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Special Issue "Wavelets, Fractals and Information Theory III"

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

Deadline for manuscript submissions: closed (31 October 2017)

Special Issue Editor

Guest Editor
Prof. Dr. Carlo Cattani

Engineering School (DEIM) University of Tuscia, 01100 Largo dell'Università, Viterbo, Italy
Website | E-Mail
Interests: wavelets; fractals; fractional calculus; dynamical systems; data analysis; time series analysis; image analysis; computer science; computational methods; composite materials; elasticity; nonlinear waves

Special Issue Information

Dear Colleagues,

Wavelet Analysis and Fractals are playing fundamental roles in Science, Engineering applications, and Information Theory. Wavelet and fractals are the most suitable methods to analyze complex systems, localized phenomena, singular solutions, non-differentiable functions, and, in general, nonlinear problems. Nonlinearity and non-regularity usually characterize the complexity of a problem; thus being the most studied features in order to approach a solution to complex problems. Wavelets, fractals, and fractional calculus might also help to improve the analysis of the entropy of a system.

In information theory, entropy encoding might be considered a sort of compression in a quantization process, and this can be further investigated by using 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 analyses, such as in data storage. In fact, the principal advantage of modeling a complex system 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 applications, the best way to model traffic in wireless communications 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.
  • Fractal and wavelet solutions of fractional differential equations
  • Wavelet Analysis, integral transforms and applications.
  • Wavelet-fractal entropy encoding and computational mathematics in data analysis and time series, including in image analysis.
  • Wavelet-fractal approach.
  • Artifical Neural Networks.

Prof. Dr. Carlo Cattani
Guest Editor

Manuscript Submission Information

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

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

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

Keywords

  • Fractal
  • Fractional
  • Wavelet
  • Data analysis
  • Image Analysis
  • Dynamical System
  • Differential Operators

Related Special Issues

Published Papers (11 papers)

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Research

Open AccessArticle How Successful Are Wavelets in Detecting Jumps?
Entropy 2017, 19(12), 638; doi:10.3390/e19120638
Received: 30 October 2017 / Revised: 22 November 2017 / Accepted: 22 November 2017 / Published: 25 November 2017
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Abstract
We evaluate the performances of wavelet jump detection tests by using simulated high-frequency data, in which jumps and some other non-standard features are present. Wavelet-based jump detection tests have a clear advantage over the alternatives, as they are capable of stating the exact
[...] Read more.
We evaluate the performances of wavelet jump detection tests by using simulated high-frequency data, in which jumps and some other non-standard features are present. Wavelet-based jump detection tests have a clear advantage over the alternatives, as they are capable of stating the exact timing and number of jumps. The results indicate that, in addition to those advantages, these detection tests also preserve desirable power and size properties even in non-standard data environments, whereas their alternatives fail to sustain their desirable properties beyond standard data features. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
Open AccessArticle The Application of Dual-Tree Complex Wavelet Transform (DTCWT) Energy Entropy in Misalignment Fault Diagnosis of Doubly-Fed Wind Turbine (DFWT)
Entropy 2017, 19(11), 587; doi:10.3390/e19110587
Received: 20 September 2017 / Revised: 23 October 2017 / Accepted: 1 November 2017 / Published: 2 November 2017
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Abstract
Misalignment is one of the common faults for the doubly-fed wind turbine (DFWT), and the normal operation of the unit will be greatly affected under this state. Because it is difficult to obtain a large number of misaligned fault samples of wind turbines
[...] Read more.
Misalignment is one of the common faults for the doubly-fed wind turbine (DFWT), and the normal operation of the unit will be greatly affected under this state. Because it is difficult to obtain a large number of misaligned fault samples of wind turbines in practice, ADAMS and MATLAB are used to simulate the various misalignment conditions of the wind turbine transmission system to obtain the corresponding stator current in this paper. Then, the dual-tree complex wavelet transform is used to decompose and reconstruct the characteristic signal, and the dual-tree complex wavelet energy entropy is obtained from the reconstructed coefficients to form the feature vector of the fault diagnosis. Support vector machine is used as classifier and particle swarm optimization is used to optimize the relevant parameters of support vector machine (SVM) to improve its classification performance. The results show that the method proposed in this paper can effectively and accurately classify the misalignment of the transmission system of the wind turbine and improve the reliability of the fault diagnosis. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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Open AccessArticle An Approximated Box Height for Differential-Box-Counting Method to Estimate Fractal Dimensions of Gray-Scale Images
Entropy 2017, 19(10), 534; doi:10.3390/e19100534
Received: 7 June 2017 / Revised: 30 August 2017 / Accepted: 30 September 2017 / Published: 10 October 2017
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Abstract
The Fractal Dimension (FD) of an image defines the roughness using a real number which is highly associated with the human perception of surface roughness. It has been applied successfully for many computer vision applications such as texture analysis, segmentation and classification. Several
[...] Read more.
The Fractal Dimension (FD) of an image defines the roughness using a real number which is highly associated with the human perception of surface roughness. It has been applied successfully for many computer vision applications such as texture analysis, segmentation and classification. Several techniques can be found in literature to estimate FD. One such technique is Differential Box Counting (DBC). Its performance is influenced by many parameters. In particular, the box height is directly related to the gray-level variations over image grid, which badly affects the performance of DBC. In this work, a new method for estimating box height is proposed without changing the other parameters of DBC. The proposed box height has been determined empirically and depends only on the image size. All the experiments have been performed on simulated Fractal Brownian Motion (FBM) Database and Brodatz Database. It has been proved experimentally that the proposed box height allow to improve the performance of DBC, Shifting DBC, Improved DBC and Improved Triangle DBC, which are closer to actual FD values of the simulated FBM images. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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Open AccessArticle Evaluation of Uncertainties in the Design Process of Complex Mechanical Systems
Entropy 2017, 19(9), 475; doi:10.3390/e19090475
Received: 20 July 2017 / Revised: 30 August 2017 / Accepted: 31 August 2017 / Published: 6 September 2017
Cited by 1 | PDF Full-text (235 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the problem of the evaluation of the uncertainties that originate in the complex design process of a new system is analyzed, paying particular attention to multibody mechanical systems. To this end, the Wiener-Shannon’s axioms are extended to non-probabilistic events and
[...] Read more.
In this paper, the problem of the evaluation of the uncertainties that originate in the complex design process of a new system is analyzed, paying particular attention to multibody mechanical systems. To this end, the Wiener-Shannon’s axioms are extended to non-probabilistic events and a theory of information for non-repetitive events is used as a measure of the reliability of data. The selection of the solutions consistent with the values of the design constraints is performed by analyzing the complexity of the relation matrix and using the idea of information in the metric space. Comparing the alternatives in terms of the amount of entropy resulting from the various distribution, this method is capable of finding the optimal solution that can be obtained with the available resources. In the paper, the algorithmic steps of the proposed method are discussed and an illustrative numerical example is provided. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
Open AccessArticle Incipient Fault Diagnosis of Rolling Bearings Based on Impulse-Step Impact Dictionary and Re-Weighted Minimizing Nonconvex Penalty Lq Regular Technique
Entropy 2017, 19(8), 421; doi:10.3390/e19080421
Received: 31 July 2017 / Revised: 15 August 2017 / Accepted: 16 August 2017 / Published: 18 August 2017
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Abstract
The periodical transient impulses caused by localized faults are sensitive and important characteristic information for rotating machinery fault diagnosis. However, it is very difficult to accurately extract transient impulses at the incipient fault stage because the fault impulse features are rather weak and
[...] Read more.
The periodical transient impulses caused by localized faults are sensitive and important characteristic information for rotating machinery fault diagnosis. However, it is very difficult to accurately extract transient impulses at the incipient fault stage because the fault impulse features are rather weak and always corrupted by heavy background noise. In this paper, a new transient impulse extraction methodology is proposed based on impulse-step dictionary and re-weighted minimizing nonconvex penalty Lq regular (R-WMNPLq, q = 0.5) for the incipient fault diagnosis of rolling bearings. Prior to the sparse representation, the original vibration signal is preprocessed by the variational mode decomposition (VMD) technique. Due to the physical mechanism of periodic double impacts, including step-like and impulse-like impacts, an impulse-step impact dictionary atom could be designed to match the natural waveform structure of vibration signals. On the other hand, the traditional sparse reconstruction approaches such as orthogonal matching pursuit (OMP), L1-norm regularization treat all vibration signal values equally and thus ignore the fact that the vibration peak value may have more useful information about periodical transient impulses and should be preserved at a larger weight value. Therefore, penalty and smoothing parameters are introduced on the reconstructed model to guarantee the reasonable distribution consistence of peak vibration values. Lastly, the proposed technique is applied to accelerated lifetime testing of rolling bearings, where it achieves a more noticeable and higher diagnostic accuracy compared with OMP, L1-norm regularization and traditional spectral Kurtogram (SK) method. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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Open AccessArticle Tidal Analysis Using Time–Frequency Signal Processing and Information Clustering
Entropy 2017, 19(8), 390; doi:10.3390/e19080390
Received: 11 June 2017 / Revised: 13 July 2017 / Accepted: 26 July 2017 / Published: 29 July 2017
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Abstract
Geophysical time series have a complex nature that poses challenges to reaching assertive conclusions, and require advanced mathematical and computational tools to unravel embedded information. In this paper, time–frequency methods and hierarchical clustering (HC) techniques are combined for processing and visualizing tidal information.
[...] Read more.
Geophysical time series have a complex nature that poses challenges to reaching assertive conclusions, and require advanced mathematical and computational tools to unravel embedded information. In this paper, time–frequency methods and hierarchical clustering (HC) techniques are combined for processing and visualizing tidal information. In a first phase, the raw data are pre-processed for estimating missing values and obtaining dimensionless reliable time series. In a second phase, the Jensen–Shannon divergence is adopted for measuring dissimilarities between data collected at several stations. The signals are compared in the frequency and time–frequency domains, and the HC is applied to visualize hidden relationships. In a third phase, the long-range behavior of tides is studied by means of power law functions. Numerical examples demonstrate the effectiveness of the approach when dealing with a large volume of real-world data. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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Open AccessArticle Incipient Fault Feature Extraction for Rotating Machinery Based on Improved AR-Minimum Entropy Deconvolution Combined with Variational Mode Decomposition Approach
Entropy 2017, 19(7), 317; doi:10.3390/e19070317
Received: 4 May 2017 / Revised: 16 June 2017 / Accepted: 25 June 2017 / Published: 29 June 2017
Cited by 2 | PDF Full-text (6951 KB) | HTML Full-text | XML Full-text
Abstract
Aiming at the issue of extracting the incipient single-fault and multi-fault of rotating machinery from the nonlinear and non-stationary vibration signals with a strong background noise, a new fault diagnosis method based on improved autoregressive-Minimum entropy deconvolution (improved AR-MED) and variational mode decomposition
[...] Read more.
Aiming at the issue of extracting the incipient single-fault and multi-fault of rotating machinery from the nonlinear and non-stationary vibration signals with a strong background noise, a new fault diagnosis method based on improved autoregressive-Minimum entropy deconvolution (improved AR-MED) and variational mode decomposition (VMD) is proposed. Due to the complexity of rotating machinery systems, the periodic transient impulses of single-fault and multiple-faults always emerge in the acquired vibration signals. The improved autoregressive minimum entropy deconvolution (AR-MED) technique can effectively deconvolve the influence of the background noise, which aims to enhance the peak value of the multiple transient impulses. Nevertheless, the envelope spectrum of simulation and experimental in this work shows that there are many interference components exist on both left and right of fault characteristic frequencies when the background noise is strong. To overcome this shortcoming, the VMD is thus applied to adaptively decompose the filtered output vibration signal into a number of quasi-orthogonal intrinsic modes so as to better detect the single- and multiple-faults via those sub-band signals. The experimental and engineering application results demonstrate that the proposed method dramatically sharpens the fault characteristic frequencies (FCFs) from the impacts of bearing outer race and gearbox faults compared to the traditional methods, which show a significant improvement in early incipient faults of rotating machinery. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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Open AccessArticle Entropic Measure of Epistemic Uncertainties in Multibody System Models by Axiomatic Design
Entropy 2017, 19(7), 291; doi:10.3390/e19070291
Received: 14 May 2017 / Revised: 12 June 2017 / Accepted: 15 June 2017 / Published: 26 June 2017
Cited by 4 | PDF Full-text (1524 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the use of the MaxInf Principle in real optimization problems is investigated for engineering applications, where the current design solution is actually an engineering approximation. In industrial manufacturing, multibody system simulations can be used to develop new machines and mechanisms
[...] Read more.
In this paper, the use of the MaxInf Principle in real optimization problems is investigated for engineering applications, where the current design solution is actually an engineering approximation. In industrial manufacturing, multibody system simulations can be used to develop new machines and mechanisms by using virtual prototyping, where an axiomatic design can be employed to analyze the independence of elements and the complexity of connections forming a general mechanical system. In the classic theories of Fisher and Wiener-Shannon, the idea of information is a measure of only probabilistic and repetitive events. However, this idea is broader than the probability alone field. Thus, the Wiener-Shannon’s axioms can be extended to non-probabilistic events and it is possible to introduce a theory of information for non-repetitive events as a measure of the reliability of data for complex mechanical systems. To this end, one can devise engineering solutions consistent with the values of the design constraints analyzing the complexity of the relation matrix and using the idea of information in the metric space. The final solution gives the entropic measure of epistemic uncertainties which can be used in multibody system models, analyzed with an axiomatic design. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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Open AccessArticle Weak Fault Diagnosis of Wind Turbine Gearboxes Based on MED-LMD
Entropy 2017, 19(6), 277; doi:10.3390/e19060277
Received: 12 April 2017 / Revised: 2 June 2017 / Accepted: 7 June 2017 / Published: 15 June 2017
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Abstract
In view of the problem that the fault signal of the rolling bearing is weak and the fault feature is difficult to extract in the strong noise environment, a method based on minimum entropy deconvolution (MED) and local mean deconvolution (LMD) is proposed
[...] Read more.
In view of the problem that the fault signal of the rolling bearing is weak and the fault feature is difficult to extract in the strong noise environment, a method based on minimum entropy deconvolution (MED) and local mean deconvolution (LMD) is proposed to extract the weak fault features of the rolling bearing. Through the analysis of the simulation signal, we find that LMD has many limitations for the feature extraction of weak signals under strong background noise. In order to eliminate the noise interference and extract the characteristics of the weak fault, MED is employed as the pre-filter to remove noise. This method is applied to the weak fault feature extraction of rolling bearings; that is, using MED to reduce the noise of the wind turbine gearbox test bench under strong background noise, and then using the LMD method to decompose the denoised signals into several product functions (PFs), and finally analyzing the PF components that have strong correlation by a cyclic autocorrelation function. The finding is that the failure of the wind power gearbox is generated from the micro-bending of the high-speed shaft and the pitting of the #10 bearing outer race at the output end of the high-speed shaft. This method is compared with LMD, which shows the effectiveness of this method. This paper provides a new method for the extraction of multiple faults and weak features in strong background noise. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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Open AccessArticle A Novel Distance Metric: Generalized Relative Entropy
Entropy 2017, 19(6), 269; doi:10.3390/e19060269
Received: 26 May 2017 / Revised: 7 June 2017 / Accepted: 7 June 2017 / Published: 13 June 2017
Cited by 1 | PDF Full-text (1133 KB) | HTML Full-text | XML Full-text
Abstract
Information entropy and its extension, which are important generalizations of entropy, are currently applied to many research domains. In this paper, a novel generalized relative entropy is constructed to avoid some defects of traditional relative entropy. We present the structure of generalized relative
[...] Read more.
Information entropy and its extension, which are important generalizations of entropy, are currently applied to many research domains. In this paper, a novel generalized relative entropy is constructed to avoid some defects of traditional relative entropy. We present the structure of generalized relative entropy after the discussion of defects in relative entropy. Moreover, some properties of the provided generalized relative entropy are presented and proved. The provided generalized relative entropy is proved to have a finite range and is a finite distance metric. Finally, we predict nucleosome positioning of fly and yeast based on generalized relative entropy and relative entropy respectively. The experimental results show that the properties of generalized relative entropy are better than relative entropy. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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Open AccessArticle Cauchy Principal Value Contour Integral with Applications
Entropy 2017, 19(5), 215; doi:10.3390/e19050215
Received: 28 March 2017 / Revised: 28 April 2017 / Accepted: 3 May 2017 / Published: 10 May 2017
PDF Full-text (278 KB) | HTML Full-text | XML Full-text
Abstract
Cauchy principal value is a standard method applied in mathematical applications by which an improper, and possibly divergent, integral is measured in a balanced way around singularities or at infinity. On the other hand, entropy prediction of systems behavior from a thermodynamic perspective
[...] Read more.
Cauchy principal value is a standard method applied in mathematical applications by which an improper, and possibly divergent, integral is measured in a balanced way around singularities or at infinity. On the other hand, entropy prediction of systems behavior from a thermodynamic perspective commonly involves contour integrals. With the aim of facilitating the calculus of such integrals in this entropic scenario, we revisit the generalization of Cauchy principal value to complex contour integral, formalize its definition and—by using residue theory techniques—provide an useful way to evaluate them. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)

Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Tentative title: Jumps and structural breaks in financial markets
Author: Ramazan Gencay and Ege Yazgan
Abstract: We evaluate the performance of jump detection tests by using simulated and real time high-frequency data in which frequent jumps and structural breaks are present. We define jumps as more than 3 standard deviation departures in the data with rapid mean reversion property; and structural breaks as without mean reversion (permanent breaks) or returning mean very sluggishly after initial departure (temporary breaks). We pay special attention to the performance of wavelet based jump test and find that the wavelet based test, especially when augmented by wavestrrapping, provide better performance.

Tentative title: Information Entropy in Metal Fractal Electrodeposits
Author: Rabih Sultan
Affiliation: Department of Chemistry, American University of Beirut

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