-generated system of Oseen-type vortices for which the theoretical solution is known, used for bench marking and evaluation of the technique; and (ii) experimental two-dimensional, particle image velocimetry data. The mother wavelet, a two-dimensional Ricker wavelet, can be dilated to infinitely large or infinitesimally small scales. We approached the problem of coherent structure detection by means of continuous wavelet transform (CWT) and decomposition (or Shannon) entropy. The main conclusion of this study is that the encoding of coherent secondary flow structures can be achieved by an optimal number of binary digits (or bits) corresponding to an optimal wavelet scale. The optimal wavelet-scale search was driven by a decomposition entropy-based algorithmic approach and led to a threshold-free coherent structure detection method. The method presented in this paper was successfully utilized in the detection of secondary flow structures in three clinically-relevant blood flow scenarios involving the curved artery model under a carotid artery-inspired, pulsatile inflow condition. These scenarios were: (i) a clean curved artery; (ii) stent-implanted curved artery; and (iii) an idealized Type IV stent fracture within the curved artery. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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Open AccessArticle
Wavelet Entropy as a Measure of Ventricular Beat Suppression from the Electrocardiogram in Atrial Fibrillation
Entropy 2015, 17(9), 6397-6411; https://doi.org/10.3390/e17096397 - 17 Sep 2015
Cited by 9
Abstract
A novel method of quantifying the effectiveness of the suppression of ventricular activity from electrocardiograms (ECGs) in atrial fibrillation is proposed. The temporal distribution of the energy of wavelet coefficients is quantified by wavelet entropy at each ventricular beat. More effective ventricular activity [...] Read more.
A novel method of quantifying the effectiveness of the suppression of ventricular activity from electrocardiograms (ECGs) in atrial fibrillation is proposed. The temporal distribution of the energy of wavelet coefficients is quantified by wavelet entropy at each ventricular beat. More effective ventricular activity suppression yields increased entropies at scales dominated by the ventricular and atrial components of the ECG. Two studies are undertaken to demonstrate the efficacy of the method: first, using synthesised ECGs with controlled levels of residual ventricular activity, and second, using patient recordings with ventricular activity suppressed by an average beat template subtraction algorithm. In both cases wavelet entropy is shown to be a good measure of the effectiveness of ventricular beat suppression. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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Open AccessArticle
Wavelet Entropy Automatically Detects Episodes of Atrial Fibrillation from Single-Lead Electrocardiograms
Entropy 2015, 17(9), 6179-6199; https://doi.org/10.3390/e17096179 - 07 Sep 2015
Cited by 33
Abstract
This work introduces for the first time the application of wavelet entropy (WE) to detect episodes of the most common cardiac arrhythmia, atrial fibrillation (AF), automatically from the electrocardiogram (ECG). Given that AF is often asymptomatic and usually presents very brief initial episodes, [...] Read more.
This work introduces for the first time the application of wavelet entropy (WE) to detect episodes of the most common cardiac arrhythmia, atrial fibrillation (AF), automatically from the electrocardiogram (ECG). Given that AF is often asymptomatic and usually presents very brief initial episodes, its early automatic detection is clinically relevant to improve AF treatment and prevent risks for the patients. After discarding noisy TQ intervals from the ECG, the WE has been computed over the median TQ segment obtained from the 10 previous noise-free beats under study. In this way, the P-waves or the fibrillatory waves present in the recording were highlighted or attenuated, respectively, thus enabling the patient’s rhythm identification (sinus rhythm or AF). Results provided a discriminant ability of about 95%, which is comparable to previous works. However, in contrast to most of them, which are mainly based on quantifying RR series variability, the proposed algorithm is able to deal with patients under rate-control therapy or with a reduced heart rate variability during AF. Additionally, it also presents interesting properties, such as the lowest delay in detecting AF or sinus rhythm, the ability to detect episodes as brief as five beats in length or its integration facilities under real-time beat-by-beat ECG monitoring systems. Consequently, this tool may help clinicians in the automatic detection of a wide variety of AF episodes, thus gaining further knowledge about the mechanisms initiating this arrhythmia. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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Open AccessArticle
Optimal Base Wavelet Selection for ECG Noise Reduction Using a Comprehensive Entropy Criterion
Entropy 2015, 17(9), 6093-6109; https://doi.org/10.3390/e17096093 - 01 Sep 2015
Cited by 17
Abstract
The selection of an appropriate wavelet is an essential issue that should be addressed in the wavelet-based filtering of electrocardiogram (ECG) signals. Since entropy can measure the features of uncertainty associated with the ECG signal, a novel comprehensive entropy criterion Ecom based [...] Read more.
The selection of an appropriate wavelet is an essential issue that should be addressed in the wavelet-based filtering of electrocardiogram (ECG) signals. Since entropy can measure the features of uncertainty associated with the ECG signal, a novel comprehensive entropy criterion Ecom based on multiple criteria related to entropy and energy is proposed in this paper to search for an optimal base wavelet for a specific ECG signal. Taking account of the decomposition capability of wavelets and the similarity in information between the decomposed coefficients and the analyzed signal, the proposed Ecom criterion integrates eight criteria, i.e., energy, entropy, energy-to-entropy ratio, joint entropy, conditional entropy, mutual information, relative entropy, as well as comparison information entropy for optimal wavelet selection. The experimental validation is conducted on the basis of ECG signals of sixteen subjects selected from the MIT-BIH Arrhythmia Database. The Ecom is compared with each of these eight criteria through four filtering performance indexes, i.e., output signal to noise ratio (SNRo), root mean square error (RMSE), percent root mean-square difference (PRD) and correlation coefficients. The filtering results of ninety-six ECG signals contaminated by noise have verified that Ecom has outperformed the other eight criteria in the selection of best base wavelets for ECG signal filtering. The wavelet identified by the Ecom has achieved the best filtering performance than the other comparative criteria. A hypothesis test also validates that SNRo, RMSE, PRD and correlation coefficients of Ecom are significantly different from those of the shape-matched approach (α = 0.05 , two-sided t- test). Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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Open AccessArticle
Combined Power Quality Disturbances Recognition Using Wavelet Packet Entropies and S-Transform
Entropy 2015, 17(8), 5811-5828; https://doi.org/10.3390/e17085811 - 12 Aug 2015
Cited by 20
Abstract
Aiming at the combined power quality +disturbance recognition, an automated recognition method based on wavelet packet entropy (WPE) and modified incomplete S-transform (MIST) is proposed in this paper. By combining wavelet packet Tsallis singular entropy, energy entropy and MIST, a 13-dimension vector of [...] Read more.
Aiming at the combined power quality +disturbance recognition, an automated recognition method based on wavelet packet entropy (WPE) and modified incomplete S-transform (MIST) is proposed in this paper. By combining wavelet packet Tsallis singular entropy, energy entropy and MIST, a 13-dimension vector of different power quality (PQ) disturbances including single disturbances and combined disturbances is extracted. Then, a ruled decision tree is designed to recognize the combined disturbances. The proposed method is tested and evaluated using a large number of simulated PQ disturbances and some real-life signals, which include voltage sag, swell, interruption, oscillation transient, impulsive transient, harmonics, voltage fluctuation and their combinations. In addition, the comparison of the proposed recognition approach with some existing techniques is made. The experimental results show that the proposed method can effectively recognize the single and combined PQ disturbances. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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Open AccessArticle
A Pilot Directional Protection for HVDC Transmission Line Based on Relative Entropy of Wavelet Energy
Entropy 2015, 17(8), 5257-5273; https://doi.org/10.3390/e17085257 - 27 Jul 2015
Cited by 10
Abstract
On the basis of analyzing high-voltage direct current (HVDC) transmission system and its fault superimposed circuit, the direction of the fault components of the voltage and the current measured at one end of transmission line is certified to be different for internal faults [...] Read more.
On the basis of analyzing high-voltage direct current (HVDC) transmission system and its fault superimposed circuit, the direction of the fault components of the voltage and the current measured at one end of transmission line is certified to be different for internal faults and external faults. As an estimate of the differences between two signals, relative entropy is an effective parameter for recognizing transient signals in HVDC transmission lines. In this paper, the relative entropy of wavelet energy is applied to distinguish internal fault from external fault. For internal faults, the directions of fault components of voltage and current are opposite at the two ends of the transmission line, indicating a huge difference of wavelet energy relative entropy; for external faults, the directions are identical, indicating a small difference. The simulation results based on PSCAD/EMTDC show that the proposed pilot protection system acts accurately for faults under different conditions, and its performance is not affected by fault type, fault location, fault resistance and noise. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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Open AccessArticle
Neural Network Reorganization Analysis During an Auditory Oddball Task in Schizophrenia Using Wavelet Entropy
Entropy 2015, 17(8), 5241-5256; https://doi.org/10.3390/e17085241 - 27 Jul 2015
Cited by 17
Abstract
The aim of the present study was to characterize the neural network reorganization during a cognitive task in schizophrenia (SCH) by means of wavelet entropy (WE). Previous studies suggest that the cognitive impairment in patients with SCH could be related to the disrupted [...] Read more.
The aim of the present study was to characterize the neural network reorganization during a cognitive task in schizophrenia (SCH) by means of wavelet entropy (WE). Previous studies suggest that the cognitive impairment in patients with SCH could be related to the disrupted integrative functions of neural circuits. Nevertheless, further characterization of this effect is needed, especially in the time-frequency domain. This characterization is sensitive to fast neuronal dynamics and their synchronization that may be an important component of distributed neuronal interactions; especially in light of the disconnection hypothesis for SCH and its electrophysiological correlates. In this work, the irregularity dynamics elicited by an auditory oddball paradigm were analyzed through synchronized-averaging (SA) and single-trial (ST) analyses. They provide complementary information on the spatial patterns involved in the neural network reorganization. Our results from 20 healthy controls and 20 SCH patients showed a WE decrease from baseline to response both in controls and SCH subjects. These changes were significantly more pronounced for healthy controls after ST analysis, mainly in central and frontopolar areas. On the other hand, SA analysis showed more widespread spatial differences than ST results. These findings suggest that the activation response is weakly phase-locked to stimulus onset in SCH and related to the default mode and salience networks. Furthermore, the less pronounced changes in WE from baseline to response for SCH patients suggest an impaired ability to reorganize neural dynamics during an oddball task. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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Open AccessArticle
An Integrated Index for the Identification of Focal Electroencephalogram Signals Using Discrete Wavelet Transform and Entropy Measures
Entropy 2015, 17(8), 5218-5240; https://doi.org/10.3390/e17085218 - 27 Jul 2015
Cited by 83
Abstract
The dynamics of brain area influenced by focal epilepsy can be studied using focal and non-focal electroencephalogram (EEG) signals. This paper presents a new method to detect focal and non-focal EEG signals based on an integrated index, termed the focal and non-focal index [...] Read more.
The dynamics of brain area influenced by focal epilepsy can be studied using focal and non-focal electroencephalogram (EEG) signals. This paper presents a new method to detect focal and non-focal EEG signals based on an integrated index, termed the focal and non-focal index (FNFI), developed using discrete wavelet transform (DWT) and entropy features. The DWT decomposes the EEG signals up to six levels, and various entropy measures are computed from approximate and detail coefficients of sub-band signals. The computed entropy measures are average wavelet, permutation, fuzzy and phase entropies. The proposed FNFI developed using permutation, fuzzy and Shannon wavelet entropies is able to clearly discriminate focal and non-focal EEG signals using a single number. Furthermore, these entropy measures are ranked using different techniques, namely the Bhattacharyya space algorithm, Student’s t-test, the Wilcoxon test, the receiver operating characteristic (ROC) and entropy. These ranked features are fed to various classifiers, namely k-nearest neighbour (KNN), probabilistic neural network (PNN), fuzzy classifier and least squares support vector machine (LS-SVM), for automated classification of focal and non-focal EEG signals using the minimum number of features. The identification of the focal EEG signals can be helpful to locate the epileptogenic focus. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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Open AccessArticle
2D Anisotropic Wavelet Entropy with an Application to Earthquakes in Chile
Entropy 2015, 17(6), 4155-4172; https://doi.org/10.3390/e17064155 - 16 Jun 2015
Cited by 4
Abstract
We propose a wavelet-based approach to measure the Shannon entropy in the context of spatial point patterns. The method uses the fully anisotropic Morlet wavelet to estimate the energy distribution at different directions and scales. The spatial heterogeneity and complexity of spatial point [...] Read more.
We propose a wavelet-based approach to measure the Shannon entropy in the context of spatial point patterns. The method uses the fully anisotropic Morlet wavelet to estimate the energy distribution at different directions and scales. The spatial heterogeneity and complexity of spatial point patterns is then analyzed using the multiscale anisotropic wavelet entropy. The efficacy of the approach is shown through a simulation study. Finally, an application to the catalog of earthquake events in Chile is considered. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
Open AccessArticle
Operational Reliability Assessment of Compressor Gearboxes with Normalized Lifting Wavelet Entropy from Condition Monitoring Information
Entropy 2015, 17(5), 3479-3500; https://doi.org/10.3390/e17053479 - 20 May 2015
Abstract
Classical reliability assessment methods have predominantly focused on probability and statistical theories, which are insufficient in assessing the operational reliability of individual mechanical equipment with time-varying characteristics. A new approach to assess machinery operational reliability with normalized lifting wavelet entropy from condition monitoring [...] Read more.
Classical reliability assessment methods have predominantly focused on probability and statistical theories, which are insufficient in assessing the operational reliability of individual mechanical equipment with time-varying characteristics. A new approach to assess machinery operational reliability with normalized lifting wavelet entropy from condition monitoring information is proposed, which is different from classical reliability assessment methods depending on probability and statistics analysis. The machinery vibration signals with time-varying operational characteristics are firstly decomposed and reconstructed by means of a lifting wavelet package transform. The relative energy of every reconstructed signal is computed as an energy percentage of the reconstructed signal in the whole signal energy. Moreover, a normalized lifting wavelet entropy is defined by the relative energy to reveal the machinery operational uncertainty. Finally, operational reliability degree is defined by the quantitative value obtained by the normalized lifting wavelet entropy belonging to the range of [0, 1]. The proposed method is applied in the operational reliability assessment of the gearbox in an oxy-generator compressor to validate the effectiveness. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
Open AccessArticle
Preclinical Diagnosis of Magnetic Resonance (MR) Brain Images via Discrete Wavelet Packet Transform with Tsallis Entropy and Generalized Eigenvalue Proximal Support Vector Machine (GEPSVM)
Entropy 2015, 17(4), 1795-1813; https://doi.org/10.3390/e17041795 - 30 Mar 2015
Cited by 133
Abstract
Background: Developing an accurate computer-aided diagnosis (CAD) system of MR brain images is essential for medical interpretation and analysis. In this study, we propose a novel automatic CAD system to distinguish abnormal brains from normal brains in MRI scanning. Methods: The [...] Read more.
Background: Developing an accurate computer-aided diagnosis (CAD) system of MR brain images is essential for medical interpretation and analysis. In this study, we propose a novel automatic CAD system to distinguish abnormal brains from normal brains in MRI scanning. Methods: The proposed method simplifies the task to a binary classification problem. We used discrete wavelet packet transform (DWPT) to extract wavelet packet coefficients from MR brain images. Next, Shannon entropy (SE) and Tsallis entropy (TE) were harnessed to obtain entropy features from DWPT coefficients. Finally, generalized eigenvalue proximate support vector machine (GEPSVM), and GEPSVM with radial basis function (RBF) kernel, were employed as classifier. We tested the four proposed diagnosis methods (DWPT + SE + GEPSVM, DWPT + TE + GEPSVM, DWPT + SE + GEPSVM + RBF, and DWPT + TE + GEPSVM + RBF) on three benchmark datasets of Dataset-66, Dataset-160, and Dataset-255. Results: The 10 repetition of K-fold stratified cross validation results showed the proposed DWPT + TE + GEPSVM + RBF method excelled not only other three proposed classifiers but also existing state-of-the-art methods in terms of classification accuracy. In addition, the DWPT + TE + GEPSVM + RBF method achieved accuracy of 100%, 100%, and 99.53% on Dataset-66, Dataset-160, and Dataset-255, respectively. For Dataset-255, the offline learning cost 8.4430s and online prediction cost merely 0.1059s. Conclusions: We have proved the effectiveness of the proposed method, which achieved nearly 100% accuracy over three benchmark datasets. Full article
(This article belongs to the Special Issue Wavelet Entropy: Computation and Applications)
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