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Entropy
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Wavelets, Fractals and Information Theory III
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Wavelets, Fractals and Information Theory III

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

Deadline for manuscript submissions: closed (30 June 2018) | Viewed by 108995

Special Issue Editor

Prof. Dr. Carlo Cattani

E-Mail Website
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 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 submissions that pass pre-check are 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 2600 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

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Published Papers (21 papers)

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Research

15 pages, 2795 KiB  
Article
Identity Vector Extraction by Perceptual Wavelet Packet Entropy and Convolutional Neural Network for Voice Authentication
by Lei Lei and Kun She
Entropy 2018, 20(8), 600; https://0-doi-org.brum.beds.ac.uk/10.3390/e20080600 - 13 Aug 2018
Cited by 9 | Viewed by 4380
Abstract
Recently, the accuracy of voice authentication system has increased significantly due to the successful application of the identity vector (i-vector) model. This paper proposes a new method for i-vector extraction. In the method, a perceptual wavelet packet transform (PWPT) is designed to convert [...] Read more.
Recently, the accuracy of voice authentication system has increased significantly due to the successful application of the identity vector (i-vector) model. This paper proposes a new method for i-vector extraction. In the method, a perceptual wavelet packet transform (PWPT) is designed to convert speech utterances into wavelet entropy feature vectors, and a Convolutional Neural Network (CNN) is designed to estimate the frame posteriors of the wavelet entropy feature vectors. In the end, i-vector is extracted based on those frame posteriors. TIMIT and VoxCeleb speech corpus are used for experiments and the experimental results show that the proposed method can extract appropriate i-vector which reduces the equal error rate (EER) and improve the accuracy of voice authentication system in clean and noisy environment. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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7 pages, 810 KiB  
Article
The Use of LPC and Wavelet Transform for Influenza Disease Modeling
by Khaled Daqrouq and Mohammed Ajour
Entropy 2018, 20(8), 590; https://0-doi-org.brum.beds.ac.uk/10.3390/e20080590 - 09 Aug 2018
Cited by 2 | Viewed by 2793
Abstract
In this paper, we investigated the modeling of the pathological features of the influenza disease on the human speech. The presented work is novel research based on a real database and a new combination of previously used methods, discrete wavelet transform (DWT) and [...] Read more.
In this paper, we investigated the modeling of the pathological features of the influenza disease on the human speech. The presented work is novel research based on a real database and a new combination of previously used methods, discrete wavelet transform (DWT) and linear prediction coding (LPC). Three verification system experiments, Normal/Influenza, Smokers/Influenza, and Normal/Smokers, were studied. For testing the proposed pathological system, several classification scores were calculated for the recorded database, from which we can see that the proposed method achieved very high scores, particularly for the Normal with Influenza verification system. The performance of the proposed system was also compared with other published recognition systems. The experiments of these schemes show that the proposed method is superior. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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15 pages, 3961 KiB  
Article
Entropy SVM–Based Recognition of Transient Surges in HVDC Transmissions
by Guomin Luo, Changyuan Yao, Yinglin Liu, Yingjie Tan and Jinghan He
Entropy 2018, 20(6), 421; https://0-doi-org.brum.beds.ac.uk/10.3390/e20060421 - 31 May 2018
Cited by 8 | Viewed by 3799
Abstract
Protection based on transient information is the primary protection of high voltage direct current (HVDC) transmission systems. As a major part of protection function, accurate identification of transient surges is quite crucial to ensure the performance and accuracy of protection algorithms. Recognition of [...] Read more.
Protection based on transient information is the primary protection of high voltage direct current (HVDC) transmission systems. As a major part of protection function, accurate identification of transient surges is quite crucial to ensure the performance and accuracy of protection algorithms. Recognition of transient surges in an HVDC system faces two challenges: signal distortion and small number of samples. Entropy, which is stable in representing frequency distribution features, and support vector machine (SVM), which is good at dealing with samples with limited numbers, are adopted and combined in this paper to solve the transient recognition problems. Three commonly detected transient surges—single-pole-to-ground fault (GF), lightning fault (LF), and lightning disturbance (LD)—are simulated in various scenarios and recognized with the proposed method. The proposed method is proved to be effective in both feature extraction and type classification and shows great potential in protection applications. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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14 pages, 4733 KiB  
Article
Research on Weak Fault Extraction Method for Alleviating the Mode Mixing of LMD
by Lin Zhang, Zhijian Wang and Long Quan
Entropy 2018, 20(5), 387; https://0-doi-org.brum.beds.ac.uk/10.3390/e20050387 - 21 May 2018
Cited by 18 | Viewed by 3643
Abstract
Compared with the strong background noise, the energy entropy of early fault signals of bearings are weak under actual working conditions. Therefore, extracting the bearings’ early fault features has always been a major difficulty in fault diagnosis of rotating machinery. Based on the [...] Read more.
Compared with the strong background noise, the energy entropy of early fault signals of bearings are weak under actual working conditions. Therefore, extracting the bearings’ early fault features has always been a major difficulty in fault diagnosis of rotating machinery. Based on the above problems, the masking method is introduced into the Local Mean Decomposition (LMD) decomposition process, and a weak fault extraction method based on LMD and mask signal (MS) is proposed. Due to the mode mixing of the product function (PF) components decomposed by LMD in the noisy background, it is difficult to distinguish the authenticity of the fault frequency. Therefore, the MS method is introduced to deal with the PF components that are decomposed by the LMD and have strong correlation with the original signal, so as to suppress the modal aliasing phenomenon and extract the fault frequencies. In this paper, the actual fault signal of the rolling bearing is analyzed. By combining the MS method with the LMD method, the fault signal mixed with the noise is processed. The kurtosis value at the fault frequency is increased by eight-fold, and the signal-to-noise ratio (SNR) is increased by 19.1%. The fault signal is successfully extracted by the proposed composite method. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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23 pages, 14490 KiB  
Article
Chaotic Attractors with Fractional Conformable Derivatives in the Liouville–Caputo Sense and Its Dynamical Behaviors
by Jesús Emmanuel Solís Pérez, José Francisco Gómez-Aguilar, Dumitru Baleanu and Fairouz Tchier
Entropy 2018, 20(5), 384; https://0-doi-org.brum.beds.ac.uk/10.3390/e20050384 - 20 May 2018
Cited by 42 | Viewed by 5251
Abstract
This paper deals with a numerical simulation of fractional conformable attractors of type Rabinovich–Fabrikant, Thomas’ cyclically symmetric attractor and Newton–Leipnik. Fractional conformable and β -conformable derivatives of Liouville–Caputo type are considered to solve the proposed systems. A numerical method based on the Adams–Moulton [...] Read more.
This paper deals with a numerical simulation of fractional conformable attractors of type Rabinovich–Fabrikant, Thomas’ cyclically symmetric attractor and Newton–Leipnik. Fractional conformable and β -conformable derivatives of Liouville–Caputo type are considered to solve the proposed systems. A numerical method based on the Adams–Moulton algorithm is employed to approximate the numerical simulations of the fractional-order conformable attractors. The results of the new type of fractional conformable and β -conformable attractors are provided to illustrate the effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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10 pages, 3500 KiB  
Article
A New Local Fractional Entropy-Based Model for Kidney MRI Image Enhancement
by Ala’a R. Al-Shamasneh, Hamid A. Jalab, Shivakumara Palaiahnakote, Unaizah Hanum Obaidellah, Rabha W. Ibrahim and Moumen T. El-Melegy
Entropy 2018, 20(5), 344; https://0-doi-org.brum.beds.ac.uk/10.3390/e20050344 - 05 May 2018
Cited by 33 | Viewed by 4849
Abstract
Kidney image enhancement is challenging due to the unpredictable quality of MRI images, as well as the nature of kidney diseases. The focus of this work is on kidney images enhancement by proposing a new Local Fractional Entropy (LFE)-based model. The proposed model [...] Read more.
Kidney image enhancement is challenging due to the unpredictable quality of MRI images, as well as the nature of kidney diseases. The focus of this work is on kidney images enhancement by proposing a new Local Fractional Entropy (LFE)-based model. The proposed model estimates the probability of pixels that represent edges based on the entropy of the neighboring pixels, which results in local fractional entropy. When there is a small change in the intensity values (indicating the presence of edge in the image), the local fractional entropy gives fine image details. Similarly, when no change in intensity values is present (indicating smooth texture), the LFE does not provide fine details, based on the fact that there is no edge information. Tests were conducted on a large dataset of different, poor-quality kidney images to show that the proposed model is useful and effective. A comparative study with the classical methods, coupled with the latest enhancement methods, shows that the proposed model outperforms the existing methods. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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15 pages, 2619 KiB  
Article
Applying Discrete Homotopy Analysis Method for Solving Fractional Partial Differential Equations
by Figen Özpınar
Entropy 2018, 20(5), 332; https://0-doi-org.brum.beds.ac.uk/10.3390/e20050332 - 01 May 2018
Cited by 12 | Viewed by 3538
Abstract
In this paper we developed a space discrete version of the homotopy analysis method (DHAM) to find the solutions of linear and nonlinear fractional partial differential equations with time derivative α   ( 0 < α ≤ 1 ) . The DHAM contains [...] Read more.
In this paper we developed a space discrete version of the homotopy analysis method (DHAM) to find the solutions of linear and nonlinear fractional partial differential equations with time derivative α   ( 0 < α ≤ 1 ) . The DHAM contains the auxiliary parameter ℏ , which provides a simple way to guarantee the convergence region of solution series. The efficiency and accuracy of the proposed method is demonstrated by test problems with initial conditions. The results obtained are compared with the exact solutions when α = 1 . It is shown they are in good agreement with each other. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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20 pages, 17751 KiB  
Article
Multichannel Signals Reconstruction Based on Tunable Q-Factor Wavelet Transform-Morphological Component Analysis and Sparse Bayesian Iteration for Rotating Machines
by Qing Li, Wei Hu, Erfei Peng and Steven Y. Liang
Entropy 2018, 20(4), 263; https://0-doi-org.brum.beds.ac.uk/10.3390/e20040263 - 10 Apr 2018
Cited by 7 | Viewed by 3663
Abstract
High-speed remote transmission and large-capacity data storage are difficult issues in signals acquisition of rotating machines condition monitoring. To address these concerns, a novel multichannel signals reconstruction approach based on tunable Q-factor wavelet transform-morphological component analysis (TQWT-MCA) and sparse Bayesian iteration algorithm [...] Read more.
High-speed remote transmission and large-capacity data storage are difficult issues in signals acquisition of rotating machines condition monitoring. To address these concerns, a novel multichannel signals reconstruction approach based on tunable Q-factor wavelet transform-morphological component analysis (TQWT-MCA) and sparse Bayesian iteration algorithm combined with step-impulse dictionary is proposed under the frame of compressed sensing (CS). To begin with, to prevent the periodical impulses loss and effectively separate periodical impulses from the external noise and additive interference components, the TQWT-MCA method is introduced to divide the raw vibration signal into low-resonance component (LRC, i.e., periodical impulses) and high-resonance component (HRC), thus, the periodical impulses are preserved effectively. Then, according to the amplitude range of generated LRC, the step-impulse dictionary atom is designed to match the physical structure of periodical impulses. Furthermore, the periodical impulses and HRC are reconstructed by the sparse Bayesian iteration combined with step-impulse dictionary, respectively, finally, the final reconstructed raw signals are obtained by adding the LRC and HRC, meanwhile, the fidelity of the final reconstructed signals is tested by the envelop spectrum and error analysis, respectively. In this work, the proposed algorithm is applied to simulated signal and engineering multichannel signals of a gearbox with multiple faults. Experimental results demonstrate that the proposed approach significantly improves the reconstructive accuracy compared with the state-of-the-art methods such as non-convex Lq (q = 0.5) regularization, spatiotemporal sparse Bayesian learning (SSBL) and L1-norm, etc. Additionally, the processing time, i.e., speed of storage and transmission has increased dramatically, more importantly, the fault characteristics of the gearbox with multiple faults are detected and saved, i.e., the bearing outer race fault frequency at 170.7 Hz and its harmonics at 341.3 Hz, ball fault frequency at 7.344 Hz and its harmonics at 15.0 Hz, and the gear fault frequency at 23.36 Hz and its harmonics at 47.42 Hz are identified in the envelope spectrum. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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9 pages, 266 KiB  
Article
An Efficient Computational Technique for Fractal Vehicular Traffic Flow
by Devendra Kumar, Fairouz Tchier, Jagdev Singh and Dumitru Baleanu
Entropy 2018, 20(4), 259; https://0-doi-org.brum.beds.ac.uk/10.3390/e20040259 - 09 Apr 2018
Cited by 76 | Viewed by 4030
Abstract
In this work, we examine a fractal vehicular traffic flow problem. The partial differential equations describing a fractal vehicular traffic flow are solved with the aid of the local fractional homotopy perturbation Sumudu transform scheme and the local fractional reduced differential transform method. [...] Read more.
In this work, we examine a fractal vehicular traffic flow problem. The partial differential equations describing a fractal vehicular traffic flow are solved with the aid of the local fractional homotopy perturbation Sumudu transform scheme and the local fractional reduced differential transform method. Some illustrative examples are taken to describe the success of the suggested techniques. The results derived with the aid of the suggested schemes reveal that the present schemes are very efficient for obtaining the non-differentiable solution to fractal vehicular traffic flow problem. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
8 pages, 653 KiB  
Article
Anomalous Advection-Dispersion Equations within General Fractional-Order Derivatives: Models and Series Solutions
by Xin Liang, Yu-Gui Yang, Feng Gao, Xiao-Jun Yang and Yi Xue
Entropy 2018, 20(1), 78; https://0-doi-org.brum.beds.ac.uk/10.3390/e20010078 - 22 Jan 2018
Cited by 5 | Viewed by 4943
Abstract
In this paper, an anomalous advection-dispersion model involving a new general Liouville–Caputo fractional-order derivative is addressed for the first time. The series solutions of the general fractional advection-dispersion equations are obtained with the aid of the Laplace transform. The results are given to [...] Read more.
In this paper, an anomalous advection-dispersion model involving a new general Liouville–Caputo fractional-order derivative is addressed for the first time. The series solutions of the general fractional advection-dispersion equations are obtained with the aid of the Laplace transform. The results are given to demonstrate the efficiency of the proposed formulations to describe the anomalous advection dispersion processes. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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2157 KiB  
Article
Chaos in a Cancer Model via Fractional Derivatives with Exponential Decay and Mittag-Leffler Law
by José Francisco Gómez-Aguilar, María Guadalupe López-López, Victor Manuel Alvarado-Martínez, Dumitru Baleanu and Hasib Khan
Entropy 2017, 19(12), 681; https://0-doi-org.brum.beds.ac.uk/10.3390/e19120681 - 19 Dec 2017
Cited by 75 | Viewed by 5624
Abstract
In this paper, a three-dimensional cancer model was considered using the Caputo-Fabrizio-Caputo and the new fractional derivative with Mittag-Leffler kernel in Liouville-Caputo sense. Special solutions using an iterative scheme via Laplace transform, Sumudu-Picard integration method and Adams-Moulton rule were obtained. We studied the [...] Read more.
In this paper, a three-dimensional cancer model was considered using the Caputo-Fabrizio-Caputo and the new fractional derivative with Mittag-Leffler kernel in Liouville-Caputo sense. Special solutions using an iterative scheme via Laplace transform, Sumudu-Picard integration method and Adams-Moulton rule were obtained. We studied the uniqueness and existence of the solutions. Novel chaotic attractors with total order less than three are obtained. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory III)
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284 KiB  
Article
How Successful Are Wavelets in Detecting Jumps?
by Burak Alparslan Eroğlu, Ramazan Gençay and M. Ege Yazgan
Entropy 2017, 19(12), 638; https://0-doi-org.brum.beds.ac.uk/10.3390/e19120638 - 25 Nov 2017
Cited by 1 | Viewed by 3648
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)
3965 KiB  
Article
The Application of Dual-Tree Complex Wavelet Transform (DTCWT) Energy Entropy in Misalignment Fault Diagnosis of Doubly-Fed Wind Turbine (DFWT)
by Yancai Xiao, Yi Hong, Xiuhai Chen and Weijia Chen
Entropy 2017, 19(11), 587; https://0-doi-org.brum.beds.ac.uk/10.3390/e19110587 - 02 Nov 2017
Cited by 27 | Viewed by 5568
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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3134 KiB  
Article
An Approximated Box Height for Differential-Box-Counting Method to Estimate Fractal Dimensions of Gray-Scale Images
by Chinmaya Panigrahy, Angel Garcia-Pedrero, Ayan Seal, Dionisio Rodríguez-Esparragón, Nihar Kumar Mahato and Consuelo Gonzalo-Martín
Entropy 2017, 19(10), 534; https://0-doi-org.brum.beds.ac.uk/10.3390/e19100534 - 10 Oct 2017
Cited by 30 | Viewed by 6102
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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235 KiB  
Article
Evaluation of Uncertainties in the Design Process of Complex Mechanical Systems
by Francesco Villecco and Arcangelo Pellegrino
Entropy 2017, 19(9), 475; https://0-doi-org.brum.beds.ac.uk/10.3390/e19090475 - 06 Sep 2017
Cited by 47 | Viewed by 4581
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)
4910 KiB  
Article
Tidal Analysis Using Time–Frequency Signal Processing and Information Clustering
by Antonio M. Lopes and Jose A. Tenreiro Machado
Entropy 2017, 19(8), 390; https://0-doi-org.brum.beds.ac.uk/10.3390/e19080390 - 29 Jul 2017
Cited by 7 | Viewed by 5940
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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6951 KiB  
Article
Incipient Fault Feature Extraction for Rotating Machinery Based on Improved AR-Minimum Entropy Deconvolution Combined with Variational Mode Decomposition Approach
by Qing Li, Xia Ji and Steven Y. Liang
Entropy 2017, 19(7), 317; https://0-doi-org.brum.beds.ac.uk/10.3390/e19070317 - 29 Jun 2017
Cited by 35 | Viewed by 6541
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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1524 KiB  
Article
Entropic Measure of Epistemic Uncertainties in Multibody System Models by Axiomatic Design
by Francesco Villecco and Arcangelo Pellegrino
Entropy 2017, 19(7), 291; https://0-doi-org.brum.beds.ac.uk/10.3390/e19070291 - 26 Jun 2017
Cited by 46 | Viewed by 6440
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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2861 KiB  
Article
Weak Fault Diagnosis of Wind Turbine Gearboxes Based on MED-LMD
by Zhijian Wang, Junyuan Wang, Yanfei Kou, Jiping Zhang, Shaohui Ning and Zhifang Zhao
Entropy 2017, 19(6), 277; https://0-doi-org.brum.beds.ac.uk/10.3390/e19060277 - 15 Jun 2017
Cited by 30 | Viewed by 5390
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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1133 KiB  
Article
A Novel Distance Metric: Generalized Relative Entropy
by Shuai Liu, Mengye Lu, Gaocheng Liu and Zheng Pan
Entropy 2017, 19(6), 269; https://0-doi-org.brum.beds.ac.uk/10.3390/e19060269 - 13 Jun 2017
Cited by 51 | Viewed by 5948
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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278 KiB  
Article
Cauchy Principal Value Contour Integral with Applications
by Matilde Legua and Luis M. Sánchez-Ruiz
Entropy 2017, 19(5), 215; https://0-doi-org.brum.beds.ac.uk/10.3390/e19050215 - 10 May 2017
Cited by 9 | Viewed by 11052
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)
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