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Entropy, Volume 21, Issue 2 (February 2019) – 119 articles

Cover Story (view full-size image): The compositionally complex alloy Al10Co25Cr8Fe15Ni36Ti6 stands out for its potential application around 800 °C. Its mechanical properties are influenced by the casting procedure and heat treatment. Directional solidification leads to better and reproducible high-temperature tensile test results. The microstructure and tensile properties can also be modified by adding Mo and Hf. The microstructure, consisting of fcc-matrix, γ' and Heusler phase, is retained, but shape and volume fractions of the phases change with these elements. After optimization of heat treatment, Hf promotes more cubic γ' precipitates, a globular Heusler phase, and better tensile strength than the base alloy. Mo has no influence on the Heusler phase but makes γ' precipitates rounder and decreases the tensile strength. View this paper.
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23 pages, 23141 KiB  
Article
Assessment of Landslide Susceptibility Using Integrated Ensemble Fractal Dimension with Kernel Logistic Regression Model
by Tingyu Zhang, Ling Han, Jichang Han, Xian Li, Heng Zhang and Hao Wang
Entropy 2019, 21(2), 218; https://doi.org/10.3390/e21020218 - 24 Feb 2019
Cited by 39 | Viewed by 4260
Abstract
The main aim of this study was to compare and evaluate the performance of fractal dimension as input data in the landslide susceptibility mapping of the Baota District, Yan’an City, China. First, a total of 632 points, including 316 landslide points and 316 [...] Read more.
The main aim of this study was to compare and evaluate the performance of fractal dimension as input data in the landslide susceptibility mapping of the Baota District, Yan’an City, China. First, a total of 632 points, including 316 landslide points and 316 non-landslide points, were located in the landslide inventory map. All points were divided into two parts according to the ratio of 70%:30%, with 70% (442) of the points used as the training dataset to train the models, and the remaining, namely the validation dataset, applied for validation. Second, 13 predisposing factors, including slope aspect, slope angle, altitude, lithology, mean annual precipitation (MAP), distance to rivers, distance to faults, distance to roads, normalized differential vegetation index (NDVI), topographic wetness index (TWI), plan curvature, profile curvature, and terrain roughness index (TRI), were selected. Then, the original numerical data, box-counting dimension, and correlation dimension corresponding to each predisposing factor were calculated to generate the input data and build three classification models, namely the kernel logistic regression model (KLR), kernel logistic regression based on box-counting dimension model (KLRbox-counting), and the kernel logistic regression based on correlation dimension model (KLRcorrelation). Next, the statistical indexes and the receiver operating characteristic (ROC) curve were employed to evaluate the models’ performance. Finally, the KLRcorrelation model had the highest area under the curve (AUC) values of 0.8984 and 0.9224, obtained by the training and validation datasets, respectively, indicating that the fractal dimension can be used as the input data for landslide susceptibility mapping with a better effect. Full article
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Figure 1

Figure 1
<p>The location of the study area and the landslide inventory map.</p> Full article ">Figure 2
<p>Landslide predisposing factor maps involving: (<b>a</b>) Slope aspect; (<b>b</b>) Slope angle; (<b>c</b>) Altitude; (<b>d</b>) Lithology; (<b>e</b>) Distance to faults; (<b>f</b>) Distance to rivers; (<b>g</b>) Distance to roads; (<b>h</b>) Mean annual precipitation (MAP); (<b>i</b>) Normalized differential vegetation index (NDVI); (<b>j</b>) Profile curvature; (<b>k</b>) plan curvature; (<b>l</b>) Topographic wetness index (TWI); (<b>m</b>) Terrain roughness index (TRI).</p> Full article ">Figure 2 Cont.
<p>Landslide predisposing factor maps involving: (<b>a</b>) Slope aspect; (<b>b</b>) Slope angle; (<b>c</b>) Altitude; (<b>d</b>) Lithology; (<b>e</b>) Distance to faults; (<b>f</b>) Distance to rivers; (<b>g</b>) Distance to roads; (<b>h</b>) Mean annual precipitation (MAP); (<b>i</b>) Normalized differential vegetation index (NDVI); (<b>j</b>) Profile curvature; (<b>k</b>) plan curvature; (<b>l</b>) Topographic wetness index (TWI); (<b>m</b>) Terrain roughness index (TRI).</p> Full article ">Figure 2 Cont.
<p>Landslide predisposing factor maps involving: (<b>a</b>) Slope aspect; (<b>b</b>) Slope angle; (<b>c</b>) Altitude; (<b>d</b>) Lithology; (<b>e</b>) Distance to faults; (<b>f</b>) Distance to rivers; (<b>g</b>) Distance to roads; (<b>h</b>) Mean annual precipitation (MAP); (<b>i</b>) Normalized differential vegetation index (NDVI); (<b>j</b>) Profile curvature; (<b>k</b>) plan curvature; (<b>l</b>) Topographic wetness index (TWI); (<b>m</b>) Terrain roughness index (TRI).</p> Full article ">Figure 3
<p>Landslide susceptibility map derived from: (<b>a</b>) the kernel logistic regression model (KLR), and; (<b>b</b>) the kernel logistic regression based on box-counting dimension model (KLR<sub>box-counting</sub>); and (<b>c</b>) the kernel logistic regression based on correlation dimension model (KLR<sub>correlation</sub>).</p> Full article ">Figure 4
<p>The receiver operating characteristic (ROC) curves of models: (<b>a</b>) Training dataset; and (<b>b</b>) validation dataset.</p> Full article ">Figure 5
<p>The variation trend of the fractal dimension.</p> Full article ">
16 pages, 1033 KiB  
Article
Secrecy Performance Enhancement for Underlay Cognitive Radio Networks Employing Cooperative Multi-Hop Transmission with and without Presence of Hardware Impairments
by Phu Tran Tin, Dang The Hung, Tan N. Nguyen, Tran Trung Duy and Miroslav Voznak
Entropy 2019, 21(2), 217; https://doi.org/10.3390/e21020217 - 24 Feb 2019
Cited by 36 | Viewed by 4347
Abstract
In this paper, we consider a cooperative multi-hop secured transmission protocol to underlay cognitive radio networks. In the proposed protocol, a secondary source attempts to transmit its data to a secondary destination with the assistance of multiple secondary relays. In addition, there exists [...] Read more.
In this paper, we consider a cooperative multi-hop secured transmission protocol to underlay cognitive radio networks. In the proposed protocol, a secondary source attempts to transmit its data to a secondary destination with the assistance of multiple secondary relays. In addition, there exists a secondary eavesdropper who tries to overhear the source data. Under a maximum interference level required by a primary user, the secondary source and relay nodes must adjust their transmit power. We first formulate effective signal-to-interference-plus-noise ratio (SINR) as well as secrecy capacity under the constraints of the maximum transmit power, the interference threshold and the hardware impairment level. Furthermore, when the hardware impairment level is relaxed, we derive exact and asymptotic expressions of end-to-end secrecy outage probability over Rayleigh fading channels by using the recursive method. The derived expressions were verified by simulations, in which the proposed scheme outperformed the conventional multi-hop direct transmission protocol. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>System model of the proposed protocol.</p> Full article ">Figure 2
<p>End-to-end secrecy outage probability (SOP) as function of <span class="html-italic">P</span> in dB when <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mrow> <mo>−</mo> <mn>15</mn> <mspace width="0.166667em"/> <mi>dB</mi> <mo>,</mo> <mspace width="0.166667em"/> <mn>20</mn> <mspace width="0.166667em"/> <mi>dB</mi> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi mathvariant="normal">S</mi> </msub> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi>x</mi> <mi>PU</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>PU</mi> </msub> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mrow> <mo>−</mo> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi>x</mi> <mi mathvariant="normal">E</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi mathvariant="normal">E</mi> </msub> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mrow> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.5</mn> </mrow> </mfenced> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>End-to-end secrecy outage probability (SOP) as function of <span class="html-italic">P</span> in dB when <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mrow> <mo>−</mo> <mn>15</mn> <mspace width="0.166667em"/> <mi>dB</mi> <mo>,</mo> <mspace width="0.166667em"/> <mn>25</mn> <mspace width="0.166667em"/> <mi>dB</mi> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi mathvariant="normal">S</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>∈</mo> <mfenced separators="" open="{" close="}"> <mrow> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.2</mn> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi>x</mi> <mi>PU</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>PU</mi> </msub> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mrow> <mo>−</mo> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi>x</mi> <mi mathvariant="normal">E</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi mathvariant="normal">E</mi> </msub> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mrow> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.5</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi mathvariant="normal">D</mi> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mi mathvariant="normal">P</mi> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mi mathvariant="normal">E</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>End-to-end secrecy outage probability (SOP) as function of <span class="html-italic">M</span> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>5</mn> <mspace width="0.166667em"/> <mi>dB</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mrow> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>10</mn> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi mathvariant="normal">S</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>∈</mo> <mfenced separators="" open="{" close="}"> <mrow> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.1</mn> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi>x</mi> <mi>PU</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>PU</mi> </msub> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mrow> <mo>−</mo> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mo>−</mo> <mn>0.5</mn> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi>x</mi> <mi mathvariant="normal">E</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi mathvariant="normal">E</mi> </msub> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mrow> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.5</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi mathvariant="normal">D</mi> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mi mathvariant="normal">P</mi> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mi mathvariant="normal">E</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>End-to-end secrecy outage probability (SOP) as function of <math display="inline"><semantics> <mi>κ</mi> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>0</mn> <mspace width="0.166667em"/> <mi>dB</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi mathvariant="normal">S</mi> </msub> <mo>∈</mo> <mfenced separators="" open="{" close="}"> <mrow> <mn>0.25</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.75</mn> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mrow> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>1</mn> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi>x</mi> <mi>PU</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>PU</mi> </msub> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mrow> <mo>−</mo> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi>x</mi> <mi mathvariant="normal">E</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi mathvariant="normal">E</mi> </msub> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mrow> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.5</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi mathvariant="normal">D</mi> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mi mathvariant="normal">P</mi> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mi mathvariant="normal">E</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>End-to-end secrecy outage probability (SOP) as function of <math display="inline"><semantics> <msub> <mi>x</mi> <mi mathvariant="normal">E</mi> </msub> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>=</mo> <mn>10</mn> <mspace width="0.166667em"/> <mi>dB</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>M</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>R</mi> <mi mathvariant="normal">S</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <mrow> <msub> <mi>x</mi> <mi>PU</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>PU</mi> </msub> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mrow> <mo>−</mo> <mn>0.5</mn> <mo>,</mo> <mspace width="0.166667em"/> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>y</mi> <mi mathvariant="normal">E</mi> </msub> <mo>∈</mo> <mfenced separators="" open="{" close="}"> <mrow> <mn>0.3</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.7</mn> </mrow> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mi mathvariant="normal">D</mi> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mi mathvariant="normal">P</mi> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mi mathvariant="normal">E</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">
26 pages, 721 KiB  
Article
Unification of Epistemic and Ontic Concepts of Information, Probability, and Entropy, Using Cognizers-System Model
by Toshiyuki Nakajima
Entropy 2019, 21(2), 216; https://doi.org/10.3390/e21020216 - 24 Feb 2019
Cited by 7 | Viewed by 4301
Abstract
Information and probability are common words used in scientific investigations. However, information and probability both involve epistemic (subjective) and ontic (objective) interpretations under the same terms, which causes controversy within the concept of entropy in physics and biology. There is another issue regarding [...] Read more.
Information and probability are common words used in scientific investigations. However, information and probability both involve epistemic (subjective) and ontic (objective) interpretations under the same terms, which causes controversy within the concept of entropy in physics and biology. There is another issue regarding the circularity between information (or data) and reality: The observation of reality produces phenomena (or events), whereas the reality is confirmed (or constituted) by phenomena. The ordinary concept of information presupposes reality as a source of information, whereas another type of information (known as it-from-bit) constitutes the reality from data (bits). In this paper, a monistic model, called the cognizers-system model (CS model), is employed to resolve these issues. In the CS model, observations (epistemic) and physical changes (ontic) are both unified as “cognition”, meaning a related state change. Information and probability, epistemic and ontic, are formalized and analyzed systematically using a common theoretical framework of the CS model or a related model. Based on the results, a perspective for resolving controversial issues of entropy originating from information and probability is presented. Full article
(This article belongs to the Special Issue The 20th Anniversary of Entropy - Recent Advances in Entropy and Information-Theoretic Concepts and Their Applications)
►▼ Show Figures

Figure 1

Figure 1
<p>Externalist model of the world using the cognizers-system model (CS model). The meta-observer describes a model of the world (squared area). Dots denote cognizers in the world. The world is the whole cognizers system that can harbor partial systems (e.g., system <span class="html-italic">A</span>). There are two types of cognizers functioning as observers: External and internal observers. External observers, e.g., external observer <span class="html-italic">A</span>, do not belong to the system they observe, whereas internal observers, e.g., internal observer <span class="html-italic">a</span>, belong to the system they observe.</p> Full article ">Figure 2
<p>A two-cognizer system composed of a focal cognizer <span class="html-italic">C<sub>1</sub></span> with state space <b>C<sub>1</sub></b> and its environmental cognizer <span class="html-italic">E</span> with state space <b>E</b>. The environmental cognizer may be composed of many cognizers such as <span class="html-italic">C<sub>2</sub>, C<sub>3</sub>,</span> …, <span class="html-italic">C<sub>n</sub></span>. Arrows indicate temporal state-changes of component cognizers by cognition.</p> Full article ">Figure 3
<p>The degree of certainty of an event occurring to an external observer (cognizer) <span class="html-italic">C</span> with state-space <b>C</b>. <span class="html-italic">C</span> may include a measurement device (cognizer). <span class="html-italic">c<sub>x</sub></span> → <span class="html-italic">c<sub>y</sub></span> indicates an observational cognition of the system, and <span class="html-italic">c<sub>y</sub></span> → <span class="html-italic">c<sub>z1</sub></span> or <span class="html-italic">c<sub>z2</sub></span> indicates resultant cognitions of the external cognizer <span class="html-italic">C</span>. <span class="html-italic">s<sub>xi</sub></span> → <span class="html-italic">s<sub>yi</sub></span> (1 ≤ <span class="html-italic">i</span> ≤ <span class="html-italic">n</span>) represents a cognition (state-change) of the entire system <span class="html-italic">S</span> with state-space <b>S</b> observed by the external observer. Arrows indicate state-changes, which may include intermediate states between a given state and the next state.</p> Full article ">Figure 4
<p>Degree of certainty of events (internal P<sub>cog</sub>) occurring to a focal internal observer (cognizer) <span class="html-italic">C</span> with state-space <b>C</b>. <span class="html-italic">c<sub>x</sub></span> → <span class="html-italic">c<sub>y</sub></span> indicates an observational cognition of the environment and <span class="html-italic">c<sub>y</sub></span> → <span class="html-italic">c<sub>z1</sub></span> or <span class="html-italic">c<sub>z2</sub></span> indicates resultant cognitions of the internal cognizer <span class="html-italic">C</span>. <span class="html-italic">e<sub>xi</sub></span> → <span class="html-italic">e<sub>yi</sub></span> (1 ≤ <span class="html-italic">i</span> ≤ <span class="html-italic">n</span>) represents a cognition of the environment, where <span class="html-italic">e<sub>yi</sub></span> = <span class="html-italic">f<sub>E</sub></span> (<span class="html-italic">c<sub>x</sub>, e<sub>xi</sub></span>). Arrows indicate state changes, which may include intermediate states between a given state and the next state.</p> Full article ">Figure 5
<p>(<b>a</b>) The circularity of entailment between epistemic (“phenomena”) and ontic (“external reality”) fields, i.e., the E-O circularity. (<b>b</b>) The internalist model representing a possible way for the subject to construct a model of the external reality within based on phenomena or data.</p> Full article ">Figure 6
<p>Observation (blue arrows) of an object system <span class="html-italic">S</span> by an external observer using inverse causality processing (ICM). (<b>a</b>) State transition of an object system <span class="html-italic">S</span>, <span class="html-italic">s<sub>1</sub></span> → <span class="html-italic">s<sub>2</sub></span> → <span class="html-italic">s<sub>3</sub></span>, …, <span class="html-italic">s<sub>n</sub></span>; and that of its observer, (<span class="html-italic">0, µ<sub>0</sub></span>) → (<span class="html-italic">1, µ<sub>0</sub></span>) → (<span class="html-italic">0, µ<sub>1</sub></span>) → (<span class="html-italic">2, µ<sub>1</sub></span>) → (<span class="html-italic">0, µ<sub>2</sub></span>), …, (・, ・). The observer’s states are represented as (sensor state, memory state). The sensor changes from its basal state (<span class="html-italic">0</span>) to another state (<span class="html-italic">1, 2, …</span>) by measurement. After ICM measurement, the sensor returns to the basal state <span class="html-italic">0</span>. Measurements are recorded in the memory, such as <span class="html-italic">µ<sub>0</sub>, µ<sub>1</sub>, µ<sub>2</sub>.</span> (<b>b</b>) The observer’s state transition is modified to obtain a state transition synchronizing with the system, such as (<span class="html-italic">0, µ<sub>0</sub></span>) → (<span class="html-italic">0, µ<sub>1</sub></span>) → (<span class="html-italic">0, µ<sub>2</sub></span>), by removing intermediate sensor-states that are not in the basal state.</p> Full article ">
18 pages, 3018 KiB  
Article
The Choice of an Appropriate Information Dissimilarity Measure for Hierarchical Clustering of River Streamflow Time Series, Based on Calculated Lyapunov Exponent and Kolmogorov Measures
by Dragutin T. Mihailović, Emilija Nikolić-Đorić, Slavica Malinović-Milićević, Vijay P. Singh, Anja Mihailović, Tatijana Stošić, Borko Stošić and Nusret Drešković
Entropy 2019, 21(2), 215; https://doi.org/10.3390/e21020215 - 23 Feb 2019
Cited by 11 | Viewed by 4445
Abstract
The purpose of this paper was to choose an appropriate information dissimilarity measure for hierarchical clustering of daily streamflow discharge data, from twelve gauging stations on the Brazos River in Texas (USA), for the period 1989–2016. For that purpose, we selected and compared [...] Read more.
The purpose of this paper was to choose an appropriate information dissimilarity measure for hierarchical clustering of daily streamflow discharge data, from twelve gauging stations on the Brazos River in Texas (USA), for the period 1989–2016. For that purpose, we selected and compared the average-linkage clustering hierarchical algorithm based on the compression-based dissimilarity measure (NCD), permutation distribution dissimilarity measure (PDDM), and Kolmogorov distance (KD). The algorithm was also compared with K-means clustering based on Kolmogorov complexity (KC), the highest value of Kolmogorov complexity spectrum (KCM), and the largest Lyapunov exponent (LLE). Using a dissimilarity matrix based on NCD, PDDM, and KD for daily streamflow, the agglomerative average-linkage hierarchical algorithm was applied. The key findings of this study are that: (i) The KD clustering algorithm is the most suitable among others; (ii) ANOVA analysis shows that there exist highly significant differences between mean values of four clusters, confirming that the choice of the number of clusters was suitably done; and (iii) from the clustering we found that the predictability of streamflow data of the Brazos River given by the Lyapunov time (LT), corrected for randomness by Kolmogorov time (KT) in days, lies in the interval from two to five days. Full article
(This article belongs to the Special Issue Application of Information Measures for Analysis and Predictability of Renewable Energy Time Series)
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<p>Geographical locations of the gauging stations on the Brazos River used in this study.</p> Full article ">Figure 2
<p>Frequency counts of daily discharge data for the USGS 08082500 Brazos River station at Seymour, Texas (USA) for the period 1989–2016.</p> Full article ">Figure 3
<p>Dendrogram for hierarchical clustering of daily streamflow based on applied dissimilarity measure: (<b>a</b>) Compression-based dissimilarity measure; (<b>b</b>) permutation distribution dissimilarity measure; and (<b>c</b>) Kolmogorov complexity distance.</p> Full article ">Figure 4
<p>Map of hierarchical clustering of daily streamflow based on the applied dissimilarity measure: (<b>a</b>) Compression-based dissimilarity measure; (<b>b</b>) permutation distribution dissimilarity measure; and (<b>c</b>) Kolmogorov complexity distance.</p> Full article ">Figure 4 Cont.
<p>Map of hierarchical clustering of daily streamflow based on the applied dissimilarity measure: (<b>a</b>) Compression-based dissimilarity measure; (<b>b</b>) permutation distribution dissimilarity measure; and (<b>c</b>) Kolmogorov complexity distance.</p> Full article ">Figure 5
<p>3D scatter plot specified by the vectors KC, KCM, and LLE.</p> Full article ">Figure 6
<p>Plot of means for all clusters.</p> Full article ">Figure 7
<p>Predictability of the standardized daily discharge data of the Brazos River, given by the Lyapunov time (LT) corrected for randomness (in days).</p> Full article ">
20 pages, 41767 KiB  
Article
Macroscopic Cluster Organizations Change the Complexity of Neural Activity
by Jihoon Park, Koki Ichinose, Yuji Kawai, Junichi Suzuki, Minoru Asada and Hiroki Mori
Entropy 2019, 21(2), 214; https://doi.org/10.3390/e21020214 - 23 Feb 2019
Cited by 12 | Viewed by 5431
Abstract
In this study, simulations are conducted using a network model to examine how the macroscopic network in the brain is related to the complexity of activity for each region. The network model is composed of multiple neuron groups, each of which consists of [...] Read more.
In this study, simulations are conducted using a network model to examine how the macroscopic network in the brain is related to the complexity of activity for each region. The network model is composed of multiple neuron groups, each of which consists of spiking neurons with different topological properties of a macroscopic network based on the Watts and Strogatz model. The complexity of spontaneous activity is analyzed using multiscale entropy, and the structural properties of the network are analyzed using complex network theory. Experimental results show that a macroscopic structure with high clustering and high degree centrality increases the firing rates of neurons in a neuron group and enhances intraconnections from the excitatory neurons to inhibitory neurons in a neuron group. As a result, the intensity of the specific frequency components of neural activity increases. This decreases the complexity of neural activity. Finally, we discuss the research relevance of the complexity of the brain activity. Full article
(This article belongs to the Special Issue Information Dynamics in Brain and Physiological Networks)
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<p>Hypothesis and assumptions about relationships among the fundamental network, synaptic network and the complexity of neural activity in this study. The black node and empty black circle represent a neuron group and a neuron in the group, respectively. The black and green lines indicate an intraconnection between neurons in a neuron group and an interconnection between neurons in different neuron groups, respectively. The red and blue dots show neuron groups with and without high clustering coefficient and high shortest path length, respectively.</p> Full article ">Figure 2
<p>Overview of the spiking neural network model. A network is created using 100 neuron groups with macroscopic connections between neuron groups based on the Watts and Strogatz model [<a href="#B21-entropy-21-00214" class="html-bibr">21</a>]. The black nodes and green edge represent the neuron groups and macroscopic connections, respectively. (<b>a</b>) a lattice network where each node connected with neighboring nodes has local over-connectivity. All connections are rewired with rewiring probability <math display="inline"><semantics> <msub> <mi>p</mi> <mrow> <mi>W</mi> <mi>S</mi> </mrow> </msub> </semantics></math>, and larger <math display="inline"><semantics> <msub> <mi>p</mi> <mrow> <mi>W</mi> <mi>S</mi> </mrow> </msub> </semantics></math> yields more random network; (<b>b</b>) a small-world network with a large number of clusters and shorter path length compared with other networks; (<b>c</b>) a random network where nodes are completely randomly connected to each other; (<b>d</b>) each neuron group contains 800 excitatory (red circles) and 200 inhibitory (blue circles) spiking neurons, and each neuron has intra- (black arrow) and inter-connections (green arrow).</p> Full article ">Figure 3
<p>Time schedule for simulation. Each colored area indicates the period for which the event occurred. Neural activities during 1110 s to 1200 s were analyzed to determine the relationship among the structural properties of the synaptic network and neural activity.</p> Full article ">Figure 4
<p>Relationship between sample entropy and <math display="inline"><semantics> <msub> <mi>p</mi> <mi>WS</mi> </msub> </semantics></math> of the WS model. The <span class="html-italic">x</span>-axis in all graphs represents <math display="inline"><semantics> <msub> <mi>p</mi> <mi>WS</mi> </msub> </semantics></math>. (<b>a</b>–<b>f</b>) average sample entropy of all neuron groups with ten independent simulations at scale factors (<math display="inline"><semantics> <mi>ϵ</mi> </semantics></math> in Equation (<a href="#FD1-entropy-21-00214" class="html-disp-formula">1</a>)) of multiscale entropy (MSE) at 1, 10, 20, 40, 60, and 80. The error bars indicate the standard deviation.</p> Full article ">Figure 5
<p>Amplitude of each frequency spectrum sampled from the local averaged potentials of the 10 neuron groups with low complexity (blue) and high complexity (red) in a network. The peak envelopes is used to plot the curve in the figure. Color curves and color-shaded areas represent average and standard deviation values for ten simulations, respectively. (<b>a</b>) the lattice network (<math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>WS</mi> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math>) during self-organization by spike-timing-dependent plasticity (STDP) (0–100 s); (<b>b</b>) the random network (<math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>WS</mi> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>) during self-organization by STDP (0–100 s); (<b>c</b>) the lattice network (<math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>WS</mi> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math>) after self-organization by STDP (1100–1200 s); (<b>d</b>) the random network (<math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>WS</mi> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>) after self-organization by STDP (1100–1200 s).</p> Full article ">Figure 6
<p>Relationship between the connectivity structure and complexity of neural activity. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The <span class="html-italic">x</span>-axis indicates the degree centrality, and the <span class="html-italic">y</span>-axis indicates the clustering coefficient.</p> Full article ">Figure 7
<p>Relationship between the connectivity structure and firing rate of excitatory and inhibitory neurons. Each marker corresponds to a neuron group in the network, and its color indicates the average firing rate of excitatory and inhibitory neurons. The <span class="html-italic">x</span>-axis indicates the degree centrality, and the <span class="html-italic">y</span>-axis indicates the clustering coefficient. (<b>a</b>) relationship between structural properties and firing rate of excitatory neurons; (<b>b</b>) relationship between structural properties and firing rate of inhibitory neurons.</p> Full article ">Figure 8
<p>Relationship among the weight of intraconnection, structural properties of the synaptic network, and complexity of neural activity. (<b>a</b>) relationship among the average weight of intraconnections from excitatory neurons to excitatory neurons, clustering coefficient based on the interconnection, and complexity; (<b>b</b>) relationship among the average weight of intraconnections from excitatory neurons to inhibitory neurons, clustering coefficient based on the interconnection, and complexity; (<b>c</b>) relationship among the average weight of intraconnections from excitatory neurons to excitatory neurons, degree centrality based on the interconnection, and complexity; (<b>d</b>) relationship among the average weight of intraconnections from excitatory neurons to inhibitory neurons, degree centrality based on the interconnection, and complexity. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The <span class="html-italic">x</span>-axis indicates the average weight of interconnection, and the <span class="html-italic">y</span>-axis the structural properties of interconnection.</p> Full article ">Figure A1
<p>Multiscale entropy (MSE)-based complexity curves of each neuron group in a synaptic network. (<b>a</b>) lattice network (<math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>WS</mi> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math>); (<b>b</b>) a small-world network (<math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>WS</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>); (<b>c</b>) a random network (<math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mi>WS</mi> </msub> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>). The <span class="html-italic">y</span>-axis indicates sample entropy, and the <span class="html-italic">x</span>-axis indicates scale factor <math display="inline"><semantics> <mi>ϵ</mi> </semantics></math>.</p> Full article ">Figure A2
<p>The differences of peak amplitude of spontaneous neural activity in some frequency bands between neuron groups with low and high complexity when <math display="inline"><semantics> <mrow> <msub> <mi>p</mi> <mrow> <mi>W</mi> <mi>S</mi> </mrow> </msub> <mo>=</mo> <mn>0.0</mn> </mrow> </semantics></math>. We used 10 neuron groups with high and low complexity in each simulation as comparison data. The number on above each violin plot denotes the average value for ten simulations. Wilcoxon signed-rank test was used for statistical test. (<b>a</b>) amplitude in the 20–40 Hz band (Wilcoxon signed-rank test, statistic = 6.0, <span class="html-italic">p</span>-value = 4.6706 × <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>18</mn> </mrow> </msup> </semantics></math>); (<b>b</b>) amplitude in the 40–60 Hz band (Wilcoxon signed-rank test, statistic = 11.0, <span class="html-italic">p</span>-value = 5.4302 × <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>18</mn> </mrow> </msup> </semantics></math>).</p> Full article ">Figure A3
<p>Relationship between the connectivity structure and the complexity of neural activity. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The <span class="html-italic">x</span>-axis indicates the degree centrality, and the <span class="html-italic">y</span>-axis indicates the clustering coefficient.</p> Full article ">Figure A4
<p>Relationship between the peak frequency and the complexity of neural activity. (<b>a</b>) relationship in the 0–20 Hz band; (<b>b</b>) relationship in the 20–40 Hz band; (<b>c</b>) relationship in the 40–60 Hz band. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The <span class="html-italic">x</span>-axis indicates the peak frequency of the neural activity, and the <span class="html-italic">y</span>-axis indicates the amplitude. Figure shows that the amplitude in increases as the complexity of neural activity decreases in 20–40 Hz and 40–60 Hz bands. Furthermore, the peak frequency increases as the complexity of neural activity decreases in the 40–60 Hz band. However, amplitude decreases as the complexity of neural activity decreases in the 0–20 Hz band. This result indicates that the robust frequency components in neural activity of the neuron groups with low complexity shifted to the high frequency bands (20–40 Hz and 40–60Hz) from the low frequency bands (0–20 Hz).</p> Full article ">Figure A5
<p>Relationship between the connectivity structure and the firing rate of excitatory and inhibitory neurons. Each marker corresponds to a neuron group in the network, and its color indicates the average firing rate of excitatory and inhibitory neurons. The <span class="html-italic">x</span>-axis indicates the degree centrality, and the <span class="html-italic">y</span>-axis indicates the clustering coefficient. (<b>a</b>) relationship between structural properties and firing rate of excitatory neurons; (<b>b</b>) relationship between structural properties and firing rate of inhibitory neurons.</p> Full article ">Figure A6
<p>Relationship among the weight of intraconnection, structural properties, and complexity of neural activity. (<b>a</b>) relationship among the weight of intraconnection from excitatory to excitatory neuron, clustering coefficient based on the interconnection, and complexity; (<b>b</b>) relationship among the weight of intraconnection from excitatory to inhibitory neuron, clustering coefficient based on the interconnection, and complexity; (<b>c</b>) relationship among the weight of intraconnection from excitatory to excitatory neuron, degree centrality based on the interconnection, and complexity; (<b>d</b>) relationship among the weight of intraconnection from excitatory to inhibitory neuron, degree centrality based on the interconnection, and complexity. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The <span class="html-italic">x</span>-axis indicates the average of weight of interconnection, and the <span class="html-italic">y</span>-axis the structural properties of interconnection.</p> Full article ">Figure A7
<p>Relationship between shortest path length and complexity for each neuron group. The <span class="html-italic">x</span>-axis is the shortest path length, and the <span class="html-italic">y</span>-axis is the summation of the sample entropy for all 80 scale factors.</p> Full article ">Figure A8
<p>Relationship between the connectivity structure without STDP and the complexity of neural activity with the tonic input. Duration for tonic input was set as 100 s. Here, we used the same initial weights of the synaptic networks in <a href="#entropy-21-00214-f0A3" class="html-fig">Figure A3</a> and fixed the weights during the tonic input. Each marker corresponds to a neuron group in the network, and its color indicates the summation of the sample entropy for all 80 scale factors. The <span class="html-italic">x</span>-axis indicates the degree centrality, and the <span class="html-italic">y</span>-axis indicates the clustering coefficient.</p> Full article ">
14 pages, 327 KiB  
Article
Asymptotic Rate-Distortion Analysis of Symmetric Remote Gaussian Source Coding: Centralized Encoding vs. Distributed Encoding
by Yizhong Wang, Li Xie, Siyao Zhou, Mengzhen Wang and Jun Chen
Entropy 2019, 21(2), 213; https://doi.org/10.3390/e21020213 - 23 Feb 2019
Cited by 4 | Viewed by 3315
Abstract
Consider a symmetric multivariate Gaussian source with ℓ components, which are corrupted by independent and identically distributed Gaussian noises; these noisy components are compressed at a certain rate, and the compressed version is leveraged to reconstruct the source subject to a mean squared [...] Read more.
Consider a symmetric multivariate Gaussian source with ℓ components, which are corrupted by independent and identically distributed Gaussian noises; these noisy components are compressed at a certain rate, and the compressed version is leveraged to reconstruct the source subject to a mean squared error distortion constraint. The rate-distortion analysis is performed for two scenarios: centralized encoding (where the noisy source components are jointly compressed) and distributed encoding (where the noisy source components are separately compressed). It is shown, among other things, that the gap between the rate-distortion functions associated with these two scenarios admits a simple characterization in the large ℓ limit. Full article
(This article belongs to the Special Issue Information Theory for Data Communications and Processing)
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<p>Symmetric remote Gaussian source coding with centralized encoding.</p> Full article ">Figure 2
<p>Symmetric remote Gaussian source coding with distributed encoding.</p> Full article ">Figure 3
<p>Illustration of <math display="inline"><semantics> <mrow> <mi>ψ</mi> <mo>(</mo> <mi>d</mi> <mo>)</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mi>X</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mi>Z</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> for different <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi>X</mi> </msub> </semantics></math>.</p> Full article ">Figure 4
<p>Illustration of <math display="inline"><semantics> <mrow> <mi>ψ</mi> <mo>(</mo> <mi>d</mi> <mo>)</mo> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>γ</mi> <mi>X</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>X</mi> </msub> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> for different <math display="inline"><semantics> <msub> <mi>γ</mi> <mi>Z</mi> </msub> </semantics></math>.</p> Full article ">
8 pages, 431 KiB  
Article
Attack Algorithm for a Keystore-Based Secret Key Generation Method
by Seungjae Chae, Young-Sik Kim, Jong-Seon No and Young-Han Kim
Entropy 2019, 21(2), 212; https://doi.org/10.3390/e21020212 - 23 Feb 2019
Viewed by 3073
Abstract
A new attack algorithm is proposed for a secure key generation and management method introduced by Yang and Wu. It was previously claimed that the key generation method of Yang and Wu using a keystore seed was information-theoretically secure and could solve the [...] Read more.
A new attack algorithm is proposed for a secure key generation and management method introduced by Yang and Wu. It was previously claimed that the key generation method of Yang and Wu using a keystore seed was information-theoretically secure and could solve the long-term key storage problem in cloud systems, thanks to the huge number of secure keys that the keystone seed can generate. Their key generation method, however, is considered to be broken if an attacker can recover the keystore seed. The proposed attack algorithm in this paper reconstructs the keystore seed of the Yang–Wu key generation method from a small number of collected keys. For example, when t = 5 and l = 2 7 , it was previously claimed that more than 2 53 secure keys could be generated, but the proposed attack algorithm can reconstruct the keystone seed based on only 84 collected keys. Hence, the Yang–Wu key generation method is not information-theoretically secure when the attacker can gather multiple keys and a critical amount of information about the keystone seed is leaked. Full article
(This article belongs to the Special Issue Information-Theoretic Security II)
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<p>Matrix operation to find keystore seed.</p> Full article ">Figure 2
<p>Successful attack probability of the proposed attack algorithm when: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <msup> <mn>2</mn> <mn>12</mn> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>l</mi> <mo>=</mo> <msup> <mn>2</mn> <mn>7</mn> </msup> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <msup> <mn>2</mn> <mn>14</mn> </msup> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>l</mi> <mo>=</mo> <msup> <mn>2</mn> <mn>8</mn> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Key generation by subkeys.</p> Full article ">
16 pages, 377 KiB  
Article
An Intuitionistic Evidential Method for Weight Determination in FMEA Based on Belief Entropy
by Zeyi Liu and Fuyuan Xiao
Entropy 2019, 21(2), 211; https://doi.org/10.3390/e21020211 - 22 Feb 2019
Cited by 17 | Viewed by 3377
Abstract
Failure Mode and Effects Analysis (FMEA) has been regarded as an effective analysis approach to identify and rank the potential failure modes in many applications. However, how to determine the weights of team members appropriately, with the impact factor of domain experts’ uncertainty [...] Read more.
Failure Mode and Effects Analysis (FMEA) has been regarded as an effective analysis approach to identify and rank the potential failure modes in many applications. However, how to determine the weights of team members appropriately, with the impact factor of domain experts’ uncertainty in decision-making of FMEA, is still an open issue. In this paper, a new method to determine the weights of team members, which combines evidence theory, intuitionistic fuzzy sets (IFSs) and belief entropy, is proposed to analyze the failure modes. One of the advantages of the presented model is that the uncertainty of experts in the decision-making process is taken into consideration. The proposed method is data driven with objective and reasonable properties, which considers the risk of weights more completely. A numerical example is shown to illustrate the feasibility and availability of the proposed method. Full article
(This article belongs to the Special Issue Entropy-Based Fault Diagnosis)
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<p>The flow chart of our proposed method.</p> Full article ">
29 pages, 579 KiB  
Review
An Overview on Denial-of-Service Attacks in Control Systems: Attack Models and Security Analyses
by Ahmet Cetinkaya, Hideaki Ishii and Tomohisa Hayakawa
Entropy 2019, 21(2), 210; https://doi.org/10.3390/e21020210 - 22 Feb 2019
Cited by 122 | Viewed by 9885
Abstract
In this paper, we provide an overview of recent research efforts on networked control systems under denial-of-service attacks. Our goal is to discuss the utility of different attack modeling and analysis techniques proposed in the literature for addressing feedback control, state estimation, and [...] Read more.
In this paper, we provide an overview of recent research efforts on networked control systems under denial-of-service attacks. Our goal is to discuss the utility of different attack modeling and analysis techniques proposed in the literature for addressing feedback control, state estimation, and multi-agent consensus problems in the face of jamming attacks in wireless channels and malicious packet drops in multi-hop networks. We discuss several modeling approaches that are employed for capturing the uncertainty in denial-of-service attack strategies. We give an outlook on deterministic constraint-based modeling ideas, game-theoretic and optimization-based techniques and probabilistic modeling approaches. A special emphasis is placed on tail-probability based failure models, which have been recently used for describing jamming attacks that affect signal to interference-plus-noise ratios of wireless channels as well as transmission failures on multi-hop networks due to packet-dropping attacks and non-malicious issues. We explain the use of attack models in the security analysis of networked systems. In addition to the modeling and analysis problems, a discussion is provided also on the recent developments concerning the design of attack-resilient control and communication protocols. Full article
(This article belongs to the Special Issue Entropy in Networked Control)
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<p>Operation of networked control system under denial-of-service attack: (<b>Left</b>) wireless networked control system facing jamming attacks; and (<b>Right</b>) multi-hop networked control system that faces packet-dropping attacks by malicious nodes in the networks.).</p> Full article ">Figure 2
<p>Operation of networked state estimation subject to DoS attacks.</p> Full article ">Figure 3
<p>Multi-agent consensus in the presence of jamming attackers: (<b>Left</b>) single jamming attacker causes transmission failures on all inter-agent communication links; and (<b>Right</b>) multiple jamming attackers cause failures on different links).</p> Full article ">Figure 4
<p>Sequence of DoS attack intervals. Transmission attempts that occur in any of the DoS attack intervals (represented with pink regions) fail.</p> Full article ">Figure 5
<p>Ranges of <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> values for the class <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Λ</mi> <mi>ρ</mi> </msub> </semantics></math>. The ranges are different, when different transmission issues are considered in the network.</p> Full article ">Figure 6
<p>The transmission failure probability function <span class="html-italic">p</span> and a concave upper-bounding function <math display="inline"><semantics> <mover accent="true"> <mi>p</mi> <mo stretchy="false">^</mo> </mover> </semantics></math> for an example wireless channel under jamming attacks.</p> Full article ">Figure 7
<p>A multi-hop network between the plant (<math display="inline"><semantics> <msub> <mi>v</mi> <mi mathvariant="normal">P</mi> </msub> </semantics></math>) and the controller (<math display="inline"><semantics> <msub> <mi>v</mi> <mi mathvariant="normal">C</mi> </msub> </semantics></math>). State measurement packets on this network are transmitted over two paths <math display="inline"><semantics> <msub> <mi mathvariant="script">P</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi mathvariant="script">P</mi> <mn>2</mn> </msub> </semantics></math>, which are both subject to malicious packet-dropping attacks.</p> Full article ">Figure 8
<p>Illustration of the event-triggering approach in [<a href="#B25-entropy-21-00210" class="html-bibr">25</a>] for the cases of successful and failed transmissions at time <math display="inline"><semantics> <msub> <mi>t</mi> <mi>i</mi> </msub> </semantics></math>. A transmission is triggered at time <math display="inline"><semantics> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </semantics></math> when the state is predicted to make the move indicated with red dotted lines. In the case of successful transmissions (<b>Left</b>), the state is guaranteed to stay inside the level set <math display="inline"><semantics> <mrow> <mo>{</mo> <mi>x</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> <mo lspace="0pt">:</mo> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>β</mi> <mi>V</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mi>i</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </semantics></math> in between two event-triggering instants.</p> Full article ">Figure 9
<p>Illustration of the randomized communication protocol, where each agent attempts communicating with its neighbors at random time instants. At time instants denoted inside rectangles with green dashed borders, the communication attempts can avoid DoS attacks.</p> Full article ">
14 pages, 824 KiB  
Article
Quantum Pumping with Adiabatically Modulated Barriers in Three-Band Pseudospin-1 Dirac–Weyl Systems
by Xiaomei Chen and Rui Zhu
Entropy 2019, 21(2), 209; https://doi.org/10.3390/e21020209 - 22 Feb 2019
Cited by 2 | Viewed by 3542
Abstract
In this work, pumped currents of the adiabatically-driven double-barrier structure based on the pseudospin-1 Dirac–Weyl fermions are studied. As a result of the three-band dispersion and hence the unique properties of pseudospin-1 Dirac–Weyl quasiparticles, sharp current-direction reversal is found at certain parameter settings [...] Read more.
In this work, pumped currents of the adiabatically-driven double-barrier structure based on the pseudospin-1 Dirac–Weyl fermions are studied. As a result of the three-band dispersion and hence the unique properties of pseudospin-1 Dirac–Weyl quasiparticles, sharp current-direction reversal is found at certain parameter settings especially at the Dirac point of the band structure, where apexes of the two cones touch at the flat band. Such a behavior can be interpreted consistently by the Berry phase of the scattering matrix and the classical turnstile mechanism. Full article
(This article belongs to the Special Issue Quantum Transport in Mesoscopic Systems)
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Figure 1

Figure 1
<p>(<b>a</b>) schematics of the adiabatic quantum pump. Two time-dependent gate voltages with identical width <span class="html-italic">d</span> and equilibrium strength <math display="inline"><semantics> <msub> <mi>V</mi> <mn>0</mn> </msub> </semantics></math> are applied to the conductor. Time variation of the two potentials <math display="inline"><semantics> <msub> <mi>V</mi> <mn>1</mn> </msub> </semantics></math> and and <math display="inline"><semantics> <msub> <mi>V</mi> <mn>2</mn> </msub> </semantics></math> is shown in panel (<b>b</b>). <math display="inline"><semantics> <msub> <mi>V</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>V</mi> <mn>2</mn> </msub> </semantics></math> have a phase difference giving rise to a looped trajectory after one driving period; (<b>c</b>) two-dimensional band structure of the pseudospin-1 Dirac–Weyl fermions with a flat band intersected two Dirac cones at the apexes; (<b>d</b>) conductivity of the pseudospin-1 Dirac–Weyl fermions measured by [<a href="#B54-entropy-21-00209" class="html-bibr">54</a>] <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="false"> <mfrac> <mrow> <msup> <mi>e</mi> <mn>2</mn> </msup> <msub> <mi>k</mi> <mi>F</mi> </msub> <mi>d</mi> </mrow> <mrow> <mi>π</mi> <mi>h</mi> </mrow> </mfrac> </mstyle> <msubsup> <mo>∫</mo> <mrow> <mo>−</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> <mrow> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </msubsup> <mrow> <msup> <mrow> <mfenced separators="" open="|" close="|"> <mrow> <mi>t</mi> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mi>F</mi> </msub> <mo>,</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> </mrow> <mn>2</mn> </msup> <mo form="prefix">cos</mo> <mi>θ</mi> <mi>d</mi> <mi>θ</mi> </mrow> </mrow> </semantics></math> in single-barrier tunneling junction as a function of the Fermi energy for three different values of barrier height <math display="inline"><semantics> <msub> <mi>V</mi> <mn>0</mn> </msub> </semantics></math>. <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>F</mi> </msub> <mo>=</mo> <msub> <mi>E</mi> <mi>F</mi> </msub> <mo>/</mo> <mo>ℏ</mo> <msub> <mi>v</mi> <mi>g</mi> </msub> </mrow> </semantics></math> is the Fermi wavevector and <span class="html-italic">t</span> is the transmission amplitude defined in Equation (<a href="#FD5-entropy-21-00209" class="html-disp-formula">5</a>). It can be seen that higher barrier allowing larger conductivity occurs at the Dirac point <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi>F</mi> </msub> <mo>=</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> </mrow> </semantics></math> and around <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi>F</mi> </msub> <mo>=</mo> <msub> <mi>V</mi> <mn>0</mn> </msub> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math> (see the text).</p> Full article ">Figure 2
<p>(<b>a</b>–<b>c</b>): angular dependence of the pumped for different Fermi energies with the driving phase difference <math display="inline"><semantics> <mi>φ</mi> </semantics></math> fixed; (<b>d</b>) angle-averaged pumped current as a function of the Fermi energy. Its inset is the zoom-in close to the Dirac point to show that the large value of the pumped current does not diverge. Other parameters are <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> meV, <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mrow> <mn>1</mn> <mi>ω</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow> <mn>2</mn> <mi>ω</mi> </mrow> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> meV, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> nm, <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> nm, and <math display="inline"><semantics> <mrow> <mi>φ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Contours of the Berry curvature <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Ω</mi> <mfenced open="(" close=")"> <mi>l</mi> </mfenced> </mrow> </semantics></math> and the eight derivatives on the right-hand side of Equation (<a href="#FD10-entropy-21-00209" class="html-disp-formula">10</a>) in the <math display="inline"><semantics> <msub> <mi>V</mi> <mn>1</mn> </msub> </semantics></math>-<math display="inline"><semantics> <msub> <mi>V</mi> <mn>2</mn> </msub> </semantics></math> parameter space. For all the subfigures, the horizonal and vertical axes are <math display="inline"><semantics> <msub> <mi>V</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>V</mi> <mn>2</mn> </msub> </semantics></math> in the unit of meV, respectively. The magnitudes of the contours are in the scale of (<b>a</b>) <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>4</mn> </mrow> </msup> </semantics></math>; (<b>d</b>) <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </semantics></math>; (<b>e</b>) <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </semantics></math>; (<b>f</b>) <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </semantics></math>; (<b>g</b>) <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </semantics></math>; (<b>h)</b><math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </semantics></math>; (<b>i</b>) <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </semantics></math>; and (<b>j</b>) <math display="inline"><semantics> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </semantics></math>, respectively. Other parameters are <math display="inline"><semantics> <mrow> <msub> <mi>V</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> meV, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> nm, <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> nm, <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mi>F</mi> </msub> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math> meV, and <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> in radians. For convenience of discussion, the parameter space in the nine panels is divided into four blocks: I (<math display="inline"><semantics> <mrow> <mo>−</mo> <mn>1</mn> <mo>&lt;</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>&lt;</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mn>1</mn> </mrow> </semantics></math>), II (<math display="inline"><semantics> <mrow> <mn>0</mn> <mo>&lt;</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>&lt;</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>0</mn> <mo>&lt;</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mn>1</mn> </mrow> </semantics></math>), III (<math display="inline"><semantics> <mrow> <mo>−</mo> <mn>1</mn> <mo>&lt;</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>1</mn> <mo>&lt;</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>), and IV (<math display="inline"><semantics> <mrow> <mn>0</mn> <mo>&lt;</mo> <msub> <mi>V</mi> <mn>1</mn> </msub> <mo>&lt;</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>1</mn> <mo>&lt;</mo> <msub> <mi>V</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mn>0</mn> </mrow> </semantics></math>). The four blocks are illustrated in (<b>a</b>).</p> Full article ">Figure A1
<p>Static transmission probabilities <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>t</mi> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> </mrow> </semantics></math> as a function of the incident angle (<b>a</b>) and the Fermi energy (<b>b</b>), respectively [<a href="#B57-entropy-21-00209" class="html-bibr">57</a>]. Parameters <math display="inline"><semantics> <msub> <mi>V</mi> <mn>0</mn> </msub> </semantics></math>, <span class="html-italic">d</span>, and <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>−</mo> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math> are the same as those in <a href="#entropy-21-00209-f002" class="html-fig">Figure 2</a>.</p> Full article ">
22 pages, 10466 KiB  
Article
Study on Asphalt Pavement Surface Texture Degradation Using 3-D Image Processing Techniques and Entropy Theory
by Yinghao Miao, Jiaqi Wu, Yue Hou, Linbing Wang, Weixiao Yu and Sudi Wang
Entropy 2019, 21(2), 208; https://doi.org/10.3390/e21020208 - 21 Feb 2019
Cited by 24 | Viewed by 4854
Abstract
Surface texture is a very important factor affecting the anti-skid performance of pavements. In this paper, entropy theory is introduced to study the decay behavior of the three-dimensional macrotexture and microtexture of road surfaces in service based on the field test data collected [...] Read more.
Surface texture is a very important factor affecting the anti-skid performance of pavements. In this paper, entropy theory is introduced to study the decay behavior of the three-dimensional macrotexture and microtexture of road surfaces in service based on the field test data collected over more than 2 years. Entropy is found to be feasible for evaluating the three-dimensional macrotexture and microtexture of an asphalt pavement surface. The complexity of the texture increases with the increase of entropy. Under the polishing action of the vehicle load, the entropy of the surface texture decreases gradually. The three-dimensional macrotexture decay characteristics of asphalt pavement surfaces are significantly different for different mixture designs. The macrotexture decay performance of asphalt pavement can be improved by designing appropriate mixtures. Compared with the traditional macrotexture parameter Mean Texture Depth (MTD) index, entropy contains more physical information and has a better correlation with the pavement anti-skid performance index. It has significant advantages in describing the relationship between macrotexture characteristics and the anti-skid performance of asphalt pavement. Full article
(This article belongs to the Special Issue Entropy in Image Analysis)
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Figure 1

Figure 1
<p>Field tests: (<b>a</b>) the 3-D scanner; (<b>b</b>) sand patch test for the mean texture depth (MTD); (<b>c</b>) Scanning test; and (<b>d</b>) dynamic friction tester (DFT) test.</p> Full article ">Figure 1 Cont.
<p>Field tests: (<b>a</b>) the 3-D scanner; (<b>b</b>) sand patch test for the mean texture depth (MTD); (<b>c</b>) Scanning test; and (<b>d</b>) dynamic friction tester (DFT) test.</p> Full article ">Figure 2
<p>Results of a typical 3-D macrotexture: (<b>a</b>) dense asphalt concrete (DAC); (<b>b</b>) stone matrix asphalt (SMA); (<b>c</b>) rubber asphalt concrete (RAC); and (<b>d</b>) ultra-thin wearing course (UTWC).</p> Full article ">Figure 2 Cont.
<p>Results of a typical 3-D macrotexture: (<b>a</b>) dense asphalt concrete (DAC); (<b>b</b>) stone matrix asphalt (SMA); (<b>c</b>) rubber asphalt concrete (RAC); and (<b>d</b>) ultra-thin wearing course (UTWC).</p> Full article ">Figure 3
<p>Results of a typical 3-D microtexture: (<b>a</b>) DAC, measured; (<b>b</b>) DAC, filtered; (<b>c</b>) SMA, measured; and (<b>d</b>) SMA, filtered.</p> Full article ">Figure 4
<p>Grey-level images corresponding to the macrotexture shown in <a href="#entropy-21-00208-f002" class="html-fig">Figure 2</a>: (<b>a</b>) DAC; (<b>b</b>) SMA; (<b>c</b>) RAC; and (<b>d</b>) UTWC.</p> Full article ">Figure 4 Cont.
<p>Grey-level images corresponding to the macrotexture shown in <a href="#entropy-21-00208-f002" class="html-fig">Figure 2</a>: (<b>a</b>) DAC; (<b>b</b>) SMA; (<b>c</b>) RAC; and (<b>d</b>) UTWC.</p> Full article ">Figure 5
<p>Grey-level images corresponding to the microtexture shown in <a href="#entropy-21-00208-f003" class="html-fig">Figure 3</a>: (<b>a</b>) SMA and (<b>b</b>) DAC.</p> Full article ">Figure 6
<p>Distribution of entropy of the macrotexture and microtexture: (<b>a</b>) Macrotexture and (<b>b</b>) Microtexture.</p> Full article ">Figure 7
<p>Change of the macrotexture entropy with the traffic volume: (<b>a</b>) DAC; (<b>b</b>) SMA; (<b>c</b>) RAC; and (<b>d</b>) UTWC.</p> Full article ">Figure 7 Cont.
<p>Change of the macrotexture entropy with the traffic volume: (<b>a</b>) DAC; (<b>b</b>) SMA; (<b>c</b>) RAC; and (<b>d</b>) UTWC.</p> Full article ">Figure 7 Cont.
<p>Change of the macrotexture entropy with the traffic volume: (<b>a</b>) DAC; (<b>b</b>) SMA; (<b>c</b>) RAC; and (<b>d</b>) UTWC.</p> Full article ">Figure 8
<p>Changing trends of the macrotexture entropy.</p> Full article ">Figure 9
<p>Scatter plots of DFT60 against <span class="html-italic">E</span> and MTD: (<b>a</b>) DFT60-<span class="html-italic">E</span> and (<b>b</b>) DFT60-MTD.</p> Full article ">Figure 9 Cont.
<p>Scatter plots of DFT60 against <span class="html-italic">E</span> and MTD: (<b>a</b>) DFT60-<span class="html-italic">E</span> and (<b>b</b>) DFT60-MTD.</p> Full article ">Figure 10
<p>Change of the microtexture entropy with the traffic volume: (<b>a</b>) DAC; (<b>b</b>) SMA; (<b>c</b>) RAC; and (<b>d</b>) UTWC.</p> Full article ">Figure 10 Cont.
<p>Change of the microtexture entropy with the traffic volume: (<b>a</b>) DAC; (<b>b</b>) SMA; (<b>c</b>) RAC; and (<b>d</b>) UTWC.</p> Full article ">Figure 11
<p>Changing trends of the microtexture entropy.</p> Full article ">
12 pages, 2107 KiB  
Article
Adaptive Synchronization of Fractional-Order Complex Chaotic system with Unknown Complex Parameters
by Ruoxun Zhang, Yongli Liu and Shiping Yang
Entropy 2019, 21(2), 207; https://doi.org/10.3390/e21020207 - 21 Feb 2019
Cited by 16 | Viewed by 3643
Abstract
This paper investigates the problem of synchronization of fractional-order complex-variable chaotic systems (FOCCS) with unknown complex parameters. Based on the complex-variable inequality and stability theory for fractional-order complex-valued system, a new scheme is presented for adaptive synchronization of FOCCS with unknown complex parameters. [...] Read more.
This paper investigates the problem of synchronization of fractional-order complex-variable chaotic systems (FOCCS) with unknown complex parameters. Based on the complex-variable inequality and stability theory for fractional-order complex-valued system, a new scheme is presented for adaptive synchronization of FOCCS with unknown complex parameters. The proposed scheme not only provides a new method to analyze fractional-order complex-valued system but also significantly reduces the complexity of computation and analysis. Theoretical proof and simulation results substantiate the effectiveness of the presented synchronization scheme. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)
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Figure 1

Figure 1
<p>Dynamic behaviors of the fractional-order complex Lorenz-like System with commensurate order <math display="inline"><semantics> <mi>α</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>10</mn> <mo>+</mo> <mi>i</mi> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo> </mo> <mi>c</mi> <mo>=</mo> <mn>16</mn> <mo>+</mo> <mn>0.3</mn> <mi>i</mi> </mrow> </semantics></math>). (<b>a</b>) maximal Lyapunov exponent; (<b>b</b>) bifurcation diagram.</p> Full article ">Figure 2
<p>Chaotic attractors of fractional-order complex Lorenz-like system with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>10</mn> <mo>+</mo> <mi>i</mi> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo> </mo> <mi>c</mi> <mo>=</mo> <mn>16</mn> <mo>+</mo> <mn>0.3</mn> <mi>i</mi> </mrow> </semantics></math> and commensurate order <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Dynamic behaviors of the fractional-order complex Lorenz-like System with commensurate order 0.95 (<math display="inline"><semantics> <mrow> <msup> <mi>a</mi> <mi>r</mi> </msup> <mo>=</mo> <mn>10</mn> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo> </mo> <mi>c</mi> <mo>=</mo> <mn>16</mn> <mo>+</mo> <mn>0.3</mn> <mi>i</mi> </mrow> </semantics></math>). (<b>a</b>) maximal Lyapunov exponent; (<b>b</b>) bifurcation diagram.</p> Full article ">Figure 4
<p>The state trajectories of fractional-order complex Lorenz-like system with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>10</mn> <mo>+</mo> <mi>i</mi> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo> </mo> <mi>c</mi> <mo>=</mo> <mn>16</mn> <mo>+</mo> <mn>0.3</mn> <mi>i</mi> </mrow> </semantics></math> and commensurate order <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Synchronization errors <span class="html-italic">e</span><sub>1</sub>, <span class="html-italic">e</span><sub>2</sub>, <span class="html-italic">e</span><sub>3</sub> of fractional-order complex Lorenz-like chaotic system.</p> Full article ">Figure 6
<p>Estimated complex parameters of fractional-order complex Lorenz-like chaotic system.</p> Full article ">
17 pages, 5258 KiB  
Article
Coherent Structure of Flow Based on Denoised Signals in T-junction Ducts with Vertical Blades
by Jing He, Xiaoyu Wang and Mei Lin
Entropy 2019, 21(2), 206; https://doi.org/10.3390/e21020206 - 21 Feb 2019
Cited by 3 | Viewed by 3873
Abstract
The skin friction consumes some of the energy when a train is running, and the coherent structure plays an important role in the skin friction. In this paper, we focus on the coherent structure generated near the vent of a train. The intention [...] Read more.
The skin friction consumes some of the energy when a train is running, and the coherent structure plays an important role in the skin friction. In this paper, we focus on the coherent structure generated near the vent of a train. The intention is to investigate the effect of the vent on the generation of coherent structures. The ventilation system of a high-speed train is reasonably simplified as a T-junction duct with vertical blades. The velocity signal of the cross duct was measured in three different sections (upstream, mid-center and downstream), and then the coherent structure of the denoised signals was analyzed by continuous wavelet transform (CWT). The analysis indicates that the coherent structure frequencies become abundant and the energy peak decreases with the increase of the velocity ratio. As a result, we conclude that a higher velocity ratio is preferable to reduce the skin friction of the train. Besides, with the increase of velocity ratio, the dimensionless frequency St of the high-energy coherent structure does not change obviously and St = 3.09 × 10−4–4.51 × 10−4. Full article
(This article belongs to the Special Issue Intermittency and Self-Organisation in Turbulence and Statistical Mechanics)
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<p>Schematic diagram of experimental system: 1-Entrance, 2-Settling chamber, 3-Contraction section, 4-Front section of cross duct, 5-Back section of cross duct, 6-Connect section, 7-Expansion section, 8-Fan, 9-Blades, 10-Branch duct, 11-Valve, 12- Glass rotameter, 13-Fan.</p> Full article ">Figure 2
<p>Schematic diagram of the blades.</p> Full article ">Figure 3
<p>Sketch of the measurement points.</p> Full article ">Figure 4
<p>(<b>a</b>) Comparison of velocity distribution between our experiment and Gessner’s et al.; (<b>b</b>) Comparison of wall velocity distribution between our experimental data and results of Schultz and Flack.</p> Full article ">Figure 5
<p>Wavelet power spectrum at <span class="html-italic">x</span>/D = 0, <span class="html-italic">R</span> = 0.13, <span class="html-italic">u</span><sub>c</sub> = 40 m/s, <span class="html-italic">y</span>/L = 0.0070.</p> Full article ">Figure 5 Cont.
<p>Wavelet power spectrum at <span class="html-italic">x</span>/D = 0, <span class="html-italic">R</span> = 0.13, <span class="html-italic">u</span><sub>c</sub> = 40 m/s, <span class="html-italic">y</span>/L = 0.0070.</p> Full article ">Figure 6
<p>Wavelet power spectrum at <span class="html-italic">x</span>/D = 0, <span class="html-italic">R</span> = 0.13, <span class="html-italic">u</span><sub>c</sub> = 40 m/s.</p> Full article ">Figure 7
<p>Wavelet power spectrum at <span class="html-italic">u</span><sub>c</sub> = 40 m/s, <span class="html-italic">R</span> = 0.13, <span class="html-italic">y</span>/L = 0.0070.</p> Full article ">Figure 8
<p>Wavelet power spectrum at <span class="html-italic">x</span>/D = 0, R = 0.13, <span class="html-italic">y</span>/L = 0.0070.</p> Full article ">Figure 8 Cont.
<p>Wavelet power spectrum at <span class="html-italic">x</span>/D = 0, R = 0.13, <span class="html-italic">y</span>/L = 0.0070.</p> Full article ">Figure 9
<p>Wavelet power spectrum at <span class="html-italic">x</span>/D = 0, <span class="html-italic">u</span><sub>c</sub> = 40m/s, <span class="html-italic">y</span>/L = 0.0070.</p> Full article ">
26 pages, 737 KiB  
Article
Matching Users’ Preference under Target Revenue Constraints in Data Recommendation Systems
by Shanyun Liu, Yunquan Dong, Pingyi Fan, Rui She and Shuo Wan
Entropy 2019, 21(2), 205; https://doi.org/10.3390/e21020205 - 21 Feb 2019
Cited by 5 | Viewed by 3715
Abstract
This paper focuses on the problem of finding a particular data recommendation strategy based on the user preference and a system expected revenue. To this end, we formulate this problem as an optimization by designing the recommendation mechanism as close to the user [...] Read more.
This paper focuses on the problem of finding a particular data recommendation strategy based on the user preference and a system expected revenue. To this end, we formulate this problem as an optimization by designing the recommendation mechanism as close to the user behavior as possible with a certain revenue constraint. In fact, the optimal recommendation distribution is the one that is the closest to the utility distribution in the sense of relative entropy and satisfies expected revenue. We show that the optimal recommendation distribution follows the same form as the message importance measure (MIM) if the target revenue is reasonable, i.e., neither too small nor too large. Therefore, the optimal recommendation distribution can be regarded as the normalized MIM, where the parameter, called importance coefficient, presents the concern of the system and switches the attention of the system over data sets with different occurring probability. By adjusting the importance coefficient, our MIM based framework of data recommendation can then be applied to systems with various system requirements and data distributions. Therefore, the obtained results illustrate the physical meaning of MIM from the data recommendation perspective and validate the rationality of MIM in one aspect. Full article
(This article belongs to the Special Issue Entropy and Information in Networks, from Societies to Cities)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>System model.</p> Full article ">Figure 2
<p>Probability simplex and optimal recommendation. Region ⓘ denotes Case ⓘ in <a href="#entropy-21-00205-t003" class="html-table">Table 3</a> for <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mn>8</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Function <math display="inline"><semantics> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ϖ</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs. importance coefficient <math display="inline"><semantics> <mi>ϖ</mi> </semantics></math>, when the utility distribution is <math display="inline"><semantics> <mrow> <mi>U</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mn>0.03</mn> <mo>,</mo> <mn>0.07</mn> <mo>,</mo> <mn>0.12</mn> <mo>,</mo> <mn>0.24</mn> <mo>,</mo> <mn>0.25</mn> <mo>,</mo> <mn>0.29</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p><math display="inline"><semantics> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>ϖ</mi> <mo>,</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> vs. <math display="inline"><semantics> <mi>ϖ</mi> </semantics></math>.</p> Full article ">Figure 5
<p>The optimal recommendation distribution vs. minimum average revenue. The parameters set <math display="inline"><semantics> <mrow> <mo stretchy="false">{</mo> <msub> <mi>C</mi> <mi>p</mi> </msub> <mo>,</mo> <msub> <mi>R</mi> <mi>p</mi> </msub> <mo>,</mo> <msub> <mi>C</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>R</mi> <mrow> <mi>a</mi> <mi>d</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mi>m</mi> </msub> <mo stretchy="false">}</mo> </mrow> </semantics></math> is denoted by D1 and D2, where D1 <math display="inline"><semantics> <mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mn>4.5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math> and D2 <math display="inline"><semantics> <mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math>. The utility distributions are U1 <math display="inline"><semantics> <mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.2</mn> <mo>,</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.4</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math> and U2 <math display="inline"><semantics> <mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mn>0.05</mn> <mo>,</mo> <mn>0.15</mn> <mo>,</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.5</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Minimum KL distance between recommendation distribution and utility distribution vs. minimum average revenue. The parameters set <math display="inline"><semantics> <mrow> <mo stretchy="false">{</mo> <msub> <mi>C</mi> <mi>p</mi> </msub> <mo>,</mo> <msub> <mi>R</mi> <mi>p</mi> </msub> <mo>,</mo> <msub> <mi>C</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>R</mi> <mrow> <mi>a</mi> <mi>d</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mi>m</mi> </msub> <mo stretchy="false">}</mo> </mrow> </semantics></math> is denoted by D1 and D2, where D1 <math display="inline"><semantics> <mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mn>4.5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math> and D1 <math display="inline"><semantics> <mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math>. The utility distribution is U1 <math display="inline"><semantics> <mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.2</mn> <mo>,</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.4</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math> and U2 <math display="inline"><semantics> <mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mn>0.05</mn> <mo>,</mo> <mn>0.15</mn> <mo>,</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.5</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math>.</p> Full article ">
10 pages, 260 KiB  
Article
On Entropic Framework Based on Standard and Fractional Phonon Boltzmann Transport Equations
by Shu-Nan Li and Bing-Yang Cao
Entropy 2019, 21(2), 204; https://doi.org/10.3390/e21020204 - 21 Feb 2019
Cited by 9 | Viewed by 2814
Abstract
Generalized expressions of the entropy and related concepts in non-Fourier heat conduction have attracted increasing attention in recent years. Based on standard and fractional phonon Boltzmann transport equations (BTEs), we study entropic functionals including entropy density, entropy flux and entropy production rate. Using [...] Read more.
Generalized expressions of the entropy and related concepts in non-Fourier heat conduction have attracted increasing attention in recent years. Based on standard and fractional phonon Boltzmann transport equations (BTEs), we study entropic functionals including entropy density, entropy flux and entropy production rate. Using the relaxation time approximation and power series expansion, macroscopic approximations are derived for these entropic concepts. For the standard BTE, our results can recover the entropic frameworks of classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT) as if there exists a well-defined effective thermal conductivity. For the fractional BTEs corresponding to the generalized Cattaneo equation (GCE) class, the entropy flux and entropy production rate will deviate from the forms in CIT and EIT. In these cases, the entropy flux and entropy production rate will contain fractional-order operators, which reflect memory effects. Full article
(This article belongs to the Special Issue Entropy Generation and Heat Transfer)
11 pages, 1426 KiB  
Article
Entropy Value-Based Pursuit Projection Cluster for the Teaching Quality Evaluation with Interval Number
by Ming Zhang, Jinpeng Wang and Runjuan Zhou
Entropy 2019, 21(2), 203; https://doi.org/10.3390/e21020203 - 21 Feb 2019
Cited by 16 | Viewed by 2965
Abstract
The issue motivating the paper is the quantification of students’ academic performance and learning achievement regarding teaching quality, under interval number condition, in order to establish a novel model for identifying, evaluating, and monitoring the major factors of the overall teaching quality. We [...] Read more.
The issue motivating the paper is the quantification of students’ academic performance and learning achievement regarding teaching quality, under interval number condition, in order to establish a novel model for identifying, evaluating, and monitoring the major factors of the overall teaching quality. We propose a projection pursuit cluster evaluation model, with entropy value method on the model weights. The weights of the model can then be obtained under the traditional real number conditions after a simulation process by Monte Carlo for transforming interval number to real number. This approach can not only simplify the evaluation of the interval number indicators but also give the weight of each index objectively. This model is applied to 5 teacher data collected from a China college with 4 primary indicators and 15 secondary sub-indicators. Results from the proposed approach are compared with the ones obtained by two alternative evaluating methods. The analysis carried out has contributed to having a better understanding of the education processes in order to promote performance in teaching. Full article
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<p>Box-plot of each index value under 1000 simulation times.</p> Full article ">Figure 2
<p>Entropy value of the weight distribution under each random simulation times. (Note: 1* is the entropy value calculated by the index weights from literature [<a href="#B12-entropy-21-00203" class="html-bibr">12</a>]).</p> Full article ">Figure 3
<p>Scatter plot of projection eigenvalues of random samples for each scheme. (<b>a</b>) 5 simulation times results; (<b>b</b>) 10 simulation times results; (<b>c</b>) 100 simulation times results; and, (<b>d</b>) 1000 simulation times results.</p> Full article ">Figure 3 Cont.
<p>Scatter plot of projection eigenvalues of random samples for each scheme. (<b>a</b>) 5 simulation times results; (<b>b</b>) 10 simulation times results; (<b>c</b>) 100 simulation times results; and, (<b>d</b>) 1000 simulation times results.</p> Full article ">
15 pages, 3587 KiB  
Article
Multimode Decomposition and Wavelet Threshold Denoising of Mold Level Based on Mutual Information Entropy
by Zhufeng Lei, Wenbin Su and Qiao Hu
Entropy 2019, 21(2), 202; https://doi.org/10.3390/e21020202 - 21 Feb 2019
Cited by 30 | Viewed by 4769
Abstract
The continuous casting process is a continuous, complex phase transition process. The noise components of the continuous casting process are complex, the model is difficult to establish, and it is difficult to separate the noise and clear signals effectively. Owing to these demerits, [...] Read more.
The continuous casting process is a continuous, complex phase transition process. The noise components of the continuous casting process are complex, the model is difficult to establish, and it is difficult to separate the noise and clear signals effectively. Owing to these demerits, a hybrid algorithm combining Variational Mode Decomposition (VMD) and Wavelet Threshold denoising (WTD) is proposed, which involves multiscale resolution and adaptive features. First of all, the original signal is decomposed into several Intrinsic Mode Functions (IMFs) by Empirical Mode Decomposition (EMD), and the model parameter K of the VMD is obtained by analyzing the EMD results. Then, the original signal is decomposed by VMD based on the number of IMFs K, and the Mutual Information Entropy (MIE) between IMFs is calculated to identify the noise dominant component and the information dominant component. Next, the noise dominant component is denoised by WTD. Finally, the denoised noise dominant component and all information dominant components are reconstructed to obtain the denoised signal. In this paper, a comprehensive comparative analysis of EMD, Ensemble Empirical Mode Decomposition (EEMD), Complementary Empirical Mode Decomposition (CEEMD), EMD-WTD, Empirical Wavelet Transform (EWT), WTD, VMD, and VMD-WTD is carried out, and the denoising performance of the various methods is evaluated from four perspectives. The experimental results show that the hybrid algorithm proposed in this paper has a better denoising effect than traditional methods and can effectively separate noise and clear signals. The proposed denoising algorithm is shown to be able to effectively recognize different cast speeds. Full article
(This article belongs to the Collection Wavelets, Fractals and Information Theory)
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<p>Mold level model.</p> Full article ">Figure 2
<p>VMD-WTD denoising flowchart. EMD: empirical mode decomposition; VMD: variational mode decomposition; IMF: intrinsic mode functions; WTD: wavelet threshold denoising; MIE: mutual information entropy.</p> Full article ">Figure 3
<p>(<b>a</b>) Noisy signal. (<b>b</b>) Clear signal.</p> Full article ">Figure 4
<p>Results of EMD.</p> Full article ">Figure 5
<p>Results of VMD.</p> Full article ">Figure 6
<p>Results of WTD.</p> Full article ">Figure 7
<p>Comparison after VMD-WTD decomposition.</p> Full article ">Figure 8
<p>Mold level decomposition results of EMD.</p> Full article ">Figure 9
<p>Mold level decomposition result by VMD.</p> Full article ">Figure 10
<p>WTD result of IMF1–IMF5.</p> Full article ">Figure 11
<p>Root-Mean-Square Error (RMSE) indicator for denoising results of multiple algorithms. EEMD: ensemble empirical mode decomposition; CEEMD: complete ensemble empirical mode decomposition; EWT: empirical wavelet transform.</p> Full article ">Figure 12
<p>SNR indicator for denoising results of multiple algorithms. EEMD: ensemble empirical mode decomposition; CEEMD: complete ensemble empirical mode decomposition; EWT: empirical wavelet transform.</p> Full article ">Figure 13
<p>Distribution of maximum energy IMF center frequency.</p> Full article ">
15 pages, 300 KiB  
Article
The Ordering of Shannon Entropies for the Multivariate Distributions and Distributions of Eigenvalues
by Ming-Tien Tsai, Feng-Ju Hsu and Chia-Hsuan Tsai
Entropy 2019, 21(2), 201; https://doi.org/10.3390/e21020201 - 20 Feb 2019
Cited by 2 | Viewed by 3002
Abstract
In this paper, we prove the Shannon entropy inequalities for the multivariate distributions via the notion of convex ordering of two multivariate distributions. We further characterize the multivariate totally positive of order 2 ( M T P 2 ) property of the distribution [...] Read more.
In this paper, we prove the Shannon entropy inequalities for the multivariate distributions via the notion of convex ordering of two multivariate distributions. We further characterize the multivariate totally positive of order 2 ( M T P 2 ) property of the distribution functions of eigenvalues of both central Wishart and central MANOVA models, and of both noncentral Wishart and noncentral MANOVA models under the general population covariance matrix set-up. These results can be directly applied to both the comparisons of two Shannon entropy measures and the power monotonicity problem for the MANOVA problem. Full article
33 pages, 486 KiB  
Article
Amplitude Constrained MIMO Channels: Properties of Optimal Input Distributions and Bounds on the Capacity
by Alex Dytso, Mario Goldenbaum, H. Vincent Poor and Shlomo Shamai (Shitz)
Entropy 2019, 21(2), 200; https://doi.org/10.3390/e21020200 - 19 Feb 2019
Cited by 11 | Viewed by 4227
Abstract
In this work, the capacity of multiple-input multiple-output channels that are subject to constraints on the support of the input is studied. The paper consists of two parts. The first part focuses on the general structure of capacity-achieving input distributions. Known results are [...] Read more.
In this work, the capacity of multiple-input multiple-output channels that are subject to constraints on the support of the input is studied. The paper consists of two parts. The first part focuses on the general structure of capacity-achieving input distributions. Known results are surveyed and several new results are provided. With regard to the latter, it is shown that the support of a capacity-achieving input distribution is a small set in both a topological and a measure theoretical sense. Moreover, explicit conditions on the channel input space and the channel matrix are found such that the support of a capacity-achieving input distribution is concentrated on the boundary of the input space only. The second part of this paper surveys known bounds on the capacity and provides several novel upper and lower bounds for channels with arbitrary constraints on the support of the channel input symbols. As an immediate practical application, the special case of multiple-input multiple-output channels with amplitude constraints is considered. The bounds are shown to be within a constant gap to the capacity if the channel matrix is invertible and are tight in the high amplitude regime for arbitrary channel matrices. Moreover, in the regime of high amplitudes, it is shown that the capacity scales linearly with the minimum between the number of transmit and receive antennas, similar to the case of average power-constrained inputs. Full article
(This article belongs to the Special Issue Information Theory for Data Communications and Processing)
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<p>An example of a support of an optimal input distribution for the special case <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>n</mi> <mi>r</mi> </msub> <mo>=</mo> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Comparison of the upper and lower bounds of Theorems 9, 11, 15, and 16 evaluated for a <math display="inline"><semantics> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </semantics></math> MIMO system with per-antenna amplitude constraints <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>A</mi> </mrow> </semantics></math> (i.e., <math display="inline"><semantics> <mrow> <mi mathvariant="bold">a</mi> <mo>=</mo> <mo>(</mo> <mi>A</mi> <mo>,</mo> <mi>A</mi> <mo>)</mo> </mrow> </semantics></math>) and channel matrix <math display="inline"><semantics> <mrow> <mi mathvariant="bold-sans-serif">H</mi> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mtable> <mtr> <mtd> <mrow> <mn>0.3</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mn>0.1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </semantics></math>. The nested figure represents a zoom into the region <math display="inline"><semantics> <mrow> <mn>0</mn> <mspace width="0.166667em"/> <mi>dB</mi> <mo>≤</mo> <mi>A</mi> <mo>≤</mo> <mn>5</mn> <mspace width="0.166667em"/> <mi>dB</mi> </mrow> </semantics></math> to visualize the differences between the bounds at small amplitude constraints.</p> Full article ">Figure 3
<p>Example of a pulse-amplitude modulation constellation with <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> points and amplitude constraint <span class="html-italic">A</span> (i.e., <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">PAM</mi> <mo>(</mo> <mn>4</mn> <mo>,</mo> <mi>A</mi> <mo>)</mo> </mrow> </semantics></math>), where <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mo>:</mo> <mo>=</mo> <mi>A</mi> <mo>/</mo> <mo>(</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> denotes half the Euclidean distance between two adjacent constellation points. In the case <span class="html-italic">N</span> is odd, 0 is a constellation point.</p> Full article ">Figure 4
<p>Comparison of the upper bound in Theorem 2 with the lower bounds of Theorem 20 for a <math display="inline"><semantics> <mrow> <mn>3</mn> <mo>×</mo> <mn>1</mn> </mrow> </semantics></math> MIMO system with amplitude constraints <math display="inline"><semantics> <mrow> <msub> <mi>A</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>A</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>A</mi> <mn>3</mn> </msub> <mo>=</mo> <mi>A</mi> </mrow> </semantics></math> (i.e., <math display="inline"><semantics> <mrow> <mi mathvariant="bold">a</mi> <mo>=</mo> <mo>(</mo> <mi>A</mi> <mo>,</mo> <mi>A</mi> <mo>,</mo> <mi>A</mi> <mo>)</mo> </mrow> </semantics></math>) and channel matrix <math display="inline"><semantics> <mrow> <mi mathvariant="bold-sans-serif">h</mi> <mo>=</mo> <mo>(</mo> <mn>0.6557</mn> <mo>,</mo> <mn>0.0357</mn> <mo>,</mo> <mn>0.8491</mn> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Comparison of upper and lower bounds on the capacity of a SISO channel with amplitude constraint <span class="html-italic">A</span>. The capacity of this channel is known for amplitudes smaller than <math display="inline"><semantics> <mrow> <mi>A</mi> <mo>≈</mo> <mn>10</mn> <msub> <mo form="prefix">log</mo> <mn>10</mn> </msub> <mrow> <mo>(</mo> <mn>1.665</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>2.214</mn> <mspace width="0.166667em"/> <mi>dB</mi> </mrow> </semantics></math> only (i.e., to the left of the gray vertical line) and unknown elsewhere. The nested figure represents a zoom into the region <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>1.9</mn> <mspace width="0.166667em"/> <mi>dB</mi> <mo>≤</mo> <mi>A</mi> <mo>≤</mo> <mo>−</mo> <mn>1.88</mn> <mspace width="0.166667em"/> <mi>dB</mi> </mrow> </semantics></math> to highlight the differences between the Moment upper bound (<a href="#FD51-entropy-21-00200" class="html-disp-formula">51</a>), the Rassouli–Clerckx upper bound in Equation (<a href="#FD50-entropy-21-00200" class="html-disp-formula">50</a>), and the lower bound with binary inputs in Equation (<a href="#FD56-entropy-21-00200" class="html-disp-formula">56</a>).</p> Full article ">
16 pages, 6206 KiB  
Article
Multiscale Entropy Quantifies the Differential Effect of the Medium Embodiment on Older Adults Prefrontal Cortex during the Story Comprehension: A Comparative Analysis
by Soheil Keshmiri, Hidenobu Sumioka, Ryuji Yamazaki and Hiroshi Ishiguro
Entropy 2019, 21(2), 199; https://doi.org/10.3390/e21020199 - 19 Feb 2019
Cited by 8 | Viewed by 4340
Abstract
Todays’ communication media virtually impact and transform every aspect of our daily communication and yet the extent of their embodiment on our brain is unexplored. The study of this topic becomes more crucial, considering the rapid advances in such fields as socially assistive [...] Read more.
Todays’ communication media virtually impact and transform every aspect of our daily communication and yet the extent of their embodiment on our brain is unexplored. The study of this topic becomes more crucial, considering the rapid advances in such fields as socially assistive robotics that envision the use of intelligent and interactive media for providing assistance through social means. In this article, we utilize the multiscale entropy (MSE) to investigate the effect of the physical embodiment on the older people’s prefrontal cortex (PFC) activity while listening to stories. We provide evidence that physical embodiment induces a significant increase in MSE of the older people’s PFC activity and that such a shift in the dynamics of their PFC activation significantly reflects their perceived feeling of fatigue. Our results benefit researchers in age-related cognitive function and rehabilitation who seek for the adaptation of these media in robot-assistive cognitive training of the older people. In addition, they offer a complementary information to the field of human-robot interaction via providing evidence that the use of MSE can enable the interactive learning algorithms to utilize the brain’s activation patterns as feedbacks for improving their level of interactivity, thereby forming a stepping stone for rich and usable human mental model. Full article
(This article belongs to the Special Issue Multiscale Entropy Approaches and Their Applications)
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<p>(<b>A</b>) Grand-average MSE of older people’s Left-hemispheric PFC activation in speaker (S), video-chat (V), Telenoid (T), and face-to-face (F) settings. In these plots, scale factors 10 and 20 correspond to the one-second and two-second data acquisition intervals, given the sampling rate of our device (i.e., 10.0 Hz). (<b>B</b>) Descriptive Statistics of the older people’s left-hemispheric MSE in speaker (S), video-chat (V), Telenoid (T), and face-to-face (F). Asterisks mark the significant differences between these media settings.</p> Full article ">Figure 2
<p>Spearman correlation between MSEs of the older people left-hemispheric PFC activation and their self-assessed responses to feeling of fatigue. (<b>S</b>) speaker; (<b>V</b>) video-chat; (<b>T</b>) Telenoid; (<b>F</b>) face-to-face.</p> Full article ">Figure 3
<p>Older people’s left-hemispheric MSE clusters. The boundaries associated with these clusters (i.e., clusters’ mean) are shown (black line segments) in the figure.</p> Full article ">Figure 4
<p>Left-hemispheric MSE vs. self-assessed responses to the perceived feeling of fatigue. In this figure, “Fatigue” and “No Fatigue” refer to the number of older people whose self-assessed responses to the feeling of fatigue was &gt;4.0 and ≤4.0, respectively.</p> Full article ">Figure 5
<p>(<b>a</b>) Telenoid setting; (<b>b</b>) Telenoid medium; (<b>c</b>) Speaker setting (<b>d</b>) Video-chat setting; and (<b>e</b>) face-to-face setting. In these figures, an experimenter demonstrates the experimental setup.</p> Full article ">Figure 6
<p>(<b>a</b>) fNIRS device in present study. Bottom subplot on left shows arrangement of source-detector of four channels of this device. Distances between short (i.e., 1.0 cm) and long (i.e., 3.0 cm) source and detector of left and right channels are shown. (<b>b</b>) Arrangement of 10–20 International Standard System: In this figure, relative locations of channels of fNIRS device in our study (i.e., <math display="inline"><semantics> <mrow> <mi>L</mi> <mn>1</mn> <mo>,</mo> <mi>L</mi> <mn>3</mn> <mo>,</mo> <mi>R</mi> <mn>1</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>3</mn> </mrow> </semantics></math>) are depicted in red (i.e., sources) and green (i.e., detectors) squares. <math display="inline"><semantics> <mrow> <mi>L</mi> <mn>1</mn> <mo>,</mo> <mi>R</mi> <mn>1</mn> <mo>,</mo> <mi>L</mi> <mn>3</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>R</mi> <mn>3</mn> </mrow> </semantics></math> are channels with short (i.e., 1.0 cm) and long (i.e., 3.0 cm) source-detector distances.</p> Full article ">Figure A1
<p>(<b>A</b>) Grand-average MSE of older people’s right-hemispheric PFC activation in speaker (S), video-chat (V), Telenoid (T), and face-to-face (F) settings. In these plots, scale factors 10 and 20 correspond to the one-second and two-second data acquisition intervals, given the sampling rate of our device (i.e., 10.0 Hz). (<b>B</b>) Descriptive Statistics of the older people’s right-hemispheric MSE in speaker (S), video-chat (V), Telenoid (T), and face-to-face (F). Asterisks mark the significant differences between these media settings.</p> Full article ">Figure A2
<p>Spearman correlation between MSEs of the older people’s right-hemispheric PFC activation and their self-assessed responses to feeling of fatigue. (<b>S</b>) speaker; (<b>V</b>) video-chat; (<b>T</b>) Telenoid; (<b>F</b>) face-to-face.</p> Full article ">Figure A3
<p>Older people’s right-hemispheric MSE clusters. The boundaries associated with these clusters (i.e., clusters’ mean) are shown (black line segments) in the figure.</p> Full article ">Figure A4
<p>Right-hemispheric MSE vs. self-assessed responses to the feeling of fatigue. In this figure, “Fatigue” and “No Fatigue” refer to the number of older people whose self-assessed responses to the feeling of fatigue was &gt;4 and ≤4.0, respectively.</p> Full article ">
17 pages, 3224 KiB  
Article
Attribute Selection Based on Constraint Gain and Depth Optimal for a Decision Tree
by Huaining Sun, Xuegang Hu and Yuhong Zhang
Entropy 2019, 21(2), 198; https://doi.org/10.3390/e21020198 - 19 Feb 2019
Cited by 6 | Viewed by 4029
Abstract
Uncertainty evaluation based on statistical probabilistic information entropy is a commonly used mechanism for a heuristic method construction of decision tree learning. The entropy kernel potentially links its deviation and decision tree classification performance. This paper presents a decision tree learning algorithm based [...] Read more.
Uncertainty evaluation based on statistical probabilistic information entropy is a commonly used mechanism for a heuristic method construction of decision tree learning. The entropy kernel potentially links its deviation and decision tree classification performance. This paper presents a decision tree learning algorithm based on constrained gain and depth induction optimization. Firstly, the calculation and analysis of single- and multi-value event uncertainty distributions of information entropy is followed by an enhanced property of single-value event entropy kernel and multi-value event entropy peaks as well as a reciprocal relationship between peak location and the number of possible events. Secondly, this study proposed an estimated method for information entropy whose entropy kernel is replaced with a peak-shift sine function to establish a decision tree learning (CGDT) algorithm on the basis of constraint gain. Finally, by combining branch convergence and fan-out indices under an inductive depth of a decision tree, we built a constraint gained and depth inductive improved decision tree (CGDIDT) learning algorithm. Results show the benefits of the CGDT and CGDIDT algorithms. Full article
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<p>Entropy of single-value event and two-value event. (<b>a</b>) Comparison of <span class="html-italic">I</span>(<span class="html-italic">x</span>) and <span class="html-italic">H</span>(<span class="html-italic">x</span>) for a single-value event; (<b>b</b>) Entropy, <span class="html-italic">H</span>(<span class="html-italic">x</span>), of two-value event.</p> Full article ">Figure 2
<p>Distribution analysis of multi-value event entropy. (<b>a</b>) Entropy of three-value event; (<b>b</b>) Entropy of four-value event. Note: (1) If three possible probabilities of the event are <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>1</sub>), <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>2</sub>), and <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>3</sub>), respectively, let parameter <span class="html-italic">k</span> exist to make <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>3</sub>) = <span class="html-italic">k P</span>(<span class="html-italic">X<sub>j</sub></span><sub>2</sub>), then, <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>2</sub>) = [1 − <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>1</sub>)]/(1 + <span class="html-italic">k</span>) in which <span class="html-italic">k</span> &gt; 0. (2) If four possible probabilities of the event are <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>1</sub>), <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>2</sub>), <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>3</sub>), and <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>4</sub>), let parameter <span class="html-italic">k</span><sub>1</sub>and <span class="html-italic">k</span><sub>2</sub> exist to make <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>3</sub>) = <span class="html-italic">k</span><sub>1</sub> <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>2</sub>) and <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>4</sub>) = <span class="html-italic">k</span><sub>2</sub> <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>2</sub>), then, <span class="html-italic">P</span>(<span class="html-italic">X<sub>j</sub></span><sub>2</sub>) = [1 − <span class="html-italic">P</span> (<span class="html-italic">X<sub>j</sub></span><sub>1</sub>)]/(1 + <span class="html-italic">k</span><sub>1</sub> + <span class="html-italic">k</span><sub>2</sub>), in which <span class="html-italic">k</span><sub>1</sub> &gt; 0 and <span class="html-italic">k</span><sub>2</sub> &gt; 0. Sign “1,1” similar to Figure (<b>b</b>), the number at the front is <span class="html-italic">k</span><sub>1</sub>, and the number after it is <span class="html-italic">k</span><sub>2</sub>.</p> Full article ">Figure 3
<p>Comparison of <span class="html-italic">I</span><sub>sin</sub>(<span class="html-italic">x</span>) for different values events, analysis of <span class="html-italic">H</span>(<span class="html-italic">x</span>) and <span class="html-italic">H<sub>s</sub></span>(<span class="html-italic">x</span>) for two-value event. Note: Is(x) is <span class="html-italic">I</span><sub>sin</sub>(<span class="html-italic">x</span>) in the figure.</p> Full article ">Figure 4
<p>Comparison analysis of multi-value event <span class="html-italic">H</span><sub>s</sub>(<span class="html-italic">x</span>) entropy. Note: Same as <a href="#entropy-21-00198-f002" class="html-fig">Figure 2</a>.</p> Full article ">Figure 5
<p>Relationship diagrams of the classification performance of the decision tree of the Balance dataset and the location of the entropy kernel peak. (<b>a</b>) Relationship between the accuracy rate and entropy kernel peak location; (<b>b</b>) Relationship between the scale and entropy kernel peak location.</p> Full article ">Figure 6
<p>Relationship diagrams of the classification performance of the decision tree of the Tic-tac-toe dataset and the location of the entropy kernel peak. (<b>a</b>) Relationship between the accuracy rate and entropy kernel peak location; (<b>b</b>) Relationship between the scale and entropy kernel peak location.</p> Full article ">Figure 7
<p>Relationship diagrams of the classification performance of the decision tree of the Dermatology dataset and the location of entropy kernel peak.</p> Full article ">
15 pages, 4137 KiB  
Article
Magnetotelluric Signal-Noise Identification and Separation Based on ApEn-MSE and StOMP
by Jin Li, Jin Cai, Yiqun Peng, Xian Zhang, Cong Zhou, Guang Li and Jingtian Tang
Entropy 2019, 21(2), 197; https://doi.org/10.3390/e21020197 - 19 Feb 2019
Cited by 6 | Viewed by 3978
Abstract
Natural magnetotelluric signals are extremely weak and susceptible to various types of noise pollution. To obtain more useful magnetotelluric data for further analysis and research, effective signal-noise identification and separation is critical. To this end, we propose a novel method of magnetotelluric signal-noise [...] Read more.
Natural magnetotelluric signals are extremely weak and susceptible to various types of noise pollution. To obtain more useful magnetotelluric data for further analysis and research, effective signal-noise identification and separation is critical. To this end, we propose a novel method of magnetotelluric signal-noise identification and separation based on ApEn-MSE and Stagewise orthogonal matching pursuit (StOMP). Parameters with good irregularity metrics are introduced: Approximate entropy (ApEn) and multiscale entropy (MSE), in combination with k-means clustering, can be used to accurately identify the data segments that are disturbed by noise. Stagewise orthogonal matching pursuit (StOMP) is used for noise suppression only in data segments identified as containing strong interference. Finally, we reconstructed the signal. The results show that the proposed method can better preserve the low-frequency slow-change information of the magnetotelluric signal compared with just using StOMP, thus avoiding the loss of useful information due to over-processing, while producing a smoother and more continuous apparent resistivity curve. Moreover, the results more accurately reflect the inherent electrical structure information of the measured site itself. Full article
(This article belongs to the Special Issue The 20th Anniversary of Entropy - Approximate and Sample Entropy)
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<p>Clustering effect diagrams of sample library signals with (<b>a</b>) Approximate entropy (ApEn) and (<b>b</b>) Multiscale entropy MSE.</p> Full article ">Figure 1 Cont.
<p>Clustering effect diagrams of sample library signals with (<b>a</b>) Approximate entropy (ApEn) and (<b>b</b>) Multiscale entropy MSE.</p> Full article ">Figure 2
<p>Simulated interference to add to the test site signal with (<b>a</b>) square wave interference, (<b>b</b>) triangular wave interference, and (<b>c</b>) pulse interference.</p> Full article ">Figure 2 Cont.
<p>Simulated interference to add to the test site signal with (<b>a</b>) square wave interference, (<b>b</b>) triangular wave interference, and (<b>c</b>) pulse interference.</p> Full article ">Figure 3
<p>The effect of signal-noise identification and separation for measured magnetotelluric (MT) data with (<b>a</b>) square wave interference, (<b>b</b>) triangular wave interference, and (<b>c</b>) pulse interference.</p> Full article ">Figure 3 Cont.
<p>The effect of signal-noise identification and separation for measured magnetotelluric (MT) data with (<b>a</b>) square wave interference, (<b>b</b>) triangular wave interference, and (<b>c</b>) pulse interference.</p> Full article ">Figure 4
<p>Comparison of the apparent resistivity-phase curves for site 2535BOAC; the black diamonds, blue triangles, and red circles show the apparent resistivity curves obtained from the original data, the overall processing of Stagewise orthogonal matching pursuit (StOMP), and the proposed method, respectively.</p> Full article ">Figure 5
<p>Comparison of the apparent resistivity-phase curves for site 2535BOAF; the black diamonds, blue triangles, and red circles show the apparent resistivity curves obtained from the original data, the overall processing of StOMP, and the proposed method, respectively.</p> Full article ">Figure 6
<p>Comparison of the polarization direction results at 5.2 Hz for site 2535BOAC, derived from the original data (<b>a</b>) and data processed by the proposed method (<b>b</b>).</p> Full article ">Figure 7
<p>Comparison of the polarization direction results at 2.3 Hz for site 2535BOAF, derived from the original data (<b>a</b>) and data processed by the proposed method (<b>b</b>).</p> Full article ">
19 pages, 956 KiB  
Article
Centroid-Based Clustering with αβ-Divergences
by Auxiliadora Sarmiento, Irene Fondón, Iván Durán-Díaz and Sergio Cruces
Entropy 2019, 21(2), 196; https://doi.org/10.3390/e21020196 - 19 Feb 2019
Cited by 13 | Viewed by 4378
Abstract
Centroid-based clustering is a widely used technique within unsupervised learning algorithms in many research fields. The success of any centroid-based clustering relies on the choice of the similarity measure under use. In recent years, most studies focused on including several divergence measures in [...] Read more.
Centroid-based clustering is a widely used technique within unsupervised learning algorithms in many research fields. The success of any centroid-based clustering relies on the choice of the similarity measure under use. In recent years, most studies focused on including several divergence measures in the traditional hard k-means algorithm. In this article, we consider the problem of centroid-based clustering using the family of α β -divergences, which is governed by two parameters, α and β . We propose a new iterative algorithm, α β -k-means, giving closed-form solutions for the computation of the sided centroids. The algorithm can be fine-tuned by means of this pair of values, yielding a wide range of the most frequently used divergences. Moreover, it is guaranteed to converge to local minima for a wide range of values of the pair ( α , β ). Our theoretical contribution has been validated by several experiments performed with synthetic and real data and exploring the ( α , β ) plane. The numerical results obtained confirm the quality of the algorithm and its suitability to be used in several practical applications. Full article
(This article belongs to the Special Issue Information Theory Applications in Signal Processing)
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<p>Analysis of the convergence region of the <math display="inline"><semantics> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </semantics></math>-<span class="html-italic">k</span>-means algorithm. The region in blue shows the convex cone that guarantee the convergence of the algorithm to a local minimum for any dataset. Blue lines represent the boundaries of the convergence region for some values of the function <math display="inline"><semantics> <mrow> <msub> <mo form="prefix">exp</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </msub> <mfenced separators="" open="(" close=")"> <mfrac> <mn>1</mn> <mrow> <mi>β</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mfenced> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Generative models for dataset used in experiment 1. Each of the four mixture models have three components of Gaussian, Log-Gaussian, Poisson, and Binomial distribution, respectively.</p> Full article ">Figure 3
<p>Average ACC obtained with the right centroid <math display="inline"><semantics> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </semantics></math>-<span class="html-italic">k</span>-means algorithm for four different datasets.</p> Full article ">Figure 4
<p>Average ACC obtained with the left-centroid <math display="inline"><semantics> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </semantics></math>-<span class="html-italic">k</span>-means algorithm for four different datasets.</p> Full article ">Figure 5
<p>Performance of the <math display="inline"><semantics> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </semantics></math>-<span class="html-italic">k</span>-means algorithm in terms of accuracy for DFT-based descriptors considering <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> classes and <math display="inline"><semantics> <mrow> <mi>K</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> classes.</p> Full article ">Figure 6
<p>Performance of the <math display="inline"><semantics> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </semantics></math>-<span class="html-italic">k</span>-means algorithm in terms of accuracy for acoustic descriptors considering <span class="html-italic">K</span> = 3 classes and <span class="html-italic">K</span> = 5 classes.</p> Full article ">Figure 7
<p>Performance of the <math display="inline"><semantics> <mrow> <mi>α</mi> <mi>β</mi> </mrow> </semantics></math>-<span class="html-italic">k</span>-means algorithm in terms of average accuracy over 50 trials for two UCI datasets: Iris and Wine.</p> Full article ">
11 pages, 2238 KiB  
Article
Spatial Organization of the Gene Regulatory Program: An Information Theoretical Approach to Breast Cancer Transcriptomics
by Guillermo de Anda-Jáuregui, Jesús Espinal-Enriquez and Enrique Hernández-Lemus
Entropy 2019, 21(2), 195; https://doi.org/10.3390/e21020195 - 19 Feb 2019
Cited by 12 | Viewed by 3681
Abstract
Gene regulation may be studied from an information-theoretic perspective. Gene regulatory programs are representations of the complete regulatory phenomenon associated to each biological state. In diseases such as cancer, these programs exhibit major alterations, which have been associated with the spatial organization of [...] Read more.
Gene regulation may be studied from an information-theoretic perspective. Gene regulatory programs are representations of the complete regulatory phenomenon associated to each biological state. In diseases such as cancer, these programs exhibit major alterations, which have been associated with the spatial organization of the genome into chromosomes. In this work, we analyze intrachromosomal, or cis-, and interchromosomal, or trans-gene regulatory programs in order to assess the differences that arise in the context of breast cancer. We find that using information theoretic approaches, it is possible to differentiate cis-and trans-regulatory programs in terms of the changes that they exhibit in the breast cancer context, indicating that in breast cancer there is a loss of trans-regulation. Finally, we use these programs to reconstruct a possible spatial relationship between chromosomes. Full article
(This article belongs to the Special Issue Information Theory in Complex Systems)
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<p>A scatterplot, where each point represents a subregulatory program for a pair of chromosomes, comprised of all mutual information function (MI) values for each pair of genes in Chromosome i and Chromosome j. By comparing the MI between gene pairs in tumor and control, in terms of gain loss score (<math display="inline"><semantics> <mi mathvariant="script">GLS</mi> </semantics></math>) (whether there are more losses or gains in MI) and gain loss ratio (<math display="inline"><semantics> <mi mathvariant="script">GLR</mi> </semantics></math>) (whether MI losses or MI gains have a higher magnitude), we identify that interchromosomal interactions between genes in any pair of chromosomes have more losses than gains of MI in disease, with an average MI loss greater than the average MI gain. Meanwhile, intrachromosomal interactions may exhibit three different behaviors: (i) they have more losses with higher average MI loss, although with higher <math display="inline"><semantics> <mi mathvariant="script">GLS</mi> </semantics></math> and <math display="inline"><semantics> <mi mathvariant="script">GLR</mi> </semantics></math> values than the interchromosomal interactions (chromosomes 1, 2, 5, 6, 11, 17, 19, X); (ii) they have more losses, but the average MI gain is higher (chromosomes 3, 4, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 22) or (iii) they have more gains, with a higher average MI gain (21).</p> Full article ">Figure 2
<p>A heatmap showing the differences between GRPs in health and disease. In each square, the color intensity is proportional to <math display="inline"><semantics> <mrow> <mo>−</mo> <mo form="prefix">log</mo> <mo stretchy="false">(</mo> <mi>k</mi> <msub> <mi>s</mi> <mrow> <mi>t</mi> <mi>c</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math>, the Kolmogorov-Smirnov (KS) distance between the subregulatory program for <math display="inline"><semantics> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </semantics></math> in cancer vs the subregulatory program for <math display="inline"><semantics> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </semantics></math> in control. We may observe that in general, the distances between trans-GRPs in control and cancer are greater than the distances between cis-GRPs in health and disease.</p> Full article ">Figure 3
<p>A heatmap showing the differences between cis-<math display="inline"><semantics> <mrow> <mi>G</mi> <mi>R</mi> <msub> <mi>P</mi> <mi>k</mi> </msub> </mrow> </semantics></math> and trans-<math display="inline"><semantics> <mrow> <mi>G</mi> <mi>R</mi> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mrow> </semantics></math> in terms of KS statistic. Notice that in tumors, each trans-<math display="inline"><semantics> <mrow> <mi>G</mi> <mi>R</mi> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mrow> </semantics></math> is almost equidistant to cis-<math display="inline"><semantics> <mrow> <mi>G</mi> <mi>R</mi> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> (that is, each column has virtually the same color intensity in all rows), which is not the case in controls.</p> Full article ">Figure 4
<p>A network visualization of spatial behavior in terms of MI. Each node represents a chromosome. Each directed link has as weight the Hellinger distance (as calculated with the <span class="html-italic">textmineR</span> R package) between the probability density functions (PDFs) of cis-<math display="inline"><semantics> <mrow> <mi>G</mi> <mi>R</mi> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> and trans-<math display="inline"><semantics> <mrow> <mi>G</mi> <mi>R</mi> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mrow> </semantics></math>. The intensity of each link (transparency and thickness) is inversely proportional to the Hellinger distance. The nodes are arranged using a prefuse force-directed layout algorithm, considering the inverse of the Hellinger distance. This pushes nodes where cis-<math display="inline"><semantics> <mrow> <mi>G</mi> <mi>R</mi> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> and trans-<math display="inline"><semantics> <mrow> <mi>G</mi> <mi>R</mi> <msub> <mi>P</mi> <mrow> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mrow> </semantics></math> are similar together. Notice that the position of chromosomes is different in tumors and controls. Also notice that, overall, links are thicker (that is, PDFs are closer) in controls. <a href="#app1-entropy-21-00195" class="html-app">Supplementary Figure S1</a> provides a force-directed visualization that shows some cases in tumor networks where chromosomes are "pushed together" (such as 2 and X, or 3 and 8).</p> Full article ">
17 pages, 3489 KiB  
Article
Entropy Mapping Approach for Functional Reentry Detection in Atrial Fibrillation: An In-Silico Study
by Juan P. Ugarte, Catalina Tobón and Andrés Orozco-Duque
Entropy 2019, 21(2), 194; https://doi.org/10.3390/e21020194 - 18 Feb 2019
Cited by 12 | Viewed by 4181
Abstract
Catheter ablation of critical electrical propagation sites is a promising tool for reducing the recurrence of atrial fibrillation (AF). The spatial identification of the arrhythmogenic mechanisms sustaining AF requires the evaluation of electrograms (EGMs) recorded over the atrial surface. This work aims to [...] Read more.
Catheter ablation of critical electrical propagation sites is a promising tool for reducing the recurrence of atrial fibrillation (AF). The spatial identification of the arrhythmogenic mechanisms sustaining AF requires the evaluation of electrograms (EGMs) recorded over the atrial surface. This work aims to characterize functional reentries using measures of entropy to track and detect a reentry core. To this end, different AF episodes are simulated using a 2D model of atrial tissue. Modified Courtemanche human action potential and Fenton–Karma models are implemented. Action potential propagation is modeled by a fractional diffusion equation, and virtual unipolar EGM are calculated. Episodes with stable and meandering rotors, figure-of-eight reentry, and disorganized propagation with multiple reentries are generated. Shannon entropy ( S h E n ), approximate entropy ( A p E n ), and sample entropy ( S a m p E n ) are computed from the virtual EGM, and entropy maps are built. Phase singularity maps are implemented as references. The results show that A p E n and S a m p E n maps are able to detect and track the reentry core of rotors and figure-of-eight reentry, while the S h E n results are not satisfactory. Moreover, A p E n and S a m p E n consistently highlight a reentry core by high entropy values for all of the studied cases, while the ability of S h E n to characterize the reentry core depends on the propagation dynamics. Such features make the A p E n and S a m p E n maps attractive tools for the study of AF reentries that persist for a period of time that is similar to the length of the observation window, and reentries could be interpreted as AF-sustaining mechanisms. Further research is needed to determine and fully understand the relation of these entropy measures with fibrillation mechanisms other than reentries. Full article
(This article belongs to the Special Issue The 20th Anniversary of Entropy - Approximate and Sample Entropy)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>(<b>a</b>) Stimulation protocol (i): the S2 stimulus occurs after S1, and it is applied to the inferior left corner of the domain. (<b>b</b>) Stimulation protocol (ii): S2 occurs after S1 and consists of two stimuli applied to the middle portion of the domain. For both protocols, S1 is a plane stimulus applied to the left boundary, and it generates a plane wave traveling from left to right.</p> Full article ">Figure 2
<p>Rotor propagation patterns generated by applying stimulation protocol (i) and (<b>a</b>) the paroxysmal atrial fibrillation (pAF) condition, (<b>b</b>) chronic AF 1 (cAF1) condition, (<b>c</b>) cAF2 condition. The maps correspond to the last 1000 ms of each simulation.</p> Full article ">Figure 3
<p>Reentry dynamics characterization maps corresponding to the rotors generated by applying stimulation protocol (i) and (<b>a</b>) pAF condition, (<b>b</b>) cAF1 condition, (<b>c</b>) cAF2 condition. The phase singularity (PS) trajectory is shown in the first column; Shannon entropy (<math display="inline"><semantics> <mrow> <mi>S</mi> <mi>h</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math>), approximate entropy (<math display="inline"><semantics> <mrow> <mi>A</mi> <mi>p</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math>), and sample entropy (<math display="inline"><semantics> <mrow> <mi>S</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math>) maps are shown from the second to the last column, respectively.</p> Full article ">Figure 4
<p>Figure-of-eight propagation patterns generated by applying stimulation protocol (ii) and (<b>a</b>) cAF2 condition, (<b>b</b>) cAF3 condition.</p> Full article ">Figure 5
<p>Reentry dynamics characterization maps corresponding to the figure-of-eight reentries generated by applying stimulation protocol (ii) and (<b>a</b>) cAF2 condition, (<b>b</b>) cAF3 condition. The PS trajectory is shown in the first column; <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>h</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>p</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math> maps are shown from the second to the last column, respectively.</p> Full article ">Figure 6
<p>Multiple-reentry propagation patterns generated by applying stimulation protocol (i) and (<b>a</b>) cAF4 condition, (<b>b</b>) cAF5 condition. Four consecutive frames are presented in each case.</p> Full article ">Figure 7
<p>Characterization of multiple-reentry dynamics resulting from the implementation of stimulation protocol (i) and the cAF4 condition. From (<b>a</b>) to (<b>d</b>), the maps correspond to four consecutive 1000 ms intervals. The PS trajectory is shown in the first column; <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>h</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>p</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math> maps are shown from the second to the last column, respectively. Red circles in the PS maps mark the occurrence of reentries. Black circles in the entropy maps mark detections matching the reentries defined in the PS maps.</p> Full article ">Figure 8
<p>Characterization of reentry breakup dynamics resulting from the implementation of stimulation protocol (i) and the cAF5 condition. From (<b>a</b>) to (<b>d</b>), the maps correspond to four consecutive 1000 ms intervals. The PS trajectory is shown in the first column; <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>h</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>A</mi> <mi>p</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>a</mi> <mi>m</mi> <mi>p</mi> <mi>E</mi> <mi>n</mi> </mrow> </semantics></math> maps are shown from the second to the last column, respectively. Red contours in the PS maps mark the occurrence of reentries. Black contours in the entropy maps mark reentries detections.</p> Full article ">
29 pages, 748 KiB  
Review
From Spin Glasses to Negative-Weight Percolation
by Alexander K. Hartmann, Oliver Melchert and Christoph Norrenbrock
Entropy 2019, 21(2), 193; https://doi.org/10.3390/e21020193 - 18 Feb 2019
Viewed by 4640
Abstract
Spin glasses are prototypical random systems modelling magnetic alloys. One important way to investigate spin glass models is to study domain walls. For two dimensions, this can be algorithmically understood as the calculation of a shortest path, which allows for negative distances or [...] Read more.
Spin glasses are prototypical random systems modelling magnetic alloys. One important way to investigate spin glass models is to study domain walls. For two dimensions, this can be algorithmically understood as the calculation of a shortest path, which allows for negative distances or weights. This led to the creation of the negative weight percolation (NWP) model, which is presented here along with all necessary basics from spin glasses, graph theory and corresponding algorithms. The algorithmic approach involves a mapping to the classical matching problem for graphs. In addition, a summary of results is given, which were obtained during the past decade. This includes the study of percolation transitions in dimension from d = 2 up to and beyond the upper critical dimension d u = 6 , also for random graphs. It is shown that NWP is in a different universality class than standard percolation. Furthermore, the question of whether NWP exhibits properties of Stochastic–Loewner Evolution is addressed and recent results for directed NWP are presented. Full article
(This article belongs to the Special Issue Phase Transitions and Critical Phenomena in Frustrated Systems and Thin Films)
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Figure 1

Figure 1
<p>Energy of two spins placed at distance <span class="html-italic">r</span> coupled through a cloud of conducting electrons, yielding the RKKY (Ruderman, Kittel, Kasuya, Yosida) interaction.</p> Full article ">Figure 2
<p>The ground state of a frustrated system consisting of four spins. Ferromagnetic interactions (“bonds”) are represented by straight lines. Thus, the interactions favour parallel orientation of the spins. An antiferromagnetic interaction is shown as a zigzag line. Independent of the orientation spin 2, one of its incident bonds is not satisfied. On the other hand, bonds 3-4 and 1-4 are satisfied.</p> Full article ">Figure 3
<p>Example for a realisation of a two-dimensional spin glass (free boundary conditions). The Spins are located on the sites of a square lattice. Nearest neighbour spins interact either through a ferromagnetic (straight blue line) or antiferromagnetic (jagged red line) bond. For this example, a bimodal bond distribution is assumed, which means all bonds exhibit <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>J</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>|</mo> <mo>=</mo> <mi>J</mi> </mrow> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The same two-dimensional realisation as shown in <a href="#entropy-21-00193-f003" class="html-fig">Figure 3</a>. Spins are located on the sites denoted by open circles. Filled circle symbols denote the sites of the dual lattice. Cycles on the dual lattice correspond to closed domain walls in the original lattice. Those cycles which exhibit a positive energy are indicated by dashed lines plus a grey colour for the enclosed areas.</p> Full article ">Figure 5
<p>The same two-dimensional realisation as shown in <a href="#entropy-21-00193-f003" class="html-fig">Figure 3</a>. Spins are located on the sites denoted by open circles. Filled circles denote the sites of the dual lattice. A ground state is depicted, where spins pointing downwards are shown, all other spins point upwards. The bonds which are not satisfied are marked grey. There does not exist a closed domain wall with positive energy.</p> Full article ">Figure 6
<p><b>Top</b>: An additional column of very <span class="html-italic">strong</span> bonds (shown at the right with very thick lines) is added for the example spin glass as shown in <a href="#entropy-21-00193-f003" class="html-fig">Figure 3</a>. The bonds are <span class="html-italic">not</span> compatible with the GS configuration. This will force the spins in the first and last column to flip relative to each other and force a domain wall of minimum energy (dashed line) into the system. <b>Bottom</b>: New ground state for the modified system. The spins on the left inside the marked area have been flipped. In both cases, again, spins are located on the sites denoted by open circles. Filled circles denote the sites of the dual lattice. The bonds which are not satisfied are marked grey.</p> Full article ">Figure 7
<p>An undirected graph.</p> Full article ">Figure 8
<p>A directed graph.</p> Full article ">Figure 9
<p>Illustration of matching for a sample graph of six nodes and four edges. In the left, the matching consist of two edges <math display="inline"><semantics> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo stretchy="false">{</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">}</mo> </mrow> </semantics></math>. Matched edges and matched nodes are shown in bold lines. On the right, a perfect matching is shown, i.e., all nodes are matched.</p> Full article ">Figure 10
<p>Illustration of the NWP phase transition. Sample configurations of loops on a square grid for <math display="inline"><semantics> <mrow> <mi>L</mi> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mn>96</mn> </mrow> </semantics></math> side length, with periodic boundary conditions. (Non) percolating loops are shown in black (grey). The configurations are taken for different values of the parameter <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> which controls the disorder. (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>&lt;</mo> <msub> <mi>ρ</mi> <mi>c</mi> </msub> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>≈</mo> <msub> <mi>ρ</mi> <mi>c</mi> </msub> </mrow> </semantics></math>, and, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>&gt;</mo> <msub> <mi>ρ</mi> <mi>c</mi> </msub> </mrow> </semantics></math>. Beyond the critical point <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi>c</mi> </msub> </semantics></math>, in the thermodynamic limit <math display="inline"><semantics> <mrow> <mi>L</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>, there exist paths which span the lattice in any direction which exhibits periodic boundaries.</p> Full article ">Figure 11
<p>Examples of the main steps of the algorithmic procedure: (<b>I</b>) the original lattice <span class="html-italic">G</span> together with weights of edges; (<b>II</b>) the auxiliary graph <math display="inline"><semantics> <msub> <mi>G</mi> <mi mathvariant="normal">A</mi> </msub> </semantics></math> with corresponding assignment of weights. Black edges exhibit the same value of the weight as the corresponding edge in the original graph and grey edges exhibit a weight of zero. Diamond symbols are used to mark the “additional” sites; (<b>III</b>) minimum-weight perfect matching (MWPM) <span class="html-italic">M</span>: the matched edges are shown in bold and unmatched edges using dashed lines, and (<b>IV</b>) loop configuration (bold edges) that is the result of the MWPM shown in (<b>III</b>).</p> Full article ">Figure 12
<p>How the matching models detects loops: (<b>I</b>) the additional node 2 (shown as diamond) is matched to a duplicated node 1a (shown as circle). This means the duplicated node 1b must be matched to an additional node (either 3, 4 or 5) as well. (<b>II</b>) When 1b is matched, e.g., with node 3, the other additional nodes connected to 1a or 1b must be matched with additional nodes as well because 1a and 1b are matched already. (<b>III</b>) It is possible that around nodes 1a,1b all additional nodes are matched with additional nodes. In this case, 1a and 1b are matched with each other.</p> Full article ">Figure 13
<p>Illustration of the algorithmic procedure for s-t paths: (<b>I</b>) auxiliary graph <math display="inline"><semantics> <msub> <mi>G</mi> <mi mathvariant="normal">A</mi> </msub> </semantics></math>. A modified mapping is used to create a minimum-weight path which connects the two nodes <span class="html-italic">s</span> to <span class="html-italic">t</span>. These two nodes are not duplicated. Black edges obtain the same weight values (partially shown as numbers here) as the corresponding edge in the original graph. Edges with zero weight are shown in grey (weight value 0 not shown), (<b>II</b>) resulting MWPM, (<b>III</b>) resulting path of minimum weight (bold edges) which is obtained from the MWPM on <math display="inline"><semantics> <msub> <mi>G</mi> <mi mathvariant="normal">A</mi> </msub> </semantics></math>. For the example shown here, the path exhibits the weight <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mi>p</mi> </msub> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>. In addition, there are no additional loops in the graph but in principle they may be present.</p> Full article ">Figure 14
<p>Examples for configuration which consist of a path of minimum weight plus loops. Here, a square lattice of size <math display="inline"><semantics> <mrow> <mi>L</mi> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mn>64</mn> </mrow> </semantics></math> is shown. Dashed (solid) lines at the boundary represent free (periodic) boundary conditions. The MWP is shown in black. The selected nodes <span class="html-italic">s</span> and <span class="html-italic">t</span> are drawn with black dots. Gray lines are used to show the loops. The snapshots are taken at different values <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> for the disorder parameter; here (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.28</mn> <mo>&lt;</mo> <msub> <mi>ρ</mi> <mi mathvariant="normal">c</mi> </msub> <mo>;</mo> </mrow> </semantics></math> (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>≈</mo> <msub> <mi>ρ</mi> <mi mathvariant="normal">c</mi> </msub> <mo>=</mo> <mn>0.340</mn> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0.7</mn> </mrow> </semantics></math>, and (<b>d</b>) full coverage <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. The bottom scale indicates where the different configurations are located along the “disorder axis” in the range <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>0</mn> <mo>…</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">Figure 15
<p>(<b>I</b>) Example for a minimum-weight s-t path (bold lines); (<b>II</b>) When shifting up all weight values, such that all weights are not negative, the shortest path changes.</p> Full article ">Figure 16
<p>Illustration of the construction of the auxiliary graph <math display="inline"><semantics> <msub> <mi>G</mi> <mi mathvariant="normal">A</mi> </msub> </semantics></math> for directed graphs <span class="html-italic">G</span>: (<b>I</b>) Node in the original graph with two incoming and two outgoing edges. (<b>II</b>) corresponding auxiliary graph with a sample perfect matching, representing a path using one ingoing and one outgoing edge.</p> Full article ">Figure 17
<p>Probability of a percolating loop for two-dimensional lattices with mixed Gaussian disorder according to Equation (<a href="#FD4-entropy-21-00193" class="html-disp-formula">4</a>) in <a href="#sec4dot1-entropy-21-00193" class="html-sec">Section 4.1</a>. The inset shows the probability <math display="inline"><semantics> <msubsup> <mi>P</mi> <mi>L</mi> <mi>s</mi> </msubsup> </semantics></math> that the system exhibits a system spanning loop as a function of the disorder parameter <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>, for different system sizes <span class="html-italic">L</span>. The main plot shows the same data with the <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>-axis rescaled to determine the critical point <math display="inline"><semantics> <msub> <mi>ρ</mi> <mi>c</mi> </msub> </semantics></math> and the critical correlation length exponent <math display="inline"><semantics> <mi>ν</mi> </semantics></math> via a data collapse according to Equation (<a href="#FD5-entropy-21-00193" class="html-disp-formula">5</a>) of this section.</p> Full article ">Figure 18
<p>Distribution of the length of the non-percolating loops for two-dimensional NWP (<math display="inline"><semantics> <mrow> <mi>L</mi> <mo>=</mo> <mn>256</mn> </mrow> </semantics></math>) with mixed Gaussian disorder according to Equation (<a href="#FD4-entropy-21-00193" class="html-disp-formula">4</a>) for several values of the disorder parameter <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>. The lines show fits to the functions <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mo>ℓ</mo> </msub> <mo>∼</mo> <msup> <mo>ℓ</mo> <mrow> <mo>−</mo> <mi>τ</mi> </mrow> </msup> <mo form="prefix">exp</mo> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>T</mi> <mi>L</mi> </msub> <mo>ℓ</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>. The inset shows the behaviour of <math display="inline"><semantics> <msub> <mi>T</mi> <mi>L</mi> </msub> </semantics></math> as a function of the distance from the critical point.</p> Full article ">Figure 19
<p>Probability that a path connecting two “furthest” nodes in the <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>-regular graph have a negative weight, corresponding to the fact that they would percolate if only negative paths would be allowed. The inset shows the raw data, renormalized by the size-dependent critical point <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>. The main plot displays the collapse of data yielding the value of critical exponent <math display="inline"><semantics> <mi>ν</mi> </semantics></math>.</p> Full article ">Figure 20
<p>The left-passage probability <math display="inline"><semantics> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> was obtained numerically for various points (257 different angles) on a semicircle (<math display="inline"><semantics> <mrow> <mi>R</mi> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mn>128</mn> </mrow> </semantics></math>). The simulations were performed right at the critical density for lattices with 1025 × 256 sites and averaged over 51,200 disorder realisations. The probability is shown together with the SLE prediction <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mi>κ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>. The diffusion constant <math display="inline"><semantics> <mrow> <msup> <mi>κ</mi> <mo>★</mo> </msup> <mspace width="-0.166667em"/> <mo>=</mo> <mspace width="-0.166667em"/> <mn>3.343</mn> </mrow> </semantics></math> was chosen such that the agreement between <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mi>κ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> is the largest. In the inset, the difference of the numerical left-passage probability to the prediction from SLE is shown, indicating that the agreement is indeed very good. To keep the inset clear, the difference is shown only for a selected number of typical data points.</p> Full article ">Figure 21
<p><b>Left</b>: determination of the correlations lengths <math display="inline"><semantics> <msub> <mi>ξ</mi> <mo>∥</mo> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ξ</mi> <mo>⊥</mo> </msub> </semantics></math> which are parallel and perpendicular to the main (up to bottom) direction for directed NWP, respectively. <b>Right</b>: correlation length <math display="inline"><semantics> <msub> <mi>ξ</mi> <mo>⊥</mo> </msub> </semantics></math> for different <math display="inline"><semantics> <msub> <mi>L</mi> <mo>⊥</mo> </msub> </semantics></math> at fixed <math display="inline"><semantics> <mrow> <msub> <mi>L</mi> <mo>∥</mo> </msub> <mo>=</mo> <mn>256</mn> </mrow> </semantics></math> to determine <math display="inline"><semantics> <mrow> <mi>ξ</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mo>∥</mo> </msub> <mo>=</mo> <mn>256</mn> <mo>,</mo> <msub> <mi>L</mi> <mo>⊥</mo> </msub> <mo>→</mo> <mo>∞</mo> <mo stretchy="false">)</mo> </mrow> </semantics></math>. The inset shows the scaling of <math display="inline"><semantics> <mrow> <mi>ξ</mi> <mo stretchy="false">(</mo> <msub> <mi>L</mi> <mo>∥</mo> </msub> <mo>,</mo> <msub> <mi>L</mi> <mo>⊥</mo> </msub> <mo>→</mo> <mo>∞</mo> <mo stretchy="false">)</mo> </mrow> </semantics></math> to find the ratio <math display="inline"><semantics> <mrow> <msub> <mi>ν</mi> <mo>∥</mo> </msub> <mo>/</mo> <msub> <mi>ν</mi> <mo>⊥</mo> </msub> </mrow> </semantics></math>.</p> Full article ">
25 pages, 4056 KiB  
Article
A Simple Secret Key Generation by Using a Combination of Pre-Processing Method with a Multilevel Quantization
by Mike Yuliana, Wirawan and Suwadi
Entropy 2019, 21(2), 192; https://doi.org/10.3390/e21020192 - 18 Feb 2019
Cited by 32 | Viewed by 4133
Abstract
Limitations of the computational and energy capabilities of IoT devices provide new challenges in securing communication between devices. Physical layer security (PHYSEC) is one of the solutions that can be used to solve the communication security challenges. In this paper, we conducted an [...] Read more.
Limitations of the computational and energy capabilities of IoT devices provide new challenges in securing communication between devices. Physical layer security (PHYSEC) is one of the solutions that can be used to solve the communication security challenges. In this paper, we conducted an investigation on PHYSEC which utilizes channel reciprocity in generating a secret key, commonly known as secret key generation (SKG) schemes. Our research focused on the efforts to get a simple SKG scheme by eliminating the information reconciliation stage so as to reduce the high computational and communication cost. We exploited the pre-processing method by proposing a modified Kalman (MK) and performing a combination of the method with a multilevel quantization, i.e., combined multilevel quantization (CMQ). Our approach produces a simple SKG scheme for its significant increase in reciprocity so that an identical secret key between two legitimate users can be obtained without going through the information reconciliation stage. Full article
(This article belongs to the Special Issue Information-Theoretic Security II)
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Figure 1

Figure 1
<p>System modeling on the secret key generation (SKG) scheme.</p> Full article ">Figure 2
<p>Measurement parameters.</p> Full article ">Figure 3
<p>Our proposed SKG scheme.</p> Full article ">Figure 4
<p>The experimental scenario in the line-of-sight (LOS) environment.</p> Full article ">Figure 5
<p>The experimental scenario in the non-line of sight (NLOS) environment.</p> Full article ">Figure 6
<p>Measurement results in the LOS environment.</p> Full article ">Figure 7
<p>Measurement results in the NLOS environment.</p> Full article ">Figure 8
<p>Improvement of the correlation coefficient for each block of data in the LOS environment.</p> Full article ">Figure 9
<p>Improvement of the correlation coefficient for each block of data in the NLOS environment.</p> Full article ">Figure 10
<p>Bit disagreement rate (BDR) of the legitimate user between combined multilevel quantization (CMQ) and several existing schemes in the LOS environment.</p> Full article ">Figure 11
<p>BDR of eavesdropper (Alice–Eve) between CMQ and several existing schemes in the LOS environment.</p> Full article ">Figure 12
<p>BDR of eavesdropper (Bob–Eve) between CMQ and several existing schemes in the LOS environment.</p> Full article ">Figure 13
<p>BDR of the legitimate user between CMQ and several existing schemes in the NLOS environment.</p> Full article ">Figure 14
<p>BDR of eavesdropper (Alice–Eve) between CMQ and several existing schemes in the NLOS environment.</p> Full article ">Figure 15
<p>BDR of eavesdropper (Bob–Eve) between CMQ and several existing schemes in the NLOS environment.</p> Full article ">
15 pages, 12593 KiB  
Article
Entropy Generation and Heat Transfer Performance in Microchannel Cooling
by Jundika C. Kurnia, Desmond C. Lim, Lianjun Chen, Lishuai Jiang and Agus P. Sasmito
Entropy 2019, 21(2), 191; https://doi.org/10.3390/e21020191 - 18 Feb 2019
Cited by 11 | Viewed by 5618
Abstract
Owing to its relatively high heat transfer performance and simple configurations, liquid cooling remains the preferred choice for electronic cooling and other applications. In this cooling approach, channel design plays an important role in dictating the cooling performance of the heat sink. Most [...] Read more.
Owing to its relatively high heat transfer performance and simple configurations, liquid cooling remains the preferred choice for electronic cooling and other applications. In this cooling approach, channel design plays an important role in dictating the cooling performance of the heat sink. Most cooling channel studies evaluate the performance in view of the first thermodynamics aspect. This study is conducted to investigate flow behaviour and heat transfer performance of an incompressible fluid in a cooling channel with oblique fins with regards to first law and second law of thermodynamics. The effect of oblique fin angle and inlet Reynolds number are investigated. In addition, the performance of the cooling channels for different heat fluxes is evaluated. The results indicate that the oblique fin channel with 20° angle yields the highest figure of merit, especially at higher Re (250–1000). The entropy generation is found to be lowest for an oblique fin channel with 90° angle, which is about twice than that of a conventional parallel channel. Increasing Re decreases the entropy generation, while increasing heat flux increases the entropy generation. Full article
(This article belongs to the Special Issue Entropy in Computational Fluid Dynamics II )
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<p>Schematics of the investigated cooling channel.</p> Full article ">Figure 2
<p>Cooling channel outlet temperature and pressure drop for various mesh sizes.</p> Full article ">Figure 3
<p>Velocity contours at the middle of cooling channel (<span class="html-italic">z</span> = 5 × 10<sup>−4</sup> m) for oblique channels with various oblique angle: (<b>a</b>) 20°, (<b>b</b>) 30°, (<b>c</b>) 45°, (<b>d</b>) 60°, (<b>e</b>) 90° and (<b>f</b>) parallel channel.</p> Full article ">Figure 4
<p>Temperature distribution at the middle of cooling channel (z = 5 × 10<sup>−4</sup> m) for oblique channels with various oblique angle: (<b>a</b>) 20°, (<b>b</b>) 30°, (<b>c</b>) 45°, (<b>d</b>) 60°, (<b>e</b>) 90° and (<b>f</b>) parallel channel at <span class="html-italic">Re</span> 1000 and <span class="html-italic">Q<sub>base</sub></span> 10,000 W/m<sup>2</sup>.</p> Full article ">Figure 5
<p>Temperature distribution at the base of the solid separator (<span class="html-italic">z</span> = −1 × 10<sup>−3</sup> m) for oblique channels with various oblique angle: (<b>a</b>) 20°, (<b>b</b>) 30°, (<b>c</b>) 45°, (<b>d</b>) 60°, (<b>e</b>) 90° and (<b>f</b>) parallel channelat <span class="html-italic">Re</span> 1000 and <span class="html-italic">Q<sub>base</sub></span> 10,000 W/m<sup>2</sup>.</p> Full article ">Figure 6
<p>Contour of entropy generation at the base of the solid separator (<span class="html-italic">z</span> = −1 × 10<sup>−3</sup> m) for oblique channels with various oblique angle: (<b>a</b>) 20°, (<b>b</b>) 30°, (<b>c</b>) 45°, (<b>d</b>) 60°, (<b>e</b>) 90° and (<b>f</b>) parallel channel at <span class="html-italic">Re</span> 1000 and <span class="html-italic">Q<sub>base</sub></span> 10,000 W/m<sup>2</sup>.</p> Full article ">Figure 7
<p>(<b>a</b>) Maximum temperature, (<b>b</b>) average temperature, and (<b>c</b>) standard deviation of temperature at the base of the solid separator (<span class="html-italic">z</span> = −1 × 10<sup>−3</sup> m) and (<b>d</b>) cooling channel pressure drop for various inlet Reynolds number at constant base heat flux of 10,000 W/m<sup>2</sup>.</p> Full article ">Figure 8
<p>(<b>a</b>) Maximum temperature, (<b>b</b>) average temperature, and (<b>c</b>) standard deviation oftemperature at the base of the solid separator (<span class="html-italic">z</span> = −1 × 10<sup>−3</sup> m) and (<b>d</b>) cooling channel pressure drop for various constant base heat flux at inlet Reynolds number of 1000 of 10,000 W/m<sup>2</sup> and constant inlet Reynolds number of 1000.</p> Full article ">
14 pages, 3604 KiB  
Article
Mixture of Experts with Entropic Regularization for Data Classification
by Billy Peralta, Ariel Saavedra, Luis Caro and Alvaro Soto
Entropy 2019, 21(2), 190; https://doi.org/10.3390/e21020190 - 18 Feb 2019
Cited by 5 | Viewed by 5250
Abstract
Today, there is growing interest in the automatic classification of a variety of tasks, such as weather forecasting, product recommendations, intrusion detection, and people recognition. “Mixture-of-experts” is a well-known classification technique; it is a probabilistic model consisting of local expert classifiers weighted by [...] Read more.
Today, there is growing interest in the automatic classification of a variety of tasks, such as weather forecasting, product recommendations, intrusion detection, and people recognition. “Mixture-of-experts” is a well-known classification technique; it is a probabilistic model consisting of local expert classifiers weighted by a gate network that is typically based on softmax functions, combined with learnable complex patterns in data. In this scheme, one data point is influenced by only one expert; as a result, the training process can be misguided in real datasets for which complex data need to be explained by multiple experts. In this work, we propose a variant of the regular mixture-of-experts model. In the proposed model, the cost classification is penalized by the Shannon entropy of the gating network in order to avoid a “winner-takes-all” output for the gating network. Experiments show the advantage of our approach using several real datasets, with improvements in mean accuracy of 3–6% in some datasets. In future work, we plan to embed feature selection into this model. Full article
(This article belongs to the Special Issue Information-Theoretical Methods in Data Mining)
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<p>Mixture-of-experts (MoE) architecture.</p> Full article ">Figure 2
<p>Log-likelihood values with 20 experts for the classical MoE and the entropic MoE (EMoE) for all datasets. In these experiments, we mainly used 50 iterations.</p> Full article ">Figure 3
<p>Average entropy scores in the network gate outputs for the Ionosphere, Spectf, Sonar, and Musk datasets in the MoE and EMoE models with 10 experts.</p> Full article ">Figure 4
<p>Average entropy scores in the network gate outputs for the Arrhythmia, Secom, Pie10P, and Leukemia datasets in the MoE and EMoE models with 10 experts.</p> Full article ">
15 pages, 3221 KiB  
Article
Nonrigid Medical Image Registration Using an Information Theoretic Measure Based on Arimoto Entropy with Gradient Distributions
by Bicao Li, Huazhong Shu, Zhoufeng Liu, Zhuhong Shao, Chunlei Li, Min Huang and Jie Huang
Entropy 2019, 21(2), 189; https://doi.org/10.3390/e21020189 - 18 Feb 2019
Cited by 6 | Viewed by 3948
Abstract
This paper introduces a new nonrigid registration approach for medical images applying an information theoretic measure based on Arimoto entropy with gradient distributions. A normalized dissimilarity measure based on Arimoto entropy is presented, which is employed to measure the independence between two images. [...] Read more.
This paper introduces a new nonrigid registration approach for medical images applying an information theoretic measure based on Arimoto entropy with gradient distributions. A normalized dissimilarity measure based on Arimoto entropy is presented, which is employed to measure the independence between two images. In addition, a regularization term is integrated into the cost function to obtain the smooth elastic deformation. To take the spatial information between voxels into account, the distance of gradient distributions is constructed. The goal of nonrigid alignment is to find the optimal solution of a cost function including a dissimilarity measure, a regularization term, and a distance term between the gradient distributions of two images to be registered, which would achieve a minimum value when two misaligned images are perfectly registered using limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) optimization scheme. To evaluate the test results of our presented algorithm in non-rigid medical image registration, experiments on simulated three-dimension (3D) brain magnetic resonance imaging (MR) images, real 3D thoracic computed tomography (CT) volumes and 3D cardiac CT volumes were carried out on elastix package. Comparison studies including mutual information (MI) and the approach without considering spatial information were conducted. These results demonstrate a slight improvement in accuracy of non-rigid registration. Full article
(This article belongs to the Special Issue Entropy in Image Analysis)
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<p>Block diagram of our registration algorithm.</p> Full article ">Figure 2
<p>(<b>a</b>) MR T1 image; (<b>b</b>) MR T2 image; (<b>c</b>) deformation field; (<b>d</b>) deformation vector.</p> Full article ">Figure 3
<p>The axis slice of 10 3D cardiac CT images in one 4D sequence. (<b>a</b>–<b>j</b>) represent the 10 frames acquired from one whole cardiac cycle of one patient.</p> Full article ">Figure 4
<p>The registration results of the simulated 3D brain MR T1 &amp; MR T2, MR T1 &amp; MR PD, and MR T2 &amp; MR PD volumes using three algorithms. The red color crosses for each box represents these outliers.</p> Full article ">Figure 5
<p>The TREs obtained when employing NJAD-GD algorithm, the registration method based on JAD without gradient distribution.</p> Full article ">Figure 6
<p>Statistics of TREs before registration and after alignment exploiting the NJAD-GD, JAD methods.</p> Full article ">Figure 7
<p>HDMs obtained when employing NJAD-GD algorithm, the registration method based on JAD without gradient distribution.</p> Full article ">Figure 8
<p>Registration results of 12 groups of 3D cardiac images. (<b>a</b>–<b>l</b>) display the test results of patient 1 to 12, respectively. In each group, left image represents the checkboard before registration, and the right accounts for the result after registration.</p> Full article ">
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