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Volume 17
Issue 7
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Entropy, Volume 17, Issue 7 (July 2015) – 36 articles , Pages 4485-5144

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1803 KiB  
Article
Setting Diverging Colors for a Large-Scale Hypsometric Lunar Map Based on Entropy
by Xingguo Zeng, Lingli Mu, Jianjun Liu and Yiman Yang
Entropy 2015, 17(7), 5133-5144; https://doi.org/10.3390/e17075133 - 22 Jul 2015
Cited by 2 | Viewed by 4734
Abstract
A hypsometric map is a type of map used to represent topographic characteristics by filling different map areas with diverging colors. The setting of appropriate diverging colors is essential for the map to reveal topographic details. When lunar real environmental exploration programs are [...] Read more.
A hypsometric map is a type of map used to represent topographic characteristics by filling different map areas with diverging colors. The setting of appropriate diverging colors is essential for the map to reveal topographic details. When lunar real environmental exploration programs are performed, large-scale hypsometric maps with a high resolution and greater topographic detail are helpful. Compared to the situation on Earth, fewer lunar exploration objects are available, and the topographic waviness is smaller at a large scale, indicating that presenting the topographic details using traditional hypsometric map-making methods may be difficult. To solve this problem, we employed the Chang’E2 (CE2) topographic and imagery data with a resolution of 7 m and developed a new hypsometric map-making method by setting the diverging colors based on information entropy. The resulting map showed that this method is suitable for presenting the topographic details and might be useful for developing a better understanding of the environment of the lunar surface. Full article
(This article belongs to the Special Issue Entropy, Utility, and Logical Reasoning)
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<p>Stereoscopic camera on the CE2.</p> Full article ">
<p>Selected mapping areas.</p> Full article ">
<p>Distribution of the elevation of Map 32.</p> Full article ">
<p>Map result when applying the available diverging color list.</p> Full article ">
<p>Map results when applying the calculated diverging color list.</p> Full article ">
737 KiB  
Article
An Entropy-Based Approach to Path Analysis of Structural Generalized Linear Models: A Basic Idea
by Nobuoki Eshima, Minoru Tabata, Claudio Giovanni Borroni and Yutaka Kano
Entropy 2015, 17(7), 5117-5132; https://doi.org/10.3390/e17075117 - 22 Jul 2015
Cited by 4 | Viewed by 7236
Abstract
A path analysis method for causal systems based on generalized linear models is proposed by using entropy. A practical example is introduced, and a brief explanation of the entropy coefficient of determination is given. Direct and indirect effects of explanatory variables are discussed [...] Read more.
A path analysis method for causal systems based on generalized linear models is proposed by using entropy. A practical example is introduced, and a brief explanation of the entropy coefficient of determination is given. Direct and indirect effects of explanatory variables are discussed as log odds ratios, i.e., relative information, and a method for summarizing the effects is proposed. The example dataset is re-analyzed by using the method. Full article
(This article belongs to the Special Issue Entropy, Utility, and Logical Reasoning)
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<p>Path diagram of social class mobility.</p> Full article ">
3223 KiB  
Article
Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System
by Gennady A. Leonov, Tatyana A. Alexeeva and Nikolay V. Kuznetsov
Entropy 2015, 17(7), 5101-5116; https://doi.org/10.3390/e17075101 - 22 Jul 2015
Cited by 10 | Viewed by 5391
Abstract
In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under [...] Read more.
In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under diffeomorphism is demonstrated in the general case. This fact is used to obtain the analytic exact upper bound of the Lyapunov dimension of an attractor of the Shimizu–Morioka system. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
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<p>Self-excited local attractors in: (a) System (18) for <span class="html-italic">α</span> = 0.375, <span class="html-italic">λ</span> = 0.81; (<b>b</b>) System (20) for <span class="html-italic">α</span> = 0.375, <span class="html-italic">λ</span> = 0.81.</p> Full article ">
<p>Self-excited local attractors in: (<b>a</b>) System (18) for <span class="html-italic">α</span> = 0.191450, <span class="html-italic">λ</span> = 0.81, “Burke and Shaw-like”; (<b>b</b>) System (20) for <span class="html-italic">α</span> = 0.191450, <span class="html-italic">λ</span> = 0.81, “Burke and Shaw-like”.</p> Full article ">
915 KiB  
Article
Averaged Extended Tree Augmented Naive Classifier
by Aaron Meehan and Cassio P. De Campos
Entropy 2015, 17(7), 5085-5100; https://doi.org/10.3390/e17075085 - 21 Jul 2015
Cited by 9 | Viewed by 5119
Abstract
This work presents a new general purpose classifier named Averaged Extended Tree Augmented Naive Bayes (AETAN), which is based on combining the advantageous characteristics of Extended Tree Augmented Naive Bayes (ETAN) and Averaged One-Dependence Estimator (AODE) classifiers. We describe the main properties of [...] Read more.
This work presents a new general purpose classifier named Averaged Extended Tree Augmented Naive Bayes (AETAN), which is based on combining the advantageous characteristics of Extended Tree Augmented Naive Bayes (ETAN) and Averaged One-Dependence Estimator (AODE) classifiers. We describe the main properties of the approach and algorithms for learning it, along with an analysis of its computational time complexity. Empirical results with numerous data sets indicate that the new approach is superior to ETAN and AODE in terms of both zero-one classification accuracy and log loss. It also compares favourably against weighted AODE and hidden Naive Bayes. The learning phase of the new approach is slower than that of its competitors, while the time complexity for the testing phase is similar. Such characteristics suggest that the new classifier is ideal in scenarios where online learning is not required. Full article
(This article belongs to the Special Issue Inductive Statistical Methods)
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<p>Some examples of the different structures possible within the classifiers we have written about here. (<b>a</b>) Possible with ETAN; (<b>b</b>) Possible with naive or ETAN; (<b>c</b>) Possible with TAN or ETAN; (<b>d</b>) Possible with ETAN.</p> Full article ">
<p>An example of a new structure possible with Extended Tree Augmented Naive Bayes (ETAN)pp.</p> Full article ">
<p>These graphs show the log loss difference between AETAN and ETAN (<b>Top</b>) and AETAN and Averaging One-dependent Estimator (AODE) (<b>Bottom</b>). The values are the difference between the log loss of the first classifier minus AETAN, so higher values mean AETAN is better.</p> Full article ">
<p>A comparison of correct classification ratio between AETAN and ETAN (<b>Top</b>) and AETAN and AODE (<b>Bottom</b>), higher value means AETAN is better as these results are the accuracy of AETAN divided by the others.</p> Full article ">
<p>Comparison of computation time to learn the classifier between AETAN and ETAN, expressed as a ratio of AETAN/ETAN so higher values mean AETAN is slower.</p> Full article ">
1675 KiB  
Article
Minimal Rényi–Ingarden–Urbanik Entropy of Multipartite Quantum States
by Marco Enríquez, Zbigniew Puchała and Karol Życzkowski
Entropy 2015, 17(7), 5063-5084; https://doi.org/10.3390/e17075063 - 20 Jul 2015
Cited by 15 | Viewed by 5956
Abstract
We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over [...] Read more.
We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the Rényi–Ingarden–Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q = 0, this quantity becomes a function of the rank of the tensor representing the state, while in the limit q ? ?, the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system, the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three- and four-qubit systems. In the former case, the distribution of the three-tangle is studied and some of its moments are evaluated, while in the latter case, we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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<p>Minimal Rényi–Ingarden–Urbanik (RIU) entropy obtained numerically for several three-qubit states as a function of the Rényi parameter <span class="html-italic">q</span>. Comparison with analytical results available for states invariant with respect to permutation confirms the accuracy of the numerical procedure.</p> Full article ">
<p>Distributions of the Rényi entropy of a random probability vector describing a generic three-qubit state (□), the Rényi entropy of the corresponding co-tensor (∇) and the estimation of minimal RIU entropy (○) for (a) <span class="html-italic">q</span> = 1, (b) <span class="html-italic">q</span> = 2 and (c) <span class="html-italic">q</span> = 100.</p> Full article ">
<p>Comparison between the distributions of <math display="inline"> <mrow> <msubsup> <mi>S</mi> <mrow> <mn>100</mn></mrow> <mrow> <mi mathvariant="normal">RIU</mi></mrow></msubsup></mrow></math> (○), <math display="inline"> <mrow> <msub> <mi>S</mi> <mi mathvariant="normal">P</mi></msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">log</mi> <mspace width="0.2em"/> <msubsup> <mi>λ</mi> <mrow> <mo>max</mo></mrow> <mrow> <msup> <mrow/> <mrow> <mi mathvariant="normal">PARAFAC</mi></mrow></msup></mrow></msubsup></mrow></math> (+) and <math display="inline"> <mrow> <msub> <mi>S</mi> <mrow> <mi mathvariant="normal">LU</mi></mrow></msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">log</mi> <mspace width="0.2em"/> <msubsup> <mi>λ</mi> <mrow> <mo>max</mo></mrow> <mrow> <msup> <mrow/> <mrow> <mi mathvariant="normal">LU</mi></mrow></msup></mrow></msubsup></mrow></math> (⋄) for three-qubit random states.</p> Full article ">
<p>Distributions of the three-tangle (+) and its square (∇) for three-qubit pure random states. The solid lines are approximations with Beta distributions <a href="#fd24-entropy-17-05063" class="html-disp-formula">Equation (23)</a> on the left and <a href="#fd25-entropy-17-05063" class="html-disp-formula">Equation (24)</a> on the right.</p> Full article ">
<p>The minimal RIU entropy <math display="inline"> <mrow> <msubsup> <mi>S</mi> <mi>q</mi> <mrow> <mi mathvariant="normal">RIU</mi></mrow></msubsup></mrow></math> computed for several four-qubit states as a function of the parameter <span class="html-italic">q.</span></p> Full article ">
<p>Distributions of the Rényi entropy of a four-qubit random state probability vector (□), the Rényi entropy of the corresponding co-tensor (∇) and the minimal RIU entropy (○) for (<b>a</b>) <span class="html-italic">q</span> = 1, (<b>b</b>) <span class="html-italic">q</span> = 2 and (<b>c</b>) <span class="html-italic">q</span> = 100.</p> Full article ">
<p>Comparison between the distributions of <math display="inline"> <mrow> <msubsup> <mi>S</mi> <mrow> <mn>100</mn></mrow> <mrow> <mi mathvariant="normal">RIU</mi></mrow></msubsup></mrow></math> (○) and <math display="inline"> <mrow> <msup> <mi>S</mi> <mrow> <mi mathvariant="normal">LU</mi></mrow></msup> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">log</mi> <mspace width="0.2em"/> <msubsup> <mi>λ</mi> <mrow> <mo>max</mo></mrow> <mrow> <msup> <mrow/> <mrow> <mi mathvariant="normal">LU</mi></mrow></msup></mrow></msubsup></mrow></math> (+) for a four-qubit system.</p> Full article ">
<p>Distribution of the absolute value of the hyperdeterminant <span class="html-italic">P</span> (<span class="html-italic">T</span>) (+) for four-qubit pure random states in a log-log plot.</p> Full article ">
<p>The mean of the maximum component (left) and the geometric measure of entanglement (right) for random states of a tri-partite system as a function of the qudit size. Bullets correspond to numeric simulations: (□) stands for the greatest tensor element <span class="html-italic">λ</span><sub>max</sub>; (∇) the maximum component of the higher order singular value decomposition (HO-SVD) co-tensor <span class="html-italic">λ<sub>H</sub></span>; (○) stands for the parallel factor model (PARAFAC) overlap <span class="html-italic">λ<sub>P</sub></span>; and (+) refers to the overlap with the closest separable state maximized by LU. The solid red line (—) is the result <a href="#fd36-entropy-17-05063" class="html-disp-formula">Equation (35)</a> with <span class="html-italic">N</span> = <span class="html-italic">d</span><sup>3</sup>; the solid lines (—) and (—) are the best linear fits for HOSVD and PARAFAC, respectively.</p> Full article ">
302 KiB  
Article
Evaluation of the Atmospheric Chemical Entropy Production of Mars
by Alfonso Delgado-Bonal and F. Javier Martín-Torres
Entropy 2015, 17(7), 5047-5062; https://doi.org/10.3390/e17075047 - 20 Jul 2015
Viewed by 6313
Abstract
Thermodynamic disequilibrium is a necessary situation in a system in which complex emergent structures are created and maintained. It is known that most of the chemical disequilibrium, a particular type of thermodynamic disequilibrium, in Earth’s atmosphere is a consequence of life. We have [...] Read more.
Thermodynamic disequilibrium is a necessary situation in a system in which complex emergent structures are created and maintained. It is known that most of the chemical disequilibrium, a particular type of thermodynamic disequilibrium, in Earth’s atmosphere is a consequence of life. We have developed a thermochemical model for the Martian atmosphere to analyze the disequilibrium by chemical reactions calculating the entropy production. It follows from the comparison with the Earth atmosphere that the magnitude of the entropy produced by the recombination reaction forming O3 (O + O2 + CO2 ? O3 + CO2) in the atmosphere of the Earth is larger than the entropy produced by the dominant set of chemical reactions considered for Mars, as a consequence of the low density and the poor variety of species of the Martian atmosphere. If disequilibrium is needed to create and maintain self-organizing structures in a system, we conclude that the current Martian atmosphere is unable to support large physico-chemical structures, such as those created on Earth. Full article
(This article belongs to the Section Thermodynamics)
►▼ Show Figures

Graphical abstract

Graphical abstract
Full article ">
<p>Entropy production per second (J·mol<sup>−1</sup>·K<sup>−1</sup>) O<sub>3</sub> + CO<sub>2</sub> on Mars at the surface level for Ls 180<span class="html-italic">°</span>. The y-axis scale is linear, <span class="html-italic">i.e.</span>, values are multiplied by 10<sup>6</sup> J·mol<sup>−1</sup>·K<sup>−1</sup>.</p> Full article ">
<p>Ozone concentration (molec/cm<sup>3</sup>) in the Martian atmosphere at the surface level during one Martian day at Ls 180<span class="html-italic">°</span>.</p> Full article ">
141 KiB  
Reply
Reply to C. Tsallis’ “Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems”
by Steve Pressé, Kingshuk Ghosh, Julian Lee and Ken A. Dill
Entropy 2015, 17(7), 5043-5046; https://doi.org/10.3390/e17075043 - 17 Jul 2015
Cited by 21 | Viewed by 5120
Abstract
In a recent PRL (2013, 111, 180604), we invoked the Shore and Johnson axioms which demonstrate that the least-biased way to infer probability distributions fpig from data is to maximize the Boltzmann-Gibbs entropy. We then showed which biases are introduced in models obtained [...] Read more.
In a recent PRL (2013, 111, 180604), we invoked the Shore and Johnson axioms which demonstrate that the least-biased way to infer probability distributions fpig from data is to maximize the Boltzmann-Gibbs entropy. We then showed which biases are introduced in models obtained by maximizing nonadditive entropies. A rebuttal of our work appears in entropy (2015, 17, 2853) and argues that the Shore and Johnson axioms are inapplicable to a wide class of complex systems. Here we highlight the errors in this reasoning. Full article
(This article belongs to the Section Complexity)
3877 KiB  
Article
The Critical Point Entanglement and Chaos in the Dicke Model
by Lina Bao, Feng Pan, Jing Lu and Jerry P. Draayer
Entropy 2015, 17(7), 5022-5042; https://doi.org/10.3390/e17075022 - 16 Jul 2015
Cited by 7 | Viewed by 6351
Abstract
Ground state properties and level statistics of the Dicke model for a finite number of atoms are investigated based on a progressive diagonalization scheme (PDS). Particle number statistics, the entanglement measure and the Shannon information entropy at the resonance point in cases with [...] Read more.
Ground state properties and level statistics of the Dicke model for a finite number of atoms are investigated based on a progressive diagonalization scheme (PDS). Particle number statistics, the entanglement measure and the Shannon information entropy at the resonance point in cases with a finite number of atoms as functions of the coupling parameter are calculated. It is shown that the entanglement measure defined in terms of the normalized von Neumann entropy of the reduced density matrix of the atoms reaches its maximum value at the critical point of the quantum phase transition where the system is most chaotic. Noticeable change in the Shannon information entropy near or at the critical point of the quantum phase transition is also observed. In addition, the quantum phase transition may be observed not only in the ground state mean photon number and the ground state atomic inversion as shown previously, but also in fluctuations of these two quantities in the ground state, especially in the atomic inversion fluctuation. Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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<p>The number of steps, <span class="html-italic">k</span>, needed in the progressive diagonalization scheme (PDS) for the lowest 20 level energies <span class="html-italic">E/ω</span><sub>0</sub> at the resonance point with <span class="html-italic">ω</span> = <span class="html-italic">ω</span><sub>0</sub>, where (<b>a</b>) <span class="html-italic">j</span> = 1/2, <span class="html-italic">λ/ω</span><sub>0</sub> = 0.5, (<b>b</b>) <span class="html-italic">j</span> = 1/2, <span class="html-italic">λ</span>/<span class="html-italic">ω</span><sub>0</sub> = 1.0 and (<b>c</b>) <span class="html-italic">j</span> = 20, <span class="html-italic">λ/ω</span><sub>0</sub> = 0.5.</p> Full article ">
<p>The ground state mean photon number <math display="inline"> <mover accent="true"> <mi>n</mi> <mo>¯</mo></mover></math> (left panel) and the atomic inversion <math display="inline"> <mrow> <mover accent="true"> <mrow> <msub> <mi>J</mi> <mi>z</mi></msub></mrow> <mo stretchy="false">¯</mo></mover></mrow></math>(right panel) as functions of <span class="html-italic">λ/ω</span><sub>0</sub> at the resonance point with <span class="html-italic">ω</span> = <span class="html-italic">ω</span><sub>0</sub> for 10 (red solid dots), 15 (open circles) and 20 (solid line), where the vertical dashed line indicates the critical point position determined in the thermodynamic limit [<a href="#b10-entropy-17-05022" class="html-bibr">10</a>].</p> Full article ">
<p>The same as <a href="#f2-entropy-17-05022" class="html-fig">Figure 2</a>, but for the ground state photon number fluctuation Δ<span class="html-italic">n</span> (left panel) and the atomic inversion fluctuation Δ<span class="html-italic">J<sub>z</sub></span> (right panel) as functions of <span class="html-italic">λ/ω</span><sub>0</sub> at the resonance point with <span class="html-italic">ω</span> = <span class="html-italic">ω</span><sub>0</sub> for 10 (red solid dots), 15 (open circles) and 20 (solid line).</p> Full article ">
<p>The same as <a href="#f2-entropy-17-05022" class="html-fig">Figure 2</a>, but for the ground state entanglement (left panel) and the Shannon information entropy (right panel) as functions of <span class="html-italic">λ/ω</span><sub>0</sub> at the resonance point with <span class="html-italic">ω</span> = <span class="html-italic">ω</span><sub>0</sub> for 10 (red solid dots), 15 (open circles) and 20 (solid line).</p> Full article ">
<p>Level spacing distribution <span class="html-italic">P</span>(<span class="html-italic">S</span>) of the model with <span class="html-italic">j</span> = 20 at the resonance point with <span class="html-italic">ω</span> = <span class="html-italic">ω</span><sub>0</sub> for various coupling strengths <span class="html-italic">λ/ω</span><sub>0</sub>. In all panels, the (blue) dashed line describes the Poisson statistics, while the (red) dash-dotted line describes the GOEstatistics.</p> Full article ">
<p>The spectral rigidity <math display="inline"> <mrow> <msub> <mover accent="true"> <mi mathvariant="normal">Δ</mi> <mo>¯</mo></mover> <mn>3</mn></msub> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo></mrow></math> of the model with <span class="html-italic">j</span> = 20 at the resonance point with <span class="html-italic">ω</span> = <span class="html-italic">ω</span><sub>0</sub> for various coupling strengths <span class="html-italic">λ/ω</span><sub>0</sub>. In all of panels, the (blue) dashed line describes the Poisson statistics, while the (red) dash-dotted line describes the GOE statistics.</p> Full article ">
2851 KiB  
Article
A Novel Method for Seismogenic Zoning Based on Triclustering: Application to the Iberian Peninsula
by Francisco Martínez-Álvarez, David Gutiérrez-Avilés, Antonio Morales-Esteban, Jorge Reyes, José L. Amaro-Mellado and Cristina Rubio-Escudero
Entropy 2015, 17(7), 5000-5021; https://doi.org/10.3390/e17075000 - 16 Jul 2015
Cited by 20 | Viewed by 5261
Abstract
A previous definition of seismogenic zones is required to do a probabilistic seismic hazard analysis for areas of spread and low seismic activity. Traditional zoning methods are based on the available seismic catalog and the geological structures. It is admitted that thermal and [...] Read more.
A previous definition of seismogenic zones is required to do a probabilistic seismic hazard analysis for areas of spread and low seismic activity. Traditional zoning methods are based on the available seismic catalog and the geological structures. It is admitted that thermal and resistant parameters of the crust provide better criteria for zoning. Nonetheless, the working out of the rheological profiles causes a great uncertainty. This has generated inconsistencies, as different zones have been proposed for the same area. A new method for seismogenic zoning by means of triclustering is proposed in this research. The main advantage is that it is solely based on seismic data. Almost no human decision is made, and therefore, the method is nearly non-biased. To assess its performance, the method has been applied to the Iberian Peninsula, which is characterized by the occurrence of small to moderate magnitude earthquakes. The catalog of the National Geographic Institute of Spain has been used. The output map is checked for validity with the geology. Moreover, a geographic information system has been used for two purposes. First, the obtained zones have been depicted within it. Second, the data have been used to calculate the seismic parameters (b-value, annual rate). Finally, the results have been compared to Kohonen’s self-organizing maps. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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<p>Data transformation to serve as TriGen’s inputs.</p> Full article ">
<p>Twenty-five cells used to calculate the features for a target cell.</p> Full article ">
<p>TriGen algorithm flowchart.</p> Full article ">
<p>Seismic zones of the Iberian Peninsula (IP) obtained with triclustering.</p> Full article ">
<p>Earthquake annual rate and b-value associated with every zone.</p> Full article ">
<p>Self-organizing maps (SOM) applied to the Iberian Peninsula without using spatial coordinates as input features, for 10 clusters.</p> Full article ">
<p>SOM applied to the Iberian Peninsula using spatial coordinates as input features, for 10 clusters.</p> Full article ">
1078 KiB  
Article
Maximum Entropy Estimation of Probability Distribution of Variables in Higher Dimensions from Lower Dimensional Data
by Jayajit Das, Sayak Mukherjee and Susan E. Hodge
Entropy 2015, 17(7), 4986-4999; https://doi.org/10.3390/e17074986 - 15 Jul 2015
Cited by 4 | Viewed by 4904
Abstract
A common statistical situation concerns inferring an unknown distribution Q(x) from a known distribution P(y), where X (dimension n), and Y (dimension m) have a known functional relationship. Most commonly, n ? m, and the task is relatively straightforward for well-defined functional relationships. [...] Read more.
A common statistical situation concerns inferring an unknown distribution Q(x) from a known distribution P(y), where X (dimension n), and Y (dimension m) have a known functional relationship. Most commonly, n ? m, and the task is relatively straightforward for well-defined functional relationships. For example, if Y1 and Y2 are independent random variables, each uniform on [0, 1], one can determine the distribution of X = Y1 + Y2; here m = 2 and n = 1. However, biological and physical situations can arise where n > m and the functional relation Y?X is non-unique. In general, in the absence of additional information, there is no unique solution to Q in those cases. Nevertheless, one may still want to draw some inferences about Q. To this end, we propose a novel maximum entropy (MaxEnt) approach that estimates Q(x) based only on the available data, namely, P(y). The method has the additional advantage that one does not need to explicitly calculate the Lagrange multipliers. In this paper we develop the approach, for both discrete and continuous probability distributions, and demonstrate its validity. We give an intuitive justification as well, and we illustrate with examples. Full article
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<p>Shows the different regions used in calculating the integral for κ(y) in Example 4.</p> Full article ">
458 KiB  
Article
Lag Synchronization of Complex Lorenz System with Applications to Communication
by Fangfang Zhang
Entropy 2015, 17(7), 4974-4985; https://doi.org/10.3390/e17074974 - 15 Jul 2015
Cited by 28 | Viewed by 4652
Abstract
In communication, the signal at the receiver end at time t is the signal from the transmitter side at time t ?? (? ? 0 and it is the lag time) as the time lag of transmission. Therefore, lag synchronization (LS) is [...] Read more.
In communication, the signal at the receiver end at time t is the signal from the transmitter side at time t ?? (? ? 0 and it is the lag time) as the time lag of transmission. Therefore, lag synchronization (LS) is more accurate than complete synchronization to design communication scheme. Taking complex Lorenz system as an example, we design the LS controller according to error feedback. Using chaotic masking, we propose a communication scheme based on LS and independent component analysis (ICA). It is suitable to transmit multiple messages with all kinds of amplitudes and it has the ability of anti-noise. Numerical simulations verify the feasibility and effectiveness of the presented schemes. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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<p>The block diagram of our communication scheme.</p> Full article ">
<p>The evolution of x(<span class="html-italic">t</span>) and y (<span class="html-italic">t</span>) from 5 s to 6 s.</p> Full article ">
<p>The error vector of LS.</p> Full article ">
<p>The LS controller.</p> Full article ">
<p>The transmission process of the famous melody “Ode to joy”.</p> Full article ">
2868 KiB  
Article
Broad Niche Overlap between Invasive Nile Tilapia Oreochromis niloticus and Indigenous Congenerics in Southern Africa: Should We be Concerned?
by Tsungai A. Zengeya, Anthony J. Booth and Christian T. Chimimba
Entropy 2015, 17(7), 4959-4973; https://doi.org/10.3390/e17074959 - 14 Jul 2015
Cited by 34 | Viewed by 7674
Abstract
This study developed niche models for the native ranges of Oreochromis andersonii, O. mortimeri, and O. mossambicus, and assessed how much of their range is climatically suitable for the establishment of O. niloticus, and then reviewed the conservation implications [...] Read more.
This study developed niche models for the native ranges of Oreochromis andersonii, O. mortimeri, and O. mossambicus, and assessed how much of their range is climatically suitable for the establishment of O. niloticus, and then reviewed the conservation implications for indigenous congenerics as a result of overlap with O. niloticus based on documented congeneric interactions. The predicted potential geographical range of O. niloticus reveals a broad climatic suitability over most of southern Africa and overlaps with all the endemic congenerics. This is of major conservation concern because six of the eight river systems predicted to be suitable for O. niloticus have already been invaded and now support established populations. Oreochromis niloticus has been implicated in reducing the abundance of indigenous species through competitive exclusion and hybridisation. Despite these well-documented adverse ecological effects, O. niloticus remains one of the most widely cultured and propagated fish species in aquaculture and stock enhancements in the southern Africa sub-region. Aquaculture is perceived as a means of protein security, poverty alleviation, and economic development and, as such, any future decisions on its introduction will be based on the trade-off between socio-economic benefits and potential adverse ecological effects. Full article
(This article belongs to the Special Issue Entropy in Hydrology)
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<p>The projected distributional range for <span class="html-italic">O. mossambicus</span>, <span class="html-italic">O. mortimeri</span>, and <span class="html-italic">O. andersonii</span> within their native ranges, and <span class="html-italic">O. niloticus</span> within its potential invasive range in river systems in southern Africa. Each map represents an average of 10 replicates for each species created using the <span class="html-italic">k</span>-fold partition method. Potential distribution is indicated by shaded areas, with red and blue indicating high and low probabilities of suitable conditions, respectively.</p> Full article ">
2084 KiB  
Article
Identity Authentication over Noisy Channels
by Fanfan Zheng, Zhiqing Xiao, Shidong Zhou, Jing Wang and Lianfen Huang
Entropy 2015, 17(7), 4940-4958; https://doi.org/10.3390/e17074940 - 14 Jul 2015
Cited by 4 | Viewed by 5299
Abstract
Identity authentication is the process of verifying users’ validity. Unlike classical key-based authentications, which are built on noiseless channels, this paper introduces a general analysis and design framework for identity authentication over noisy channels. Specifically, the authentication scenarios of single time and multiple [...] Read more.
Identity authentication is the process of verifying users’ validity. Unlike classical key-based authentications, which are built on noiseless channels, this paper introduces a general analysis and design framework for identity authentication over noisy channels. Specifically, the authentication scenarios of single time and multiple times are investigated. For each scenario, the lower bound on the opponent’s success probability is derived, and it is smaller than the classical identity authentication’s. In addition, it can remain the same, even if the secret key is reused. Remarkably, the Cartesian authentication code proves to be helpful for hiding the secret key to maximize the secrecy performance. Finally, we show a potential application of this authentication technique. Full article
(This article belongs to the Special Issue Entropy-Based Applied Cryptography and Enhanced Security for Ubiquitous Computing)
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<p>The challenge-response authentication model.</p> Full article ">
<p>The challenge-response authentication over noiseless channels.</p> Full article ">
<p>The challenge-response authentication over noisy channels.</p> Full article ">
<p>(<b>a</b>) Example of the secret key agreement from wireless channels. (<b>b</b>) The secret key agreement is protected by the challenge-response authentication. (<b>c</b>) The secret key agreement is improved by our proposed authentication model.</p> Full article ">
283 KiB  
Article
Fisher Information Properties
by Pablo Zegers
Entropy 2015, 17(7), 4918-4939; https://doi.org/10.3390/e17074918 - 13 Jul 2015
Cited by 41 | Viewed by 9697
Abstract
A set of Fisher information properties are presented in order to draw a parallel with similar properties of Shannon differential entropy. Already known properties are presented together with new ones, which include: (i) a generalization of mutual information for Fisher information; (ii) a [...] Read more.
A set of Fisher information properties are presented in order to draw a parallel with similar properties of Shannon differential entropy. Already known properties are presented together with new ones, which include: (i) a generalization of mutual information for Fisher information; (ii) a new proof that Fisher information increases under conditioning; (iii) showing that Fisher information decreases in Markov chains; and (iv) bound estimation error using Fisher information. This last result is especially important, because it completes Fano’s inequality, i.e., a lower bound for estimation error, showing that Fisher information can be used to define an upper bound for this error. In this way, it is shown that Shannon’s differential entropy, which quantifies the behavior of the random variable, and the Fisher information, which quantifies the internal structure of the density function that defines the random variable, can be used to characterize the estimation error. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
2005 KiB  
Article
Informational and Causal Architecture of Discrete-Time Renewal Processes
by Sarah E. Marzen and James P. Crutchfield
Entropy 2015, 17(7), 4891-4917; https://doi.org/10.3390/e17074891 - 13 Jul 2015
Cited by 33 | Viewed by 6503
Abstract
Renewal processes are broadly used to model stochastic behavior consisting of isolated events separated by periods of quiescence, whose durations are specified by a given probability law. Here, we identify the minimal sufficient statistic for their prediction (the set of causal states), calculate [...] Read more.
Renewal processes are broadly used to model stochastic behavior consisting of isolated events separated by periods of quiescence, whose durations are specified by a given probability law. Here, we identify the minimal sufficient statistic for their prediction (the set of causal states), calculate the historical memory capacity required to store those states (statistical complexity), delineate what information is predictable (excess entropy), and decompose the entropy of a single measurement into that shared with the past, future, or both. The causal state equivalence relation defines a new subclass of renewal processes with a finite number of causal states despite having an unbounded interevent count distribution. We use the resulting formulae to analyze the output of the parametrized Simple Nonunifilar Source, generated by a simple two-state hidden Markov model, but with an infinite-state machine presentation. All in all, the results lay the groundwork for analyzing more complex processes with infinite statistical complexity and infinite excess entropy. Full article
(This article belongs to the Section Statistical Physics)
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<p>The role of maximally-predictive (prescient) models: Estimating information measures directly from trajectory data encounters a curse of dimensionality or, in other words, severe undersampling. Instead, one can calculate information measures in closed-form from (inferred) maximally-predictive models [<a href="#b25-entropy-17-04891" class="html-bibr">25</a>]. Alternate generative models that are not maximally predictive cannot be used directly, as Blackwell showed in the 1950s [<a href="#b26-entropy-17-04891" class="html-bibr">26</a>].</p> Full article ">
<p>(<b>a</b>) Minimal generative model for the Poisson process with rate λ; (<b>b</b>) a generator for the Simple Nonunifilar Source (SNS). Both generate a stationary renewal process. Transition labels <span class="html-italic">p|s</span> denote the probability <span class="html-italic">p</span> of taking a transition and emitting symbol</p> Full article ">
<p><span class="html-italic">ϵ</span>-Machine architectures for discrete-time stationary renewal processes: (<b>a</b>) not eventually Δ-Poisson with unbounded-support interevent distribution; (<b>b</b>) not eventually Δ-Poisson with bounded-support interevent distribution; (<b>c</b>) eventually Δ-Poisson with characteristic (<span class="html-italic">ñ</span>,Δ = 1) in Definition 2; (<b>d</b>) eventually Δ-Poisson with characteristic <span class="html-italic">ñ</span>,Δ &gt; 1) in Definition 2; (<b>e</b>) for comparison, a Poisson process <span class="html-italic">ϵ</span>-machine.</p> Full article ">
<p>(<b>a</b>) A (nonunifilar) hidden Markov model for the (<span class="html-italic">p,q</span>) parametrized SNS; (<b>b</b>) example interevent distributions <span class="html-italic">F</span>(<span class="html-italic">n</span>) from <a href="#fd23-entropy-17-04891" class="html-disp-formula">Equation (15)</a> for three parameter settings of (<span class="html-italic">p,q</span>).</p> Full article ">
<p>Contour plots of various information measures (in bits) as functions of SNS parameters <span class="html-italic">p</span> and <span class="html-italic">q</span>. (<b>a</b>) <span class="html-italic">C<sub>µ</sub></span>, increasing when <span class="html-italic">F</span>(<span class="html-italic">n</span>) has slower decay; (<b>b</b>) <span class="html-italic">h<sub>µ</sub></span>, higher when transition probabilities are maximally stochastic; (<b>c</b>) E, higher the closer the SNS comes to Period 2; (<b>d</b>) <span class="html-italic">b<sub>µ</sub></span>, highest between the maximally-stochastic transition probabilities that maximize <span class="html-italic">h<sub>µ</sub></span> and maximally-deterministic transition probabilities that maximize <b>E</b>.</p> Full article ">
<p>(<b>a</b>) Simple nonunifilar source information anatomy as a function of <span class="html-italic">p</span> with parameters <span class="html-italic">q = p.</span> The single-measurement entropy <span class="html-italic">H</span>[<span class="html-italic">X</span><sub>0</sub>] is the upper solid (red) line, entropy rate <span class="html-italic">h<sub>μ</sub></span> the middle solid (green) line and the bound information <span class="html-italic">b<sub>μ</sub></span> the lower solid (blue) line. Thus, the blue area corresponds to <span class="html-italic">b<sub>μ</sub></span>, the green area to the ephemeral information <span class="html-italic">r<sub>μ</sub> = h<sub>μ</sub> − b<sub>μ</sub></span>, and the red area to the single-symbol redundancy <span class="html-italic"><sup>ρ</sup><sub>μ</sub> = H</span>[<span class="html-italic">X</span><sub>0</sub>] <span class="html-italic">− h<sub>µ</sub>.</span> (<b>b</b>) The components of the predictable information (the excess entropy <b>E</b> = <span class="html-italic">σ<sub>µ</sub></span> + <span class="html-italic">b<sub>µ</sub></span> + <span class="html-italic">q<sub>µ</sub></span> in bits) also as a function of <span class="html-italic">p</span> with <span class="html-italic">q = p.</span> The lowest (blue) line is <span class="html-italic">q<sub>µ</sub>;</span> the middle (green) line is q<span class="html-italic"><sub>µ</sub></span> + <span class="html-italic">b<sub>µ</sub></span>, so that the green area denotes b<span class="html-italic"><sub>µ</sub></span>’s contribution to <b>E</b>. The upper (red) line is <b>E</b>, so that the red area denotes elusive information <span class="html-italic">σ<sub>µ</sub></span> in E. Note that for a large range of <span class="html-italic">p</span>, the co-information <span class="html-italic">q<sub>µ</sub></span> is (slightly) negative.</p> Full article ">
582 KiB  
Article
Entropy, Information and Complexity or Which Aims the Arrow of Time?
by George E. Mikhailovsky and Alexander P. Levich
Entropy 2015, 17(7), 4863-4890; https://doi.org/10.3390/e17074863 - 10 Jul 2015
Cited by 22 | Viewed by 10603
Abstract
In this article, we analyze the interrelationships among such notions as entropy, information, complexity, order and chaos and show using the theory of categories how to generalize the second law of thermodynamics as a law of increasing generalized entropy or a general law [...] Read more.
In this article, we analyze the interrelationships among such notions as entropy, information, complexity, order and chaos and show using the theory of categories how to generalize the second law of thermodynamics as a law of increasing generalized entropy or a general law of complification. This law could be applied to any system with morphisms, including all of our universe and its subsystems. We discuss how such a general law and other laws of nature drive the evolution of the universe, including physicochemical and biological evolutions. In addition, we determine eliminating selection in physicochemical evolution as an extremely simplified prototype of natural selection. Laws of nature do not allow complexity and entropy to reach maximal values by generating structures. One could consider them as a kind of “breeder” of such selection. Full article
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<p>Relation between entropy and complexity for a discrete 8 × 8 space.</p> Full article ">
<p>Dependence of entropy, information, length of description and complexity on the degrees of freedom for a discrete 8 × 8 space.</p> Full article ">
<p>Relation between entropy and complexity for a discrete 16 × 16 space.</p> Full article ">
<p>Dependence of entropy, information, length of description and complexity on the degrees of freedom for a discrete 16 × 16 space.</p> Full article ">
268 KiB  
Article
Faster Together: Collective Quantum Search
by Demosthenes Ellinas and Christos Konstandakis
Entropy 2015, 17(7), 4838-4862; https://doi.org/10.3390/e17074838 - 10 Jul 2015
Cited by 3 | Viewed by 4444
Abstract
Joining independent quantum searches provides novel collective modes of quantum search (merging) by utilizing the algorithm’s underlying algebraic structure. If n quantum searches, each targeting a single item, join the domains of their classical oracle functions and sum their Hilbert spaces (merging), instead [...] Read more.
Joining independent quantum searches provides novel collective modes of quantum search (merging) by utilizing the algorithm’s underlying algebraic structure. If n quantum searches, each targeting a single item, join the domains of their classical oracle functions and sum their Hilbert spaces (merging), instead of acting independently (concatenation), then they achieve a reduction of the search complexity by factor O(?n). Full article
(This article belongs to the Special Issue Quantum Computation and Information: Multi-Particle Aspects)
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<p>Plots for <span class="html-italic">T<sub>con</sub></span><sub>c</sub>/<span class="html-italic">T<sub>merg</sub></span>, for non decreasing and bounded above sequence of database sizes (blue curve), and an unbounded one (black curve). Here the bounded sequence <math display="inline"> <mrow> <msub> <mi>N</mi> <mi>j</mi></msub> <mo>=</mo> <msup> <mn>2</mn> <mrow> <msub> <mi>b</mi> <mi>j</mi></msub></mrow></msup></mrow></math>, <math display="inline"> <mrow> <msub> <mi>b</mi> <mi>j</mi></msub> <mo>=</mo> <mrow> <mo>⌊</mo> <mrow> <mstyle scriptlevel="+1"> <mfrac> <mrow> <mn>6</mn> <msup> <mi>j</mi> <mn>2</mn></msup> <mo>+</mo> <mi>j</mi> <mo>−</mo> <mn>1</mn></mrow> <mrow> <msup> <mi>j</mi> <mn>2</mn></msup> <mo>+</mo> <mn>4</mn></mrow></mfrac></mstyle></mrow> <mo>⌋</mo></mrow></mrow></math>,<span class="html-italic">N</span><sub>1</sub> = 2, <math display="inline"> <mrow> <mi>λ</mi> <mo>=</mo> <mrow> <mo>⌊</mo> <mrow> <mfrac> <mrow> <mstyle scriptlevel="+1"> <mfrac> <mi>π</mi> <mn>4</mn></mfrac></mstyle> <msqrt> <mi>p</mi></msqrt></mrow> <mrow> <mstyle scriptlevel="+1"> <mfrac> <mi>π</mi> <mn>4</mn></mfrac></mstyle> <msqrt> <mi>q</mi></msqrt></mrow></mfrac></mrow> <mo>⌋</mo></mrow></mrow></math>,<span class="html-italic">p</span> = 2<sup>6</sup>, <span class="html-italic">q</span> = <span class="html-italic">N</span><sub>1</sub> = 2, and the unbounded one <span class="html-italic">N<sub>j</sub></span> = 2<span class="html-italic"><sup>j</sup></span>, <span class="html-italic">N</span><sub>1</sub> =2 are used. Dashed line: <span class="html-italic">y</span> = <span class="html-italic">x.</span></p> Full article ">
<p>Young tableaux for m = 6 and the corresponding complexities T<sub>G</sub>.</p> Full article ">
<p>Jensen’s inequality for the numerical example. Round dots represent points lying on the graph, and square dots represent center of mass points.</p> Full article ">
<p>Display of the contour of equal complexity families of joined quantum algorithms. 4. Oracle Algebra and Representations</p> Full article ">
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852 KiB  
Article
Power-Type Functions of Prediction Error of Sea Level Time Series
by Ming Li, Yuanchun Li and Jianxing Leng
Entropy 2015, 17(7), 4809-4837; https://doi.org/10.3390/e17074809 - 9 Jul 2015
Cited by 11 | Viewed by 5842
Abstract
This paper gives the quantitative relationship between prediction error and given past sample size in our research of sea level time series. The present result exhibits that the prediction error of sea level time series in terms of given past sample size follows [...] Read more.
This paper gives the quantitative relationship between prediction error and given past sample size in our research of sea level time series. The present result exhibits that the prediction error of sea level time series in terms of given past sample size follows decayed power functions, providing a quantitative guideline for the quality control of sea level prediction. Full article
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)
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<p>Process of predicting sea level time series from the 5001st value to the 5040th value.</p> Full article ">
<p>Sea level series at the Station LKWF1 in 1999.</p> Full article ">
<p>Prediction results with the sample size 100 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p> Full article ">
<p>Prediction results with the sample size 200 at the Station LKWF1in 1999. Solid line: predicted values, dashed line: original values.</p> Full article ">
<p>Prediction results with the sample size 300 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p> Full article ">
<p>Prediction results with the sample size 400 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p> Full article ">
<p>Prediction results with the sample size 500 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p> Full article ">
<p>Prediction results with the sample size 600 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p> Full article ">
<p>Prediction results with the sample size 700 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.</p> Full article ">
1046 KiB  
Article
Modeling and Analysis of Entropy Generation in Light Heating of Nanoscaled Silicon and Germanium Thin Films
by José Ernesto Nájera-Carpio, Federico Vázquez and Aldo Figueroa
Entropy 2015, 17(7), 4786-4808; https://doi.org/10.3390/e17074786 - 9 Jul 2015
Viewed by 5108
Abstract
In this work, the irreversible processes in light heating of Silicon (Si) and Germanium (Ge) thin films are examined. Each film is exposed to light irradiation with radiative and convective boundary conditions. Heat, electron and hole transport and generation-recombination processes of electron-hole pairs [...] Read more.
In this work, the irreversible processes in light heating of Silicon (Si) and Germanium (Ge) thin films are examined. Each film is exposed to light irradiation with radiative and convective boundary conditions. Heat, electron and hole transport and generation-recombination processes of electron-hole pairs are studied in terms of a phenomenological model obtained from basic principles of irreversible thermodynamics. We present an analysis of the contributions to the entropy production in the stationary state due to the dissipative effects associated with electron and hole transport, generation-recombination of electron-hole pairs as well as heat transport. The most significant contribution to the entropy production comes from the interaction of light with the medium in both Si and Ge. This interaction includes two processes, namely, the generation of electron-hole pairs and the transferring of energy from the absorbed light to the lattice. In Si the following contribution in magnitude comes from the heat transport. In Ge all the remaining contributions to entropy production have nearly the same order of magnitude. The results are compared and explained addressing the differences in the magnitude of the thermodynamic forces, Onsager’s coefficients and transport properties of Si and Ge. Full article
(This article belongs to the Special Issue Entropy Generation in Thermal Systems and Processes 2015)
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Graphical abstract

Graphical abstract
Full article ">
<p>A schematic view of a Si (Ge) thin film of length <span class="html-italic">L</span> with incidence of light from the left.</p> Full article ">
<p>Absorbed light converted in heat <span class="html-italic">vs.</span> the position in the profile as given by the source term <span class="html-italic">P</span><math display="inline"> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>/</mo> <msup> <mi>m</mi> <mn>3</mn></msup></mrow> <mo>)</mo></mrow></mrow></math> in <a href="#fd49-entropy-17-04786" class="html-disp-formula">Equation (B1)</a>. Si (<b>left</b>) and Ge (<b>right</b>).</p> Full article ">
<p>Stationary temperature (K) profiles for Si (<b>left</b>) and Ge (<b>right</b>).</p> Full article ">
<p>Stationary electron density <math display="inline"> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>/</mo> <msup> <mi>m</mi> <mn>3</mn></msup></mrow> <mo>)</mo></mrow></mrow></math> profile for Si (<b>left</b>) and Ge (<b>right</b>).</p> Full article ">
<p>Stationary hole density <math display="inline"> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>/</mo> <msup> <mi>m</mi> <mn>3</mn></msup></mrow> <mo>)</mo></mrow></mrow></math> profile for Si (<b>left</b>) and Ge (<b>right</b>).</p> Full article ">
<p>Distribution in the profile of entropy production <math display="inline"> <mrow> <mrow> <mo>(</mo> <mrow> <mi>J</mi> <mo>/</mo> <mi>K</mi> <msup> <mi>m</mi> <mn>3</mn></msup> <mi>s</mi></mrow> <mo>)</mo></mrow></mrow></math> from heat transport. The figure on the left corresponds to Si and that on the right to Ge.</p> Full article ">
<p>Entropy produced <math display="inline"> <mrow> <mrow> <mo>(</mo> <mrow> <mi>J</mi> <mo>/</mo> <mi>K</mi> <msup> <mi>m</mi> <mn>3</mn></msup> <mi>s</mi></mrow> <mo>)</mo></mrow></mrow></math> by particle transport processes in Si (<b>left</b>) and Ge (<b>right</b>). Above it can be seen the entropy production due to electron flux and below that due to hole flux.</p> Full article ">
<p>Entropy produced <math display="inline"> <mrow> <mrow> <mo>(</mo> <mrow> <mi>J</mi> <mo>/</mo> <mi>K</mi> <msup> <mi>m</mi> <mn>3</mn></msup> <mi>s</mi></mrow> <mo>)</mo></mrow></mrow></math> by particle transport processes in Si (<b>left</b>) and Ge (<b>right</b>). Above it can be seen the entropy production due to electron flux and below that due to hole flux.</p> Full article ">
<p>Entropy production <math display="inline"> <mrow> <mrow> <mo>(</mo> <mrow> <mi>J</mi> <mo>/</mo> <mi>K</mi> <msup> <mi>m</mi> <mn>3</mn></msup> <mi>s</mi></mrow> <mo>)</mo></mrow></mrow></math> from the interaction of the incident light with the material. The figure on the left corresponds to Si and those on the right to Ge.</p> Full article ">
<p>Partial entropy production (<a href="#fd12-entropy-17-04786" class="html-disp-formula">Equation (11)</a>) in the profile (it does not includes the interaction light-matter processes) for Si (<b>left</b>) and Ge (<b>right</b>). The entropy production is in <math display="inline"> <mrow> <mrow> <mo>(</mo> <mrow> <mi>J</mi> <mo>/</mo> <mi>K</mi> <msup> <mi>m</mi> <mn>3</mn></msup> <mi>s</mi></mrow> <mo>)</mo></mrow></mrow></math>.</p> Full article ">
872 KiB  
Article
Fractional Differential Texture Descriptors Based on the Machado Entropy for Image Splicing Detection
by Rabha W. Ibrahim, Zahra Moghaddasi, Hamid A. Jalab and Rafidah Md Noor
Entropy 2015, 17(7), 4775-4785; https://doi.org/10.3390/e17074775 - 8 Jul 2015
Cited by 33 | Viewed by 5646
Abstract
Image splicing is a common operation in image forgery. Different techniques of image splicing detection have been utilized to regain people’s trust. This study introduces a texture enhancement technique involving the use of fractional differential masks based on the Machado entropy. The masks [...] Read more.
Image splicing is a common operation in image forgery. Different techniques of image splicing detection have been utilized to regain people’s trust. This study introduces a texture enhancement technique involving the use of fractional differential masks based on the Machado entropy. The masks slide over the tampered image, and each pixel of the tampered image is convolved with the fractional mask weight window on eight directions. Consequently, the fractional differential texture descriptors are extracted using the gray-level co-occurrence matrix for image splicing detection. The support vector machine is used as a classifier that distinguishes between authentic and spliced images. Results prove that the achieved improvements of the proposed algorithm are compatible with other splicing detection methods. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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<p>Standard deviation distributions of extracted features. Rows indicate the standard deviation distributions of features extracted from gray-scale images. The first column indicates the original features. The second column shows the features after applying kernel PCA.</p> Full article ">
<p>Samples of DVMM image dataset.</p> Full article ">
<p>Selection of α value.</p> Full article ">
<p>Comparison between the features with 1764-D and features with Kernel PCA in 40-D.</p> Full article ">
436 KiB  
Article
H? Control for Markov Jump Systems with Nonlinear Noise Intensity Function and Uncertain Transition Rates
by Xiaonian Wang and Yafeng Guo
Entropy 2015, 17(7), 4762-4774; https://doi.org/10.3390/e17074762 - 6 Jul 2015
Cited by 2 | Viewed by 4107
Abstract
The problem of robust H? control is investigated for Markov jump systems with nonlinear noise intensity function and uncertain transition rates. A robust H? performance criterion is developed for the given systems for the first time. Based on the developed performance [...] Read more.
The problem of robust H? control is investigated for Markov jump systems with nonlinear noise intensity function and uncertain transition rates. A robust H? performance criterion is developed for the given systems for the first time. Based on the developed performance criterion, the desired H? state-feedback controller is also designed, which guarantees the robust H? performance of the closed-loop system. All the conditions are in terms of linear matrix inequalities (LMIs), and hence they can be readily solved by any LMI solver. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed methods. Full article
(This article belongs to the Special Issue Complex and Fractional Dynamics)
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<p>State response of the open-loop system with 1000 random samplings.</p> Full article ">
<p>State response of the closed-loop system with 1000 random samplings.</p> Full article ">
<p>Functional cost <span class="html-italic">J</span>(<span class="html-italic">T<sub>f</sub></span>) with 1000 random samplings.</p> Full article ">
1677 KiB  
Article
Energetic and Exergetic Analysis of an Ejector-Expansion Refrigeration Cycle Using the Working Fluid R32
by Zhenying Zhang, Lirui Tong, Li Chang, Yanhua Chen and Xingguo Wang
Entropy 2015, 17(7), 4744-4761; https://doi.org/10.3390/e17074744 - 6 Jul 2015
Cited by 20 | Viewed by 7259
Abstract
The performance characteristics of an ejector-expansion refrigeration cycle (EEC) using R32 have been investigated in comparison with that using R134a. The coefficient of performance (COP), the exergy destruction, the exergy efficiency and the suction nozzle pressure drop (SNPD) are discussed. The results show [...] Read more.
The performance characteristics of an ejector-expansion refrigeration cycle (EEC) using R32 have been investigated in comparison with that using R134a. The coefficient of performance (COP), the exergy destruction, the exergy efficiency and the suction nozzle pressure drop (SNPD) are discussed. The results show that the application of an ejector instead of a throttle valve in R32 cycle decreases the cycle’s total exergy destruction by 8.84%–15.84% in comparison with the basic cycle (BC). The R32 EEC provides 5.22%–13.77% COP improvement and 5.13%–13.83% exergy efficiency improvement respectively over the BC for the given ranges of evaporating and condensing temperatures. There exists an optimum suction nozzle pressure drop (SNPD) which gives a maximum system COP and volumetric cooling capacity (VCC) under a specified condition. The value of the optimum SNPD mainly depends on the efficiencies of the ejector components, but is virtually independent of evaporating temperature and condensing temperature. In addition, the improvement of the component efficiency, especially the efficiencies of diffusion nozzle and the motive nozzle, can enhance the EEC performance. Full article
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<p>Schematic and P-h diagram of the ejector-expansion refrigeration cycle.</p> Full article ">
<p>Comparison of the present simulation results with Li <span class="html-italic">et al.</span> [<a href="#b19-entropy-17-04744" class="html-bibr">19</a>] results.</p> Full article ">
<p>Influence of SNPD on COP and iCOP.</p> Full article ">
<p>Influence of SNPD on VCC and iVCC.</p> Full article ">
<p>Influence of SNPD on CPR and PLR.</p> Full article ">
<p>Influence of SNPD on μ.</p> Full article ">
<p>Influence of AR on COP and iCOP.</p> Full article ">
<p>Influence of condensing temperature on COP and iCOP.</p> Full article ">
<p>Influence of evaporating temperature on COP and iCOP.</p> Full article ">
399 KiB  
Article
Asymptotic Description of Neural Networks with Correlated Synaptic Weights
by Olivier Faugeras and James MacLaurin
Entropy 2015, 17(7), 4701-4743; https://doi.org/10.3390/e17074701 - 6 Jul 2015
Cited by 9 | Viewed by 5531
Abstract
We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network [...] Read more.
We study the asymptotic law of a network of interacting neurons when the number of neurons becomes infinite. Given a completely connected network of neurons in which the synaptic weights are Gaussian correlated random variables, we describe the asymptotic law of the network when the number of neurons goes to infinity. We introduce the process-level empirical measure of the trajectories of the solutions to the equations of the finite network of neurons and the averaged law (with respect to the synaptic weights) of the trajectories of the solutions to the equations of the network of neurons. The main result of this article is that the image law through the empirical measure satisfies a large deviation principle with a good rate function which is shown to have a unique global minimum. Our analysis of the rate function allows us also to characterize the limit measure as the image of a stationary Gaussian measure defined on a transformed set of trajectories. Full article
(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)
1597 KiB  
Article
Geometric Interpretation of Surface Tension Equilibrium in Superhydrophobic Systems
by Michael Nosonovsky and Rahul Ramachandran
Entropy 2015, 17(7), 4684-4700; https://doi.org/10.3390/e17074684 - 6 Jul 2015
Cited by 34 | Viewed by 13075
Abstract
Surface tension and surface energy are closely related, although not identical concepts. Surface tension is a generalized force; unlike a conventional mechanical force, it is not applied to any particular body or point. Using this notion, we suggest a simple geometric interpretation of [...] Read more.
Surface tension and surface energy are closely related, although not identical concepts. Surface tension is a generalized force; unlike a conventional mechanical force, it is not applied to any particular body or point. Using this notion, we suggest a simple geometric interpretation of the Young, Wenzel, Cassie, Antonoff and Girifalco–Good equations for the equilibrium during wetting. This approach extends the traditional concept of Neumann’s triangle. Substances are presented as points, while tensions are vectors connecting the points, and the equations and inequalities of wetting equilibrium obtain simple geometric meaning with the surface roughness effect interpreted as stretching of corresponding vectors; surface heterogeneity is their linear combination, and contact angle hysteresis is rotation. We discuss energy dissipation mechanisms during wetting due to contact angle hysteresis, the superhydrophobicity and the possible entropic nature of the surface tension. Full article
(This article belongs to the Special Issue Geometry in Thermodynamics)
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<p>(<b>a</b>) The popular, yet ambiguous description of the origin of surface tension; the imbalance of cohesive forces between molecules due to the lack of bonds at the surface results in surface tension. (<b>b</b>) The network of tetrahedral water molecules at the air-water interface is more ordered than that in the bulk. Consequently, the decrease of entropy <span class="html-italic">TΔS</span> due to the additional orderliness in the surface layer may be partially responsible for the surface tension. (<b>c</b>) When the interface between A and B is displaced along the vector <math display="inline"> <mrow> <mspace width="0.2em"/> <mover accent="true"> <mrow> <mi>d</mi> <mi>r</mi></mrow> <mo stretchy="true">→</mo></mover></mrow></math>, the surface tension force acts on the three-phase line <span class="html-italic">l</span> in the direction of the normal <math display="inline"> <mrow> <mspace width="0.2em"/> <mover accent="true"> <mi>n</mi> <mo>→</mo></mover></mrow></math>.</p> Full article ">
<p>(<b>a</b>) Surface tension forces at the three-phase line on a non-deformable solid surface. The vertical components of the forces remain unbalanced. (<b>b</b>) The equilibrium of surface tensions at the three-phase line of liquid Phases A, B and C. Note the deformation of the interface between A and C.</p> Full article ">
<p>(<b>a</b>) A liquid droplet in the Wenzel state, with a homogenous solid to liquid interface below the droplet; (<b>b</b>) a liquid droplet in the Cassie–Baxter state, with a composite solid to liquid to vapor interface below the droplet; (<b>c</b>) contact angle hysteresis (<span class="html-italic">CAH</span>) measurement by tilting the droplet. The maximum or advancing (<span class="html-italic">θ<sub>adv</sub></span>) and minimum or receding (<span class="html-italic">θ<sub>rec</sub></span>) contact angles are measured at the front and rear of a moving droplet, respectively.</p> Full article ">
<p>(<b>a</b>) Equilibrium of surface tension vectors at the three-phase line; (<b>b</b>) Neumann’s triangle for a three-phase system; (<b>c</b>) when Antonoff’s rule is an exact equality, the surface tension vectors lie on the same line; this corresponds to complete wetting; (<b>d</b>) geometric interpretation of the Girifalco and Good equation.</p> Full article ">
<p>Neumann’s triangle for a three-phase system in the Wenzel state.</p> Full article ">
<p>(<b>a</b>,<b>b</b>) Neumann’s triangles for three-phase systems S1AW and S2AW, respectively; (<b>c</b>) Neumann’s triangle for a three-phase Cassie–Baxter state.</p> Full article ">
<p>(<b>a</b>) Contact angle hysteresis represented using Neumann’s triangle as a rotation of the vector <math display="inline"> <mrow> <mover accent="true"> <mrow> <msub> <mi>γ</mi> <mrow> <mi>w</mi> <mi>a</mi></mrow></msub></mrow> <mo stretchy="true">→</mo></mover></mrow></math>. A range of contact angles are possible under the constraint that <math display="inline"> <mrow> <mrow> <mo stretchy="false">|</mo> <mrow> <mspace width="0.2em"/> <mover accent="true"> <mrow> <msub> <mi>γ</mi> <mrow> <mi>w</mi> <mi>a</mi></mrow></msub></mrow> <mo stretchy="true">→</mo></mover></mrow> <mo stretchy="false">|</mo></mrow></mrow></math> remains constant. (<b>b</b>) The forces involved in contact angle hysteresis. For a droplet on a tilted surface, the pressure inside acting normal to the surface is the sum of Laplace, as well as hydrostatic pressures. (<b>c</b>) The components of the surface tension vectors normal to the surface balance the pressure force, while the components of the surface tension vectors along the surface balance the friction force.</p> Full article ">
<p>(<b>a</b>) Tetrahedron of surface tension vectors in 3D space for a four-phase system; (<b>b</b>) a four-phase system on aluminum-graphite composite, which consists of a vegetable oil (γ<sub>o</sub> = 0.032 Jm<sup>−2</sup>) droplet (volume about 5 μL) on Al-C composite immersed in water (γ<sub>w</sub> = 0.072 Jm<sup>−2</sup>) with pockets of air trapped on the Al-C surface forming the fourth phase; (<b>c</b>) the tetrahedron of surface tension vectors for the four-phase system.</p> Full article ">
1278 KiB  
Article
A New Feature Extraction Method Based on the Information Fusion of Entropy Matrix and Covariance Matrix and Its Application in Face Recognition
by Shunfang Wang and Ping Liu
Entropy 2015, 17(7), 4664-4683; https://doi.org/10.3390/e17074664 - 3 Jul 2015
Cited by 7 | Viewed by 5215
Abstract
The classic principal components analysis (PCA), kernel PCA (KPCA) and linear discriminant analysis (LDA) feature extraction methods evaluate the importance of components according to their covariance contribution, not considering the entropy contribution, which is important supplementary information for the covariance. To further improve [...] Read more.
The classic principal components analysis (PCA), kernel PCA (KPCA) and linear discriminant analysis (LDA) feature extraction methods evaluate the importance of components according to their covariance contribution, not considering the entropy contribution, which is important supplementary information for the covariance. To further improve the covariance-based methods such as PCA (or KPCA), this paper firstly proposed an entropy matrix to load the uncertainty information of random variables similar to the covariance matrix loading the variation information in PCA. Then an entropy-difference matrix was used as a weighting matrix for transforming the original training images. This entropy-difference weighting (EW) matrix not only made good use of the local information of the training samples, contrast to the global method of PCA, but also considered the category information similar to LDA idea. Then the EW method was integrated with PCA (or KPCA), to form new feature extracting method. The new method was used for face recognition with the nearest neighbor classifier. The experimental results based on the ORL and Yale databases showed that the proposed method with proper threshold parameters reached higher recognition rates than the usual PCA (or KPCA) methods. Full article
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<p>The comparison of the original ORL images and the images ideally processed by EW method. (a) the original ORL images; (b) the processed images by EW ( <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mn>52</mn></mrow></math>).</p> Full article ">
<p>The comparison of the originally ORL images and the processed images by EW method with different threshold values. <b>(a)</b> the original ORL images; <b>(b)</b> the processed images by EW ( <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mn>52</mn></mrow></math>); <b>(c)</b> the processed images by EW ( <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mn>42</mn></mrow></math>); <b>(d)</b> the processed images by EW ( <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mn>62</mn></mrow></math>).</p> Full article ">
<p>The comparison of the originally Yale images and the images ideally processed by EW method. (a) the original Yale images; (b) the processed images by EW ( <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mn>24</mn></mrow></math>).</p> Full article ">
<p>The comparison of the originally Yale images and the images processed by EW method. (a) the original Yale images; (b) the processed images by EW ( <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mn>14</mn></mrow></math>); (c) the processed images by EW ( <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mn>24</mn></mrow></math>); (d) the processed images by EW ( <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mn>34</mn></mrow></math>).</p> Full article ">
<p>Comparison of EW-PCA and PCA with ORL and Program 1.</p> Full article ">
<p>Comparison of EW-PCA and PCA with ORL and Program 2.</p> Full article ">
<p>Comparison of EW-PCA and PCA with ORL and Program 3.</p> Full article ">
<p>Comparison of EW-PCA and PCA with Yale and Program 1.</p> Full article ">
<p>Comparison of EW-PCA and PCA with Yale and Program 2.</p> Full article ">
1350 KiB  
Article
Continuous Variable Quantum Key Distribution with a Noisy Laser
by Christian S. Jacobsen, Tobias Gehring and Ulrik L. Andersen
Entropy 2015, 17(7), 4654-4663; https://doi.org/10.3390/e17074654 - 3 Jul 2015
Cited by 22 | Viewed by 5675 | Correction
Abstract
Existing experimental implementations of continuous-variable quantum key distribution require shot-noise limited operation, achieved with shot-noise limited lasers. However, loosening this requirement on the laser source would allow for cheaper, potentially integrated systems. Here, we implement a theoretically proposed prepare-and-measure continuous-variable protocol and experimentally [...] Read more.
Existing experimental implementations of continuous-variable quantum key distribution require shot-noise limited operation, achieved with shot-noise limited lasers. However, loosening this requirement on the laser source would allow for cheaper, potentially integrated systems. Here, we implement a theoretically proposed prepare-and-measure continuous-variable protocol and experimentally demonstrate the robustness of it against preparation noise stemming for instance from technical laser noise. Provided that direct reconciliation techniques are used in the post-processing we show that for small distances large amounts of preparation noise can be tolerated in contrast to reverse reconciliation where the key rate quickly drops to zero. Our experiment thereby demonstrates that quantum key distribution with non-shot-noise limited laser diodes might be feasible. Full article
(This article belongs to the Special Issue Quantum Cryptography)
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<p>Equivalent entanglement based model used in the security proof. Alice produces a noisy Einstein–Podolsky–Rosen (EPR) state which she sends to Bob. The quantum channel with transmission <span class="html-italic">T</span> is controlled by the eavesdropper who injects an EPR state with variance <span class="html-italic">W</span>.</p> Full article ">
<p>Contour plots of the secure key generation rate for varying preparation noise in shot-noise units (SNUs) and transmission <span class="html-italic">T</span> for (<b>a</b>) reverse reconciliation and (<b>b</b>) direct reconciliation. The error reconciliation efficiency was set to <span class="html-italic">β</span> = 95%, the modulation variance was 32 SNUs and the channel excess noise 0.11. The dashed lines indicate the minimal possible transmission of a channel where a positive secret key rate can still be obtained, in the ideal case for <span class="html-italic">β</span> = 1, no channel excess noise and in the limit of high modulation variance. (a) For no preparation noise (<span class="html-italic">κ</span> = 0), the rate decreases asymptotically to zero as the transmission approaches zero. When the preparation noise increases the security of reverse reconciliation is quickly compromised, to the point where almost unity transmission is required to achieve security. (b) For heterodyne detection and no preparation noise the rate goes to zero at about 79% transmission, due to the extra unit of vacuum introduced by heterodyne detection. The plot shows the robustness of direct reconciliation to preparation noise.</p> Full article ">
<p>Schematic representation of the experiment. A shot-noise limited laser is amplitude- and phase-modulated with two independent white-noise sources to simulate a noisy laser. Subsequently, Alice modulates the noisy laser beam in amplitude and phase using a known modulation and sends it to Bob through the quantum channel who performs heterodyne detection. The quantum channel’s transmission was simulated by an (for coherent states) equivalent reduction of the modulation variances. AM: Amplitude Modulation. PM: Phase Modulation. PD: Photo Detector.</p> Full article ">
<p>Measured data and theory curves for different levels of preparation noise using (<b>a</b>) reverse reconciliation and (<b>b</b>) direct reconciliation in the post-processing. Error reconciliation efficiency <span class="html-italic">β</span> = 95%. Due to our simulation of losses (see main text) the error bars on the channel loss are negligibly small and, thus, not shown in the figure.</p> Full article ">
3093 KiB  
Article
Quantifying Redundant Information in Predicting a Target Random Variable
by Virgil Griffith and Tracey Ho
Entropy 2015, 17(7), 4644-4653; https://doi.org/10.3390/e17074644 - 2 Jul 2015
Cited by 24 | Viewed by 5941
Abstract
We consider the problem of defining a measure of redundant information that quantifies how much common information two or more random variables specify about a target random variable. We discussed desired properties of such a measure, and propose new measures with some desirable [...] Read more.
We consider the problem of defining a measure of redundant information that quantifies how much common information two or more random variables specify about a target random variable. We discussed desired properties of such a measure, and propose new measures with some desirable properties. Full article
(This article belongs to the Special Issue Information Processing in Complex Systems)
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<p>PI-diagrams for <span class="html-italic">n</span> = 2 predictors, showing the amount of redundant (yellow/bottom), unique (magenta/left and right) and synergistic (cyan/top) information with respect to the target <span class="html-italic">Y</span>.</p> Full article ">
<p>Example UNQ. <span class="html-italic">X</span><sub>1</sub> and <span class="html-italic">X</span><sub>2</sub> each uniquely carry one bit of information about <span class="html-italic">Y</span>. I(<span class="html-italic">X</span><sub>1</sub><span class="html-italic">X</span><sub>2</sub>:<span class="html-italic">Y</span>) = H(<span class="html-italic">Y</span>) = 2 bits.</p> Full article ">
<p>Example RDNXOR. This is the canonical example of redundancy and synergy coexisting. I<sub>min</sub> and I<sub>Λ</sub> each reach the desired decomposition of one bit of redundancy and one bit of synergy. This example demonstrates I<sub>Λ</sub> correctly extracting the embedded redundant bit within <span class="html-italic">X</span><sub>1</sub> and <span class="html-italic">X</span><sub>2</sub>.</p> Full article ">
<p>Example AND. It is universally agreed that the redundant information is between [0, 0.311] bits. The most compelling argument is from [<a href="#b15-entropy-17-04644" class="html-bibr">15</a>] arguing for 0.311 bits of redundant information.</p> Full article ">
<p>Example IMPERFECTRDN. I<sub>Λ</sub> is blind to the noisy correlation between <span class="html-italic">X</span><sub>1</sub> and <span class="html-italic">X</span><sub>2</sub> and calculates zero redundant information. An ideal I<sub>∩</sub> measure would detect that all of the information <span class="html-italic">X</span><sub>2</sub> specifies about <span class="html-italic">Y</span> is also specified by <span class="html-italic">X</span><sub>1</sub> to calculate I<sub>∩</sub>({<span class="html-italic">X</span><sub>1</sub>, <span class="html-italic">X</span><sub>2</sub>}:<span class="html-italic">Y</span>) = 0.99 bits.</p> Full article ">
<p>Example SUBTLE.</p> Full article ">
1470 KiB  
Article
Differentiating Interictal and Ictal States in Childhood Absence Epilepsy through Permutation Rényi Entropy
by Nadia Mammone, Jonas Duun-Henriksen, Troels W. Kjaer and Francesco C. Morabito
Entropy 2015, 17(7), 4627-4643; https://doi.org/10.3390/e17074627 - 2 Jul 2015
Cited by 44 | Viewed by 7721
Abstract
Permutation entropy (PE) has been widely exploited to measure the complexity of the electroencephalogram (EEG), especially when complexity is linked to diagnostic information embedded in the EEG. Recently, the authors proposed a spatial-temporal analysis of the EEG recordings of absence epilepsy patients based [...] Read more.
Permutation entropy (PE) has been widely exploited to measure the complexity of the electroencephalogram (EEG), especially when complexity is linked to diagnostic information embedded in the EEG. Recently, the authors proposed a spatial-temporal analysis of the EEG recordings of absence epilepsy patients based on PE. The goal here is to improve the ability of PE in discriminating interictal states from ictal states in absence seizure EEG. For this purpose, a parametrical definition of permutation entropy is introduced here in the field of epileptic EEG analysis: the permutation Rényi entropy (PEr). PEr has been extensively tested against PE by tuning the involved parameters (order, delay time and alpha). The achieved results demonstrate that PEr outperforms PE, as there is a statistically-significant, wider gap between the PEr levels during the interictal states and PEr levels observed in the ictal states compared to PE. PEr also outperformed PE as the input to a classifier aimed at discriminating interictal from ictal states. Full article
(This article belongs to the Special Issue Entropy in Human Brain Networks)
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<p>The flowchart of the procedure. (1) The <span class="html-italic">n</span>-channel EEG is recorded and stored on a computer; (2) the EEG is partitioned into <span class="html-italic">m</span> overlapping windows and processed window by window; (3) given a window under analysis, permutation entropy (PE) is estimated channel by channel, and the <span class="html-italic">n</span> values are arranged in an <span class="html-italic">n ×</span> 1 PE vector; (4) once the <span class="html-italic">m</span> PE vectors were estimated, the <span class="html-italic">n × m</span> matrix showing the PE trend of each channel can be displayed. PEr, permutation Rényi entropy.</p> Full article ">
<p>The transient logistic map data <span class="html-italic">x</span> (top), variations of PE and PEr with <span class="html-italic">r</span> for <span class="html-italic">m</span> = 5, <span class="html-italic">L</span> = 1 (middle), and <span class="html-italic">m</span> = 5, <span class="html-italic">L</span> = 2 (bottom).</p> Full article ">
<p>R <span class="html-italic">vs</span>. D in every possible parameter setting for either PE (blue circles) or PEr (red circles). The parameter setting that ensured the highest <span class="html-italic">Di</span> and <span class="html-italic">Ri</span> for PE was <span class="html-italic">m</span> = 4, <span class="html-italic">L</span> = 8 (bold blue circle), whereas the parameters that ensured the best performance for PEr with respect to either D or R was <span class="html-italic">m</span> = 4, <span class="html-italic">L</span> = 7 and <span class="html-italic">α</span> = 7 (red bold circle; the configuration ensured 96.5% of both the highest <span class="html-italic">D</span> and the highest <span class="html-italic">R</span>).</p> Full article ">
<p>Effect of alpha on the behavior of PEr. The Figure shows the PE and PEr profiles of an EEG trace (from Channel Fp2), estimated in the same parameter configuration (<span class="html-italic">m</span> = 4 and <span class="html-italic">L</span> = 8, which is optimal for PE, but not for PEr). In the PEr estimation, <span class="html-italic">α</span> ranged from 2 to 7. The seizure onset is marked with a vertical dashed line.</p> Full article ">
<p>PEr profiles estimated for all of the channels, computed with <span class="html-italic">m</span> = 4, <span class="html-italic">L</span> = 7 and <span class="html-italic">α</span> = 7.</p> Full article ">
<p>Boxplot of R and D vectors. On each box, the central mark is the median; the edges of the box are the 25th and 75th percentiles; the whiskers extend to the most extreme data points not considered as outliers.</p> Full article ">
<p>Boxplot of <b>avgPEr(interictal)</b> and <b>avgPEr(ictal)</b> vectors. On each box, the central mark is the median; the edges of the box are the 25th and 75th percentiles; the whiskers extend to the most extreme data points not considered as outliers.</p> Full article ">
<p>Investigation on the effect of the EEG channel on the discrimination ability of PEr for ictal/interictal segments. Given a patient, PEr profiles are averaged over every cerebral area of interest (frontal, temporal, parietal, central, occipital), then the elements of <b>R</b> vector are calculated as avgPEr(interictal/avgPEr(ictal). At the end of the process, we have one R vectors with <span class="html-italic">n</span> elements, where <span class="html-italic">n</span> is the number of patients, for every specific area. The boxplot of such <b>R</b> vectors is shown. On each box, the central mark is the median; the edges of the box are the 25th and 75th percentiles; the whiskers extend to the most extreme data points not considered as outliers.</p> Full article ">
<p>Comparison of the accuracy in the cerebral state classification (interictal or ictal) provided by PE + learning vector quantization (LVQ) and by PEr + LVQ.</p> Full article ">
2403 KiB  
Article
A New Robust Regression Method Based on Minimization of Geodesic Distances on a Probabilistic Manifold: Application to Power Laws
by Geert Verdoolaege
Entropy 2015, 17(7), 4602-4626; https://doi.org/10.3390/e17074602 - 1 Jul 2015
Cited by 10 | Viewed by 5805
Abstract
In regression analysis for deriving scaling laws that occur in various scientific disciplines, usually standard regression methods have been applied, of which ordinary least squares (OLS) is the most popular. In many situations, the assumptions underlying OLS are not fulfilled, and several other [...] Read more.
In regression analysis for deriving scaling laws that occur in various scientific disciplines, usually standard regression methods have been applied, of which ordinary least squares (OLS) is the most popular. In many situations, the assumptions underlying OLS are not fulfilled, and several other approaches have been proposed. However, most techniques address only part of the shortcomings of OLS. We here discuss a new and more general regression method, which we call geodesic least squares regression (GLS). The method is based on minimization of the Rao geodesic distance on a probabilistic manifold. For the case of a power law, we demonstrate the robustness of the method on synthetic data in the presence of significant uncertainty on both the data and the regression model. We then show good performance of the method in an application to a scaling law in magnetic confinement fusion. Full article
(This article belongs to the Special Issue Information, Entropy and Their Geometric Structures)
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<p>(<b>a</b>) Illustration of the Poincaré half-plane with several half-circle geodesics in the background, together with the geodesic between the points <span class="html-italic">p</span><sub>1</sub> and <span class="html-italic">p</span><sub>2</sub> and between <span class="html-italic">p</span><sub>3</sub> and <span class="html-italic">p</span><sub>4</sub>, defined in the main text. (<b>b</b>) Probability densities corresponding to the points <span class="html-italic">p</span><sub>1</sub>, <span class="html-italic">p</span><sub>2</sub>, <span class="html-italic">p</span><sub>3</sub> and <span class="html-italic">p</span><sub>4</sub> indicated in (a). The densities associated with some intermediate points on the geodesics between <span class="html-italic">p</span><sub>1</sub> and <span class="html-italic">p</span><sub>2</sub> and between <span class="html-italic">p</span><sub>3</sub> and <span class="html-italic">p</span><sub>4</sub> are also drawn. (<b>c</b>) Rendering of one blade of the tractroid, again with the two geodesics superimposed. The parallels of the tractroid are lines of constant standard deviation σ, while the meridians (the tractrices) are lines of constant mean µ. This representation of the normal manifold is periodic in the µ-direction, and a rescaled version (longer period along µ) is shown in (<b>d</b>).</p> Full article ">
<p>A portion of the pseudosphere together with the regression results on synthetic data with an outlier, as described in the main text.</p> Full article ">
<p>(<b>a</b>) Histograms of the relative error in estimating the regression coefficients β<span class="html-italic"><sub>i</sub></span> by means of OLS, MAP and GLS for a linear regression problem with outliers. Horizontal axes represent the error in percent and vertical axes probability, normalized to one. (<b>b</b>) Similar, for TLS and ROB.</p> Full article ">
<p>Histograms of the relative error in estimating the regression coefficients β<span class="html-italic"><sub>i</sub></span> by means of OLS, MAP and GLS for a power-law regression problem after a logarithmic transformation. Horizontal axes represent the error in percent and vertical axes probability, normalized to one. (<b>b</b>) Similar, for TLS and ROB.</p> Full article ">
522 KiB  
Article
Applications of the Fuzzy Sumudu Transform for the Solution of First Order Fuzzy Differential Equations
by Norazrizal Aswad Abdul Rahman and Muhammad Zaini Ahmad
Entropy 2015, 17(7), 4582-4601; https://doi.org/10.3390/e17074582 - 1 Jul 2015
Cited by 19 | Viewed by 5924
Abstract
In this paper, we study the classical Sumudu transform in fuzzy environment, referred to as the fuzzy Sumudu transform (FST). We also propose some results on the properties of the FST, such as linearity, preserving, fuzzy derivative, shifting and convolution theorem. In order [...] Read more.
In this paper, we study the classical Sumudu transform in fuzzy environment, referred to as the fuzzy Sumudu transform (FST). We also propose some results on the properties of the FST, such as linearity, preserving, fuzzy derivative, shifting and convolution theorem. In order to show the capability of the FST, we provide a detailed procedure to solve fuzzy differential equations (FDEs). A numerical example is provided to illustrate the usage of the FST. Full article
(This article belongs to the Special Issue Dynamical Equations and Causal Structures from Observations)
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<p>The solution of <a href="#fd80-entropy-17-04582" class="html-disp-formula">Equation (12)</a> for Case 1 when <span class="html-italic">Y</span>(<span class="html-italic">t</span><sub>0</sub>) = (<span class="html-italic">−</span>1, 0, 1).</p> Full article ">
<p>The solution of <a href="#fd80-entropy-17-04582" class="html-disp-formula">Equation (12)</a> for Case 2 when <span class="html-italic">Y</span>(<span class="html-italic">t</span><sub>0</sub>) = (<span class="html-italic">−</span>1, 0, 1).</p> Full article ">
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