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Wavelets, Fractals and Information Theory I

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

Deadline for manuscript submissions: closed (20 December 2015) | Viewed by 133914

Special Issue Editor


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Guest Editor
Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Wavelet Analysis and Fractals are playing fundamental roles in various applications in Science, Engineering, and Information Theory.

In information theory, the entropy encoding might be considered a sort of compression in a quantization process, and this can be further investigated by using the wavelet compression. There are many types of entropy definitions that are very useful in the Engineering and Applied Sciences, such as the Shannon-Fano entropy, the Kolmogorov entropy, etc. However, only entropy encoding is optimal for the complexity of large data analysis, such as in data storage. In fact, the principal advantage of modeling a complex problem via wavelet analysis is the minimization of the memory space for storage or transmission. Moreover, this kind of approach reveals some new aspects and promising perspectives in many other kinds of applied and theoretical problems. For instance, in engineering, the best way to model the traffic in wireless communication is based on fractal geometry, whereas the data are efficiently studied through wavelet basis.

This Special Issue will also be an opportunity for extending the research fields of image processing, differential/integral equations, number theory and special functions, image segmentation, the sparse component analysis approach, generalized multiresolution analysis, and entropy as a measure in all aspects of the theoretical and practical studies of Mathematics, Physics, and Engineering.

The main topics of this Special Issue include (but are not limited to):

  • Entropy encoding, wavelet compression, and information theory.
  • Fractals, Non-differentiable functions. Theoretical and applied analytical problems of fractal type, fractional equations.
  • Wavelet Analysis, integral transforms and applications.
  • Wavelet-fractal entropy encoding and computational mathematics, including in image processing.
  • Wavelet-fractal approach.

Prof. Dr. Carlo Cattani
Guest Editor

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Published Papers (18 papers)

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906 KiB  
Article
Entropy and Fractal Antennas
by Emanuel Guariglia
Entropy 2016, 18(3), 84; https://doi.org/10.3390/e18030084 - 4 Mar 2016
Cited by 176 | Viewed by 14616
Abstract
The entropies of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape [...] Read more.
The entropies of Shannon, Rényi and Kolmogorov are analyzed and compared together with their main properties. The entropy of some particular antennas with a pre-fractal shape, also called fractal antennas, is studied. In particular, their entropy is linked with the fractal geometrical shape and the physical performance. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Rényi and Shannon entropies for a binomial distribution with N = 20: they converge for <math display="inline"> <mrow> <mi>α</mi> <mo>→</mo> <mn>1</mn> </mrow> </math>, in accord with Equation (<a href="#FD3-entropy-18-00084" class="html-disp-formula">3</a>). The computation of both entropies was done for <math display="inline"> <mrow> <mi>b</mi> <mo>=</mo> <mn>2</mn> </mrow> </math>.</p>
Full article ">Figure 2
<p>Kolmogorov entropy for 1D-regular, chaotic-deterministic and random systems. The attractor is the classical Lorentz attractor [<a href="#B27-entropy-18-00084" class="html-bibr">27</a>] with <math display="inline"> <mrow> <mi>σ</mi> <mo>=</mo> <mn>10</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>R</mi> <mo>=</mo> <mn>28</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>b</mi> <mo>=</mo> <mn>8</mn> <mo>/</mo> <mn>3</mn> </mrow> </math> and initial values <math display="inline"> <mrow> <mi>x</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mi>z</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>y</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math>, while the random motion is given by a 2D-random walk from <math display="inline"> <mrow> <mo>-</mo> <mn>1</mn> <mo>-</mo> <mi>j</mi> </mrow> </math> to <math display="inline"> <mrow> <mn>1</mn> <mo>+</mo> <mi>j</mi> </mrow> </math> of 500 elements in which <span class="html-italic">j</span> is the imaginary unit.</p>
Full article ">Figure 3
<p>Graph of the <math display="inline"> <mrow> <msub> <mi mathvariant="script">M</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <mi>A</mi> <mo>)</mo> </mrow> </mrow> </math>, where <span class="html-italic">A</span> is a bounded subset of the Euclidean metric space <math display="inline"> <msup> <mrow> <mi>R</mi> </mrow> <mi>n</mi> </msup> </math>. It takes only two possible values, and the Hausdorff–Besicovitch dimension of <span class="html-italic">A</span> is given by the value of <span class="html-italic">s</span> in which there is the jump from <span class="html-italic">∞</span> to zero.</p>
Full article ">Figure 4
<p>Here, the first steps of the box-counting procedure about England’s coastline are represented.</p>
Full article ">Figure 5
<p>Here, the von Kock curve (on the left) and the middle third Cantor set (on the right) are shown: <math display="inline"> <msub> <mi>A</mi> <mn>0</mn> </msub> </math> is the initiator of length equal to one in both cases; in the generator <math display="inline"> <msub> <mi>A</mi> <mn>1</mn> </msub> </math> for the von Kock curve, the middle third of the unit interval is replaced by the other two sides of an equilateral triangle, while that of the middle third Cantor set is obtained removing the middle third of the interval.</p>
Full article ">Figure 6
<p>Archimedean spiral antenna (on the left) and commercial log-periodic dipole antenna of 16 elements (on the right) [<a href="#B7-entropy-18-00084" class="html-bibr">7</a>].</p>
Full article ">Figure 7
<p>A Sierpinski triangle (on the left) and a Hilbert curve (on the right) are shown: as in <a href="#entropy-18-00084-f005" class="html-fig">Figure 5</a>, <math display="inline"> <msub> <mi>A</mi> <mn>0</mn> </msub> </math> is the initiator, and <math display="inline"> <msub> <mi>A</mi> <mn>1</mn> </msub> </math> is the generator. The Sierpinski triangle is constructed using the iterated function system (IFS), while the other construction is that of David Hilbert. The two relative antennas are shown below with their feed points.</p>
Full article ">Figure 8
<p>Examples of non-fractal antennas that offer similar performance over their fractal counterparts. Three different non-fractal antennas are presented above: they outperform their fractal counterparts, while the current distribution on the Sierpinski gasket antenna at the first three resonance frequencies is shown in the middle of the page. The modified Parany antenna (starting from the classical Sierpinski gasket antenna) is represented below.</p>
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<p>The Rényi entropy <math display="inline"> <msub> <mi>H</mi> <mi>α</mi> </msub> </math> of a Sierpinski gasket (<a href="#entropy-18-00084-f007" class="html-fig">Figure 7</a>) with <math display="inline"> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math>: the plot shows us that Hartley entropy <math display="inline"> <msub> <mi>H</mi> <mn>0</mn> </msub> </math> is an upper to both Shannon entropy <math display="inline"> <msub> <mi>H</mi> <mn>1</mn> </msub> </math> and collision entropy <math display="inline"> <msub> <mi>H</mi> <mn>2</mn> </msub> </math>. The main limit of this procedure is clearly the precision of the triangulation.</p>
Full article ">
4556 KiB  
Article
Wavelet Entropy-Based Traction Inverter Open Switch Fault Diagnosis in High-Speed Railways
by Keting Hu, Zhigang Liu and Shuangshuang Lin
Entropy 2016, 18(3), 78; https://doi.org/10.3390/e18030078 - 1 Mar 2016
Cited by 27 | Viewed by 6785
Abstract
In this paper, a diagnosis plan is proposed to settle the detection and isolation problem of open switch faults in high-speed railway traction system traction inverters. Five entropy forms are discussed and compared with the traditional fault detection methods, namely, discrete wavelet transform [...] Read more.
In this paper, a diagnosis plan is proposed to settle the detection and isolation problem of open switch faults in high-speed railway traction system traction inverters. Five entropy forms are discussed and compared with the traditional fault detection methods, namely, discrete wavelet transform and discrete wavelet packet transform. The traditional fault detection methods cannot efficiently detect the open switch faults in traction inverters because of the low resolution or the sudden change of the current. The performances of Wavelet Packet Energy Shannon Entropy (WPESE), Wavelet Packet Energy Tsallis Entropy (WPETE) with different non-extensive parameters, Wavelet Packet Energy Shannon Entropy with a specific sub-band (WPESE3,6), Empirical Mode Decomposition Shannon Entropy (EMDESE), and Empirical Mode Decomposition Tsallis Entropy (EMDETE) with non-extensive parameters in detecting the open switch fault are evaluated by the evaluation parameter. Comparison experiments are carried out to select the best entropy form for the traction inverter open switch fault detection. In addition, the DC component is adopted to isolate the failure Isolated Gate Bipolar Transistor (IGBT). The simulation experiments show that the proposed plan can diagnose single and simultaneous open switch faults correctly and timely. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Schematic diagram of the vector controlled traction system.</p>
Full article ">Figure 2
<p>(<b>a</b>) A-phase torque in normal state; (<b>b</b>) A-phase currents in normal state; (<b>c</b>) A-phase torque when <span class="html-italic">S</span>1 open switch fault occurs; (<b>d</b>) A-phase currents when <span class="html-italic">S</span>1 open switch fault occurs.</p>
Full article ">Figure 3
<p>Wavelet decomposition.</p>
Full article ">Figure 4
<p>(<b>a</b>) A-phase fault current with <span class="html-italic">S</span>1 open (50 kHz sampling frequency); (<b>b</b>) First level detail coefficient; (<b>c</b>) Second level detail coefficient; (<b>d</b>) Third level detail coefficient.</p>
Full article ">Figure 5
<p>(<b>a</b>) spectrum of fault free current (50 kHz sampling frequency and 0.2 s acquisition time); (<b>b</b>) spectrum of A-phase fault current with <span class="html-italic">S</span>1 open (50 kHz sampling frequency and 0.2 s acquisition time).</p>
Full article ">Figure 6
<p>Frequency of each sub-band after wavelet packet decomposition.</p>
Full article ">Figure 7
<p>(<b>a</b>) A-phase fault current with <span class="html-italic">S</span>1 open; (<b>b</b>) Coefficients of the sixth node in the third level.</p>
Full article ">Figure 8
<p>(<b>a</b>) WPESE of A-phase fault current with <span class="html-italic">S</span>1 open; (<b>b</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.1; (<b>c</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.5; (<b>d</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.9.</p>
Full article ">Figure 9
<p>(<b>a</b>) WPESE of A-phase fault current with <span class="html-italic">S</span>1 open; (<b>b</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 2; (<b>c</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 5; (<b>d</b>) WPETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 9.</p>
Full article ">Figure 10
<p>WPETE<sub>3,6</sub> of fault current.</p>
Full article ">Figure 11
<p>EMD results of A-phase fault current with <span class="html-italic">S</span>1 open.</p>
Full article ">Figure 12
<p>(<b>a</b>) EMDESE of A-phase fault current with <span class="html-italic">S</span>1 open; (<b>b</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.1; (<b>c</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.5; (<b>d</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 0.9.</p>
Full article ">Figure 13
<p>(<b>a</b>) EMDESE of A-phase fault current with <span class="html-italic">S</span>1 open; (<b>b</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 2; (<b>c</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 5; (<b>d</b>) EMDETE of A-phase fault current with <span class="html-italic">S</span>1 open when <span class="html-italic">q</span> = 9.</p>
Full article ">Figure 14
<p>(<b>a</b>) WPETE of fault current when <span class="html-italic">q</span> is in the range of [0.1,0.9]; (<b>b</b>) WPETE of fault current when <span class="html-italic">q</span> is in the range of [2,9]; (<b>c</b>) EMDETE of fault current when <span class="html-italic">q</span> is in the range of [0.1,0.9]; (<b>d</b>) EMDETE of fault current when <span class="html-italic">q</span> is in the range of [2,9].</p>
Full article ">Figure 15
<p>(<b>a</b>) Current flow of USF; Current wave when USF occurs in(<b>b</b>) negative current flow; (<b>c</b>) positive current flow; (<b>d</b>) Current flow of LSF; Current wave when LSF occurs in (<b>e</b>) positive current flow; (<b>f</b>) negative current flow.</p>
Full article ">Figure 16
<p>Flow chart of the proposed method.</p>
Full article ">Figure 17
<p>(<b>a</b>) A-phase current in normal condition; (<b>b</b>) A-phase WPESE<sub>3,6</sub>; (<b>c</b>) B-phase current in normal condition; (<b>d</b>) B-phase WPESE<sub>3,6</sub>; (<b>e</b>) C-phase current in normal condition; (<b>f</b>) C-phase WPESE<sub>3,6</sub>.</p>
Full article ">Figure 18
<p>Normal condition (<b>a</b>) Torque; (<b>b</b>) Speed of the traction motor; (<b>c</b>) Three phase current.</p>
Full article ">Figure 19
<p>Fault A condition (<b>a</b>) A-phase WPESE<sub>3,6</sub>; (<b>b</b>) B-phase WPESE<sub>3,6</sub>; (<b>c</b>) A-phase DC component; (<b>d</b>) C-phase WPESE<sub>3,6</sub>.</p>
Full article ">Figure 20
<p>Fault B condition (<b>a</b>) A-phase WPESE<sub>3,6</sub>; (<b>b</b>) B-phase WPESE<sub>3,6</sub>; (<b>c</b>) A-phase DC component; (<b>d</b>) B-phase DC component; (<b>d</b>) C-phase WPESE<sub>3,6</sub>; (<b>f</b>) C-phase DC component.</p>
Full article ">
1838 KiB  
Article
Improved LMD, Permutation Entropy and Optimized K-Means to Fault Diagnosis for Roller Bearings
by Zongli Shi, Wanqing Song and Saied Taheri
Entropy 2016, 18(3), 70; https://doi.org/10.3390/e18030070 - 25 Feb 2016
Cited by 54 | Viewed by 6410
Abstract
A novel bearing vibration signal fault feature extraction and recognition method based on the improved local mean decomposition (LMD), permutation entropy (PE) and the optimized K-means clustering algorithm is put forward in this paper. The improved LMD is proposed based on the self-similarity [...] Read more.
A novel bearing vibration signal fault feature extraction and recognition method based on the improved local mean decomposition (LMD), permutation entropy (PE) and the optimized K-means clustering algorithm is put forward in this paper. The improved LMD is proposed based on the self-similarity of roller bearing vibration signal extending the right and left side of the original signal to suppress its edge effect. After decomposing the extended signal into a set of product functions (PFs), the PE is utilized to display the complexity of the PF component and extract the fault feature meanwhile. Then, the optimized K-means algorithm is used to cluster analysis as a new pattern recognition approach, which uses the probability density distribution (PDD) to identify the initial centroid selection and has the priority of recognition accuracy compared with the classic one. Finally, the experiment results show the proposed method is effectively to fault extraction and recognition for roller bearing. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>The local mean decomposition (LMD) decomposition of <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> .</p>
Full article ">Figure 2
<p>The time-frequency representation of the product functions (PFs) derived from the LMD.</p>
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<p>The improved LMD decomposition results of <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics> </math> .</p>
Full article ">Figure 4
<p>The time-frequency representation of the PFs derived from the improved LMD.</p>
Full article ">Figure 5
<p>The sample data sets for clustering.</p>
Full article ">Figure 6
<p>The roller bearing experiment system’s sketch.</p>
Full article ">Figure 7
<p>The temporal distributions of roller bearing vibration signal in different sample frequency. (<b>a</b>) 5000 Hz (<b>b</b>) 10,000 Hz (<b>c</b>) 20,000 Hz.</p>
Full article ">Figure 8
<p>The permutation entropy (PE) values for all of the four conditions.</p>
Full article ">
1572 KiB  
Article
Modelling the Spread of River Blindness Disease via the Caputo Fractional Derivative and the Beta-derivative
by Abdon Atangana and Rubayyi T. Alqahtani
Entropy 2016, 18(2), 40; https://doi.org/10.3390/e18020040 - 26 Jan 2016
Cited by 122 | Viewed by 7118
Abstract
Information theory is used in many branches of science and technology. For instance, to inform a set of human beings living in a particular region about the fatality of a disease, one makes use of existing information and then converts it into a [...] Read more.
Information theory is used in many branches of science and technology. For instance, to inform a set of human beings living in a particular region about the fatality of a disease, one makes use of existing information and then converts it into a mathematical equation for prediction. In this work, a model of the well-known river blindness disease is created via the Caputo and beta derivatives. A partial study of stability analysis was presented. The extended system describing the spread of this disease was solved via two analytical techniques: the Laplace perturbation and the homotopy decomposition methods. Summaries of the iteration methods used were provided to derive special solutions to the extended systems. Employing some theoretical parameters, we present some numerical simulations. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Female black fly biting a human being.</p>
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<p>Approximate solution for alpha = 0.106.</p>
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<p>Approximate solution for alpha = 0.106.</p>
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<p>Approximate solution for mu = 1.</p>
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<p>Approximate solution for mu =0.6.</p>
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<p>Approximation for mu = 0.4.</p>
Full article ">Figure 7
<p>Approximate solution for mu = 0.25.</p>
Full article ">
2450 KiB  
Article
Wavelet Energy and Wavelet Coherence as EEG Biomarkers for the Diagnosis of Parkinson’s Disease-Related Dementia and Alzheimer’s Disease
by Dong-Hwa Jeong, Young-Do Kim, In-Uk Song, Yong-An Chung and Jaeseung Jeong
Entropy 2016, 18(1), 8; https://doi.org/10.3390/e18010008 - 29 Dec 2015
Cited by 52 | Viewed by 9624
Abstract
Parkinson’s disease (PD) and Alzheimer’s disease (AD) can coexist in severely affected; elderly patients. Since they have different pathological causes and lesions and consequently require different treatments; it is critical to distinguish PD-related dementia (PD-D) from AD. Conventional electroencephalograph (EEG) analysis has produced [...] Read more.
Parkinson’s disease (PD) and Alzheimer’s disease (AD) can coexist in severely affected; elderly patients. Since they have different pathological causes and lesions and consequently require different treatments; it is critical to distinguish PD-related dementia (PD-D) from AD. Conventional electroencephalograph (EEG) analysis has produced poor results. This study investigated the possibility of using relative wavelet energy (RWE) and wavelet coherence (WC) analysis to distinguish between PD-D patients; AD patients and healthy elderly subjects. In EEG signals; we found that low-frequency wavelet energy increased and high-frequency wavelet energy decreased in PD-D patients and AD patients relative to healthy subjects. This result suggests that cognitive decline in both diseases is potentially related to slow EEG activity; which is consistent with previous studies. More importantly; WC values were lower in AD patients and higher in PD-D patients compared with healthy subjects. In particular; AD patients exhibited decreased WC primarily in the γ band and in links related to frontal regions; while PD-D patients exhibited increased WC primarily in the α and β bands and in temporo-parietal links. Linear discriminant analysis (LDA) of RWE produced a maximum accuracy of 79.18% for diagnosing PD-D and 81.25% for diagnosing AD. The discriminant accuracy was 73.40% with 78.78% sensitivity and 69.47% specificity. In distinguishing between the two diseases; the maximum performance of LDA using WC was 80.19%. We suggest that using a wavelet approach to evaluate EEG results may facilitate discrimination between PD-D and AD. In particular; RWE is useful for differentiating individuals with and without dementia and WC is useful for differentiating between PD-D and AD. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>The mean RWE values in different brain regions in the PD-D, AD, and control groups. (<b>a</b>) Frontal region (Fp1, Fp2, F3, F4, and Fz); (<b>b</b>) Parietal region (Fz, C3, C4, Cz, and Pz); (<b>c</b>) Occipital region (P3, P4, Pz, O1, and O2); (<b>d</b>) Temporal region (F7, F8, T3, T4, T5, and T6). Larger RWE values were found in the β frequency band in the PD-D and AD groups relative to the control group in the frontal, parietal, and occipital regions (a–c). There were significantly larger decreases in RWE value in the θ frequency band in the PD-D group <span class="html-italic">vs.</span> the control group (c). (* denotes a Bonferroni-corrected <span class="html-italic">p</span>-value &lt; 0.05.)</p>
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<p>WC values corresponding to between-channel distances. Larger WC values were found in lower frequency bands, indicating that low-frequency components of EEG signals were more influenced by volume conduction effects. (blue line = δ, red line = θ, yellow line = α, purple line = β, green line = γ). (<b>a</b>) In the anterior-posterior comparison, WC values were largest between adjacent channel pairs and decreased as channels became more distant; (<b>b</b>) In the lateral-medial comparison, the largest WC values were found between symmetric channels; other WC values were similar regardless of distance (dist. = distance between channels).</p>
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<p>Connections with significant differences between the two patient groups for each band. For Bonferroni-corrected <span class="html-italic">p</span>-value &lt; 0.05, the channel pairs were connected with different colored lines to indicate channel topography and marked in black on the adjacency matrix (CS: Control subjects, AD: AD patients, PD-D: PD-D patients). (<b>a</b>) In the α band, the connections in the PD-D group had larger WC values than those in the AD group; (<b>b</b>) In the β band, many of the connections in the PD-D group had larger WC values than those in the AD and control groups; (<b>c</b>) In the γ band, the AD group had lower WC values than those of the control groups.</p>
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<p>Diagnostic performance of LDA. (<b>a</b>) Diagnostic accuracy when using RWE values; (<b>b</b>) Diagnostic accuracy when using WC values; (<b>c</b>) Diagnostic accuracy when using WC values while limiting the number of features to 95. (<b>Left</b>: Total Performance; <b>Middle</b>: Sensitivity; <b>Right</b>: Specificity).</p>
Full article ">
3204 KiB  
Article
Mechanical Fault Diagnosis of High Voltage Circuit Breakers Based on Wavelet Time-Frequency Entropy and One-Class Support Vector Machine
by Nantian Huang, Huaijin Chen, Shuxin Zhang, Guowei Cai, Weiguo Li, Dianguo Xu and Lihua Fang
Entropy 2016, 18(1), 7; https://doi.org/10.3390/e18010007 - 26 Dec 2015
Cited by 54 | Viewed by 5943
Abstract
Mechanical faults of high voltage circuit breakers (HVCBs) are one of the most important factors that affect the reliability of power system operation. Because of the limitation of a lack of samples of each fault type; some fault conditions can be recognized as [...] Read more.
Mechanical faults of high voltage circuit breakers (HVCBs) are one of the most important factors that affect the reliability of power system operation. Because of the limitation of a lack of samples of each fault type; some fault conditions can be recognized as a normal condition. The fault diagnosis results of HVCBs seriously affect the operation reliability of the entire power system. In order to improve the fault diagnosis accuracy of HVCBs; a method for mechanical fault diagnosis of HVCBs based on wavelet time-frequency entropy (WTFE) and one-class support vector machine (OCSVM) is proposed. In this method; the S-transform (ST) is proposed to analyze the energy time-frequency distribution of HVCBs’ vibration signals. Then; WTFE is selected as the feature vector that reflects the information characteristics of vibration signals in the time and frequency domains. OCSVM is used for judging whether a mechanical fault of HVCBs has occurred or not. In order to improve the fault detection accuracy; a particle swarm optimization (PSO) algorithm is employed to optimize the parameters of OCSVM; including the window width of the kernel function and error limit. If the mechanical fault is confirmed; a support vector machine (SVM)-based classifier will be used to recognize the fault type. The experiments carried on a real SF6 HVCB demonstrated the improved effectiveness of the new approach. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>The partition of time-frequency plane.</p>
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<p>The principle of OCSVM.</p>
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<p>Comparison between the principles of OCSVM and SVM.</p>
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<p>The flow of the fault diagnosis process.</p>
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<p>The vibration signal acquisition system for a circuit breaker.</p>
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<p>(<b>a</b>) The normal signal and its STMM contour plot; (<b>b</b>) The signal of fault type I and its STMM contour plot; (<b>c</b>) The signal of fault type II and its STMM contour plot; (<b>d</b>) The signal of fault type III and its STMM contour plot.</p>
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<p>(<b>a</b>) WTFE feature distribution of the normal signals; (<b>b</b>) WTFE feature distribution of the iron core jam fault signals; (<b>c</b>) WTFE feature distribution of the base screw looseness fault signals; (<b>d</b>) WTFE feature distribution of the lack of mechanical lubrication fault signals.</p>
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<p>(<b>a</b>) WSE feature distribution of the normal signals; (<b>b</b>) WSE feature distribution of the iron core jam fault signals; (<b>c</b>) WSE feature distribution of the base screw looseness fault signals; (<b>d</b>) WSE feature distribution of the lack of mechanical lubrication fault signals.</p>
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<p>Fitness curve of PSO.</p>
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1777 KiB  
Article
Pathological Brain Detection by a Novel Image Feature—Fractional Fourier Entropy
by Shuihua Wang, Yudong Zhang, Xiaojun Yang, Ping Sun, Zhengchao Dong, Aijun Liu and Ti-Fei Yuan
Entropy 2015, 17(12), 8278-8296; https://doi.org/10.3390/e17127877 - 17 Dec 2015
Cited by 83 | Viewed by 7473
Abstract
Aim: To detect pathological brain conditions early is a core procedure for patients so as to have enough time for treatment. Traditional manual detection is either cumbersome, or expensive, or time-consuming. We aim to offer a system that can automatically identify pathological [...] Read more.
Aim: To detect pathological brain conditions early is a core procedure for patients so as to have enough time for treatment. Traditional manual detection is either cumbersome, or expensive, or time-consuming. We aim to offer a system that can automatically identify pathological brain images in this paper. Method: We propose a novel image feature, viz., Fractional Fourier Entropy (FRFE), which is based on the combination of Fractional Fourier Transform (FRFT) and Shannon entropy. Afterwards, the Welch’s t-test (WTT) and Mahalanobis distance (MD) were harnessed to select distinguishing features. Finally, we introduced an advanced classifier: twin support vector machine (TSVM). Results: A 10 × K-fold stratified cross validation test showed that this proposed “FRFE + WTT + TSVM” yielded an accuracy of 100.00%, 100.00%, and 99.57% on datasets that contained 66, 160, and 255 brain images, respectively. Conclusions: The proposed “FRFE + WTT + TSVM” method is superior to 20 state-of-the-art methods. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Samples of pathological brain images: (<b>a</b>) healthy brain; (<b>b</b>) AD with visual agnosia; (<b>c</b>) Meningioma; (<b>d</b>) AD; (<b>e</b>) Glioma; (<b>f</b>) Huntington’s disease; (<b>g</b>) Herpes encephalitis; (<b>h</b>) Pick’s disease; (<b>i</b>) Multiple sclerosis; (<b>j</b>) Cerebral toxoplasmosis; (<b>k</b>) Sarcoma; (<b>l</b>) Subdural hematoma.</p>
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<p>Illustration of how FRFT changes with α, whose value varies from zero to one (the real and imaginary parts are shown in black and blue lines, respectively).</p>
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<p>Diagram of our method.</p>
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<p>WFRFT of a normal brain.</p>
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7408 KiB  
Article
On the Complex and Hyperbolic Structures for the (2 + 1)-Dimensional Boussinesq Water Equation
by Figen Özpinar, Haci Mehmet Baskonus and Hasan Bulut
Entropy 2015, 17(12), 8267-8277; https://doi.org/10.3390/e17127878 - 17 Dec 2015
Cited by 44 | Viewed by 5544
Abstract
In this study, we have applied the modified exp(−Ω(ξ))-expansion function method to the (2 + 1)-dimensional Boussinesq water equation. We have obtained some new analytical solutions such as exponential function, complex function and hyperbolic function solutions. It has been observed that all analytical [...] Read more.
In this study, we have applied the modified exp(−Ω(ξ))-expansion function method to the (2 + 1)-dimensional Boussinesq water equation. We have obtained some new analytical solutions such as exponential function, complex function and hyperbolic function solutions. It has been observed that all analytical solutions have been verified to the (2 + 1)-dimensional Boussinesq water equation by using Wolfram Mathematica 9. We have constructed the two- and three-dimensional surfaces for all analytical solutions obtained in this paper using the same computer program. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>The 3D surfaces of the analytical solution, Equation (20), using the values <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mtext>  </mtext> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>B</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mtext>  </mtext> <mi>E</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo>−</mo> <mn>10</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>10</mn> <mo>,</mo> <mtext> </mtext> <mo>−</mo> <mn>10</mn> <mo>&lt;</mo> <mi>t</mi> <mo>&lt;</mo> <mn>10</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math> for 2D transect.</p>
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<p>The 3D surfaces of the imaginary and real part of the analytical solution, Equation (21), using the values <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mo>−</mo> <mn>0.3</mn> <mo>,</mo> <mtext>  </mtext> <msub> <mi>A</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>B</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mtext>  </mtext> <mi>E</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo>−</mo> <mn>6</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>6</mn> <mo>,</mo> <mtext> </mtext> <mo>−</mo> <mn>5</mn> <mo>&lt;</mo> <mi>t</mi> <mo>&lt;</mo> <mn>5</mn> </mrow> </semantics></math>. (<b>a</b>) Imaginary part; (<b>b</b>) Real part.</p>
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<p>The 2D transect of the imaginary and real part of the analytical solution, Equation (21), using the values <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mtext>  </mtext> <mi>μ</mi> <mo>=</mo> <mo>−</mo> <mn>0.3</mn> <mo>,</mo> <mtext>  </mtext> <msub> <mi>A</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext>  </mtext> <msub> <mi>B</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mtext>  </mtext> <mi>E</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mtext>  </mtext> <mi>y</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mtext>  </mtext> <mi>t</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mtext> </mtext> <mo>−</mo> <mn>6</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>6</mn> </mrow> </semantics></math>. (<b>a</b>) Imaginary part; (<b>b</b>) Real part.</p>
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<p>The 3D surface and 2D transect of the analytical solution, Equation (22), using the values <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mo>−</mo> <mn>0.4</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mn>0.5</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>B</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.7</mn> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>0.3</mn> <mo>,</mo> <mo>−</mo> <mn>4</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>6</mn> <mo>,</mo> <mtext> </mtext> <mn>0</mn> <mo>&lt;</mo> <mi>t</mi> <mo>&lt;</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.09</mn> </mrow> </semantics></math> for 2D transect.</p>
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<p>The 3D surface of the analytical solution, Equation (23), using the values <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>B</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.7</mn> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>40</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>40</mn> <mo>,</mo> <mtext> </mtext> <mo>−</mo> <mn>1</mn> <mo>&lt;</mo> <mi>t</mi> <mo>&lt;</mo> <mn>1</mn> <mo>,</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> for 2D transect.</p>
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<p>The 3D surfaces of the imaginary and real part of the analytical solution, Equation (28), using the values <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mo>−</mo> <mn>0.3</mn> <mo>,</mo> <mtext>  </mtext> <msub> <mi>A</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>B</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mo>−</mo> <mn>10</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>10</mn> <mo>,</mo> <mtext> </mtext> <mo>−</mo> <mn>10</mn> <mo>&lt;</mo> <mi>t</mi> <mo>&lt;</mo> <mn>10</mn> </mrow> </semantics></math>. (<b>a</b>) Imaginary part; (<b>b</b>) Real part.</p>
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<p>The 2D transect of the imaginary and real part of the analytical solution, Equation (28), using values <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mo>−</mo> <mn>0.3</mn> <mo>,</mo> <mtext>  </mtext> <msub> <mi>A</mi> <mn>4</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>B</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mn>0.4</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>y</mi> <mo>=</mo> <mn>0.1</mn> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mo>−</mo> <mn>10</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>10</mn> </mrow> </semantics></math>. (<b>a</b>) Imaginary part; (<b>b</b>) Real part.</p>
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<p>The 3D surfaces of the imaginary and real part of the analytical solution, Equation (29), using the values <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>−</mo> <mn>0.3</mn> <mo>,</mo> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mtext>  </mtext> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>A</mi> <mn>4</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>B</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mo>−</mo> <mn>0.2</mn> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>30</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>30</mn> <mo>,</mo> <mtext> </mtext> <mo>−</mo> <mn>30</mn> <mo>&lt;</mo> <mi>t</mi> <mo>&lt;</mo> <mn>30</mn> </mrow> </semantics></math>. (<b>a</b>) Imaginary part; (<b>b</b>) Real part.</p>
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<p>The 2D transects of the imaginary and real part of the analytical solution, Equation (29), using the values <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>−</mo> <mn>0.3</mn> <mo>,</mo> <mtext> </mtext> <mi>μ</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mtext>  </mtext> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>A</mi> <mn>4</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msub> <mi>B</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.3</mn> <mo>,</mo> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <msub> <mi>B</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.6</mn> <mo>,</mo> <mi>E</mi> <mo>=</mo> <mo>−</mo> <mn>0.2</mn> <mo>,</mo> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>0.1</mn> <mo>,</mo> </mrow> </semantics></math> <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>30</mn> <mo>&lt;</mo> <mi>x</mi> <mo>&lt;</mo> <mn>30</mn> </mrow> </semantics></math>. (<b>a</b>) Imaginary part; (<b>b</b>) Real part.</p>
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9485 KiB  
Article
An Optimal Segmentation Method Using Jensen–Shannon Divergence via a Multi-Size Sliding Window Technique
by Qutaibeh D. Katatbeh, José Martínez-Aroza, Juan Francisco Gómez-Lopera and David Blanco-Navarro
Entropy 2015, 17(12), 7996-8006; https://doi.org/10.3390/e17127858 - 4 Dec 2015
Cited by 10 | Viewed by 6301
Abstract
In this paper we develop a new procedure for entropic image edge detection. The presented method computes the Jensen–Shannon divergence of the normalized grayscale histogram of a set of multi-sized double sliding windows over the entire image. The procedure presents a good performance [...] Read more.
In this paper we develop a new procedure for entropic image edge detection. The presented method computes the Jensen–Shannon divergence of the normalized grayscale histogram of a set of multi-sized double sliding windows over the entire image. The procedure presents a good performance in images with textures, contrast variations and noise. We illustrate our procedure in the edge detection of medical images. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Histograms of square samples from several images. In the first case (<b>top left</b>), there are only two gray levels with the same absolute frequency. The second case (<b>top right</b>) corresponds to the previous image affected by gaussian noise, so that a strong scattering of the histogram can be appreciated. In the image of Lenna, the first histogram (<b>left</b>) corresponds to shoulders and the right histogram to feathers.</p>
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<p><b>Left</b>: Picture of Lenna, a real scene. <b>Middle</b>: smoothed <math display="inline"> <mrow> <mi>J</mi> <mspace width="-0.166667em"/> <mi>S</mi> </mrow> </math> matrix of Lenna processed with semicircular sliding windows of radius 3. <b>Right</b>: the same as the previous image, but using semicircular sliding windows of radius 20. Smoothing procedure is explained below in this Section.</p>
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<p>Four test images (<b>top row</b>) for edge detection. <b>Image 1</b>: a synthetic image made of four homogeneous regions with different, equally-spaced gray levels. <b>Image 2</b>: the same image as the first corrupted by salt and pepper noise over <math display="inline"> <mrow> <mn>10</mn> <mo>%</mo> </mrow> </math> of the image pixels. <b>Image 3</b>: an image composed of four different gray levels, with different distances between gray levels. <b>Image 4</b>: an image composed of four different textures. <b>Second row</b>: representation, along a horizontal line in the middle of each image, of the respective gradient modules of the images, using the Sobel mask. <b>Third row</b>: corresponding representation of the <math display="inline"> <mrow> <mi>J</mi> <mspace width="-0.166667em"/> <mi>S</mi> </mrow> </math> of the images using two square semi-windows with a <math display="inline"> <mrow> <mn>5</mn> <mo>×</mo> <mn>5</mn> </mrow> </math> size.</p>
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<p>In this figure, we compute divergence matrices for radii of <math display="inline"> <mrow> <mo>{</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>16</mn> <mo>,</mo> <mn>23</mn> <mo>,</mo> <mn>32</mn> <mo>,</mo> <mn>45</mn> <mo>,</mo> <mn>64</mn> <mo>}</mo> </mrow> </math>, respectively from top to down, left to right. In the last picture, we used multi-size window.</p>
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<p>Variation of the maximum and mean divergence in the matrix for the experiment of <a href="#entropy-17-07858-f004" class="html-fig">Figure 4</a> (Brain), depending on the radius.</p>
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<p>In this figure we see on the top row an original image of an angiogram with its edges detected by using the multi-window procedure with the same set of radii as in <a href="#entropy-17-07858-f004" class="html-fig">Figure 4</a>; on the second row, we see the results obtained using the Canny method (<b>left</b>) and the Sobel method (<b>right</b>); both methods fail at detecting the edges due to noise and vertical textures.</p>
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Article
Wavelet-Tsallis Entropy Detection and Location of Mean Level-Shifts in Long-Memory fGn Signals
by Julio César Ramírez-Pacheco, Luis Rizo-Domínguez and Joaquin Cortez-González
Entropy 2015, 17(12), 7979-7995; https://doi.org/10.3390/e17127856 - 4 Dec 2015
Cited by 2 | Viewed by 5776
Abstract
Long-memory processes, in particular fractional Gaussian noise processes, have been applied as models for many phenomena occurring in nature. Non-stationarities, such as trends, mean level-shifts, etc., impact the accuracy of long-memory parameter estimators, giving rise to biases and misinterpretations of the phenomena. [...] Read more.
Long-memory processes, in particular fractional Gaussian noise processes, have been applied as models for many phenomena occurring in nature. Non-stationarities, such as trends, mean level-shifts, etc., impact the accuracy of long-memory parameter estimators, giving rise to biases and misinterpretations of the phenomena. In this article, a novel methodology for the detection and location of mean level-shifts in stationary long-memory fractional Gaussian noise (fGn) signals is proposed. It is based on a joint application of the wavelet-Tsallis q-entropy as a preprocessing technique and a peak detection methodology. Extensive simulation experiments in synthesized fGn signals with mean level-shifts confirm that the proposed methodology not only detects, but also locates level-shifts with high accuracy. A comparative study against standard techniques of level-shift detection and location shows that the technique based on wavelet-Tsallis q-entropy outperforms the one based on trees and the Bai and Perron procedure, as well. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Wavelet-Tsallis <span class="html-italic">q</span>-entropy planes for <math display="inline"> <mrow> <mi>α</mi> <mo>∈</mo> <mo>(</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </math> and <math display="inline"> <mrow> <mi>M</mi> <mo>∈</mo> <mo>(</mo> <mn>5</mn> <mo>,</mo> <mn>15</mn> <mo>)</mo> </mrow> </math>. The left plot displays Tsallis planes for <math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>5</mn> </mrow> </math> and the right one for <math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>10</mn> </mrow> </math>. Variations of the entropy planes for different values of the wavelet moment <span class="html-italic">p</span> are also shown stacked. Within each plot, the lower graph corresponds to <math display="inline"> <msubsup> <mi mathvariant="script">H</mi> <mrow> <mi>p</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>5</mn> </mrow> <mi>T</mi> </msubsup> </math>, the next on top to <math display="inline"> <mrow> <msubsup> <mi mathvariant="script">H</mi> <mrow> <mi>p</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>4</mn> </mrow> <mi>T</mi> </msubsup> <mo>+</mo> <mn>0</mn> <mo>.</mo> <mn>33</mn> </mrow> </math>, the next to <math display="inline"> <mrow> <msubsup> <mi mathvariant="script">H</mi> <mrow> <mi>p</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>3</mn> </mrow> <mi>T</mi> </msubsup> <mo>+</mo> <mn>0</mn> <mo>.</mo> <mn>66</mn> </mrow> </math> and the upper one to <math display="inline"> <mrow> <msubsup> <mi mathvariant="script">H</mi> <mrow> <mi>p</mi> <mo>=</mo> <mn>5</mn> <mo>,</mo> <mi>q</mi> <mo>=</mo> <mn>2</mn> </mrow> <mi>T</mi> </msubsup> <mo>+</mo> <mn>1</mn> </mrow> </math>.</p>
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<p>Wavelet-Tsallis planes for fixed length <span class="html-italic">M</span>. The left plot displays the wavelet-Tsallis entropies for <math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>10</mn> </mrow> </math> and different values of the parameter <span class="html-italic">p</span>. The right plot depicts the wavelet-Tsallis entropies for <math display="inline"> <mrow> <mi>q</mi> <mo>=</mo> <mn>5</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>p</mi> <mo>=</mo> <mo>{</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>}</mo> </mrow> </math>.</p>
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<p>Fractional Gaussian noise (fGn) signals with two mean level-shifts. The first column displays the signal with two opposite breaks and their corresponding wavelet-Tsallis <span class="html-italic">q</span>-entropy. The second column displays the same type of analysis using two positive breaks within the fGn signal with <math display="inline"> <mrow> <mi>H</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>6</mn> </mrow> </math>.</p>
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<p>fGn signals with multiple mean level-shifts. The first column displays the signal with three breaks and their corresponding wavelet-Tsallis <span class="html-italic">q</span>-entropy. The second column displays the same type of analysis using four multiple mean breaks within the fGn signal with <math display="inline"> <mrow> <mi>H</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>6</mn> </mrow> </math>.</p>
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<p>Average number of breaks reported by the Bai and Perron, wavelet-Tsallis <span class="html-italic">q</span>-entropy and atheoretical regression (ART) techniques as a function of break size and length of series (regime). The regime is the exponent <span class="html-italic">y</span> of the length of the signal <span class="html-italic">M</span>, obtained as <math display="inline"> <mrow> <mi>M</mi> <mo>=</mo> <msup> <mn>2</mn> <mi>y</mi> </msup> </mrow> </math>. Two types of dependence structures are considered, low dependence (<math display="inline"> <mrow> <mi>H</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>20</mn> </mrow> </math>) and long-range dependence (<math display="inline"> <mrow> <mi>H</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>60</mn> </mrow> </math>).</p>
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<p>Average number of breaks reported by ART and the technique based on wavelet-Tsallis <span class="html-italic">q</span>-entropy. The top row shows the ART results, while the bottom one displays the ones of wavelet-Tsallis <span class="html-italic">q</span>-entropy.</p>
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<p>Average number of breaks reported by ART and the technique based on wavelet-Tsallis <span class="html-italic">q</span>-entropy and a single peak detection and location algorithm. The top row shows the results of ART, while the bottom row shows the results of the wavelet-Tsallis <span class="html-italic">q</span>-entropy technique.</p>
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<p>Average number of breaks reported by ART and the proposed technique based on Tsallis entropies in fGn signals with a single break and Gaussian noise. The top row shows the results of ART, while the bottom row shows the results of the wavelet-Tsallis <span class="html-italic">q</span>-entropy technique.</p>
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<p>Computation times for ART and the technique based on wavelet-Tsallis <span class="html-italic">q</span>-entropy as a function of signal length <span class="html-italic">M</span>. The times are given in seconds.</p>
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1404 KiB  
Article
A Novel Method for PD Feature Extraction of Power Cable with Renyi Entropy
by Jikai Chen, Yanhui Dou, Zhenhao Wang and Guoqing Li
Entropy 2015, 17(11), 7698-7712; https://doi.org/10.3390/e17117698 - 13 Nov 2015
Cited by 12 | Viewed by 5779
Abstract
Partial discharge (PD) detection can effectively achieve the status maintenance of XLPE (Cross Linked Polyethylene) cable, so it is the direction of the development of equipment maintenance in power systems. At present, a main method of PD detection is the broadband electromagnetic coupling [...] Read more.
Partial discharge (PD) detection can effectively achieve the status maintenance of XLPE (Cross Linked Polyethylene) cable, so it is the direction of the development of equipment maintenance in power systems. At present, a main method of PD detection is the broadband electromagnetic coupling with a high-frequency current transformer (HFCT). Due to the strong electromagnetic interference (EMI) generated among the mass amount of cables in a tunnel and the impedance mismatching of HFCT and the data acquisition equipment, the features of the pulse current generated by PD are often submerged in the background noise. The conventional method for the stationary signal analysis cannot analyze the PD signal, which is transient and non-stationary. Although the algorithm of Shannon wavelet singular entropy (SWSE) can be used to analyze the PD signal at some level, its precision and anti-interference capability of PD feature extraction are still insufficient. For the above problem, a novel method named Renyi wavelet packet singular entropy (RWPSE) is proposed and applied to the PD feature extraction on power cables. Taking a three-level system as an example, we analyze the statistical properties of Renyi entropy and the intrinsic correlation with Shannon entropy under different values of α . At the same time, discrete wavelet packet transform (DWPT) is taken instead of discrete wavelet transform (DWT), and Renyi entropy is combined to construct the RWPSE algorithm. Taking the grounding current signal from the shielding layer of XLPE cable as the research object, which includes the current pulse feature of PD, the effectiveness of the novel method is tested. The theoretical analysis and experimental results show that compared to SWSE, RWPSE can not only improve the feature extraction accuracy for PD, but also can suppress EMI effectively. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Relation between Renyi entropy with different values of <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> and the probability distribution: (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics> </math>; (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.99</mn> </mrow> </semantics> </math>; (<b>d</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math>.</p>
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<p>Relation between Renyi entropy with different values of <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> and the probability distribution: (<b>a</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics> </math>; (<b>b</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics> </math>; (<b>c</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.99</mn> </mrow> </semantics> </math>; (<b>d</b>) <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math>.</p>
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<p>Relation between Shannon entropy and the probability distribution.</p>
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<p>Framework of discrete wavelet packet transform (DWPT).</p>
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<p>Partial discharge (PD) detection process. (<b>a</b>) Working principle diagram of the high-frequency current transformer (HFCT); (<b>b</b>) Schematic diagram of the PD detection system.</p>
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<p>On-site installation of the HFCT.</p>
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<p>Data collection on the spot.</p>
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<p>PD signal collected by HFCT.</p>
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<p>Performance comparison of feature extraction between Renyi wavelet packet singular entropy (RWPSE) and Shannon wavelet singular entropy (SWSE): (<b>a</b>) RWPSE; (<b>b</b>) SWSE.</p>
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<p>Anti-interference performance comparison between RWPSE and SWSE. (<b>a</b>) RWPSE; (<b>b</b>) SWSE.</p>
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<p>Low-frequency reconstructed signal in 0~0.78 MHz.</p>
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<p>High-frequency reconstructed signals: (<b>a</b>) in 12.5~18.75 MHz; (<b>b</b>) in 25~31.25 MHz.</p>
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952 KiB  
Article
Multi-Level Wavelet Shannon Entropy-Based Method for Single-Sensor Fault Location
by Qiaoning Yang and Jianlin Wang
Entropy 2015, 17(10), 7101-7117; https://doi.org/10.3390/e17107101 - 20 Oct 2015
Cited by 28 | Viewed by 8806
Abstract
In actual application, sensors are prone to failure because of harsh environments, battery drain, and sensor aging. Sensor fault location is an important step for follow-up sensor fault detection. In this paper, two new multi-level wavelet Shannon entropies (multi-level wavelet time [...] Read more.
In actual application, sensors are prone to failure because of harsh environments, battery drain, and sensor aging. Sensor fault location is an important step for follow-up sensor fault detection. In this paper, two new multi-level wavelet Shannon entropies (multi-level wavelet time Shannon entropy and multi-level wavelet time-energy Shannon entropy) are defined. They take full advantage of sensor fault frequency distribution and energy distribution across multi-subband in wavelet domain. Based on the multi-level wavelet Shannon entropy, a method is proposed for single sensor fault location. The method firstly uses a criterion of maximum energy-to-Shannon entropy ratio to select the appropriate wavelet base for signal analysis. Then multi-level wavelet time Shannon entropy and multi-level wavelet time-energy Shannon entropy are used to locate the fault. The method is validated using practical chemical gas concentration data from a gas sensor array. Compared with wavelet time Shannon entropy and wavelet energy Shannon entropy, the experimental results demonstrate that the proposed method can achieve accurate location of a single sensor fault and has good anti-noise ability. The proposed method is feasible and effective for single-sensor fault location. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Flowchart of sensor fault location method.</p>
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<p>Normal sensor data and fault sensor data. (<b>a</b>) Normal data, (<b>b</b>) bias, (<b>c</b>) stuck-at, and (<b>d</b>) drift.</p>
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<p>Magnitude spectrum of four signals. (<b>a</b>) Normal data, (<b>b</b>) bias, (<b>c)</b> stuck-at, and (<b>d</b>) Drift.</p>
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<p>Multi-level wavelet time Shannon entropy (MWTE) of faults. (<b>a</b>) Bias, (<b>b</b>) stuck-at, and (<b>c</b>) drift.</p>
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<p>Multi-level wavelet time-energy Shannon entropy (MWTEE) of faults. (<b>a</b>) Bias, (<b>b</b>) stuck-at, and (<b>c</b>) drift.</p>
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<p>Multi-level wavelet time-energy Shannon entropy (MWTEE) of faults. (<b>a</b>) Bias, (<b>b</b>) stuck-at, and (<b>c</b>) drift.</p>
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<p>Wavelet Time Shannon Entropy (WTE) of faults. (<b>a</b>) Bias, (<b>b</b>) stuck-at, and (<b>c</b>) drift.</p>
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<p>Wavelet Energy Shannon Entropy (WEE) of faults. (<b>a</b>) Bias, (<b>b</b>) stuck-at, and (<b>c</b>) drift.</p>
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<p>Wavelet Energy Shannon Entropy (WEE) of faults and Wavelet Time Shannon Entropy (WTE) of stuck-at fault when SNR = 45 dB. (<b>a</b>) Bias, (<b>b</b>) stuck-at, (<b>c</b>) drift, and (<b>d</b>) WTE of stuck-at.</p>
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<p>Multi-level wavelet time Shannon entropy (MWTE) of faults when SNR = 45 dB. (<b>a</b>) Bias, (<b>b</b>) stuck-at, and (<b>c</b>) drift.</p>
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<p>Multi-level wavelet time-energy Shannon entropy (MWTEE) of three type fault when SNR = 45 dB. (<b>a</b>) Bias, (<b>b</b>) stuck-at, and (<b>c</b>) drift.</p>
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744 KiB  
Article
Modified Legendre Wavelets Technique for Fractional Oscillation Equations
by Syed Tauseef Mohyud-Din, Muhammad Asad Iqbal and Saleh M. Hassan
Entropy 2015, 17(10), 6925-6936; https://doi.org/10.3390/e17106925 - 16 Oct 2015
Cited by 25 | Viewed by 5850
Abstract
Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among [...] Read more.
Physical Phenomena’s located around us are primarily nonlinear in nature and their solutions are of highest significance for scientists and engineers. In order to have a better representation of these physical models, fractional calculus is used. Fractional order oscillation equations are included among these nonlinear phenomena’s. To tackle with the nonlinearity arising, in these phenomena’s we recommend a new method. In the proposed method, Picard’s iteration is used to convert the nonlinear fractional order oscillation equation into a fractional order recurrence relation and then Legendre wavelets method is applied on the converted problem. In order to check the efficiency and accuracy of the suggested modification, we have considered three problems namely: fractional order force-free Duffing–van der Pol oscillator, forced Duffing–van der Pol oscillator and higher order fractional Duffing equations. The obtained results are compared with the results obtained via other techniques. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Comparison of solutions for different fractional values by RK-4 solution for single well case.</p>
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<p>Comparison of solutions for different fractional values by RK-4 solution for double well case.</p>
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<p>Comparison of solutions for different fractional values by RK-4 solution for double hump case.</p>
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<p>Comparison of solutions for different fractional values by RK-4 solution for Problem 2.</p>
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<p>Comparison of solutions for different fractional values by RK-4 solution for Problem 3.</p>
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1699 KiB  
Article
Identification of Green, Oolong and Black Teas in China via Wavelet Packet Entropy and Fuzzy Support Vector Machine
by Shuihua Wang, Xiaojun Yang, Yudong Zhang, Preetha Phillips, Jianfei Yang and Ti-Fei Yuan
Entropy 2015, 17(10), 6663-6682; https://doi.org/10.3390/e17106663 - 25 Sep 2015
Cited by 100 | Viewed by 8668
Abstract
To develop an automatic tea-category identification system with a high recall rate, we proposed a computer-vision and machine-learning based system, which did not require expensive signal acquiring devices and time-consuming procedures. We captured 300 tea images using a 3-CCD digital camera, and then [...] Read more.
To develop an automatic tea-category identification system with a high recall rate, we proposed a computer-vision and machine-learning based system, which did not require expensive signal acquiring devices and time-consuming procedures. We captured 300 tea images using a 3-CCD digital camera, and then extracted 64 color histogram features and 16 wavelet packet entropy (WPE) features to obtain color information and texture information, respectively. Principal component analysis was used to reduce features, which were fed into a fuzzy support vector machine (FSVM). Winner-take-all (WTA) was introduced to help the classifier deal with this 3-class problem. The 10 × 10-fold stratified cross-validation results show that the proposed FSVM + WTA method yields an overall recall rate of 97.77%, higher than 5 existing methods. In addition, the number of reduced features is only five, less than or equal to existing methods. The proposed method is effective for tea identification. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Computer vision based system to obtain the tea image database.</p>
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<p>Flowchart of feature processing.</p>
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<p>Diagram of two-level one-dimensional wavelet packet transform. Here, <span class="html-italic">a</span> and <span class="html-italic">b</span> represent the low-pass and high-pass filters, respectively. L and H represent the low-frequency and high-frequency subbands, respectively.</p>
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<p>Diagram of the proposed automatic tea classification system.</p>
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<p>Curve of accumulated variance explained against number of principal components (PCs).</p>
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<p>Decomposition functions for bior4.4, (<b>a</b>) Scaling Function, (<b>b</b>) Wavelet Function, (<b>c</b>) low-pass filter (LPF), (<b>d</b>) high-pass filter (HPF).</p>
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<p>Reconstruction functions for bior4.4, (<b>a</b>) Scaling Function, (<b>b</b>) Wavelet Function, (<b>c</b>), low-pass filter (LPF), (<b>d</b>) high-pass filter (HPF).</p>
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612 KiB  
Article
On the Exact Solution of Wave Equations on Cantor Sets
by Dumitru Baleanu, Hasib Khan, Hossien Jafari and Rahmat Ali Khan
Entropy 2015, 17(9), 6229-6237; https://doi.org/10.3390/e17096229 - 8 Sep 2015
Cited by 29 | Viewed by 5444
Abstract
The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and [...] Read more.
The transfer of heat due to the emission of electromagnetic waves is called thermal radiations. In local fractional calculus, there are numerous contributions of scientists, like Mandelbrot, who described fractal geometry and its wide range of applications in many scientific fields. Christianto and Rahul gave the derivation of Proca equations on Cantor sets. Hao et al. investigated the Helmholtz and diffusion equations in Cantorian and Cantor-Type Cylindrical Coordinates. Carpinteri and Sapora studied diffusion problems in fractal media in Cantor sets. Zhang et al. studied local fractional wave equations under fixed entropy. In this paper, we are concerned with the exact solutions of wave equations by the help of local fractional Laplace variation iteration method (LFLVIM). We develop an iterative scheme for the exact solutions of local fractional wave equations (LFWEs). The efficiency of the scheme is examined by two illustrative examples. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Exact solution of Equation (23) for <math display="inline"> <semantics> <mrow> <mtext>γ</mtext> <mo>=</mo> <mfrac> <mrow> <mi>ln</mi> <mn>2</mn> </mrow> <mrow> <mi>ln</mi> <mn>3</mn> </mrow> </mfrac> </mrow> </semantics> </math>.</p>
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<p>Exact solution of Equation (32) for <math display="inline"> <semantics> <mrow> <mtext>γ</mtext> <mo>=</mo> <mfrac> <mrow> <mi>ln</mi> <mn>2</mn> </mrow> <mrow> <mi>ln</mi> <mn>3</mn> </mrow> </mfrac> <mo>.</mo> </mrow> </semantics> </math></p>
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359 KiB  
Article
Active Control of a Chaotic Fractional Order Economic System
by Haci Mehmet Baskonus, Toufik Mekkaoui, Zakia Hammouch and Hasan Bulut
Entropy 2015, 17(8), 5771-5783; https://doi.org/10.3390/e17085771 - 11 Aug 2015
Cited by 128 | Viewed by 7157
Abstract
In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth [...] Read more.
In this paper, a fractional order economic system is studied. An active control technique is applied to control chaos in this system. The stabilization of equilibria is obtained by both theoretical analysis and the simulation result. The numerical simulations, via the improved Adams–Bashforth algorithm, show the effectiveness of the proposed controller. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Stability region of the fractional order system (<a href="#FD3-entropy-17-05771" class="html-disp-formula">3</a>).</p>
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<p>The time histories of variables (<b>a</b>) <span class="html-italic">x(t)</span>, (<b>b</b>) <span class="html-italic">y(t)</span> and (<b>c</b>) <span class="html-italic">z(t)</span> for <span class="html-italic">α</span> = 0.9.</p>
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<p>Phase portraits: (<b>a</b>) <span class="html-italic">x − y</span>, (<b>b</b>) <span class="html-italic">x − z</span> and (<b>c</b>) <span class="html-italic">y − z</span> for System (<a href="#FD10-entropy-17-05771" class="html-disp-formula">10</a>) when <span class="html-italic">α</span> = 0.9.</p>
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<p>Chaotic attractor <span class="html-italic">xyz</span> for System (<a href="#FD10-entropy-17-05771" class="html-disp-formula">10</a>) when <span class="html-italic">α</span> = 0.9.</p>
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<p>Time histories of System (<a href="#FD11-entropy-17-05771" class="html-disp-formula">11</a>) for <span class="html-italic">x</span> signal at the equilibrium <span class="html-italic">E</span><sub>0</sub> with <span class="html-italic">α</span> = 0.9: (<b>a</b>) <span class="html-italic">t<sub>max</sub></span> = 100, (<b>b</b>) <span class="html-italic">t<sub>max</sub></span> = 300.</p>
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<p>Time histories of system (<a href="#FD11-entropy-17-05771" class="html-disp-formula">11</a>) for <span class="html-italic">y</span> signal at the equilibrium <span class="html-italic">E</span><sub>0</sub> with <span class="html-italic">α</span> = 0.9: (<b>a</b>) <span class="html-italic">t<sub>max</sub></span> = 100, (<b>b</b>) <span class="html-italic">t<sub>max</sub></span> = 300.</p>
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<p>Time histories of System (<a href="#FD11-entropy-17-05771" class="html-disp-formula">11</a>) for <span class="html-italic">z</span> signal at the equilibrium <span class="html-italic">E</span><sub>0</sub> with <span class="html-italic">α</span> = 0.9: (<b>a</b>) <span class="html-italic">t<sub>max</sub></span> = 100, (<b>b</b>) <span class="html-italic">t<sub>max</sub></span> = 300.</p>
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<p>Time histories of System (<a href="#FD11-entropy-17-05771" class="html-disp-formula">11</a>) for <span class="html-italic">x</span> signal at the equilibrium <span class="html-italic">E</span><sub>1</sub> with <span class="html-italic">α</span> = 0.9: (<b>a</b>) <span class="html-italic">t<sub>max</sub></span> = 100, (<b>b</b>) <span class="html-italic">t<sub>max</sub></span> = 300.</p>
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<p>Time histories of System (<a href="#FD11-entropy-17-05771" class="html-disp-formula">11</a>) for <span class="html-italic">y</span> signal at the equilibrium <span class="html-italic">E</span><sub>1</sub> with <span class="html-italic">α</span> = 0.9: (<b>a</b>) <span class="html-italic">t<sub>max</sub></span> = 100, (<b>b</b>) <span class="html-italic">t<sub>max</sub></span> = 300.</p>
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<p>Time histories of System (<a href="#FD11-entropy-17-05771" class="html-disp-formula">11</a>) for <span class="html-italic">z</span> signal at the equilibrium <span class="html-italic">E</span><sub>1</sub> with <span class="html-italic">α</span> = 0.9: (<b>a</b>) <span class="html-italic">t<sub>max</sub></span> = 100, (<b>b</b>) <span class="html-italic">t<sub>max</sub></span> = 300.</p>
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<p>Time histories of System (<a href="#FD11-entropy-17-05771" class="html-disp-formula">11</a>) for <span class="html-italic">x</span> signal at the equilibrium <span class="html-italic">E</span><sub>2</sub> with <span class="html-italic">α</span> = 0.9: (<b>a</b>) <span class="html-italic">t<sub>max</sub></span> = 100, (<b>b</b>) <span class="html-italic">t<sub>max</sub></span> = 300.</p>
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<p>Time histories of System (<a href="#FD11-entropy-17-05771" class="html-disp-formula">11</a>) for <span class="html-italic">y</span> signal at the equilibrium <span class="html-italic">E</span><sub>2</sub> with <span class="html-italic">α</span> = 0.9: (<b>a</b>) <span class="html-italic">t<sub>max</sub></span> = 100, (<b>b</b>) <span class="html-italic">t<sub>max</sub></span> = 300.</p>
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<p>Time histories of System (<a href="#FD11-entropy-17-05771" class="html-disp-formula">11</a>) for <span class="html-italic">z</span> signal at the equilibrium <span class="html-italic">E</span><sub>2</sub> with <span class="html-italic">α</span> = 0.9: (<b>a</b>) <span class="html-italic">t<sub>max</sub></span> = 100, (<b>b</b>) <span class="html-italic">t<sub>max</sub></span> = 300.</p>
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852 KiB  
Article
Power-Type Functions of Prediction Error of Sea Level Time Series
by Ming Li, Yuanchun Li and Jianxing Leng
Entropy 2015, 17(7), 4809-4837; https://doi.org/10.3390/e17074809 - 9 Jul 2015
Cited by 11 | Viewed by 5769
Abstract
This paper gives the quantitative relationship between prediction error and given past sample size in our research of sea level time series. The present result exhibits that the prediction error of sea level time series in terms of given past sample size follows [...] Read more.
This paper gives the quantitative relationship between prediction error and given past sample size in our research of sea level time series. The present result exhibits that the prediction error of sea level time series in terms of given past sample size follows decayed power functions, providing a quantitative guideline for the quality control of sea level prediction. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Process of predicting sea level time series from the 5001st value to the 5040th value.</p>
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<p>Sea level series at the Station LKWF1 in 1999.</p>
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<p>Prediction results with the sample size 100 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p>
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<p>Prediction results with the sample size 200 at the Station LKWF1in 1999. Solid line: predicted values, dashed line: original values.</p>
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<p>Prediction results with the sample size 300 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p>
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<p>Prediction results with the sample size 400 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p>
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<p>Prediction results with the sample size 500 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p>
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<p>Prediction results with the sample size 600 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p>
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<p>Prediction results with the sample size 700 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p>
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1582 KiB  
Article
Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel
by Abdon Atangana and Badr Saad T. Alkahtani
Entropy 2015, 17(6), 4439-4453; https://doi.org/10.3390/e17064439 - 23 Jun 2015
Cited by 260 | Viewed by 9019
Abstract
Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence [...] Read more.
Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Numerical simulation of concentration of chemical substance for alpha = 0.5.</p>
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<p>Numerical simulation of concentration of amoebae for alpha = 0.5.</p>
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<p>Numerical simulation of chemical substance for alpha = 0.95.</p>
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<p>Numerical simulation of concentration of amoebae for alpha = 0.95.</p>
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