E-Mail Alert

Add your e-mail address to receive forthcoming issues of this journal:

Journal Browser

Journal Browser

Special Issue "Wavelets, Fractals and Information Theory II"

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

Deadline for manuscript submissions: closed (30 November 2016)

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 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

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.

Related Special Issues

Published Papers (21 papers)

View options order results:
result details:
Displaying articles 1-21
Export citation of selected articles as:

Research

Jump to: Review

Open AccessArticle Off-Line Handwritten Signature Recognition by Wavelet Entropy and Neural Network
Entropy 2017, 19(6), 252; doi:10.3390/e19060252
Received: 7 March 2017 / Revised: 7 May 2017 / Accepted: 10 May 2017 / Published: 31 May 2017
PDF Full-text (2358 KB) | HTML Full-text | XML Full-text
Abstract
Handwritten signatures are widely utilized as a form of personal recognition. However, they have the unfortunate shortcoming of being easily abused by those who would fake the identification or intent of an individual which might be very harmful. Therefore, the need for an
[...] Read more.
Handwritten signatures are widely utilized as a form of personal recognition. However, they have the unfortunate shortcoming of being easily abused by those who would fake the identification or intent of an individual which might be very harmful. Therefore, the need for an automatic signature recognition system is crucial. In this paper, a signature recognition approach based on a probabilistic neural network (PNN) and wavelet transform average framing entropy (AFE) is proposed. The system was tested with a wavelet packet (WP) entropy denoted as a WP entropy neural network system (WPENN) and with a discrete wavelet transform (DWT) entropy denoted as a DWT entropy neural network system (DWENN). Our investigation was conducted over several wavelet families and different entropy types. Identification tasks, as well as verification tasks, were investigated for a comprehensive signature system study. Several other methods used in the literature were considered for comparison. Two databases were used for algorithm testing. The best recognition rate result was achieved by WPENN whereby the threshold entropy reached 92%. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle Classification of Fractal Signals Using Two-Parameter Non-Extensive Wavelet Entropy
Entropy 2017, 19(5), 224; doi:10.3390/e19050224
Received: 3 March 2017 / Revised: 28 April 2017 / Accepted: 9 May 2017 / Published: 15 May 2017
PDF Full-text (1217 KB) | HTML Full-text | XML Full-text
Abstract
This article proposes a methodology for the classification of fractal signals as stationary or nonstationary. The methodology is based on the theoretical behavior of two-parameter wavelet entropy of fractal signals. The wavelet (q,q)-entropy is a wavelet-based extension
[...] Read more.
This article proposes a methodology for the classification of fractal signals as stationary or nonstationary. The methodology is based on the theoretical behavior of two-parameter wavelet entropy of fractal signals. The wavelet ( q , q ) -entropy is a wavelet-based extension of the ( q , q ) -entropy of Borges and is based on the entropy planes for various q and q ; it is theoretically shown that it constitutes an efficient and effective technique for fractal signal classification. Moreover, the second parameter q provides further analysis flexibility and robustness in the sense that different ( q , q ) pairs can analyze the same phenomena and increase the range of dispersion of entropies. A comparison study against the standard signal summation conversion technique shows that the proposed methodology is not only comparable in accuracy but also more computationally efficient. The application of the proposed methodology to physiological and financial time series is also presented along with the classification of these as stationary or nonstationary. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle The Solution of Modified Fractional Bergman’s Minimal Blood Glucose-Insulin Model
Entropy 2017, 19(5), 114; doi:10.3390/e19050114
Received: 21 January 2017 / Revised: 4 March 2017 / Accepted: 9 March 2017 / Published: 2 May 2017
Cited by 1 | PDF Full-text (283 KB) | HTML Full-text | XML Full-text
Abstract
In the present paper, we use analytical techniques to solve fractional nonlinear differential equations systems that arise in Bergman’s minimal model, used to describe blood glucose and insulin metabolism, after intravenous tolerance testing. We also discuss the stability and uniqueness of the solution.
[...] Read more.
In the present paper, we use analytical techniques to solve fractional nonlinear differential equations systems that arise in Bergman’s minimal model, used to describe blood glucose and insulin metabolism, after intravenous tolerance testing. We also discuss the stability and uniqueness of the solution. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle Multi-Scale Permutation Entropy Based on Improved LMD and HMM for Rolling Bearing Diagnosis
Entropy 2017, 19(4), 176; doi:10.3390/e19040176
Received: 8 January 2017 / Revised: 3 March 2017 / Accepted: 14 April 2017 / Published: 19 April 2017
Cited by 8 | PDF Full-text (2881 KB) | HTML Full-text | XML Full-text
Abstract
Based on the combination of improved Local Mean Decomposition (LMD), Multi-scale Permutation Entropy (MPE) and Hidden Markov Model (HMM), the fault types of bearings are diagnosed. Improved LMD is proposed based on the self-similarity of roller bearing vibration signal by extending the right
[...] Read more.
Based on the combination of improved Local Mean Decomposition (LMD), Multi-scale Permutation Entropy (MPE) and Hidden Markov Model (HMM), the fault types of bearings are diagnosed. Improved LMD is proposed based on the self-similarity of roller bearing vibration signal by extending the right and left side of the original signal to suppress its edge effect. First, the vibration signals of the rolling bearing are decomposed into several product function (PF) components by improved LMD respectively. Then, the phase space reconstruction of the PF1 is carried out by using the mutual information (MI) method and the false nearest neighbor (FNN) method to calculate the delay time and the embedding dimension, and then the scale is set to obtain the MPE of PF1. After that, the MPE features of rolling bearings are extracted. Finally, the features of MPE are used as HMM training and diagnosis. The experimental results show that the proposed method can effectively identify the different faults of the rolling bearing. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle “Over-Learning” Phenomenon of Wavelet Neural Networks in Remote Sensing Image Classifications with Different Entropy Error Functions
Entropy 2017, 19(3), 101; doi:10.3390/e19030101
Received: 14 November 2016 / Revised: 27 February 2017 / Accepted: 27 February 2017 / Published: 8 March 2017
Cited by 1 | PDF Full-text (8692 KB) | HTML Full-text | XML Full-text
Abstract
Artificial neural networks are widely applied for prediction, function simulation, and data classification. Among these applications, the wavelet neural network is widely used in image classification problems due to its advantages of high approximation capabilities, fault-tolerant capabilities, learning capacity, its ability to effectively
[...] Read more.
Artificial neural networks are widely applied for prediction, function simulation, and data classification. Among these applications, the wavelet neural network is widely used in image classification problems due to its advantages of high approximation capabilities, fault-tolerant capabilities, learning capacity, its ability to effectively overcome local minimization issues, and so on. The error function of a network is critical to determine the convergence, stability, and classification accuracy of a neural network. The selection of the error function directly determines the network’s performance. Different error functions will correspond with different minimum error values in training samples. With the decrease of network errors, the accuracy of the image classification is increased. However, if the image classification accuracy is difficult to improve upon, or is even decreased with the decreasing of the errors, then this indicates that the network has an “over-learning” phenomenon, which is closely related to the selection of the function errors. With regards to remote sensing data, it has not yet been reported whether there have been studies conducted regarding the “over-learning” phenomenon, as well as the relationship between the “over-learning” phenomenon and error functions. This study takes SAR, hyper-spectral, high-resolution, and multi-spectral images as data sources, in order to comprehensively and systematically analyze the possibility of an “over-learning” phenomenon in the remote sensing images from the aspects of image characteristics and neural network. Then, this study discusses the impact of three typical entropy error functions (NB, CE, and SH) on the “over-learning” phenomenon of a network. The experimental results show that the “over-learning” phenomenon may be caused only when there is a strong separability between the ground features, a low image complexity, a small image size, and a large number of hidden nodes. The SH entropy error function in that case will show a good “over-learning” resistance ability. However, for remote sensing image classification, the “over-learning” phenomenon will not be easily caused in most cases, due to the complexity of the image itself, and the diversity of the ground features. In that case, the NB and CE entropy error network mainly show a good stability. Therefore, a blind selection of a SH entropy error function with a high “over-learning” resistance ability from the wavelet neural network classification of the remote sensing image will only decrease the classification accuracy of the remote sensing image. It is therefore recommended to use an NB or CE entropy error function with a stable learning effect. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle Bateman–Feshbach Tikochinsky and Caldirola–Kanai Oscillators with New Fractional Differentiation
Entropy 2017, 19(2), 55; doi:10.3390/e19020055
Received: 23 November 2016 / Revised: 23 January 2017 / Accepted: 24 January 2017 / Published: 28 January 2017
Cited by 3 | PDF Full-text (1849 KB) | HTML Full-text | XML Full-text
Abstract
In this work, the study of the fractional behavior of the Bateman–Feshbach–Tikochinsky and Caldirola–Kanai oscillators by using different fractional derivatives is presented. We obtained the Euler–Lagrange and the Hamiltonian formalisms in order to represent the dynamic models based on the Liouville–Caputo, Caputo–Fabrizio–Caputo and
[...] Read more.
In this work, the study of the fractional behavior of the Bateman–Feshbach–Tikochinsky and Caldirola–Kanai oscillators by using different fractional derivatives is presented. We obtained the Euler–Lagrange and the Hamiltonian formalisms in order to represent the dynamic models based on the Liouville–Caputo, Caputo–Fabrizio–Caputo and the new fractional derivative based on the Mittag–Leffler kernel with arbitrary order α. Simulation results are presented in order to show the fractional behavior of the oscillators, and the classical behavior is recovered when α is equal to 1. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle A New Feature Extraction Method Based on EEMD and Multi-Scale Fuzzy Entropy for Motor Bearing
Entropy 2017, 19(1), 14; doi:10.3390/e19010014
Received: 23 September 2016 / Revised: 3 December 2016 / Accepted: 26 December 2016 / Published: 31 December 2016
Cited by 3 | PDF Full-text (3905 KB) | HTML Full-text | XML Full-text
Abstract
Feature extraction is one of the most important, pivotal, and difficult problems in mechanical fault diagnosis, which directly relates to the accuracy of fault diagnosis and the reliability of early fault prediction. Therefore, a new fault feature extraction method, called the EDOMFE method
[...] Read more.
Feature extraction is one of the most important, pivotal, and difficult problems in mechanical fault diagnosis, which directly relates to the accuracy of fault diagnosis and the reliability of early fault prediction. Therefore, a new fault feature extraction method, called the EDOMFE method based on integrating ensemble empirical mode decomposition (EEMD), mode selection, and multi-scale fuzzy entropy is proposed to accurately diagnose fault in this paper. The EEMD method is used to decompose the vibration signal into a series of intrinsic mode functions (IMFs) with a different physical significance. The correlation coefficient analysis method is used to calculate and determine three improved IMFs, which are close to the original signal. The multi-scale fuzzy entropy with the ability of effective distinguishing the complexity of different signals is used to calculate the entropy values of the selected three IMFs in order to form a feature vector with the complexity measure, which is regarded as the inputs of the support vector machine (SVM) model for training and constructing a SVM classifier (EOMSMFD based on EDOMFE and SVM) for fulfilling fault pattern recognition. Finally, the effectiveness of the proposed method is validated by real bearing vibration signals of the motor with different loads and fault severities. The experiment results show that the proposed EDOMFE method can effectively extract fault features from the vibration signal and that the proposed EOMSMFD method can accurately diagnose the fault types and fault severities for the inner race fault, the outer race fault, and rolling element fault of the motor bearing. Therefore, the proposed method provides a new fault diagnosis technology for rotating machinery. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle On the Existence and Uniqueness of Solutions for Local Fractional Differential Equations
Entropy 2016, 18(11), 420; doi:10.3390/e18110420
Received: 21 June 2016 / Revised: 26 October 2016 / Accepted: 21 November 2016 / Published: 23 November 2016
Cited by 1 | PDF Full-text (229 KB) | HTML Full-text | XML Full-text
Abstract
In this manuscript, we prove the existence and uniqueness of solutions for local fractional differential equations (LFDEs) with local fractional derivative operators (LFDOs). By using the contracting mapping theorem (CMT) and increasing and decreasing theorem (IDT), existence and uniqueness results are obtained. Some
[...] Read more.
In this manuscript, we prove the existence and uniqueness of solutions for local fractional differential equations (LFDEs) with local fractional derivative operators (LFDOs). By using the contracting mapping theorem (CMT) and increasing and decreasing theorem (IDT), existence and uniqueness results are obtained. Some examples are presented to illustrate the validity of our results. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Open AccessArticle Prediction of Bearing Fault Using Fractional Brownian Motion and Minimum Entropy Deconvolution
Entropy 2016, 18(11), 418; doi:10.3390/e18110418
Received: 24 October 2016 / Revised: 12 November 2016 / Accepted: 18 November 2016 / Published: 23 November 2016
PDF Full-text (3755 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we propose a novel framework for the diagnosis of incipient bearing faults and trend prediction of weak faults which result in gradual aggravation with the bearing vibration intensity as the characteristic parameter. For the weak fault diagnosis, the proposed framework
[...] Read more.
In this paper, we propose a novel framework for the diagnosis of incipient bearing faults and trend prediction of weak faults which result in gradual aggravation with the bearing vibration intensity as the characteristic parameter. For the weak fault diagnosis, the proposed framework adopts the improved minimum entropy deconvolution (MED) theory to identify the weak fault characteristics of mechanical equipment. From a large number of actual data analysis, once a bearing shows a weak fault, the bearing vibration intensity not only has random non-stationary, but also long-range dependent (LRD) characteristics. Therefore, the stochastic model with LRD−fractional Brown motion (FBM) is proposed to evaluate and predict the condition of slowly varying bearing faults which is a gradual process from weak fault occurrence to severity. For the FBM stochastic model, we mainly implement the derivation and the parameter identification of the FBM model. This is the first study to slowly fault prediction with stochastic model FBM. Experimental results show that the proposed methods can obtain the best performance in incipient fault diagnosis and bearing condition trend prediction. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle Texture Segmentation Using Laplace Distribution-Based Wavelet-Domain Hidden Markov Tree Models
Entropy 2016, 18(11), 384; doi:10.3390/e18110384
Received: 24 May 2016 / Revised: 20 October 2016 / Accepted: 21 October 2016 / Published: 4 November 2016
PDF Full-text (6281 KB) | HTML Full-text | XML Full-text
Abstract
Multiresolution models such as the wavelet-domain hidden Markov tree (HMT) model provide a powerful approach for image modeling and processing because it captures the key features of the wavelet coefficients of real-world data. It is observed that the Laplace distribution is peakier in
[...] Read more.
Multiresolution models such as the wavelet-domain hidden Markov tree (HMT) model provide a powerful approach for image modeling and processing because it captures the key features of the wavelet coefficients of real-world data. It is observed that the Laplace distribution is peakier in the center and has heavier tails compared with the Gaussian distribution. Thus we propose a new HMT model based on the two-state, zero-mean Laplace mixture model (LMM), the LMM-HMT, which provides significantly potential for characterizing real-world textures. By using the HMT segmentation framework, we develop LMM-HMT based segmentation methods for image textures and dynamic textures. The experimental results demonstrate the effectiveness of the introduced model and segmentation methods. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle Recognition of Abnormal Uptake through 123I-mIBG Scintigraphy Entropy for Paediatric Neuroblastoma Identification
Entropy 2016, 18(10), 349; doi:10.3390/e18100349
Received: 4 August 2016 / Revised: 16 September 2016 / Accepted: 20 September 2016 / Published: 27 September 2016
PDF Full-text (1080 KB) | HTML Full-text | XML Full-text
Abstract
Whole-body 123I-Metaiodobenzylguanidine (mIBG) scintigraphy is used as primary image modality to visualize neuroblastoma tumours and metastases because it is the most sensitive and specific radioactive tracer in staging the disease and evaluating the response to treatment. However, especially in paediatric neuroblastoma,
[...] Read more.
Whole-body 123I-Metaiodobenzylguanidine (mIBG) scintigraphy is used as primary image modality to visualize neuroblastoma tumours and metastases because it is the most sensitive and specific radioactive tracer in staging the disease and evaluating the response to treatment. However, especially in paediatric neuroblastoma, information from mIBG scans is difficult to extract because of acquisition difficulties that produce low definition images, with poor contours, resolution and contrast. These problems limit physician assessment. Current oncological guidelines are based on qualitative observer-dependant analysis. This makes comparing results taken at different moments of therapy, or in different institutions, difficult. In this paper, we present a computerized method that processes an image and calculates a quantitative measurement considered as its entropy, suitable for the identification of abnormal uptake regions, for which there is enough suspicion that they may be a tumour or metastatic site. This measurement can also be compared with future scintigraphies of the same patient. Over 46 scintigraphies of 22 anonymous patients were tested; the procedure identified 96.7% of regions of abnormal uptake and it showed a low overall false negative rate of 3.3%. This method provides assistance to physicians in diagnosing tumours and also allows the monitoring of patients’ evolution. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle A Novel Operational Matrix of Caputo Fractional Derivatives of Fibonacci Polynomials: Spectral Solutions of Fractional Differential Equations
Entropy 2016, 18(10), 345; doi:10.3390/e18100345
Received: 28 July 2016 / Revised: 19 September 2016 / Accepted: 19 September 2016 / Published: 23 September 2016
Cited by 4 | PDF Full-text (1282 KB) | HTML Full-text | XML Full-text
Abstract
Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation
[...] Read more.
Herein, two numerical algorithms for solving some linear and nonlinear fractional-order differential equations are presented and analyzed. For this purpose, a novel operational matrix of fractional-order derivatives of Fibonacci polynomials was constructed and employed along with the application of the tau and collocation spectral methods. The convergence and error analysis of the suggested Fibonacci expansion were carefully investigated. Some numerical examples with comparisons are presented to ensure the efficiency, applicability and high accuracy of the proposed algorithms. Two accurate semi-analytic polynomial solutions for linear and nonlinear fractional differential equations are the result. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle Study on the Inherent Complex Features and Chaos Control of IS–LM Fractional-Order Systems
Entropy 2016, 18(9), 332; doi:10.3390/e18090332
Received: 24 June 2016 / Revised: 24 August 2016 / Accepted: 5 September 2016 / Published: 14 September 2016
Cited by 1 | PDF Full-text (3512 KB) | HTML Full-text | XML Full-text
Abstract
Based on the traditional IS–LM economic theory, which shows the relationship between interest rates and output in the goods and services market and the money market in macroeconomic. We established a four-dimensional IS–LM model involving four variables. With the Caputo fractional calculus theory,
[...] Read more.
Based on the traditional IS–LM economic theory, which shows the relationship between interest rates and output in the goods and services market and the money market in macroeconomic. We established a four-dimensional IS–LM model involving four variables. With the Caputo fractional calculus theory, we improved it into a fractional order nonlinear model, analyzed the complexity and stability of the fractional order system. The existences conditions of attractors under different order conditions are compared, and obtain the orders when the system reaches a stable state. Have the detail analysis on the dynamic phenomena, such as the strange attractor, sensitivity to initial values through phase diagram and the power spectral. The order changes in two ways: orders changes synchronously or single order changes. The results show regardless of which the order situation is, the economic system will enter into multiple states, such as strong divergence, strange attractor and the convergence, finally, system will enter into the stable state under a certain order; parameter changes have similar effects on the economic system. Therefore, selecting an appropriate order is significant for an economic system, which guarantees a steady development. Furthermore, this paper construct the chaos control to IS–LM fractional-order macroeconomic model by means of linear feedback control method, by calculating and adjusting the feedback coefficient, we make the system return to the convergence state. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle Solution of Higher Order Nonlinear Time-Fractional Reaction Diffusion Equation
Entropy 2016, 18(9), 329; doi:10.3390/e18090329
Received: 8 July 2016 / Revised: 30 August 2016 / Accepted: 31 August 2016 / Published: 8 September 2016
Cited by 1 | PDF Full-text (2008 KB) | HTML Full-text | XML Full-text
Abstract
The approximate analytical solution of fractional order, nonlinear, reaction differential equations, namely the nonlinear diffusion equations, with a given initial condition, is obtained by using the homotopy analysis method. As a demonstration of a good mathematical model, the present article gives graphical presentations
[...] Read more.
The approximate analytical solution of fractional order, nonlinear, reaction differential equations, namely the nonlinear diffusion equations, with a given initial condition, is obtained by using the homotopy analysis method. As a demonstration of a good mathematical model, the present article gives graphical presentations of the effect of the reaction terms on the solution profile for various anomalous exponents of particular cases, to predict damping of the field variable. Numerical computations of the convergence control parameter, used to evaluate the convergence of approximate series solution through minimizing error, are also presented graphically for these cases. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle Analytical Solutions of the Electrical RLC Circuit via Liouville–Caputo Operators with Local and Non-Local Kernels
Entropy 2016, 18(8), 402; doi:10.3390/e18080402
Received: 22 June 2016 / Revised: 9 August 2016 / Accepted: 17 August 2016 / Published: 20 August 2016
Cited by 9 | PDF Full-text (572 KB) | HTML Full-text | XML Full-text
Abstract
In this work we obtain analytical solutions for the electrical RLC circuit model defined with Liouville–Caputo, Caputo–Fabrizio and the new fractional derivative based in the Mittag-Leffler function. Numerical simulations of alternative models are presented for evaluating the effectiveness of these representations. Different source
[...] Read more.
In this work we obtain analytical solutions for the electrical RLC circuit model defined with Liouville–Caputo, Caputo–Fabrizio and the new fractional derivative based in the Mittag-Leffler function. Numerical simulations of alternative models are presented for evaluating the effectiveness of these representations. Different source terms are considered in the fractional differential equations. The classical behaviors are recovered when the fractional order α is equal to 1. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Open AccessArticle Noise Suppression in 94 GHz Radar-Detected Speech Based on Perceptual Wavelet Packet
Entropy 2016, 18(7), 265; doi:10.3390/e18070265
Received: 12 April 2016 / Revised: 4 July 2016 / Accepted: 13 July 2016 / Published: 19 July 2016
Cited by 1 | PDF Full-text (2644 KB) | HTML Full-text | XML Full-text
Abstract
A millimeter wave (MMW) radar sensor is employed in our laboratory to detect human speech because it provides a new non-contact speech acquisition method that is suitable for various applications. However, the speech detected by the radar sensor is often degraded by combined
[...] Read more.
A millimeter wave (MMW) radar sensor is employed in our laboratory to detect human speech because it provides a new non-contact speech acquisition method that is suitable for various applications. However, the speech detected by the radar sensor is often degraded by combined noise. This paper proposes a new perceptual wavelet packet method that is able to enhance the speech acquired using a 94 GHz MMW radar system by suppressing the noise. The process is as follows. First, the radar speech signal is decomposed using a perceptual wavelet packet. Then, an adaptive wavelet threshold and new modified thresholding function are employed to remove the noise from the detected speech. The results obtained from the speech spectrograms, listening tests and objective evaluation show that the new method significantly improves the performance of the detected speech. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Open AccessArticle Fractional-Order Grey Prediction Method for Non-Equidistant Sequences
Entropy 2016, 18(6), 227; doi:10.3390/e18060227
Received: 11 May 2016 / Revised: 6 June 2016 / Accepted: 8 June 2016 / Published: 14 June 2016
Cited by 1 | PDF Full-text (1023 KB) | HTML Full-text | XML Full-text
Abstract
There are lots of non-equidistant sequences in actual applications due to random sampling, imperfect sensors, event-triggered phenomena, and so on. A new grey prediction method for non-equidistant sequences (r-NGM(1,1)) is proposed based on the basic grey model and the developed fractional-order
[...] Read more.
There are lots of non-equidistant sequences in actual applications due to random sampling, imperfect sensors, event-triggered phenomena, and so on. A new grey prediction method for non-equidistant sequences (r-NGM(1,1)) is proposed based on the basic grey model and the developed fractional-order non-equidistant accumulated generating operation (r-NAGO), and the accumulated order is extended from the positive to the negative. The whole r-NAGO deletes the randomness of original sequences in the form of weighted accumulation and improves the exponential law of accumulated sequences. Furthermore, the Levenberg–Marquardt algorithm is used to optimize the fractional order. The optimal r-NGM(1,1) can enhance the predicting performance of the non-equidistant sequences. Results of three practical cases in engineering applications demonstrate that the proposed r-NGM(1,1) provides the significant predicting performance compared with the traditional grey model. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Open AccessArticle A Reliable Algorithm for a Local Fractional Tricomi Equation Arising in Fractal Transonic Flow
Entropy 2016, 18(6), 206; doi:10.3390/e18060206
Received: 13 April 2016 / Revised: 12 May 2016 / Accepted: 23 May 2016 / Published: 25 May 2016
Cited by 13 | PDF Full-text (204 KB) | HTML Full-text | XML Full-text
Abstract
The pivotal proposal of this work is to present a reliable algorithm based on the local fractional homotopy perturbation Sumudu transform technique for solving a local fractional Tricomi equation occurring in fractal transonic flow. The proposed technique provides the results without any transformation
[...] Read more.
The pivotal proposal of this work is to present a reliable algorithm based on the local fractional homotopy perturbation Sumudu transform technique for solving a local fractional Tricomi equation occurring in fractal transonic flow. The proposed technique provides the results without any transformation of the equation into discrete counterparts or imposing restrictive assumptions and is completely free of round-off errors. The results of the scheme show that the approach is straightforward to apply and computationally very user-friendly and accurate. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Open AccessArticle Detection of Left-Sided and Right-Sided Hearing Loss via Fractional Fourier Transform
Entropy 2016, 18(5), 194; doi:10.3390/e18050194
Received: 30 March 2016 / Revised: 12 May 2016 / Accepted: 16 May 2016 / Published: 19 May 2016
Cited by 16 | PDF Full-text (1734 KB) | HTML Full-text | XML Full-text
Abstract
In order to detect hearing loss more efficiently and accurately, this study proposed a new method based on fractional Fourier transform (FRFT). Three-dimensional volumetric magnetic resonance images were obtained from 15 patients with left-sided hearing loss (LHL), 20 healthy controls (HC), and 14
[...] Read more.
In order to detect hearing loss more efficiently and accurately, this study proposed a new method based on fractional Fourier transform (FRFT). Three-dimensional volumetric magnetic resonance images were obtained from 15 patients with left-sided hearing loss (LHL), 20 healthy controls (HC), and 14 patients with right-sided hearing loss (RHL). Twenty-five FRFT spectrums were reduced by principal component analysis with thresholds of 90%, 95%, and 98%, respectively. The classifier is the single-hidden-layer feed-forward neural network (SFN) trained by the Levenberg–Marquardt algorithm. The results showed that the accuracies of all three classes are higher than 95%. In all, our method is promising and may raise interest from other researchers. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Open AccessArticle Fractal Information by Means of Harmonic Mappings and Some Physical Implications
Entropy 2016, 18(5), 160; doi:10.3390/e18050160
Received: 25 March 2016 / Revised: 16 April 2016 / Accepted: 20 April 2016 / Published: 26 April 2016
Cited by 1 | PDF Full-text (320 KB) | HTML Full-text | XML Full-text
Abstract
Considering that the motions of the complex system structural units take place on continuous, but non-differentiable curves, in the frame of the extended scale relativity model (in its Schrödinger-type variant), it is proven that the imaginary part of a scalar potential of velocities
[...] Read more.
Considering that the motions of the complex system structural units take place on continuous, but non-differentiable curves, in the frame of the extended scale relativity model (in its Schrödinger-type variant), it is proven that the imaginary part of a scalar potential of velocities can be correlated with the fractal information and, implicitly, with a tensor of “tensions”, which is fundamental in the construction of the constitutive laws of material. In this way, a specific differential geometry based on a Poincaré-type metric of the Lobachevsky plane (which is invariant to the homographic group of transformations) and also a specific variational principle (whose field equations represent an harmonic map from the usual space into the Lobachevsky plane) are generated. Moreover, fractal information (which is made explicit at any scale resolution) is produced, so that the field variables define a gravitational field. This latter situation is specific to a variational principle in the sense of Matzner–Misner and to certain Ernst-type field equations, the fractal information being contained in the material structure and, thus, in its own space associated with it. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)

Review

Jump to: Research

Open AccessReview Application of Shannon Wavelet Entropy and Shannon Wavelet Packet Entropy in Analysis of Power System Transient Signals
Entropy 2016, 18(12), 437; doi:10.3390/e18120437
Received: 9 August 2016 / Revised: 25 November 2016 / Accepted: 2 December 2016 / Published: 7 December 2016
Cited by 2 | PDF Full-text (3823 KB) | HTML Full-text | XML Full-text
Abstract
In a power system, the analysis of transient signals is the theoretical basis of fault diagnosis and transient protection theory. Shannon wavelet entropy (SWE) and Shannon wavelet packet entropy (SWPE) are powerful mathematics tools for transient signal analysis. Combined with the recent achievements
[...] Read more.
In a power system, the analysis of transient signals is the theoretical basis of fault diagnosis and transient protection theory. Shannon wavelet entropy (SWE) and Shannon wavelet packet entropy (SWPE) are powerful mathematics tools for transient signal analysis. Combined with the recent achievements regarding SWE and SWPE, their applications are summarized in feature extraction of transient signals and transient fault recognition. For wavelet aliasing at adjacent scale of wavelet decomposition, the impact of wavelet aliasing is analyzed for feature extraction accuracy of SWE and SWPE, and their differences are compared. Meanwhile, the analyses mentioned are verified by partial discharge (PD) feature extraction of power cable. Finally, some new ideas and further researches are proposed in the wavelet entropy mechanism, operation speed and how to overcome wavelet aliasing. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)
Figures

Figure 1

Back to Top