Entropy

263 KiB

Open AccessArticle

A Large Deviation Principle and an Expression of the Rate Function for a Discrete Stationary Gaussian Process

by Olivier Faugeras and James MacLaurin

Entropy 2014, 16(12), 6722-6738; https://doi.org/10.3390/e16126722 - 22 Dec 2014

Viewed by 4731

We prove a large deviation principle for a stationary Gaussian process over R^b,indexed by Ζ^d (for some positive integers d and b), with positive definite spectral density, andprovide an expression of the corresponding rate function in terms of the [...] Read more.

We prove a large deviation principle for a stationary Gaussian process over R^b,indexed by Ζ^d (for some positive integers d and b), with positive definite spectral density, andprovide an expression of the corresponding rate function in terms of the mean of the processand its spectral density. This result is useful in applications where such an expression isneeded. Full article

(This article belongs to the Section Statistical Physics)

304 KiB

Open AccessArticle

A Representation of the Relative Entropy with Respect to a Diffusion Process in Terms of Its Infinitesimal Generator

by Oliver Faugeras and James MacLaurin

Entropy 2014, 16(12), 6705-6721; https://doi.org/10.3390/e16126705 - 22 Dec 2014

Cited by 1 | Viewed by 5044

Abstract

In this paper we derive an integral (with respect to time) representation of the relative entropy (or Kullback–Leibler Divergence) R(μ||P), where μ and P are measures on C([0,T];R^d). The underlying measure P is a weak solution to a martingale problem with [...] Read more.

In this paper we derive an integral (with respect to time) representation of the relative entropy (or Kullback–Leibler Divergence) R(μ||P), where μ and P are measures on C([0,T];R^d). The underlying measure P is a weak solution to a martingale problem with continuous coefficients. Our representation is in the form of an integral with respect to its infinitesimal generator. This representation is of use in statistical inference (particularly involving medical imaging). Since R(μ||P) governs the exponential rate of convergence of the empirical measure (according to Sanov’s theorem), this representation is also of use in the numerical and analytical investigation of finite-size effects in systems of interacting diffusions. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

1699 KiB

Open AccessArticle

Effect of the Postural Challenge on the Dependence of the Cardiovascular Control Complexity on Age

by Aparecida M. Catai, Anielle C.M. Takahashi, Natália M. Perseguini, Juliana C. Milan, Vinicius Minatel, Patrícia Rehder-Santos, Andrea Marchi, Vlasta Bari and Alberto Porta

Entropy 2014, 16(12), 6686-6704; https://doi.org/10.3390/e16126686 - 22 Dec 2014

Cited by 44 | Viewed by 6477

Abstract

Short-term complexity of heart period (HP) and systolic arterial pressure (SAP) was computed to detect age and gender influences over cardiovascular control in resting supine condition (REST) and during standing (STAND). Healthy subjects (n = 110, men = 55) were equally divided [...] Read more.

Short-term complexity of heart period (HP) and systolic arterial pressure (SAP) was computed to detect age and gender influences over cardiovascular control in resting supine condition (REST) and during standing (STAND). Healthy subjects (n = 110, men = 55) were equally divided into five groups (21–30; 31–40; 41–50; 51–60; and 61–70 years of age). HP and SAP series were recorded for 15 min at REST and during STAND. A normalized complexity index (NCI) based on conditional entropy was assessed. At REST we found that both NCI_HP and NCI_SAPdecreased with age in the overall population, but only women were responsible for this trend. During STAND we observed that both NCI_HP and NCI_SAP were unrelated to age in the overall population, even when divided by gender. When the variation of NCI in response to STAND (ΔNCI = NCI at REST-NCI during STAND) was computed individually, we found that ΔNCI_HP progressively decreased with age in the overall population, and women were again responsible for this trend. Conversely, ΔNCI_SAP was unrelated to age and gender. This study stresses that the complexity of cardiovascular control and its ability to respond to stressors are more importantly lost with age in women than in men. Full article

(This article belongs to the Special Issue Entropy and Cardiac Physics)

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Graphical abstract

830 KiB

Open AccessArticle

Fast Rate Estimation for RDO Mode Decision in HEVC

by Maxim P. Sharabayko and Oleg G. Ponomarev

Entropy 2014, 16(12), 6667-6685; https://doi.org/10.3390/e16126667 - 19 Dec 2014

Cited by 12 | Viewed by 8139

Abstract

The latter-day H.265/HEVC video compression standard is able to provide two-times higher compression efficiency compared to the current industrial standard, H.264/AVC. However, coding complexity also increased. The main bottleneck of the compression process is the rate-distortion optimization (RDO) stage, as it involves numerous [...] Read more.

The latter-day H.265/HEVC video compression standard is able to provide two-times higher compression efficiency compared to the current industrial standard, H.264/AVC. However, coding complexity also increased. The main bottleneck of the compression process is the rate-distortion optimization (RDO) stage, as it involves numerous sequential syntax-based binary arithmetic coding (SBAC) loops. In this paper, we present an entropy-based RDO estimation technique for H.265/HEVC compression, instead of the common approach based on the SBAC. Our RDO implementation reduces RDO complexity, providing an average bit rate overhead of 1.54%. At the same time, elimination of the SBAC from the RDO estimation reduces block interdependencies, thus providing an opportunity for the development of the compression system with parallel processing of multiple blocks of a video frame. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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312 KiB

Open AccessArticle

The Effects of Spontaneous Random Activity on Information Transmission in an Auditory Brain Stem Neuron Model

by Hiroyuki Mino

Entropy 2014, 16(12), 6654-6666; https://doi.org/10.3390/e16126654 - 19 Dec 2014

Cited by 8 | Viewed by 6028

Abstract

This paper presents the effects of spontaneous random activity on information transmission in an auditory brain stem neuron model. In computer simulations, the supra-threshold synaptic current stimuli ascending from auditory nerve fibers (ANFs) were modeled by a filtered inhomogeneous Poisson process modulated by [...] Read more.

This paper presents the effects of spontaneous random activity on information transmission in an auditory brain stem neuron model. In computer simulations, the supra-threshold synaptic current stimuli ascending from auditory nerve fibers (ANFs) were modeled by a filtered inhomogeneous Poisson process modulated by sinusoidal functions at a frequency of 220–3520 Hz with regard to the human speech spectrum. The stochastic sodium and stochastic high- and low-threshold potassium channels were incorporated into a single compartment model of the soma in spherical bushy neurons, so as to realize threshold fluctuations or a variation of spike firing times. The results show that the information rates estimated from the entropy of inter-spike intervals of spike trains tend toward a convex function of the spontaneous rates when the intensity of sinusoidal functions decreases. Furthermore, the results show that a convex function of the spontaneous rates tends to disappear as the frequency of the sinusoidal function increases, such that the phase-locked response can be unobserved. It is concluded that this sort of stochastic resonance (SR) phenomenon, which depends on the spontaneous rates with supra-threshold stimuli, can better enhance information transmission in a smaller intensity of sinusoidal functions within the human speech spectrum, like the situation in which the regular SR can enhance weak signals. Full article

(This article belongs to the Special Issue Entropy in Human Brain Networks)

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Electrical equivalent circuit of a spherical bushy neuron model with four kinds of stochastic ion channels. Full article ">
An illustrative example of a membrane potential (top), an excitatory synaptic conductance with an alpha function (middle) and an intensity function of the inhomogeneous Poisson process modulated by a sinusoidal function (bottom) as a function of time. Full article ">
An intensity function (top), raster plots (middle) and the estimated spike firing rates (bottom) as a function of time at λspon = 5 (left), 25 (middle) and 100 (right) s−1 at λc = 200 s−1, f = 220 Hz. The post-stimulus time histograms (PSTHs) were generated with a bin width of 0.1 ms in the bottom traces. Full article ">
Inter-spike interval histogram (ISIH) at λspon = 5 (left), 25 (middle) and 100 (right) s−1 at λc = 200 s−1, f = 220 Hz. The ISIHs were generated with a bin width of 0.5 ms. Full article ">
Information rate as a function of spontaneous spike rate in which λc is set to 100 s−1 (blue), 200 s−1 (cyan), 400 s−1 (magenta) and 800 s−1 (red) at f = 220 Hz. Full article ">
Information rate as a function of spontaneous spike rate in which f is set to 220 Hz (blue), 880 Hz (cyan), 1760 Hz (magenta) and 3520 Hz (red) at λc = 200 s−1. Full article ">
Information rate as a function of spontaneous spike rate in which the number of sodium channels is set to 8312 (blue), 27,708 (cyan), 83,126 (magenta) and 166,253 (red) at f = 220 Hz and λc = 200 s−1. Full article ">

328 KiB

Open AccessArticle

The McMillan Theorem for Colored Branching Processes and Dimensions of Random Fractals

by Victor Bakhtin

Entropy 2014, 16(12), 6624-6653; https://doi.org/10.3390/e16126624 - 19 Dec 2014

Cited by 2 | Viewed by 4810

Abstract

For the simplest colored branching process, we prove an analog to the McMillan theorem and calculate the Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting, the role of [...] Read more.

For the simplest colored branching process, we prove an analog to the McMillan theorem and calculate the Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting, the role of Shannon’s entropy is played by the Kullback–Leibler divergence, and the Hausdorff dimensions are computed by means of the so-called Billingsley–Kullback entropy, defined in the paper. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

1300 KiB

Open AccessArticle

Detection and Modeling of Cyber Attacks with Petri Nets

by Bartosz Jasiul, Marcin Szpyrka and Joanna Śliwa

Entropy 2014, 16(12), 6602-6623; https://doi.org/10.3390/e16126602 - 19 Dec 2014

Cited by 38 | Viewed by 10670

Abstract

The aim of this article is to present an approach to develop and verify a method of formal modeling of cyber threats directed at computer systems. Moreover, the goal is to prove that the method enables one to create models resembling the behavior [...] Read more.

The aim of this article is to present an approach to develop and verify a method of formal modeling of cyber threats directed at computer systems. Moreover, the goal is to prove that the method enables one to create models resembling the behavior of malware that support the detection process of selected cyber attacks and facilitate the application of countermeasures. The most common cyber threats targeting end users and terminals are caused by malicious software, called malware. The malware detection process can be performed either by matching their digital signatures or analyzing their behavioral models. As the obfuscation techniques make the malware almost undetectable, the classic signature-based anti-virus tools must be supported with behavioral analysis. The proposed approach to modeling of malware behavior is based on colored Petri nets. This article is addressed to cyber defense researchers, security architects and developers solving up-to-date problems regarding the detection and prevention of advanced persistent threats. Full article

(This article belongs to the Special Issue Entropy-Based Applied Cryptography and Enhanced Security for Ubiquitous Computing)

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Graphical abstract

Graphical abstract
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Colored Petri net (CP-net) model of PRONTOnet : Page hierarchy graph. Full article ">
CP-net model of PRONTOnet: Primary module. Full article ">
CP-net: Acquisition module. Full article ">
CP-net: verification module. Full article ">
CP-net: Virut module (classifier). Full article ">
CP-net malware modeling tool: First window. Full article ">
Malware editor window of the CP-net malware modeling tool (CPN MM). Full article ">
Editor for malware symptoms. Full article ">
Result of the detection of the Virut malware. Full article ">

1435 KiB

Open AccessArticle

Depth Image Coding Using Entropy-Based Adaptive Measurement Allocation

by Huihui Bai, Mengmeng Zhang, Meiqin Liu, Anhong Wang and Yao Zhao

Entropy 2014, 16(12), 6590-6601; https://doi.org/10.3390/e16126590 - 17 Dec 2014

Cited by 5 | Viewed by 5590

Abstract

Differently from traditional two-dimensional texture images, the depth images of three-dimensional (3D) video systems have significant sparse characteristics under the certain transform basis, which make it possible for compressive sensing to represent depth information efficiently. Therefore, in this paper, a novel depth image [...] Read more.

Differently from traditional two-dimensional texture images, the depth images of three-dimensional (3D) video systems have significant sparse characteristics under the certain transform basis, which make it possible for compressive sensing to represent depth information efficiently. Therefore, in this paper, a novel depth image coding scheme is proposed based on a block compressive sensing method. At the encoder, in view of the characteristics of depth images, the entropy of pixels in each block is employed to represent the sparsity of depth signals. Then according to the different sparsity in the pixel domain, the measurements can be adaptively allocated to each block for higher compression efficiency. At the decoder, the sparse transform can be combined to achieve the compressive sensing reconstruction. Experimental results have shown that at the same sampling rate, the proposed scheme can obtain higher PSNR values and better subjective quality of the rendered virtual views, compared with the method using a uniform sampling rate. Full article

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Basic framework of image compression based on CS. Full article ">
Block diagram of the proposed scheme. Full article ">
Example of the anti-ground noise filter. Full article ">
Flowchart of measurement allocation. Full article ">
A typical example of measurement allocation. Full article ">
A typical example of threshold determination. Full article ">
Objective quality comparison for Balloons, Kendo and Pantomime. (a), (c) and (e): for depth map; (b), (d) and (f): for synthesized virtual viewpoint. Full article ">
Subjective quality comparison of synthesized virtual viewpoint for Balloons and Kendo. (a), (c), (e) and (g): uniform sampling; (b), (d), (f) and (h): proposed scheme. Full article ">

1157 KiB

Open AccessArticle

Automatic Sleep Stages Classification Using EEG Entropy Features and Unsupervised Pattern Analysis Techniques

by Jose Luis Rodríguez-Sotelo, Alejandro Osorio-Forero, Alejandro Jiménez-Rodríguez, David Cuesta-Frau, Eva Cirugeda-Roldán and Diego Peluffo

Entropy 2014, 16(12), 6573-6589; https://doi.org/10.3390/e16126573 - 17 Dec 2014

Cited by 101 | Viewed by 11484

Abstract

Sleep is a growing area of research interest in medicine and neuroscience. Actually, one major concern is to find a correlation between several physiologic variables and sleep stages. There is a scientific agreement on the characteristics of the five stages of human sleep, [...] Read more.

Sleep is a growing area of research interest in medicine and neuroscience. Actually, one major concern is to find a correlation between several physiologic variables and sleep stages. There is a scientific agreement on the characteristics of the five stages of human sleep, based on EEG analysis. Nevertheless, manual stage classification is still the most widely used approach. This work proposes a new automatic sleep classification method based on unsupervised feature classification algorithms recently developed, and on EEG entropy measures. This scheme extracts entropy metrics from EEG records to obtain a feature vector. Then, these features are optimized in terms of relevance using the Q-α algorithm. Finally, the resulting set of features is entered into a clustering procedure to obtain a final segmentation of the sleep stages. The proposed method reached up to an average of 80% correctly classified stages for each patient separately while keeping the computational cost low. Full article

(This article belongs to the Special Issue Entropy and Electroencephalography)

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18016 KiB

Open AccessArticle

Enhanced Automatic Wavelet Independent Component Analysis for Electroencephalographic Artifact Removal

by Nadia Mammone and Francesco C. Morabito

Entropy 2014, 16(12), 6553-6572; https://doi.org/10.3390/e16126553 - 17 Dec 2014

Cited by 54 | Viewed by 12933

Abstract

Electroencephalography (EEG) is a fundamental diagnostic instrument for many neurological disorders, and it is the main tool for the investigation of the cognitive or pathological activity of the brain through the bioelectromagnetic fields that it generates. The correct interpretation of the EEG is [...] Read more.

Electroencephalography (EEG) is a fundamental diagnostic instrument for many neurological disorders, and it is the main tool for the investigation of the cognitive or pathological activity of the brain through the bioelectromagnetic fields that it generates. The correct interpretation of the EEG is misleading, both for clinicians’ visual evaluation and for automated procedures, because of artifacts. As a consequence, artifact rejection in EEG is a key preprocessing step, and the quest for reliable automatic processors has been quickly growing in the last few years. Recently, a promising automatic methodology, known as automatic wavelet-independent component analysis (AWICA), has been proposed. In this paper, a more efficient and sensitive version, called enhanced-AWICA (EAWICA), is proposed, and an extensive performance comparison is carried out by a step of tuning the different parameters that are involved in artifact detection. EAWICA is shown to minimize information loss and to outperform AWICA in artifact removal, both on simulated and real experimental EEG recordings. Full article

(This article belongs to the Special Issue Entropy and Electroencephalography)

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306 KiB

Open AccessArticle

Chaos Control and Synchronization of a Hyperchaotic Zhou System by Integral Sliding Mode control

by Yashar Toopchi and Jidong Wang

Entropy 2014, 16(12), 6539-6552; https://doi.org/10.3390/e16126539 - 12 Dec 2014

Cited by 25 | Viewed by 6772

Abstract

In this paper, an adaptive integral sliding mode control scheme is proposed for synchronization of hyperchaotic Zhou systems. In the proposed scheme, an integral sliding mode control is designed to stabilize a hyperchaotic Zhou system with known parameters to its unstable equilibrium at [...] Read more.

In this paper, an adaptive integral sliding mode control scheme is proposed for synchronization of hyperchaotic Zhou systems. In the proposed scheme, an integral sliding mode control is designed to stabilize a hyperchaotic Zhou system with known parameters to its unstable equilibrium at the origin. The control is then applied to the synchronization of two identical systems, i.e., a slave and a master hyperchaotic Zhou system with unknown parameters. The adaptive control mechanism introduced synchronizes the systems by estimating the unknown parameters. Simulation results have shown that the proposed method has an excellent convergence from both speed and accuracy points of view, and it outperforms Vaidyanathan’s scheme, which is a well-recognized scheme in this area. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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(a) Phase portrait of the hyperchaotic Zhou system; (b) time-history of states x1, x2, x3 and x4. Full article ">
Time responses of the controlled hyperchaotic Zhou system. Full article ">
Trajectories of the synchronization states x1 (a), x2 (b), x3 (c) and x4 (d) (the controller is off for t < 2 and is on for t ≥ 2). Full article ">
Trajectories of the synchronization errors of e1, e2, e3 and e4 (the controller is off for t < 2 and is on for t ≥ 2). Full article ">
Comparison between simulations results of the proposed method and the previously known method: x1 (a), x2 (b), x3 (c) and x4 (d). Full article ">
Comparison between the simulation errors of the proposed method and the previously known method e1, e2, e3 and e4. Full article ">
Comparison results of parameter convergence between two schemes: time history of parameters a, b, c and d in the proposed scheme (a) and Vaidyanathan’s scheme (b). Full article ">

209 KiB

Open AccessArticle

Consensus of Discrete Multiagent System with Various Time Delays and Environmental Disturbances

by Zheping Yan, Di Wu and Yibo Liu

Entropy 2014, 16(12), 6524-6538; https://doi.org/10.3390/e16126524 - 11 Dec 2014

Cited by 6 | Viewed by 5414

Abstract

In this paper, the consensus problem of discrete multiagent systems with time varying sampling periods is studied. Firstly, with thorough analysis of various delays among agents, the control input of each agent is designed with consideration of sending delay and receiving delay. With [...] Read more.

In this paper, the consensus problem of discrete multiagent systems with time varying sampling periods is studied. Firstly, with thorough analysis of various delays among agents, the control input of each agent is designed with consideration of sending delay and receiving delay. With construction of discrete dynamics of state error vector, it is proved by applying Halanay inequality that consensus of the system can be reached. Further, the definition of bounded consensus is proposed in the situation where environmental disturbances exist. In order to handle this problem, the Halanay inequality is extended into a more general one with boundedness property. Based on the new Halanay inequality obtained, the boundedness of consensus error is guaranteed. At last, simulation examples are presented to demonstrate the theoretical conclusions. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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199 KiB

Open AccessArticle

Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature

by Miguel Ángel García-Ariza, Merced Montesinos and Gerardo F. Torres del Castillo

Entropy 2014, 16(12), 6515-6523; https://doi.org/10.3390/e16126515 - 11 Dec 2014

Cited by 16 | Viewed by 6851

Abstract

In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying [...] Read more.

In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner’s metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold’s approach. The corresponding Ruppeiner’s metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures. Full article

(This article belongs to the Special Issue Entropy and Spacetime)

224 KiB

Open AccessArticle

Statistical Power Law due to Reservoir Fluctuations and the Universal Thermostat Independence Principle

by Tamás Sándor Biró, Péter Ván, Gergely Gábor Barnaföldi and Károly Ürmössy

Entropy 2014, 16(12), 6497-6514; https://doi.org/10.3390/e16126497 - 9 Dec 2014

Cited by 36 | Viewed by 6696

Abstract

Certain fluctuations in particle number, \(n\), at fixed total energy, \(E\), lead exactly to a cut-power law distribution in the one-particle energy, \(\omega\), via the induced fluctuations in the phase-space volume ratio, \(\Omega_n(E-\omega)/\Omega_n(E)=(1-\omega/E)^n\). The only parameters are \(1/T=\langle \beta \rangle=\langle n \rangle/E\) and [...] Read more.

Certain fluctuations in particle number, \(n\), at fixed total energy, \(E\), lead exactly to a cut-power law distribution in the one-particle energy, \(\omega\), via the induced fluctuations in the phase-space volume ratio, \(\Omega_n(E-\omega)/\Omega_n(E)=(1-\omega/E)^n\). The only parameters are \(1/T=\langle \beta \rangle=\langle n \rangle/E\) and \(q=1-1/\langle n \rangle + \Delta n^2/\langle n \rangle^2\). For the binomial distribution of \(n\) one obtains \(q=1-1/k\), for the negative binomial \(q=1+1/(k+1)\). These results also represent an approximation for general particle number distributions in the reservoir up to second order in the canonical expansion \(\omega \ll E\). For general systems the average phase-space volume ratio \(\langle e^{S(E-\omega)}/e^{S(E)}\rangle\) to second order delivers \(q=1-1/C+\Delta \beta^2/\langle \beta \rangle^2\) with \(\beta=S^{\prime}(E)\) and \(C=dE/dT\) heat capacity. However, \(q \ne 1\) leads to non-additivity of the Boltzmann–Gibbs entropy, \(S\). We demonstrate that a deformed entropy, \(K(S)\), can be constructed and used for demanding additivity, i.e., \(q_K=1\). This requirement leads to a second order differential equation for \(K(S)\). Finally, the generalized \(q\)-entropy formula, \(K(S)=\sum p_i K(-\ln p_i)\), contains the Tsallis, Rényi and Boltzmann–Gibbs–Shannon expressions as particular cases. For diverging variance, \(\Delta\beta^2\) we obtain a novel entropy formula. Full article

(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)

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1551 KiB

Open AccessArticle

An Evolutionary Algorithm for the Texture Analysis of Cubic System Materials Derived by the Maximum Entropy Principle

by Dapeng Wang, Dazhi Wang, Baolin Wu, Fu Wang and Zhide Liang

Entropy 2014, 16(12), 6477-6496; https://doi.org/10.3390/e16126477 - 9 Dec 2014

Cited by 1 | Viewed by 5566

Abstract

Based on the principle of maximum entropy method (MEM) for quantitative texture analysis, the differential evolution (DE) algorithm was effectively introduced. Using a DE-optimized algorithm with a faster but more stable convergence rate of iteration reliable complete orientation distributions (C-ODF) have [...] Read more.

Based on the principle of maximum entropy method (MEM) for quantitative texture analysis, the differential evolution (DE) algorithm was effectively introduced. Using a DE-optimized algorithm with a faster but more stable convergence rate of iteration reliable complete orientation distributions (C-ODF) have been obtained for deep-drawn IF steel sheets and the recrystallized aluminum foils after cold-rolling, which are designated as showing a macroscopic cubic-orthogonal symmetry. With special reference to the data processing, no more other assumptions are required for DE-optimized MEM except that the system entropy approach the maximum. Full article

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Constant-φ section of ODF representation for a Aluminum foil in comparison derived by different algorithms. (a) R-ODF derived by Two-step Method; (b) C-ODF derived by MEM using the damped Newton algorithm, G = 4000 iterations; (c) C-ODF derived by MEM using DE algorithm, G = 600. Full article ">
Constant-φ section of ODF representation for a Aluminum foil in comparison derived by different algorithms. (a) R-ODF derived by Two-step Method; (b) C-ODF derived by MEM using the damped Newton algorithm, G = 4000 iterations; (c) C-ODF derived by MEM using DE algorithm, G = 600. Full article ">
Constant-φ section of ODF representation for IF steel sheet in comparison derived by different algorithms. (a) R-ODF derived by Two-step Method; (b) C-ODF derived by MEM using the damped Newton algorithm, Gmax = 5000 iterations; (c) C-ODF derived by MEM using DE algorithm, Gmax = 500. Full article ">
Constant-φ section of ODF representation for IF steel sheet in comparison derived by different algorithms. (a) R-ODF derived by Two-step Method; (b) C-ODF derived by MEM using the damped Newton algorithm, Gmax = 5000 iterations; (c) C-ODF derived by MEM using DE algorithm, Gmax = 500. Full article ">
The rectangular coordinate system of a specimen in combination with a crystalline one. Full article ">
The division of 72 equivalent orientation units. Full article ">
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1505 KiB

Open AccessArticle

A Memristive Diode Bridge-Based Canonical Chua’s Circuit

by Mo Chen, Jingjing Yu, Qing Yu, Changdi Li and Bocheng Bao

Entropy 2014, 16(12), 6464-6476; https://doi.org/10.3390/e16126464 - 8 Dec 2014

Cited by 50 | Viewed by 7988

Abstract

A novel memristor circuit is presented, which is generated from the canonical Chua’s circuit by replacing the Chua’s diode with a first order memristive diode bridge. The circuit dynamical characteristics with the variations of circuit parameters are investigated both theoretically and numerically. It [...] Read more.

A novel memristor circuit is presented, which is generated from the canonical Chua’s circuit by replacing the Chua’s diode with a first order memristive diode bridge. The circuit dynamical characteristics with the variations of circuit parameters are investigated both theoretically and numerically. It can be found that the circuit has three determined equilibrium points, including a zero saddle point and two nonzero saddle-foci with index 2. Specially, the circuit is non-dissipative in the neighborhood of the zero saddle point, and there exists complex nonlinear phenomena of coexisting bifurcation modes and coexisting chaotic attractors. Experimental observations are performed to verify the numerical simulation results. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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Generalized memristor realized by memristive diode bridge with parallel RC filter [<a href="#b16-entropy-16-06464" class="html-bibr">16</a>]. Full article ">
Generalized memristor-based canonical Chua’s circuit. Full article ">
Dynamics of generalized memristor based canonical Chua’s circuit. (a) Phase portrait in v1(t)-v2(t) plane; (b) Phase portrait in v1(t)-i(t) plane and (c) Poincaré mapping in v1(t)-v2(t) plane. Full article ">
Two function curves and their intersection points. Full article ">
Dynamics with the value of L increasing. (a) Bifurcation diagram of v2(t) and (b) Lyapunov exponent spectra. Full article ">
Chaotic and periodic orbits in v1(t)-v2(t) plane. (a) Coexisting limit cycles with period-1 (L = 55 mH); (b) coexisting limit cycles with period-2 (L = 85 mH); (c) coexisting spiral chaotic attractors (L = 160 mH) and (d) double-scroll chaotic attractor (L = 220 mH). Full article ">
Time-domain waveforms of v2(t). (a) L = 85 mH and (b) L = 185 mH. Full article ">
Experimental results of phase portraits in v1(t)-v2(t) plane. (a) Coexisting limit cycles with period-1 (L = 55 mH); (b) coexisting limit cycles with period-2 (L = 85 mH); (c) coexisting spiral chaotic attractors (L = 160 mH) and (d) double-scroll chaotic attractor (L = 220 mH). Full article ">

966 KiB

Open AccessArticle

An Entropy-Based Damage Characterization

by Mehdi Amiri and Mohammad Modarres

Entropy 2014, 16(12), 6434-6463; https://doi.org/10.3390/e16126434 - 5 Dec 2014

Cited by 60 | Viewed by 8070

Abstract

This paper presents a scientific basis for the description of the causes of damage within an irreversible thermodynamic framework and the effects of damage as observable variables that signify degradation of structural integrity. The approach relies on the fundamentals of irreversible thermodynamics and [...] Read more.

This paper presents a scientific basis for the description of the causes of damage within an irreversible thermodynamic framework and the effects of damage as observable variables that signify degradation of structural integrity. The approach relies on the fundamentals of irreversible thermodynamics and specifically the notion of entropy generation as a measure of degradation and damage. We first review the state-of-the-art advances in entropic treatment of damage followed by a discussion on generalization of the entropic concept to damage characterization that may offers a better definition of damage metric commonly used for structural integrity assessment. In general, this approach provides the opportunity to described reliability and risk of structures in terms of fundamental science concepts. Over the years, many studies have focused on materials damage assessment by determining physics-based cause and affect relationships, the goal of this paper is to put this work in perspective and encourage future work of materials damage based on the entropy concept. Full article

(This article belongs to the Section Thermodynamics)

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1714 KiB

Open AccessArticle

Detecting Chaos from Agricultural Product Price Time Series

by Xin Su, Yi Wang, Shengsen Duan and Junhai Ma

Entropy 2014, 16(12), 6415-6433; https://doi.org/10.3390/e16126415 - 5 Dec 2014

Cited by 26 | Viewed by 6946

Abstract

Analysis of the characteristics of agricultural product price volatility and trend forecasting are necessary to formulate and implement agricultural price control policies. Taking wholesale cabbage prices as an example, a multiple test methodology has been adopted to identify the nonlinearity, fractality, and chaos [...] Read more.

Analysis of the characteristics of agricultural product price volatility and trend forecasting are necessary to formulate and implement agricultural price control policies. Taking wholesale cabbage prices as an example, a multiple test methodology has been adopted to identify the nonlinearity, fractality, and chaos of the data. The approaches used include the R/S analysis, the BDS test, the power spectra, the recurrence plot, the largest Lyapunov exponent, the Kolmogorov entropy, and the correlation dimension. The results show that there is chaos in agricultural wholesale price data, which provides a good theoretical basis for selecting reasonable forecasting models as prediction techniques based on chaos theory can be applied to forecasting agricultural prices. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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The logn~log(R/S)n curves of XNl and XLLD. The red curve is for XNl and the blue curve is for XLLD. Full article ">
Power spectrum for XNl and XLLD. The red curve is for XNl. The blue curve is for XLLD. The x axis represents the values of frequency, and the y axis represents the values of spectrum. Full article ">
The m~E1 and E2 plot of XNl. Full article ">
The m~E1 and E2 plot of XLLD. Full article ">
The RP plot of XNl. Full article ">
The RP plot of XLLD. Full article ">
%DET curves of XNl and XLLD. Full article ">
The XNl plot of lnr~lnC(m, N, r, τ). Full article ">
The XNl plots of lnr~lnC′(r) and lnr~lnC″(r)). Full article ">

870 KiB

Open AccessArticle

Determining Common Weights in Data Envelopment Analysis with Shannon’s Entropy

by Xiao-Guang Qi and Bo Guo

Entropy 2014, 16(12), 6394-6414; https://doi.org/10.3390/e16126394 - 4 Dec 2014

Cited by 27 | Viewed by 7708

Abstract

Data Envelopment Analysis (DEA) is a non-parametric method for evaluating the efficiency of Decision Making Units (DMUs) with multiple inputs and outputs. In the traditional DEA models, the DMU is allowed to use its most favorable multiplier weights to maximize its efficiency. There [...] Read more.

Data Envelopment Analysis (DEA) is a non-parametric method for evaluating the efficiency of Decision Making Units (DMUs) with multiple inputs and outputs. In the traditional DEA models, the DMU is allowed to use its most favorable multiplier weights to maximize its efficiency. There is usually more than one efficient DMU which cannot be further discriminated. Evaluating DMUs with different multiplier weights would also be somewhat irrational in practice. The common weights DEA model is an effective method for solving these problems. In this paper, we propose a methodology combining the common weights DEA with Shannon’s entropy. In our methodology, we propose a modified weight restricted DEA model for calculating non-zero optimal weights. Then these non-zero optimal weights would be aggregated to be the common weights using Shannon’s entropy. Compared with the traditional models, our proposed method is more powerful in discriminating DMUs, especially when the inputs and outputs are numerous. Our proposed method also keeps in accordance with the basic DEA method considering the evaluation of the most efficient and inefficient DMUs. Numerical examples are provided to examine the validity and effectiveness of our proposed methodology. Full article

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1055 KiB

Open AccessArticle

Entropy Analysis of RR and QT Interval Variability during Orthostatic and Mental Stress in Healthy Subjects

by Mathias Baumert, Barbora Czippelova, Anand Ganesan, Martin Schmidt, Sebastian Zaunseder and Michal Javorka

Entropy 2014, 16(12), 6384-6393; https://doi.org/10.3390/e16126384 - 3 Dec 2014

Cited by 21 | Viewed by 7454

Abstract

Autonomic activity affects beat-to-beat variability of heart rate and QT interval. The aim of this study was to explore whether entropy measures are suitable to detect changes in neural outflow to the heart elicited by two different stress paradigms. We recorded short-term ECG [...] Read more.

Autonomic activity affects beat-to-beat variability of heart rate and QT interval. The aim of this study was to explore whether entropy measures are suitable to detect changes in neural outflow to the heart elicited by two different stress paradigms. We recorded short-term ECG in 11 normal subjects during an experimental protocol that involved head-up tilt and mental arithmetic stress and computed sample entropy, cross-sample entropy and causal interactions based on conditional entropy from RR and QT interval time series. Head-up tilt resulted in a significant reduction in sample entropy of RR intervals and cross-sample entropy, while mental arithmetic stress resulted in a significant reduction in coupling directed from RR to QT. In conclusion, measures of entropy are suitable to detect changes in neural outflow to the heart and decoupling of repolarisation variability from heart rate variability elicited by orthostatic or mental arithmetic stress. Full article

(This article belongs to the Special Issue Entropy and Cardiac Physics)

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Illustration of the two-dimensional warping algorithm. The template beat (blue) is mapped to a warping grid (blue), which is adapted to the incoming beat (black) by moving the warping points such that the distance between template and incoming beat is minimized. The optimally fitted template and the corresponding warping points are shown in red. Full article ">
Mean and standard deviations of beat-to-beat RR and QT interval time series during baseline (bl), head-up tilt (tilt), recovery (rec), mental arithmetic task (ma) and recovery (rec) phases. Data are expressed as group mean values and standard deviations. Asterisks indicate significant changes with respect to baseline measurement. (****p < 0.0001, ***p < 0.001, **p < 0.01, *p < 0.05) Full article ">
Sample entropy and cross-sample entropy of beat-to-beat RR and QT interval time series during baseline (bl), head-up tilt (tilt), recovery (rec), mental arithmetic task (ma) and recovery (rec) phases. Data are expressed as group mean values and standard deviations. Asterisks indicate significant changes with respect to baseline measurement. (*p < 0.05) Full article ">
Conditional coupling from RR to QT and from QT to RR, respectively, with lags shown below during baseline (bl), head-up tilt (tilt), recovery (rec), mental arithmetic task (ma) and recovery (rec) phases. Data are expressed as group mean values and standard deviations. Asterisks indicate significant changes with respect to baseline measurement. (*p < 0.05) Full article ">

619 KiB

Open AccessCorrection

Correction on Iliyasu, A.M. et al. Hybrid Quantum-Classical Protocol for Storage and Retrieval of Discrete-Valued Information. Entropy, 2014, 16, 3537-3551

by Abdullah M. Iliyasu, Salvador E. Venegas-Andraca, Fei Yan and Ahmed S. Salama

Entropy 2014, 16(12), 6382-6383; https://doi.org/10.3390/e16126382 - 2 Dec 2014

Viewed by 5033

Abstract

The authors wish to make the following corrections to this paper [1]: The correct name of the fourth author is: Ahmed S. Salama. In the Acknowledgment Section, we added the research Project No. 2014/01/2079. Below is the corrected version of the section.[...] [...] Read more.

The authors wish to make the following corrections to this paper [1]: The correct name of the fourth author is: Ahmed S. Salama. In the Acknowledgment Section, we added the research Project No. 2014/01/2079. Below is the corrected version of the section.[...] Full article

438 KiB

Open AccessArticle

The Information Geometry of Bregman Divergences and Some Applications in Multi-Expert Reasoning

by Martin Adamčík

Entropy 2014, 16(12), 6338-6381; https://doi.org/10.3390/e16126338 - 1 Dec 2014

Cited by 18 | Viewed by 10050

Abstract

The aim of this paper is to develop a comprehensive study of the geometry involved in combining Bregman divergences with pooling operators over closed convex sets in a discrete probabilistic space. A particular connection we develop leads to an iterative procedure, which is [...] Read more.

The aim of this paper is to develop a comprehensive study of the geometry involved in combining Bregman divergences with pooling operators over closed convex sets in a discrete probabilistic space. A particular connection we develop leads to an iterative procedure, which is similar to the alternating projection procedure by Csiszár and Tusnády. Although such iterative procedures are well studied over much more general spaces than the one we consider, only a few authors have investigated combining projections with pooling operators. We aspire to achieve here a comprehensive study of such a combination. Besides, pooling operators combining the opinions of several rational experts allows us to discuss possible applications in multi-expert reasoning. Full article

(This article belongs to the Special Issue Maximum Entropy Applied to Inductive Logic and Reasoning)

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Graphical abstract

6384 KiB

Open AccessArticle

Landscape Analysis of Geographical Names in Hubei Province, China

by Xixi Chen, Tao Hu, Fu Ren, Deng Chen, Lan Li and Nan Gao

Entropy 2014, 16(12), 6313-6337; https://doi.org/10.3390/e16126313 - 1 Dec 2014

Cited by 14 | Viewed by 8630

Abstract

Hubei Province is the hub of communications in central China, which directly determines its strategic position in the country’s development. Additionally, Hubei Province is well-known for its diverse landforms, including mountains, hills, mounds and plains. This area is called “The Province of Thousand [...] Read more.

Hubei Province is the hub of communications in central China, which directly determines its strategic position in the country’s development. Additionally, Hubei Province is well-known for its diverse landforms, including mountains, hills, mounds and plains. This area is called “The Province of Thousand Lakes” due to the abundance of water resources. Geographical names are exclusive names given to physical or anthropogenic geographic entities at specific spatial locations and are important signs by which humans understand natural and human activities. In this study, geographic information systems (GIS) technology is adopted to establish a geodatabase of geographical names with particular characteristics in Hubei Province and extract certain geomorphologic and environmental factors. We carry out landscape analysis of mountain-related geographical names and water-related geographical names respectively. In the end, we calculate the information entropy of geographical names of each county to describe the diversity and inhomogeneity of place names in Hubei province. Our study demonstrates that geographical names represent responses to the cultural landscape and physical environment. The geographical names are more interesting in specific landscapes, such as mountains and rivers. Full article

(This article belongs to the Special Issue Entropy and Space-Time Analysis in Environment and Health)

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1044 KiB

Open AccessArticle

Ab intio Investigation of the Thermochemistry and Kinetics of the SO₂ + O₃⁻ → SO₃⁻ + O₂ Reaction in Aircraft Engines and the Environment

by Xuechao Guo, Alexey B. Nadykto, Yisheng Xu, Qingzhu Zhang and Jingtian Hu

Entropy 2014, 16(12), 6300-6312; https://doi.org/10.3390/e16126300 - 1 Dec 2014

Cited by 3 | Viewed by 7672

Abstract

In the present work, the mechanisms, thermochemistry and kinetics of the reaction of SO₂ with O₃⁻ have been studied using the CCSD(T)/6-31G(d) + CF method. It has been shown that there exist two possible pathways A and B of the [...] Read more.

In the present work, the mechanisms, thermochemistry and kinetics of the reaction of SO₂ with O₃⁻ have been studied using the CCSD(T)/6-31G(d) + CF method. It has been shown that there exist two possible pathways A and B of the SO₂ + O₃⁻ → SO₃⁻ + O₂reaction. The two pathways’ A and B barrier heights are 0.61 kcal mol⁻¹ and 3.40 kcal mol⁻¹, respectively, while the energy of the SO₂ + O₃⁻ → SO₃⁻ + O₂ reaction is −25.25 kcal mol⁻¹. The canonical variational transition state theory with small-curvature tunneling (CVT/SCT) has been applied to study the reaction kinetics. The CVT/SCT study shows that the rate constants K for pathways A and B, K_A = 1.11 × 10⁻¹²exp(−2526.13/T) and K_B = 2.7 × 10⁻¹⁴exp(−1029.25/T), respectively, grow as the temperature increases and are much larger than those of the SO₂ + O₃ → SO₃ + O₂reaction over the entire temperature range of 200–1500 K. This indicates that ionization of O₃and high temperatures are favorable for the SO₂ oxidation via the reaction with ozone. The new data obtained in the present study can be utilized directly for the evaluation of experiments and model predictions concerning SO₂ oxidation and kinetic modeling of gas-phase chemistry of pollutants/nucleation precursors formed in aircraft engines and the Earth’s atmosphere. Full article

(This article belongs to the Section Thermodynamics)

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Geometries of stationary points associated with the SO2 + O3− → SO3− + O2 reaction obtained at B3LYP/aug-cc-pvdz level of theory. Bond distances are given in Å. TS1 and TS2 denote transition states of pathway A and B, respectively. Full article ">
The potential energy surface for the SO2 + O3 → SO3− + O2 reaction. Full article ">
Comparison of the rate constants KA and KB for SO2 + O3− → SO3− + O2 reaction at 200−1500 K with the experimental and theoretical data for the SO2 + O3 → SO3 + O2 reaction [<a href="#b32-entropy-16-06300" class="html-bibr">32</a>,<a href="#b43-entropy-16-06300" class="html-bibr">43</a>,<a href="#b58-entropy-16-06300" class="html-bibr">58</a>]. Symbols A, B, exp*, theor*, NIST* refer to pathway A (present study), pathway B (present study), ref. [<a href="#b32-entropy-16-06300" class="html-bibr">32</a>], ref. [<a href="#b43-entropy-16-06300" class="html-bibr">43</a>] and NIST fitted data ref. [<a href="#b58-entropy-16-06300" class="html-bibr">58</a>], respectively. Full article ">
The possible reaction pathways of the SO2+O3−→SO3−+O2 reaction with the potential barriers E* (kcal mol−1) and reaction heats (kcal mol−1). Full article ">

1391 KiB

Open AccessArticle

Adaptive Synchronization of Fractional Neural Networks with Unknown Parameters and Time Delays

by Weiyuan Ma, Changpin Li, Yujiang Wu and Yongqing Wu

Entropy 2014, 16(12), 6286-6299; https://doi.org/10.3390/e16126286 - 1 Dec 2014

Cited by 45 | Viewed by 6762

Abstract

In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain [...] Read more.

In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain complex networks with time delays. Besides, the system parameters in the uncertain network can be identified in the process of synchronization. To demonstrate the validity of the proposed method, several illustrative examples are presented. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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266 KiB

Open AccessArticle

A Thermodynamical Selection-Based Discrete Differential Evolution for the 0-1 Knapsack Problem

by Zhaolu Guo, Xuezhi Yue, Kejun Zhang, Shenwen Wang and Zhijian Wu

Entropy 2014, 16(12), 6263-6285; https://doi.org/10.3390/e16126263 - 28 Nov 2014

Cited by 13 | Viewed by 6484

Abstract

Many problems in business and engineering can be modeled as 0-1 knapsack problems. However, the 0-1 knapsack problem is one of the classical NP-hard problems. Therefore, it is valuable to develop effective and efficient algorithms for solving 0-1 knapsack problems. Aiming at the [...] Read more.

Many problems in business and engineering can be modeled as 0-1 knapsack problems. However, the 0-1 knapsack problem is one of the classical NP-hard problems. Therefore, it is valuable to develop effective and efficient algorithms for solving 0-1 knapsack problems. Aiming at the drawbacks of the selection operator in the traditional differential evolution (DE), we present a novel discrete differential evolution (TDDE) for solving 0-1 knapsack problem. In TDDE, an enhanced selection operator inspired by the principle of the minimal free energy in thermodynamics is employed, trying to balance the conflict between the selective pressure and the diversity of population to some degree. An experimental study is conducted on twenty 0-1 knapsack test instances. The comparison results show that TDDE can gain competitive performance on the majority of the test instances. Full article

(This article belongs to the Special Issue Entropy in Bioinspired Intelligence)

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206 KiB

Open AccessArticle

On a Local Fractional Wave Equation under Fixed Entropy Arising in Fractal Hydrodynamics

by Yu Zhang, Dumitru Baleanu and Xiaojun Yang

Entropy 2014, 16(12), 6254-6262; https://doi.org/10.3390/e16126254 - 28 Nov 2014

Cited by 22 | Viewed by 5406

Abstract

In this paper, based on fixed entropy, the adiabatic equation of state in fractal flow is discussed. The local fractional wave equation for the velocity potential is also obtained by using the non-differential perturbations for the pressure and density of fractal hydrodynamics. Full article

(This article belongs to the Section Complexity)

14220 KiB

Open AccessArticle

Generation and Nonlinear Dynamical Analyses of Fractional-Order Memristor-Based Lorenz Systems

by Huiling Xi, Yuxia Li and Xia Huang

Entropy 2014, 16(12), 6240-6253; https://doi.org/10.3390/e16126240 - 28 Nov 2014

Cited by 35 | Viewed by 9448

Abstract

In this paper, four fractional-order memristor-based Lorenz systems with the flux-controlled memristor characterized by a monotone-increasing piecewise linear function, a quadratic nonlinearity, a smooth continuous cubic nonlinearity and a quartic nonlinearity are presented, respectively. The nonlinear dynamics are analyzed by using numerical simulation [...] Read more.

In this paper, four fractional-order memristor-based Lorenz systems with the flux-controlled memristor characterized by a monotone-increasing piecewise linear function, a quadratic nonlinearity, a smooth continuous cubic nonlinearity and a quartic nonlinearity are presented, respectively. The nonlinear dynamics are analyzed by using numerical simulation methods, including phase portraits, bifurcation diagrams, the largest Lyapunov exponent and power spectrum diagrams. Some interesting phenomena, such as inverse period-doubling bifurcation and intermittent chaos, are found to exist in the proposed systems. Full article

(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)

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Graphical abstract

Graphical abstract
Full article ">
Bifurcation diagram of System (7) with respect to parameter c1. Full article ">
Time history, phase portrait and power spectrum diagram of the chaotic attractor when c1 = 35. Full article ">
Bifurcation diagram of System <a href="#fd10-entropy-16-06240" class="html-disp-formula">(8)</a> with respect to parameter c2. Full article ">
Time histories, phase portraits and power spectrum diagrams of the chaotic attractor, period-5 orbit, period-3 orbit and period-1 orbit. Full article ">
Bifurcation diagram of System <a href="#fd13-entropy-16-06240" class="html-disp-formula">(11)</a> with respect to parameter c3. Full article ">
Time histories, phase portraits and power spectrum diagrams of the chaotic attractor, period-5 orbit, period-3 orbit and period-1 orbit. Full article ">
Bifurcation diagram of System <a href="#fd16-entropy-16-06240" class="html-disp-formula">(14)</a> with respect to parameter c4. Full article ">
Time histories, phase portraits and power spectrum diagrams of the chaotic attractor, period-3 orbit, quasi-period orbit and period-1 orbit. Full article ">

882 KiB

Open AccessArticle

Ordinal Patterns, Entropy, and EEG

by Karsten Keller, Anton M. Unakafov and Valentina A. Unakafova

Entropy 2014, 16(12), 6212-6239; https://doi.org/10.3390/e16126212 - 27 Nov 2014

Cited by 67 | Viewed by 11877

Abstract

In this paper we illustrate the potential of ordinal-patterns-based methods for analysis of real-world data and, especially, of electroencephalogram (EEG) data. We apply already known (empirical permutation entropy, ordinal pattern distributions) and new (empirical conditional entropy of ordinal patterns, robust to noise empirical [...] Read more.

In this paper we illustrate the potential of ordinal-patterns-based methods for analysis of real-world data and, especially, of electroencephalogram (EEG) data. We apply already known (empirical permutation entropy, ordinal pattern distributions) and new (empirical conditional entropy of ordinal patterns, robust to noise empirical permutation entropy) methods for measuring complexity, segmentation and classification of time series. Full article

(This article belongs to the Special Issue Entropy and Electroencephalography)

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2785 KiB

Open AccessArticle

Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos

by Jian Liu

Entropy 2014, 16(12), 6195-6211; https://doi.org/10.3390/e16126195 - 27 Nov 2014

Cited by 26 | Viewed by 5964

Abstract

This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based [...] Read more.

This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based on the stability theory of fractional-order systems, by employing the feedback control technique, necessary and sufficient criteria on CMHPS are derived. Furthermore, CMHPS between fractional-order real hyper-chaotic Rössler system and other two different dimensional fractional-order complex Lorenz-like chaotic systems is provided as two examples to discuss reduced order and increased order synchronization, respectively. Full article

(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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Entropy, Volume 16, Issue 12 (December 2014) – 30 articles , Pages 6195-6738

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