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# Wavelets, Fractals and Information Theory

A topical collection in Entropy (ISSN 1099-4300). This collection belongs to the section "Multidisciplinary Applications".

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## 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

## Topical Collection Information

Dear Colleagues,

Wavelet analysis and fractals are playing fundamental roles in science, engineering applications, and information theory. Wavelets and fractals are the most suitable methods for the analysis of 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; they are thus 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 and complexity 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 using wavelet compression. There are many types of definition of entropy that are very useful in the engineering and applied sciences, such as Shannon–Fano entropy, 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 modelling 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 for 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 on a wavelet basis.

This Topical Collection will also be an opportunity to extend 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 of all aspects of the theoretical and practical studies of mathematics, physics, and engineering.

The main topics of this Topical Collection 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
• Fractals, entropy and complexity
• Fractals, wavelets, fractional methods in the stochastic process, stochastic equations
• Fractal and wavelet solutions of fractional differential equations
• Wavelet analysis, integral transforms and applications
• Wavelets, fractals and fractional methods in fault diagnosis, in signal analysis, in nonlinear time series
• Wavelet-fractal entropy encoding and computational mathematics in data analysis and time series, including in image analysis
• Wavelet–fractal approach
• Fractional nonlinear equations
• Chaotic dynamics
• Artifical neural networks.

Prof. Dr. Carlo Cattani
Collection Editor

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## Keywords

• Fractal
• Fractional
• Wavelet
• Entropy
• Nonlinear time series
• Data analysis
• Image Analysis
• Dynamical System
• Chaos
• Differential Operators

# 2023

18 pages, 4976 KiB
Article
Multi-Fractal Weibull Adaptive Model for the Remaining Useful Life Prediction of Electric Vehicle Lithium Batteries
Entropy 2023, 25(4), 646; https://doi.org/10.3390/e25040646 - 12 Apr 2023
Cited by 1 | Viewed by 1045
Abstract
In this paper, an adaptive remaining useful life prediction model is proposed for electric vehicle lithium batteries. Capacity degradation of the electric car lithium batteries is modeled by the multi-fractal Weibull motion. The varying degree of long-range dependence and the 1/f characteristics in [...] Read more.
In this paper, an adaptive remaining useful life prediction model is proposed for electric vehicle lithium batteries. Capacity degradation of the electric car lithium batteries is modeled by the multi-fractal Weibull motion. The varying degree of long-range dependence and the 1/f characteristics in the frequency domain are also analyzed. The age and state-dependent degradation model is derived, with the associated adaptive drift and diffusion coefficients. The adaptive mechanism considers the quantitative relations between the drift and diffusion coefficients. The unit-to-unit variability is considered a random variable. To facilitate the application, the convergence of the RUL prediction model is proved. Replacement of the lithium battery in the electric car is recommended according to the remaining useful life prediction results. The effectiveness of the proposed model is shown in the case study. Full article
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# 2022

18 pages, 12911 KiB
Article
Natural Fractals as Irreversible Disorder: Entropy Approach from Cracks in the Semi Brittle-Ductile Lithosphere and Generalization
Entropy 2022, 24(10), 1337; https://doi.org/10.3390/e24101337 - 22 Sep 2022
Cited by 6 | Viewed by 1528
Abstract
The seismo-electromagnetic theory describes the growth of fractally distributed cracks within the lithosphere that generate the emission of magnetic anomalies prior to large earthquakes. One of the main physical properties of this theory is their consistency regarding the second law of thermodynamics. That [...] Read more.
The seismo-electromagnetic theory describes the growth of fractally distributed cracks within the lithosphere that generate the emission of magnetic anomalies prior to large earthquakes. One of the main physical properties of this theory is their consistency regarding the second law of thermodynamics. That is, the crack generation of the lithosphere corresponds to the manifestation of an irreversible process evolving from one steady state to another. Nevertheless, there is still not a proper thermodynamic description of lithospheric crack generation. That is why this work presents the derivation of the entropy changes generated by the lithospheric cracking. It is found that the growth of the fractal cracks increases the entropy prior impending earthquakes. As fractality is observed across different topics, our results are generalized by using the Onsager’s coefficient for any system characterized by fractal volumes. It is found that the growth of fractality in nature corresponds to an irreversible process. Full article
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21 pages, 6406 KiB
Article
Double-Matrix Decomposition Image Steganography Scheme Based on Wavelet Transform with Multi-Region Coverage
Entropy 2022, 24(2), 246; https://doi.org/10.3390/e24020246 - 07 Feb 2022
Cited by 18 | Viewed by 2027
Abstract
On the basis of ensuring the quality and concealment of steganographic images, this paper proposes a double-matrix decomposition image steganography scheme with multi-region coverage, to solve the problem of poor extraction ability of steganographic images under attack or interference. First of all, the [...] Read more.
On the basis of ensuring the quality and concealment of steganographic images, this paper proposes a double-matrix decomposition image steganography scheme with multi-region coverage, to solve the problem of poor extraction ability of steganographic images under attack or interference. First of all, the cover image is transformed by multi-wavelet transform, and the hidden region covering multiple wavelet sub-bands is selected in the wavelet domain of the cover image to embed the secret information. After determining the hidden region, the hidden region is processed by Arnold transform, Hessenberg decomposition, and singular-value decomposition. Finally, the secret information is embedded into the cover image by embedding intensity factor. In order to ensure robustness, the hidden region selected in the wavelet domain is used as the input of Hessenberg matrix decomposition, and the robustness of the algorithm is further enhanced by Hessenberg matrix decomposition and singular-value decomposition. Experimental results show that the proposed method has excellent performance in concealment and quality of extracted secret images, and secret information is extracted from steganographic images attacked by various image processing attacks, which proves that the proposed method has good anti-attack ability under different attacks. Full article
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# 2021

14 pages, 4533 KiB
Article
Modeling Predictability of Traffic Counts at Signalised Intersections Using Hurst Exponent
Entropy 2021, 23(2), 188; https://doi.org/10.3390/e23020188 - 03 Feb 2021
Cited by 3 | Viewed by 1827
Abstract
Predictability is important in decision-making in many fields, including transport. The ill-predictability of time-varying processes poses severe problems for traffic and transport planners. The sources of ill-predictability in traffic phenomena could be due to uncertainty and incompleteness of data and models and/or due [...] Read more.
Predictability is important in decision-making in many fields, including transport. The ill-predictability of time-varying processes poses severe problems for traffic and transport planners. The sources of ill-predictability in traffic phenomena could be due to uncertainty and incompleteness of data and models and/or due to the complexity of the processes itself. Traffic counts at intersections are typically consistent and repetitive on the one hand and yet can be less predictable on the other hand, in which on any given time, unusual circumstances such as crashes and adverse weather can dramatically change the traffic condition. Understanding the various causes of high/low predictability in traffic counts is essential for better predictions and the choice of prediction methods. Here, we utilise the Hurst exponent metric from the fractal theory to quantify fluctuations and evaluate the predictability of intersection approach volumes. Data collected from 37 intersections in Sydney, Australia for one year are used. Further, we develop a random-effects linear regression model to quantify the effect of factors such as the day of the week, special event days, public holidays, rainfall, temperature, bus stops, and parking lanes on the predictability of traffic counts. We find that the theoretical predictability of traffic counts at signalised intersections is upwards of 0.80 (i.e., 80%) for most of the days, and the predictability is strongly associated with the day of the week. Public holidays, special event days, and weekends are better predictable than typical weekdays. Rainfall decreases predictability, and intersections with more parking spaces are highly predictable. Full article
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11 pages, 1442 KiB
Article
Complexity Measures of Heart-Rate Variability in Amyotrophic Lateral Sclerosis with Alternative Pulmonary Capacities
Entropy 2021, 23(2), 159; https://doi.org/10.3390/e23020159 - 28 Jan 2021
Cited by 2 | Viewed by 2087
Abstract
Objective: the complexity of heart-rate variability (HRV) in amyotrophic lateral sclerosis (ALS) patients with different pulmonary capacities was evaluated. Methods: We set these according to their pulmonary capacity, and specifically forced vital capacity (FVC). We split the groups according to FVC (FVC > [...] Read more.
Objective: the complexity of heart-rate variability (HRV) in amyotrophic lateral sclerosis (ALS) patients with different pulmonary capacities was evaluated. Methods: We set these according to their pulmonary capacity, and specifically forced vital capacity (FVC). We split the groups according to FVC (FVC > 50% (n = 29) and FVC < 50% (n = 28)). In ALS, the presence of an FVC below 50% is indicative of noninvasive ventilation with two pressure levels and with the absence of other respiratory symptoms. As the number of subjects per group was different, we applied the unbalanced one-way analysis of variance (uANOVA1) test after three tests of normality, and effect size by Cohen’s d to assess parameter significance. Results: with regard to chaotic global analysis, CFP4 (p < 0.001; d = 0.91), CFP5 (p = 0.0022; d = 0.85), and CFP6 (p = 0.0009; d = 0.92) were enlarged. All entropies significantly increased. Shannon (p = 0.0005; d = 0.98), Renyi (p = 0.0002; d = 1.02), Tsallis (p = 0.0004; d = 0.99), approximate (p = 0.0005; d = 0.97), and sample (p < 0.0001; d = 1.22). Detrended fluctuation analysis (DFA) (p = 0.0358) and Higuchi fractal dimension (HFD) (p = 0.15) were statistically inconsequential between the two groups. Conclusions: HRV complexity in ALS subjects with different pulmonary capacities increased via chaotic global analysis, especially CFP5 and 3 out of 5 entropies. Full article
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# 2020

11 pages, 765 KiB
Article
Wavelet-Based Entropy Measures to Characterize Two-Dimensional Fractional Brownian Fields
Entropy 2020, 22(2), 196; https://doi.org/10.3390/e22020196 - 07 Feb 2020
Cited by 10 | Viewed by 2643
Abstract
The aim of this work was to extend the results of Perez et al. (Physica A (2006), 365 (2), 282–288) to the two-dimensional (2D) fractional Brownian field. In particular, we defined Shannon entropy using the wavelet spectrum from which the Hurst exponent is [...] Read more.
The aim of this work was to extend the results of Perez et al. (Physica A (2006), 365 (2), 282–288) to the two-dimensional (2D) fractional Brownian field. In particular, we defined Shannon entropy using the wavelet spectrum from which the Hurst exponent is estimated by the regression of the logarithm of the square coefficients over the levels of resolutions. Using the same methodology. we also defined two other entropies in 2D: Tsallis and the Rényi entropies. A simulation study was performed for showing the ability of the method to characterize 2D (in this case, $α = 2$ ) self-similar processes. Full article
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# 2019

13 pages, 517 KiB
Article
Biorthogonal-Wavelet-Based Method for Numerical Solution of Volterra Integral Equations
Entropy 2019, 21(11), 1098; https://doi.org/10.3390/e21111098 - 10 Nov 2019
Cited by 14 | Viewed by 2041
Abstract
Framelets theory has been well studied in many applications in image processing, data recovery and computational analysis due to the key properties of framelets such as sparse representation and accuracy in coefficients recovery in the area of numerical and computational theory. This work [...] Read more.
Framelets theory has been well studied in many applications in image processing, data recovery and computational analysis due to the key properties of framelets such as sparse representation and accuracy in coefficients recovery in the area of numerical and computational theory. This work is devoted to shedding some light on the benefits of using such framelets in the area of numerical computations of integral equations. We introduce a new numerical method for solving Volterra integral equations. It is based on pseudo-spline quasi-affine tight framelet systems generated via the oblique extension principles. The resulting system is converted into matrix equations via these generators. We present examples of the generated pseudo-splines quasi-affine tight framelet systems. Some numerical results to validate the proposed method are presented to illustrate the efficiency and accuracy of the method. Full article
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15 pages, 1915 KiB
Article
Classification of Heart Sounds Based on the Wavelet Fractal and Twin Support Vector Machine
by and
Entropy 2019, 21(5), 472; https://doi.org/10.3390/e21050472 - 06 May 2019
Cited by 42 | Viewed by 4466
Abstract
Heart is an important organ of human beings. As more and more heart diseases are caused by people’s living pressure or habits, the diagnosis and treatment of heart diseases also require technical improvement. In order to assist the heart diseases diagnosis, the heart [...] Read more.
Heart is an important organ of human beings. As more and more heart diseases are caused by people’s living pressure or habits, the diagnosis and treatment of heart diseases also require technical improvement. In order to assist the heart diseases diagnosis, the heart sound signal is used to carry a large amount of cardiac state information, so that the heart sound signal processing can achieve the purpose of heart diseases diagnosis and treatment. In order to quickly and accurately judge the heart sound signal, the classification method based on Wavelet Fractal and twin support vector machine (TWSVM) is proposed in this paper. Firstly, the original heart sound signal is decomposed by wavelet transform, and the wavelet decomposition coefficients of the signal are extracted. Then the two-norm eigenvectors of the heart sound signal are obtained by solving the two-norm values of the decomposition coefficients. In order to express the feature information more abundantly, the energy entropy of the decomposed wavelet coefficients is calculated, and then the energy entropy characteristics of the signal are obtained. In addition, based on the fractal dimension, the complexity of the signal is quantitatively described. The box dimension of the heart sound signal is solved by the binary box dimension method. So its fractal dimension characteristics can be obtained. The above eigenvectors are synthesized as the eigenvectors of the heart sound signal. Finally, the twin support vector machine (TWSVM) is applied to classify the heart sound signals. The proposed algorithm is verified on the PhysioNet/CinC Challenge 2016 heart sound database. The experimental results show that this proposed algorithm based on twin support vector machine (TWSVM) is superior to the algorithm based on support vector machine (SVM) in classification accuracy and speed. The proposed algorithm achieves the best results with classification accuracy 90.4%, sensitivity 94.6%, specificity 85.5% and F1 Score 95.2%. Full article
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18 pages, 753 KiB
Article
The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation
Entropy 2019, 21(5), 453; https://doi.org/10.3390/e21050453 - 30 Apr 2019
Cited by 14 | Viewed by 3653
Abstract
Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of the study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem [...] Read more.
Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of the study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential solutions to it. Using regular fractals as archetypes, we can reveal the causes and effects of the diversity of fractal dimension estimation results by analogy. The main factors influencing fractal dimension values of cities include prefractal structure, multi-scaling fractal patterns, and self-affine fractal growth. The solution to the problem is to substitute the real fractal dimension values with comparable fractal dimensions. The main measures are as follows. First, select a proper method for a special fractal study. Second, define a proper study area for a city according to a study aim, or define comparable study areas for different cities. These suggestions may be helpful for the students who take interest in or have already participated in the studies of fractal cities. Full article
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21 pages, 6015 KiB
Article
Local Complexity Estimation Based Filtering Method in Wavelet Domain for Magnetic Resonance Imaging Denoising
Entropy 2019, 21(4), 401; https://doi.org/10.3390/e21040401 - 16 Apr 2019
Cited by 9 | Viewed by 2986
Abstract
In this paper, we propose the local complexity estimation based filtering method in wavelet domain for MRI (magnetic resonance imaging) denoising. A threshold selection methodology is proposed in which the edge and detail preservation properties for each pixel are determined by the local [...] Read more.
In this paper, we propose the local complexity estimation based filtering method in wavelet domain for MRI (magnetic resonance imaging) denoising. A threshold selection methodology is proposed in which the edge and detail preservation properties for each pixel are determined by the local complexity of the input image. In the proposed filtering method, the current wavelet kernel is compared with a threshold to identify the signal- or noise-dominant pixels in a scale providing a good visual quality avoiding blurred and over smoothened processed images. We present a comparative performance analysis with different wavelets to find the optimal wavelet for MRI denoising. Numerical experiments and visual results in simulated MR images degraded with Rician noise demonstrate that the proposed algorithm consistently outperforms other denoising methods by balancing the tradeoff between noise suppression and fine detail preservation. The proposed algorithm can enhance the contrast between regions allowing the delineation of the regions of interest between different textures or tissues in the processed images. The proposed approach produces a satisfactory result in the case of real MRI denoising by balancing the detail preservation and noise removal, by enhancing the contrast between the regions of the image. Additionally, the proposed algorithm is compared with other approaches in the case of Additive White Gaussian Noise (AWGN) using standard images to demonstrate that the proposed approach does not need to be adapted specifically to Rician or AWGN noise; it is an advantage of the proposed approach in comparison with other methods. Finally, the proposed scheme is simple, efficient and feasible for MRI denoising. Full article
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29 pages, 10071 KiB
Article
Explicit Lump Solitary Wave of Certain Interesting (3+1)-Dimensional Waves in Physics via Some Recent Traveling Wave Methods
Entropy 2019, 21(4), 397; https://doi.org/10.3390/e21040397 - 15 Apr 2019
Cited by 53 | Viewed by 3317
Abstract
This study investigates the solitary wave solutions of the nonlinear fractional Jimbo–Miwa (JM) equation by using the conformable fractional derivative and some other distinct analytical techniques. The JM equation describes the certain interesting (3+1)-dimensional waves in physics. Moreover, it is considered as a [...] Read more.
This study investigates the solitary wave solutions of the nonlinear fractional Jimbo–Miwa (JM) equation by using the conformable fractional derivative and some other distinct analytical techniques. The JM equation describes the certain interesting (3+1)-dimensional waves in physics. Moreover, it is considered as a second equation of the famous Painlev’e hierarchy of integrable systems. The fractional conformable derivatives properties were employed to convert it into an ordinary differential equation with an integer order to obtain many novel exact solutions of this model. The conformable fractional derivative is equivalent to the ordinary derivative for the functions that has continuous derivatives up to some desired order over some domain (smooth functions). The obtained solutions for each technique were characterized and compared to illustrate the similarities and differences between them. Profound solutions were concluded to be powerful, easy and effective on the nonlinear partial differential equation. Full article
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25 pages, 13420 KiB
Article
MVMD-MOMEDA-TEO Model and Its Application in Feature Extraction for Rolling Bearings
Entropy 2019, 21(4), 331; https://doi.org/10.3390/e21040331 - 27 Mar 2019
Cited by 9 | Viewed by 3702
Abstract
In order to extract fault features of rolling bearings to characterize their operation state effectively, an improved method, based on modified variational mode decomposition (MVMD) and multipoint optimal minimum entropy deconvolution adjusted (MOMEDA), is proposed. Firstly, the MVMD method is introduced to decompose [...] Read more.
In order to extract fault features of rolling bearings to characterize their operation state effectively, an improved method, based on modified variational mode decomposition (MVMD) and multipoint optimal minimum entropy deconvolution adjusted (MOMEDA), is proposed. Firstly, the MVMD method is introduced to decompose the vibration signal into intrinsic mode functions (IMFs), and then calculate the energy ratio of each IMF component. The IMF component is selected as the effective component from high energy ratio to low in turn until the total energy proportion Esum(t) ≥ 90%. The IMF effective components are reconstructed to obtain the subsequent analysis signal x_new(t). Secondly, the MOMEDA method is introduced to analyze x_new(t), extract the fault period impulse component x_cov(t), which is submerged by noise, and demodulate the signal x_cov(t) by Teager energy operator demodulation (TEO) to calculate Teager energy spectrum. Thirdly, matching the dominant frequency in the spectrum with the fault characteristic frequency of rolling bearings, the fault feature extraction of rolling bearings are completed. Finally, the experiments have compared MVMD-MOEDA-TEO with MVMD-TEO and MOMEDA-TEO based on two different data sets to verify the superiority of the proposed method. The experimental results show that MVMD-MOMEDA-TEO method has better performance than the other two methods, and provides a new solution for condition monitoring and fault diagnosis of rolling bearings. Full article
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15 pages, 3587 KiB
Article
Multimode Decomposition and Wavelet Threshold Denoising of Mold Level Based on Mutual Information Entropy
Entropy 2019, 21(2), 202; https://doi.org/10.3390/e21020202 - 21 Feb 2019
Cited by 27 | Viewed by 4013
Abstract
The continuous casting process is a continuous, complex phase transition process. The noise components of the continuous casting process are complex, the model is difficult to establish, and it is difficult to separate the noise and clear signals effectively. Owing to these demerits, [...] Read more.
The continuous casting process is a continuous, complex phase transition process. The noise components of the continuous casting process are complex, the model is difficult to establish, and it is difficult to separate the noise and clear signals effectively. Owing to these demerits, a hybrid algorithm combining Variational Mode Decomposition (VMD) and Wavelet Threshold denoising (WTD) is proposed, which involves multiscale resolution and adaptive features. First of all, the original signal is decomposed into several Intrinsic Mode Functions (IMFs) by Empirical Mode Decomposition (EMD), and the model parameter K of the VMD is obtained by analyzing the EMD results. Then, the original signal is decomposed by VMD based on the number of IMFs K, and the Mutual Information Entropy (MIE) between IMFs is calculated to identify the noise dominant component and the information dominant component. Next, the noise dominant component is denoised by WTD. Finally, the denoised noise dominant component and all information dominant components are reconstructed to obtain the denoised signal. In this paper, a comprehensive comparative analysis of EMD, Ensemble Empirical Mode Decomposition (EEMD), Complementary Empirical Mode Decomposition (CEEMD), EMD-WTD, Empirical Wavelet Transform (EWT), WTD, VMD, and VMD-WTD is carried out, and the denoising performance of the various methods is evaluated from four perspectives. The experimental results show that the hybrid algorithm proposed in this paper has a better denoising effect than traditional methods and can effectively separate noise and clear signals. The proposed denoising algorithm is shown to be able to effectively recognize different cast speeds. Full article
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17 pages, 478 KiB
Article
Combining Multi-Scale Wavelet Entropy and Kernelized Classification for Bearing Multi-Fault Diagnosis
Entropy 2019, 21(2), 152; https://doi.org/10.3390/e21020152 - 05 Feb 2019
Cited by 22 | Viewed by 3828
Abstract
Discriminative feature extraction and rolling element bearing failure diagnostics are very important to ensure the reliability of rotating machines. Therefore, in this paper, we propose multi-scale wavelet Shannon entropy as a discriminative fault feature to improve the diagnosis accuracy of bearing fault under [...] Read more.
Discriminative feature extraction and rolling element bearing failure diagnostics are very important to ensure the reliability of rotating machines. Therefore, in this paper, we propose multi-scale wavelet Shannon entropy as a discriminative fault feature to improve the diagnosis accuracy of bearing fault under variable work conditions. To compute the multi-scale wavelet entropy, we consider integrating stationary wavelet packet transform with both dispersion (SWPDE) and permutation (SWPPE) entropies. The multi-scale entropy features extracted by our proposed methods are then passed on to the kernel extreme learning machine (KELM) classifier to diagnose bearing failure types with different severities. In the end, both the SWPDE–KELM and the SWPPE–KELM methods are evaluated on two bearing vibration signal databases. We compare these two feature extraction methods to a recently proposed method called stationary wavelet packet singular value entropy (SWPSVE). Based on our results, we can say that the diagnosis accuracy obtained by the SWPDE–KELM method is slightly better than the SWPPE–KELM method and they both significantly outperform the SWPSVE–KELM method. Full article
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15 pages, 10672 KiB
Article
Parameter Identification of Fractional-Order Discrete Chaotic Systems
Entropy 2019, 21(1), 27; https://doi.org/10.3390/e21010027 - 01 Jan 2019
Cited by 41 | Viewed by 3658
Abstract
Research on fractional-order discrete chaotic systems has grown in recent years, and chaos synchronization of such systems is a new topic. To address the deficiencies of the extant chaos synchronization methods for fractional-order discrete chaotic systems, we proposed an improved particle swarm optimization [...] Read more.
Research on fractional-order discrete chaotic systems has grown in recent years, and chaos synchronization of such systems is a new topic. To address the deficiencies of the extant chaos synchronization methods for fractional-order discrete chaotic systems, we proposed an improved particle swarm optimization algorithm for the parameter identification. Numerical simulations are carried out for the Hénon map, the Cat map, and their fractional-order form, as well as the fractional-order standard iterated map with hidden attractors. The problem of choosing the most appropriate sample size is discussed, and the parameter identification with noise interference is also considered. The experimental results demonstrate that the proposed algorithm has the best performance among the six existing algorithms and that it is effective even with random noise interference. In addition, using two samples offers the most efficient performance for the fractional-order discrete chaotic system, while the integer-order discrete chaotic system only needs one sample. Full article
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# 2018

12 pages, 3115 KiB
Article
Dynamics Analysis of a New Fractional-Order Hopfield Neural Network with Delay and Its Generalized Projective Synchronization
Entropy 2019, 21(1), 1; https://doi.org/10.3390/e21010001 - 20 Dec 2018
Cited by 16 | Viewed by 3004
Abstract
In this paper, a new three-dimensional fractional-order Hopfield-type neural network with delay is proposed. The system has a unique equilibrium point at the origin, which is a saddle point with index two, hence unstable. Intermittent chaos is found in this system. The complex [...] Read more.
In this paper, a new three-dimensional fractional-order Hopfield-type neural network with delay is proposed. The system has a unique equilibrium point at the origin, which is a saddle point with index two, hence unstable. Intermittent chaos is found in this system. The complex dynamics are analyzed both theoretically and numerically, including intermittent chaos, periodicity, and stability. Those phenomena are confirmed by phase portraits, bifurcation diagrams, and the Largest Lyapunov exponent. Furthermore, a synchronization method based on the state observer is proposed to synchronize a class of time-delayed fractional-order Hopfield-type neural networks. Full article
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