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Entropy, Volume 15, Issue 1 (January 2013) – 20 articles , Pages 1-415

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480 KiB  
Article
Expanding the Algorithmic Information Theory Frame for Applications to Earth Observation
by Daniele Cerra and Mihai Datcu
Entropy 2013, 15(1), 407-415; https://doi.org/10.3390/e15010407 - 22 Jan 2013
Cited by 11 | Viewed by 6623
Abstract
Recent years have witnessed an increased interest towards compression-based methods and their applications to remote sensing, as these have a data-driven and parameter-free approach and can be thus succesfully employed in several applications, especially in image information mining. This paper expands the algorithmic [...] Read more.
Recent years have witnessed an increased interest towards compression-based methods and their applications to remote sensing, as these have a data-driven and parameter-free approach and can be thus succesfully employed in several applications, especially in image information mining. This paper expands the algorithmic information theory frame, on which these methods are based. On the one hand, algorithms originally defined in the pattern matching domain are reformulated, allowing a better understanding of the available compression-based tools for remote sensing applications. On the other hand, the use of existing compression algorithms is proposed to store satellite images with added semantic value. Full article
(This article belongs to the Special Issue Applications of Information Theory in the Geosciences)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Visual description and hierarchical clustering of NCD (<b>a</b>), PRDC (<b>b</b>) and Normalized PRDC (<b>c</b>) distances between 60 satellite images of size <math display="inline"> <mrow> <mn>64</mn> <mo>×</mo> <mn>64</mn> </mrow> </math> belonging to 6 classes. Misplacements are circled in red.</p> Full article ">
777 KiB  
Review
Is Encephalopathy a Mechanism to Renew Sulfate in Autism?
by Stephanie Seneff, Ann Lauritzen, Robert M. Davidson and Laurie Lentz-Marino
Entropy 2013, 15(1), 372-406; https://doi.org/10.3390/e15010372 - 22 Jan 2013
Cited by 13 | Viewed by 20246
Abstract
This paper makes two claims: (1) autism can be characterized as a chronic low-grade encephalopathy, associated with excess exposure to nitric oxide, ammonia and glutamate in the central nervous system, which leads to hippocampal pathologies and resulting cognitive impairment, and (2), encephalitis is [...] Read more.
This paper makes two claims: (1) autism can be characterized as a chronic low-grade encephalopathy, associated with excess exposure to nitric oxide, ammonia and glutamate in the central nervous system, which leads to hippocampal pathologies and resulting cognitive impairment, and (2), encephalitis is provoked by a systemic deficiency in sulfate, but associated seizures and fever support sulfate restoration. We argue that impaired synthesis of cholesterol sulfate in the skin and red blood cells, catalyzed by sunlight and nitric oxide synthase enzymes, creates a state of colloidal instability in the blood manifested as a low zeta potential and increased interfacial stress. Encephalitis, while life-threatening, can result in partial renewal of sulfate supply, promoting neuronal survival. Research is cited showing how taurine may not only help protect neurons from hypochlorite exposure, but also provide a source for sulfate renewal. Several environmental factors can synergistically promote the encephalopathy of autism, including the herbicide, glyphosate, aluminum, mercury, lead, nutritional deficiencies in thiamine and zinc, and yeast overgrowth due to excess dietary sugar. Given these facts, dietary and lifestyle changes, including increased sulfur ingestion, organic whole foods, increased sun exposure, and avoidance of toxins such as aluminum, mercury, and lead, may help to alleviate symptoms or, in some instances, to prevent autism altogether. Full article
(This article belongs to the Special Issue Biosemiotic Entropy: Disorder, Disease, and Mortality)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Schematic of cascade that we believe leads to ASD, which can best be characterized as low-grade chronic encephalitis, brought on by impaired sulfate synthesis. The activities in the brain are corrective in that they produce sulfate to resupply the blood and the tissues, leading to healing. However, the damaging effects of chronic ammonia and glutamate exposure lead over time to brain impairment, which is especially apparent in the hippocampus. The four boxes indicating dysfunction (in yellow) identify phenomena that are closely associated with or directly give rise to ASD/encephalitis symptoms. HS: heparan sulfate; NMDA: <span class="html-italic">N</span>-methyl-D-aspartate.</p> Full article ">
191 KiB  
Article
Asymptotic Behavior of the Maximum Entropy Routing in Computer Networks
by Milan Tuba
Entropy 2013, 15(1), 361-371; https://doi.org/10.3390/e15010361 - 21 Jan 2013
Cited by 10 | Viewed by 5524
Abstract
Maximum entropy method has been successfully used for underdetermined systems. Network design problem, with routing and topology subproblems, is an underdetermined system and a good candidate for maximum entropy method application. Wireless ad-hoc networks with rapidly changing topology and link quality, where the [...] Read more.
Maximum entropy method has been successfully used for underdetermined systems. Network design problem, with routing and topology subproblems, is an underdetermined system and a good candidate for maximum entropy method application. Wireless ad-hoc networks with rapidly changing topology and link quality, where the speed of recalculation is of crucial importance, have been recently successfully investigated by maximum entropy method application. In this paper we prove a theorem that establishes asymptotic properties of the maximum entropy routing solution. This result, besides being theoretically interesting, can be used to direct initial approximation for iterative optimization algorithms and to speed up their convergence. Full article
(This article belongs to the Special Issue Information Theory Applied to Communications and Networking)
996 KiB  
Review
The Liang-Kleeman Information Flow: Theory and Applications
by X. San Liang
Entropy 2013, 15(1), 327-360; https://doi.org/10.3390/e15010327 - 18 Jan 2013
Cited by 62 | Viewed by 13941
Abstract
Information flow, or information transfer as it may be referred to, is a fundamental notion in general physics which has wide applications in scientific disciplines. Recently, a rigorous formalism has been established with respect to both deterministic and stochastic systems, with flow measures [...] Read more.
Information flow, or information transfer as it may be referred to, is a fundamental notion in general physics which has wide applications in scientific disciplines. Recently, a rigorous formalism has been established with respect to both deterministic and stochastic systems, with flow measures explicitly obtained. These measures possess some important properties, among which is flow or transfer asymmetry. The formalism has been validated and put to application with a variety of benchmark systems, such as the baker transformation, Hénon map, truncated Burgers-Hopf system, Langevin equation, etc. In the chaotic Burgers-Hopf system, all the transfers, save for one, are essentially zero, indicating that the processes underlying a dynamical phenomenon, albeit complex, could be simple. (Truth is simple.) In the Langevin equation case, it is found that there could be no information flowing from one certain time series to another series, though the two are highly correlated. Information flow/transfer provides a potential measure of the cause–effect relation between dynamical events, a relation usually hidden behind the correlation in a traditional sense. Full article
(This article belongs to the Special Issue Transfer Entropy)
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Figure 1

Figure 1
<p>Illustration of the Frobenius-Perron operator <math display="inline"> <mi mathvariant="script">P</mi> </math>, which takes <math display="inline"> <mrow> <mi>ρ</mi> <mo>(</mo> <mi mathvariant="bold-italic">x</mi> <mo>)</mo> </mrow> </math> to <math display="inline"> <mrow> <mi mathvariant="script">P</mi> <mi>ρ</mi> <mo>(</mo> <mi mathvariant="bold-italic">x</mi> <mo>)</mo> </mrow> </math> as Φ takes <math display="inline"> <mi mathvariant="bold-italic">x</mi> </math> to <math display="inline"> <mrow> <mo>Φ</mo> <mi mathvariant="bold-italic">x</mi> </mrow> </math>.</p> Full article ">Figure 2
<p>Illustration of the unidirectional information flow within the baker transformation.</p> Full article ">Figure 3
<p>A trajectory of the canonical H<math display="inline"> <mover accent="true"> <mi mathvariant="normal">e</mi> <mo>´</mo> </mover> </math>non map (<math display="inline"> <mrow> <mi>a</mi> <mo>=</mo> <mn>1.4</mn> </mrow> </math>, <math display="inline"> <mrow> <mi>b</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </math>) starting at <math display="inline"> <mrow> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </math>.</p> Full article ">Figure 4
<p>The invariant attractor of the truncated Burgers–Hopf system (73)–(76). Shown here is the trajectory segment for <math display="inline"> <mrow> <mn>2</mn> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <mn>20</mn> </mrow> </math> starting at <math display="inline"> <mrow> <mo>(</mo> <mn>40</mn> <mo>,</mo> <mn>40</mn> <mo>,</mo> <mn>40</mn> <mo>,</mo> <mn>40</mn> <mo>)</mo> </mrow> </math>. (a) and (b) are the 3-dimensional projections onto the subspaces <math display="inline"> <msub> <mi>x</mi> <mn>1</mn> </msub> </math>-<math display="inline"> <msub> <mi>x</mi> <mn>2</mn> </msub> </math>-<math display="inline"> <msub> <mi>x</mi> <mn>3</mn> </msub> </math> and <math display="inline"> <msub> <mi>x</mi> <mn>2</mn> </msub> </math>-<math display="inline"> <msub> <mi>x</mi> <mn>3</mn> </msub> </math>-<math display="inline"> <msub> <mi>x</mi> <mn>4</mn> </msub> </math>, respectively.</p> Full article ">Figure 5
<p>Information flows within the 4D truncated Burgers-Hopf system. The series prior to <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mn>2</mn> </mrow> </math> are not shown because some trajectories have not entered the attractor by that time.</p> Full article ">Figure 6
<p>A solution of Equation (78), the model examined in [<a href="#B54-entropy-15-00327" class="html-bibr">54</a>], with <math display="inline"> <mrow> <msub> <mi>a</mi> <mn>21</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math> and initial conditions as shown in the text: (<b>a</b>) <math display="inline"> <mi mathvariant="bold">μ</mi> </math>; (<b>b</b>) Σ; and (<b>c</b>) a sample path starting from (1,2).</p> Full article ">Figure 7
<p>The computed rates of information flow for the system (77): (a) <math display="inline"> <msub> <mi>T</mi> <mrow> <mn>2</mn> <mo>→</mo> <mn>1</mn> </mrow> </msub> </math>, (b) <math display="inline"> <msub> <mi>T</mi> <mrow> <mn>1</mn> <mo>→</mo> <mn>2</mn> </mrow> </msub> </math>.</p> Full article ">
278 KiB  
Article
Information and Metabolism in Bacterial Chemotaxis
by Gennaro Auletta
Entropy 2013, 15(1), 311-326; https://doi.org/10.3390/e15010311 - 18 Jan 2013
Cited by 22 | Viewed by 8919
Abstract
One of the most important issues in theoretical biology is to understand how control mechanisms are deployed by organisms to maintain their homeostasis and ensure their survival. A crucial issue is how organisms deal with environmental information in a way that ensures appropriate [...] Read more.
One of the most important issues in theoretical biology is to understand how control mechanisms are deployed by organisms to maintain their homeostasis and ensure their survival. A crucial issue is how organisms deal with environmental information in a way that ensures appropriate exchanges with the environment — even in the most basic of life forms (namely, bacteria). In this paper, I present an information theoretic formulation of how Escherichia coli responds to environmental information during chemotaxis and, more generally, a cybernetic model of the relationship between information and biophysical (metabolic) dynamics. Full article
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Figure 1
<p>A reformulation of Friston’s model that accounts for the studies of bacterial chemotaxis that is also quite in accordance with the model presented in [<a href="#B11-entropy-15-00311" class="html-bibr">11</a>] (see also [<a href="#B9-entropy-15-00311" class="html-bibr">9</a>], Chapters 2 and 8). The action <span class="html-italic">a</span> on the environment will change environmental states. Even a simple motion away from a certain signal source will induce a change external states and consequence input. This establishes an internal–external feedback circuit (the action <span class="html-italic">a</span> changes the external state <span class="html-italic">k</span> that determines the input <span class="html-italic">i</span>). Internally, there are two kinds of processes: (i) information transmission processes (from <span class="html-italic">s</span> to <span class="html-italic">a</span>) displaying a feedforward activity; which (ii) are coupled to internal feedback circuit (from <math display="inline"> <msup> <mi>s</mi> <mo>′</mo> </msup> </math> to <span class="html-italic">s</span>).</p> Full article ">Figure 2
<p>This diagram illustrates the bidirectional or circular causality discussed in the main text. Top: the action <span class="html-italic">a</span> of the organism on the environment resulting from a programmed goal: the maintenance of the default state <math display="inline"> <msup> <mi>s</mi> <mo>′</mo> </msup> </math>. Bottom: the action <span class="html-italic">a</span> of the organism as induced by the codification of the physical-chemical signal.</p> Full article ">Figure 3
<p>Top: on the left the instructional semiotic process, on the right the representational one. They correspond roughly to the top and bottom parts of <a href="#entropy-15-00311-f002" class="html-fig">Figure 2</a>, respectively. Bottom: the whole semiotic circle of information control that corresponds to <a href="#entropy-15-00311-f001" class="html-fig">Figure 1</a>.</p> Full article ">Figure 4
<p>The instructional information bridges between genetic information (the icon) and a given functionality (the referent) necessary for survival, while information acquisition from the exterior allows the organism to be informed about the (induced) changes of the environment (the referent) due to appropriate functional steps (the icon). Through regulation, these functionalities couple back to the genetic system allowing for expression or repression. Ensuing distributed circuit therefore displays information control. Adapted from [<a href="#B9-entropy-15-00311" class="html-bibr">9</a>].</p> Full article ">Figure 5
<p>The figure should be read first from the top to bottom (top-down, following the gray-shadowed arrows in the middle): The information-control step leads—through appropriate actions—to lower the surprisal, enabling the organism to approach and consume nutrients. Then, the second (thermodynamicit) step (represented at the bottom) starts, through which appropriate metabolic exchanges with the environment are deployed and controlled. This enables the organism to maintain its order and preserve its configuration (encoding its prior beliefs) so that it can engage in a further information-control step—to find new energy sources or, alternatively, to reproduce through binary fission. Here, we go from the bottom towards the top (bottom-up, following the gray arrows on the left).</p> Full article ">Figure 6
<p>Jacob and Monod’s cybernetic model of <span class="html-italic">lac</span> operon. Adapted from [<a href="#B9-entropy-15-00311" class="html-bibr">9</a>].</p> Full article ">
3021 KiB  
Article
Covariance-Based Measurement Selection Criterion for Gaussian-Based Algorithms
by Fernando A. Auat Cheein
Entropy 2013, 15(1), 287-310; https://doi.org/10.3390/e15010287 - 17 Jan 2013
Cited by 1 | Viewed by 5498
Abstract
Process modeling by means of Gaussian-based algorithms often suffers from redundant information which usually increases the estimation computational complexity without significantly improving the estimation performance. In this article, a non-arbitrary measurement selection criterion for Gaussian-based algorithms is proposed. The measurement selection criterion is [...] Read more.
Process modeling by means of Gaussian-based algorithms often suffers from redundant information which usually increases the estimation computational complexity without significantly improving the estimation performance. In this article, a non-arbitrary measurement selection criterion for Gaussian-based algorithms is proposed. The measurement selection criterion is based on the determination of the most significant measurement from both an estimation convergence perspective and the covariance matrix associated with the measurement. The selection criterion is independent from the nature of the measured variable. This criterion is used in conjunction with three Gaussian-based algorithms: the EIF (Extended Information Filter), the EKF (Extended Kalman Filter) and the UKF (Unscented Kalman Filter). Nevertheless, the measurement selection criterion shown herein can also be applied to other Gaussian-based algorithms. Although this work is focused on environment modeling, the results shown herein can be applied to other Gaussian-based algorithm implementations. Mathematical descriptions and implementation results that validate the proposal are also included in this work. Full article
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<p>Graphical representation of features selection procedure.</p> Full article ">Figure 2
<p>Graphical representation of the feature selection for the UKF. A same feature is extracted from the environment—<span class="html-italic">z</span>—by two different methods—or sensors—with their respective covariance matrices (<math display="inline"> <msub> <mi>R</mi> <mi>a</mi> </msub> </math> and <math display="inline"> <msub> <mi>R</mi> <mi>b</mi> </msub> </math>).</p> Full article ">Figure 3
<p>3-D images acquired by the TOF camera. <a href="#entropy-15-00287-f003" class="html-fig">Figure 3</a>(a) shows a representation of the TOF camera with the IMU used to estimate the camera’s movements; <a href="#entropy-15-00287-f003" class="html-fig">Figure 3</a>(b) shows a picture of the office-like environment from the camera plane point of view–the solid black arrows represent the corners detected in such image–; <a href="#entropy-15-00287-f003" class="html-fig">Figure 3</a>(c) and <a href="#entropy-15-00287-f003" class="html-fig">Figure 3</a>(d) show two snapshots of the TOF camera from the camera’s focus reference frame.</p> Full article ">Figure 4
<p>EKF positioning system. <a href="#entropy-15-00287-f004" class="html-fig">Figure 4</a>(a) shows a top view of the detected corners from the first acquired image, the way-points—used as ground truth—the traveled path of the camera (solid black line) and the estimated path by the IMU measurements (solid magenta line). The green ellipses represent the covariance of the IMU measurements at each way-point of the path. <a href="#entropy-15-00287-f004" class="html-fig">Figure 4</a>(b) and <a href="#entropy-15-00287-f004" class="html-fig">Figure 4</a>(c) show the EKF estimation (solid blue line). In <a href="#entropy-15-00287-f004" class="html-fig">Figure 4</a>(b) all the detected features are used within the correction stage of the EKF algorithm whereas in <a href="#entropy-15-00287-f004" class="html-fig">Figure 4</a>(c), only the two most significant features—from the optimization criterion point of view, presented in Equation (<a href="#FD11-entropy-15-00287" class="html-disp-formula">11</a>)—were used. In addition, <a href="#entropy-15-00287-f004" class="html-fig">Figure 4</a>(d) shows the error evolution of the three cases: the IMU estimation (dotted green line) and the two EKF estimations (the dotted blue line) correspond to the full features case.</p> Full article ">Figure 5
<p>EKF positioning system using the worst two features. <a href="#entropy-15-00287-f005" class="html-fig">Figure 5</a>(a) shows the path estimated by the EKF using only the worst two features–<span class="html-italic">i.e</span>., the features with the highest covariance matrices associated with them–; <a href="#entropy-15-00287-f005" class="html-fig">Figure 5</a>(b) shows the error in the estimation process: the dotted blue line corresponds to the error of the EKF estimation when using the full set of features; the dotted magenta line corresponds to the EKF localization estimation when using the best two features according to the selection criterion shown in Equation (<a href="#FD11-entropy-15-00287" class="html-disp-formula">11</a>); the dotted red line is the case when the localization process is performed by using the worst two features.</p> Full article ">Figure 6
<p>Picture of the mobile robot Pioneer 3AT used in this work. The range sensor laser is mounted on the vehicle.</p> Full article ">Figure 7
<p>Several snapshots of the EIF–SLAM algorithm with implemented feature selection criterion.</p> Full article ">Figure 8
<p>Experimental comparison of the full-featured EIF–SLAM and the EIF–SLAM with feature selection criterion. <a href="#entropy-15-00287-f008" class="html-fig">Figure 8</a>(a) shows the map constructed using all the information available, whereas <a href="#entropy-15-00287-f008" class="html-fig">Figure 8</a>(b) shows the map built using the information acquired with the feature selection criterion. In order to see the differences between both SLAM algorithms, <a href="#entropy-15-00287-f008" class="html-fig">Figure 8</a>(c) shows a zoom in of <a href="#entropy-15-00287-f008" class="html-fig">Figure 8</a>(a) and <a href="#entropy-15-00287-f008" class="html-fig">Figure 8</a>(d) shows a zoom in of <a href="#entropy-15-00287-f008" class="html-fig">Figure 8</a>(b).</p> Full article ">Figure 9
<p>Consistency tests of both the full-featured EIF–SLAM and the EIF–SLAM with feature selection criterion show in <a href="#entropy-15-00287-f008" class="html-fig">Figure 8</a>. <a href="#entropy-15-00287-f009" class="html-fig">Figure 9</a>(a) and <a href="#entropy-15-00287-f009" class="html-fig">Figure 9</a>(c) show the mobile robot localization consistency test of the full-featured EIF–SLAM; <a href="#entropy-15-00287-f009" class="html-fig">Figure 9</a>(b) and <a href="#entropy-15-00287-f009" class="html-fig">Figure 9</a>(d) show the localization consistency test for the EIF–SLAM with feature selection criterion. <a href="#entropy-15-00287-f009" class="html-fig">Figure 9</a>(e) shows the evolution of the variance associated with several trees from the environment for the full-featured SLAM, whereas <a href="#entropy-15-00287-f009" class="html-fig">Figure 9</a>(f) shows the variance evolution of the same features for the EIF–SLAM with feature selection restriction. The <span class="html-italic">number of iterations</span> corresponds to the instant in which the SLAM algorithm was performed.</p> Full article ">Figure 10
<p>Computational cost analysis. The solid red line shows the computational cost (in <math display="inline"> <mrow> <mi>s</mi> <mi>e</mi> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>d</mi> <msup> <mi>s</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math>) of implementing the EIF–SLAM using all the features detected at each correction stage. On the other hand, the solid green line shows the computational cost of implementing the EIF–SLAM for the same experiment, shown in <a href="#entropy-15-00287-f009" class="html-fig">Figure 9</a>, when using the feature selection criterion shown in Equation (<a href="#FD17-entropy-15-00287" class="html-disp-formula">17</a>). As is evident, the selection criterion reduces the computational cost of the correction stage of the EIF algorithm.</p> Full article ">
934 KiB  
Article
Using Exergy to Correlate Energy Research Investments and Efficiencies: Concept and Case Studies
by Marc A. Rosen
Entropy 2013, 15(1), 262-286; https://doi.org/10.3390/e15010262 - 16 Jan 2013
Cited by 16 | Viewed by 6406
Abstract
The use of exergy to correlate energy-utilization efficiencies and energy research investments is described. Specifically, energy and exergy losses are compared with energy research and development expenditures, demonstrating that the latter correlates with energy losses, even though it would be more sensible to [...] Read more.
The use of exergy to correlate energy-utilization efficiencies and energy research investments is described. Specifically, energy and exergy losses are compared with energy research and development expenditures, demonstrating that the latter correlates with energy losses, even though it would be more sensible to allocate energy research and development funding in line with exergy losses, as they represent the actual deviation of efficiency from the ideal. The methodology is outlined and illustrated with two case studies. The case studies consider the province of Ontario, Canada and the United States. The investigation utilizes data on the energy utilization in a country or region, including flows of energy and exergy through the main sectors of the economy. The results are expected to be of use to government and public authorities that administer research and development funding and resources and should help improve the effectiveness of such investments. Full article
(This article belongs to the Special Issue Exergy: Analysis and Applications)
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Figure 1

Figure 1
<p>Model of a region, country or the world, showing flows of resources like energy.</p> Full article ">Figure 2
<p>Energy flow diagram for Ontario (in PJ or 10<sup>15</sup> J) for 1987. The hatched region denotes losses and the note “1/” indicates steam extracted from the utility sector. Hydraulic energy is shown in kinetic energy equivalent.</p> Full article ">Figure 3
<p>Exergy flow diagram for Ontario (in PJ or 10<sup>15</sup> J) for 1987. The hatched region denotes losses (external exergy emissions and internal exergy destructions) and the note “1/” indicates steam extracted from the utility sector. Hydraulic exergy is shown in kinetic exergy equivalent.</p> Full article ">Figure 4
<p>Comparison of exergy efficiencies and margin for improvement (or actual inefficiency) for regions and countries having various attributes.</p> Full article ">
6400 KiB  
Concept Paper
Biosemiotic Entropy of the Genome: Mutations and Epigenetic Imbalances Resulting in Cancer
by Berkley E. Gryder, Chase W. Nelson and Samuel S. Shepard
Entropy 2013, 15(1), 234-261; https://doi.org/10.3390/e15010234 - 16 Jan 2013
Cited by 14 | Viewed by 15839
Abstract
Biosemiotic entropy involves the deterioration of biological sign systems. The genome is a coded sign system that is connected to phenotypic outputs through the interpretive functions of the tRNA/ribosome machinery. This symbolic sign system (semiosis) at the core of all biology has been [...] Read more.
Biosemiotic entropy involves the deterioration of biological sign systems. The genome is a coded sign system that is connected to phenotypic outputs through the interpretive functions of the tRNA/ribosome machinery. This symbolic sign system (semiosis) at the core of all biology has been termed “biosemiosis”. Layers of biosemiosis and cellular information management are analogous in varying degrees to the semiotics of computer programming, spoken, and written human languages. Biosemiotic entropy — an error or deviation from a healthy state — results from errors in copying functional information (mutations) and errors in the appropriate context or quantity of gene expression (epigenetic imbalance). The concept of biosemiotic entropy is a deeply imbedded assumption in the study of cancer biology. Cells have a homeostatic, preprogrammed, ideal or healthy state that is rooted in genomics, strictly orchestrated by epigenetic regulation, and maintained by DNA repair mechanisms. Cancer is an eminent illustration of biosemiotic entropy, in which the corrosion of genetic information via substitutions, deletions, insertions, fusions, and aberrant regulation results in malignant phenotypes. However, little attention has been given to explicitly outlining the paradigm of biosemiotic entropy in the context of cancer. Herein we distill semiotic theory (from the familiar and well understood spheres of human language and computer code) to draw analogies useful for understanding the operation of biological semiosis at the genetic level. We propose that the myriad checkpoints, error correcting mechanisms, and immunities are all systems whose primary role is to defend against the constant pressure of biosemiotic entropy, which malignancy must shut down in order to achieve advanced stages. In lieu of the narrower tumor suppressor/oncogene model, characterization of oncogenesis into the biosemiotic framework of sign, index, or object entropy may allow for more effective explanatory hypotheses for cancer diagnosis, with consequence in improving profiling and bettering therapeutic outcomes. Full article
(This article belongs to the Special Issue Biosemiotic Entropy: Disorder, Disease, and Mortality)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>(<b>A</b>) Semiotic models and (<b>B</b>) examples of semiotic systems.</p> Full article ">Figure 2
<p>Similarities between Biology and Human Language.</p> Full article ">Figure 3
<p>Three areas of entropy in biosemiotics: entropy in the sign (gene code), entropy in mapping (ribosomal translation), and entropy in the object (protein structure).</p> Full article ">Figure 4
<p>Biosemiotic Pressure Points: a qualitative depiction, suggesting an inverse relationship between biosemiotic entropy and biological fitness/function.</p> Full article ">Figure 5
<p>Sign Regulators: determining the exome. DNMT, DNA Methyltransferase. HAT, histone acetyltransferase. HMT, Histone methyltransferase. HDAC, Histone deacetylase. LSD1, Lysine specific demethylase 1. BPTF, Bromodomain plant homeodomain finger transcription factor. Bromo, bromodomain. PHD, plant homeodomain. siRNA, silencing RNA. AR, Androgen Receptor. ER, Estrogen Receptor.</p> Full article ">Figure 6
<p>Modified from C.H. Waddinton’s conception of the epigenetic landscape.</p> Full article ">Figure 7
<p>Maintaining Homeostasis by Fighting Biosemiotic Entropy.</p> Full article ">
609 KiB  
Article
Effects of Convective Heating on Entropy Generation Rate in a Channel with Permeable Walls
by Oluwole Daniel Makinde and Adetayo Samuel Eegunjobi
Entropy 2013, 15(1), 220-233; https://doi.org/10.3390/e15010220 - 11 Jan 2013
Cited by 67 | Viewed by 6472
Abstract
This study deals with the combined effects of convective heating and suction/injection on the entropy generation rate in a steady flow of an incompressible viscous fluid through a channel with permeable walls. The model equations for momentum and energy balance are solved numerically [...] Read more.
This study deals with the combined effects of convective heating and suction/injection on the entropy generation rate in a steady flow of an incompressible viscous fluid through a channel with permeable walls. The model equations for momentum and energy balance are solved numerically using shooting quadrature. Both the velocity and temperature profiles are obtained and utilized to compute the entropy generation number. The effects of the key parameters on the fluid velocity, temperature, entropy generation rate and Bejan number are depicted graphically and analyzed in detail. Full article
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Figure 1

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<p>Schematic diagram of the problem.</p> Full article ">Figure 2
<p>Effect of increasing Re on velocity profiles.</p> Full article ">Figure 3
<p>Effect of increasing Re on temperature profiles.</p> Full article ">Figure 4
<p>Effect of increasing <span class="html-italic">Bi<sub>1</sub></span> on temperature profiles.</p> Full article ">Figure 5
<p>Effect of increasing <span class="html-italic">Bi<sub>2</sub></span> on temperature profiles.</p> Full article ">Figure 6
<p>Effect of increasing <span class="html-italic">Ec</span> on temperature profiles.</p> Full article ">Figure 7
<p>Effect of increasing <span class="html-italic">Pr</span> on temperature profiles.</p> Full article ">Figure 8
<p>Effect of increasing Re on entropy generation rate.</p> Full article ">Figure 9
<p>Effect of increasing <span class="html-italic">Bi<sub>1</sub></span> on entropy generation rate.</p> Full article ">Figure 10
<p>Effect of increasing <span class="html-italic">Bi<sub>2</sub></span> on entropy generation rate.</p> Full article ">Figure 11
<p>Effect of increasing <span class="html-italic">Br</span><span class="html-italic">Ω<sup>−1</sup></span> on entropy generation rate.</p> Full article ">Figure 12
<p>Effect of increasing Re on Bejan number.</p> Full article ">Figure 13
<p>Effect of increasing <span class="html-italic">Bi<sub>1</sub></span> on Bejan number.</p> Full article ">Figure 14
<p>Effect of increasing <span class="html-italic">Bi<sub>2</sub></span> on Bejan number.</p> Full article ">Figure 15
<p>Effect of increasing <span class="html-italic">Br</span><span class="html-italic">Ω<sup>−1</sup></span> on Bejan number.</p> Full article ">
314 KiB  
Article
Compensated Transfer Entropy as a Tool for Reliably Estimating Information Transfer in Physiological Time Series
by Luca Faes, Giandomenico Nollo and Alberto Porta
Entropy 2013, 15(1), 198-219; https://doi.org/10.3390/e15010198 - 11 Jan 2013
Cited by 81 | Viewed by 10672
Abstract
We present a framework for the estimation of transfer entropy (TE) under the conditions typical of physiological system analysis, featuring short multivariate time series and the presence of instantaneous causality (IC). The framework is based on recognizing that TE can be interpreted as [...] Read more.
We present a framework for the estimation of transfer entropy (TE) under the conditions typical of physiological system analysis, featuring short multivariate time series and the presence of instantaneous causality (IC). The framework is based on recognizing that TE can be interpreted as the difference between two conditional entropy (CE) terms, and builds on an efficient CE estimator that compensates for the bias occurring for high dimensional conditioning vectors and follows a sequential embedding procedure whereby the conditioning vectors are formed progressively according to a criterion for CE minimization. The issue of IC is faced accounting for zero-lag interactions according to two alternative empirical strategies: if IC is deemed as physiologically meaningful, zero-lag effects are assimilated to lagged effects to make them causally relevant; if not, zero-lag effects are incorporated in both CE terms to obtain a compensation. The resulting compensated TE (cTE) estimator is tested on simulated time series, showing that its utilization improves sensitivity (from 61% to 96%) and specificity (from 5/6 to 0/6 false positives) in the detection of information transfer respectively when instantaneous effect are causally meaningful and non-meaningful. Then, it is evaluated on examples of cardiovascular and neurological time series, supporting the feasibility of the proposed framework for the investigation of physiological mechanisms. Full article
(This article belongs to the Special Issue Transfer Entropy)
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Example of transfer entropy analysis performed for the first simulation. <b>(a)</b> realization of the three processes generated according to (6). <b>(b)</b> TE estimation between pairs of processes based on nonuniform embedding; each panel depicts the CE estimated for the destination process through application of the non-uniform embedding procedure without considering the source process (black circles), or considering the source process (red triangles), in the definition of the set of candidates; the terms selected at each step <span class="html-italic">k</span> of the sequential embedding are indicated within the plots, while filled symbols denote each detected CE minimum. <b>(c)</b> Same of (b) for estimation of the compensated TE (<span class="html-italic">cTE′</span>).</p> Full article ">Figure 2
<p>Results of transfer entropy analysis for the first simulation. <b>(a)</b> Distribution over 100 realizations of (6) (expressed as 5th percentile, median and 95th percentile) of the information transfer estimated between each pair of processes using the traditional TE (white) and the compensated TE (black). <b>(b)</b> Percentage of realizations for which the information transfer estimated using TE (white) and compensated TE (black) was detected as statistically significant according to the test based on time-shifted surrogates.</p> Full article ">Figure 3
<p>Example of transfer entropy analysis performed for the second simulation. <b>(a)</b> Presence of coupling and absence of instantaneous mixing (<span class="html-italic">C</span> = 0.2, <span class="html-italic">ε</span> = 0) <b>(b)</b> Absence of coupling and presence of instantaneous mixing (<span class="html-italic">C</span> = 0, <span class="html-italic">ε</span> = 0.2). Panels depict a realization of the two processes <span class="html-italic">X</span> and <span class="html-italic">Y</span> generated according to (7) and (8), together with the estimation of <span class="html-italic">TE</span> and <span class="html-italic">cTE′′</span> over the two directions of interaction based on nonuniform embedding and conditional entropy (CE, see caption of <a href="#entropy-15-00198-f001" class="html-fig">Figure 1</a> for details).</p> Full article ">Figure 4
<p>Results of transfer entropy analysis for the second simulation, showing the median values over 50 realizations of (7) and (8) of the TE (first panel row) and the compensated TE (second panel row) computed along the two directions of interactions (<span class="html-italic">X</span>→<span class="html-italic">Y</span>, circles; <span class="html-italic">Y</span>→<span class="html-italic">X</span>, triangles) <b>(a)</b> at varying the parameter <span class="html-italic">C</span> with parameter <span class="html-italic">ε</span> = 0; <b>(b)</b> at varying <span class="html-italic">ε</span> with <span class="html-italic">C</span> = 0 (b); and <b>(c)</b> varying <span class="html-italic">ε</span> with <span class="html-italic">C</span> = 0.2. Filled symbols denote statistically significant values of <span class="html-italic">TE</span> or <span class="html-italic">cTE′′</span> assessed by means of the permutation test.</p> Full article ">Figure 5
<p>Measurement of heart period (series <span class="html-italic">z</span>), systolic arterial pressure (series <span class="html-italic">y</span>) and respiratory flow (series <span class="html-italic">x</span>) variability series from the electrocardiogram, arterial blood pressure and nasal flow signals.</p> Full article ">Figure 6
<p>Transfer entropy analysis in cardiovascular and cardiorespiratory variability performed. <b>(a)</b> during spontaneous breathing and <b>(b)</b> during paced breathing. Plots depict the analyzed time series of respiratory flow (<span class="html-italic">x<sub>n</sub></span>, system <span class="html-italic">X</span>), systolic arterial pressure (<span class="html-italic">y<sub>n</sub></span>, system <span class="html-italic">Y</span>) and heart period (<span class="html-italic">z<sub>n</sub></span>, system <span class="html-italic">Z</span>) together with the corresponding TE (circles) and compensated TE (triangles) estimated between each pair of series. The gray symbols indicate the values of TE/cTE obtained over 40 pairs of time-shifted surrogates; filled symbols denote statistically significant <span class="html-italic">TE</span> or <span class="html-italic">cTE′</span>.</p> Full article ">Figure 7
<p>Transfer entropy analysis in magnetoencephalography performed before (left) and during (right) presentation of the combined visuo-tactile stimuli. <b>(a)</b> Representative MEG signals acquired from the somatosensory cortex (<span class="html-italic">x<sub>n</sub></span>, system <span class="html-italic">X</span>) and the visual cortex (<span class="html-italic">y<sub>n</sub></span>, system <span class="html-italic">Y</span>) for one of the experiment trials (<span class="html-italic">n</span> ranges from 1 to 293 samples before and during simulation). <b>(b)</b> Median over the 60 trials of TE (circles) and compensated TE (triangles) estimated for the two directions of interaction between <span class="html-italic">X</span> and <span class="html-italic">Y</span> before and during stimulation; gray symbols indicate the values of <span class="html-italic">TE</span>/<span class="html-italic">cTE′</span>′ obtained over 100 trial permutations; filled symbols denote statistically significant <span class="html-italic">TE</span> or <span class="html-italic">cTE′′</span>.</p> Full article ">
493 KiB  
Article
Moving Frames of Reference, Relativity and Invariance in Transfer Entropy and Information Dynamics
by Joseph T. Lizier and John R. Mahoney
Entropy 2013, 15(1), 177-197; https://doi.org/10.3390/e15010177 - 10 Jan 2013
Cited by 12 | Viewed by 7167
Abstract
We present a new interpretation of a local framework for informationdynamics, including the transfer entropy, by defining a moving frame of reference for theobserver of dynamics in lattice systems. This formulation is inspired by the idea ofinvestigating “relativistic” effects on observing the dynamics [...] Read more.
We present a new interpretation of a local framework for informationdynamics, including the transfer entropy, by defining a moving frame of reference for theobserver of dynamics in lattice systems. This formulation is inspired by the idea ofinvestigating “relativistic” effects on observing the dynamics of information - in particular,we investigate a Galilean transformation of the lattice system data. In applying thisinterpretation to elementary cellular automata, we demonstrate that using a moving frameof reference certainly alters the observed spatiotemporal measurements of informationdynamics, yet still returns meaningful results in this context. We find that, as expected,an observer will report coherent spatiotemporal structures that are moving in their frame asinformation transfer, and structures that are stationary in their frame as information storage.Crucially, the extent to which the shifted frame of reference alters the results dependson whether the shift of frame retains, adds or removes relevant information regarding thesource-destination interaction. Full article
(This article belongs to the Special Issue Transfer Entropy)
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<p>Measures of information dynamics applied to ECA Rule 54 with a stationary frame of reference (all units in <a href="#entropy-15-00177-f001" class="html-fig">(b)</a>–<a href="#entropy-15-00177-f001" class="html-fig">(d)</a> are in bits). Time increases down the page for all plots. (<b>a</b>) Raw CA; (<b>b</b>) Local active information storage <math display="inline"> <mrow> <mi>a</mi> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>16</mn> <mo>)</mo> </mrow> </math>; (<b>c</b>) Local apparent transfer entropy <math display="inline"> <mrow> <mi>t</mi> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>16</mn> <mo>)</mo> </mrow> </math>; (<b>d</b>) Local complete transfer entropy <math display="inline"> <mrow> <msup> <mi>t</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow> </math>.</p> Full article ">Figure 2
<p>Local information dynamics for a lattice system with speed of light <math display="inline"> <mrow> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> unit per time step: (<b>a</b>) (left) with stationary frame of reference (<math display="inline"> <mrow> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mrow> </math>); (<b>b</b>) (right) with moving frame of reference <math display="inline"> <mrow> <mi>f</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> (<span class="html-italic">i.e</span>., at one cell to the right per unit time step). Red double-headed arrow represents active information storage <math display="inline"> <mrow> <mi>a</mi> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>f</mi> <mo>)</mo> </mrow> </math> from the frame of reference; the blue single-headed arrow represent transfer entropy <math display="inline"> <mrow> <mi>t</mi> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>f</mi> <mo>)</mo> </mrow> </math> from each source orthogonal to the frame of reference. Note that the frame of reference in the figures is the path of the moving observer through space-time.</p> Full article ">Figure 3
<p>Measures of local information dynamics applied to ECA rule 54, computed in frame of reference <math display="inline"> <mrow> <mi>f</mi> <mo>=</mo> <mn>1</mn> </mrow> </math>, <span class="html-italic">i.e</span>., moving 1 cell to the right per unit time (all units in <a href="#entropy-15-00177-f003" class="html-fig">(b)</a>–<a href="#entropy-15-00177-f003" class="html-fig">(f)</a> are in bits). Note that raw states are the same as in <a href="#entropy-15-00177-f001" class="html-fig">Figure 1</a>. (<b>a</b>) Raw CA; (<b>b</b>) Local active information storage <math display="inline"> <mrow> <mi>a</mi> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>16</mn> <mo>,</mo> <mi>f</mi> <mo>=</mo> <mn>1</mn> <mo>)</mo> </mrow> </math>; (<b>c</b>) Local apparent transfer entropy <math display="inline"> <mrow> <mi>t</mi> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>16</mn> <mo>,</mo> <mi>f</mi> <mo>=</mo> <mn>1</mn> <mo>)</mo> </mrow> </math>; (<b>d</b>) Local complete transfer entropy <math display="inline"> <mrow> <msup> <mi>t</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>16</mn> <mo>,</mo> <mi>f</mi> <mo>=</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </math>; (<b>e</b>) Local apparent transfer entropy <math display="inline"> <mrow> <mi>t</mi> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>16</mn> <mo>,</mo> <mi>f</mi> <mo>=</mo> <mn>1</mn> <mo>)</mo> </mrow> </math>; (<b>f</b>) Local complete transfer entropy <math display="inline"> <mrow> <msup> <mi>t</mi> <mi>c</mi> </msup> <mrow> <mo>(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>16</mn> <mo>,</mo> <mi>f</mi> <mo>=</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </math>.</p> Full article ">
216 KiB  
Review
Conformal Gravity: Dark Matter and Dark Energy
by Robert K. Nesbet
Entropy 2013, 15(1), 162-176; https://doi.org/10.3390/e15010162 - 9 Jan 2013
Cited by 24 | Viewed by 6495
Abstract
This short review examines recent progress in understanding dark matter, dark energy, and galactic halos using theory that departs minimally from standard particle physics and cosmology. Strict conformal symmetry (local Weyl scaling covariance), postulated for all elementary massless fields, retains standard fermion and [...] Read more.
This short review examines recent progress in understanding dark matter, dark energy, and galactic halos using theory that departs minimally from standard particle physics and cosmology. Strict conformal symmetry (local Weyl scaling covariance), postulated for all elementary massless fields, retains standard fermion and gauge boson theory but modifies Einstein–Hilbert general relativity and the Higgs scalar field model, with no new physical fields. Subgalactic phenomenology is retained. Without invoking dark matter, conformal gravity and a conformal Higgs model fit empirical data on galactic rotational velocities, galactic halos, and Hubble expansion including dark energy. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
147 KiB  
Article
The Thermal Entropy Density of Spacetime
by Rongjia Yang
Entropy 2013, 15(1), 156-161; https://doi.org/10.3390/e15010156 - 8 Jan 2013
Cited by 9 | Viewed by 7136
Abstract
Introducing the notion of thermal entropy density via the first law of thermodynamics and assuming the Einstein equation as an equation of thermal state, we obtain the thermal entropy density of any arbitrary spacetime without assuming a temperature or a horizon. The results [...] Read more.
Introducing the notion of thermal entropy density via the first law of thermodynamics and assuming the Einstein equation as an equation of thermal state, we obtain the thermal entropy density of any arbitrary spacetime without assuming a temperature or a horizon. The results confirm that there is a profound connection between gravity and thermodynamics. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
614 KiB  
Article
Numerical Study of Entropy Generation in a Flowing Nanofluid Used in Micro- and Minichannels
by Mohammadreza Hassan, Rad Sadri, Goodarz Ahmadi, Mahidzal B. Dahari, Salim N. Kazi, Mohammad R. Safaei and Emad Sadeghinezhad
Entropy 2013, 15(1), 144-155; https://doi.org/10.3390/e15010144 - 7 Jan 2013
Cited by 71 | Viewed by 8087
Abstract
This article mainly concerns theoretical research on entropy generation influences due to heat transfer and flow in nanofluid suspensions. A conventional nanofluid of alumina-water (Al2O3-H2O) was considered as the fluid model. Due to the sensitivity of entropy [...] Read more.
This article mainly concerns theoretical research on entropy generation influences due to heat transfer and flow in nanofluid suspensions. A conventional nanofluid of alumina-water (Al2O3-H2O) was considered as the fluid model. Due to the sensitivity of entropy to duct diameter, mini- and microchannels with diameters of 3 mm and 0.05 mm were considered, and a laminar flow regime was assumed. The conductivity and viscosity of two different nanofluid models were examined with the help of theoretical and experimentally determined parameter values. It was shown that order of the magnitude analysis can be used for estimating entropy generation characteristics of nanofluids in mini- and microchannels. It was found that using highly viscous alumina-water nanofluid under laminar flow regime in microchannels was not desirable. Thus, there is a need for the development of low viscosity alumina-water (Al2O3-H2O) nanofluids for use in microchannels under laminar flow condition. On the other hand, Al2O3-H2O nanofluid was a superior coolant under laminar flow regime in minichannels. The presented results also indicate that flow friction and thermal irreversibility are, respectively, more significant at lower and higher tube diameters. Full article
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Figure 1
<p>Variation of conductivity ratio with volume fraction based on the data of [<a href="#B21-entropy-15-00144" class="html-bibr">21</a>,<a href="#B22-entropy-15-00144" class="html-bibr">22</a>,<a href="#B23-entropy-15-00144" class="html-bibr">23</a>,<a href="#B24-entropy-15-00144" class="html-bibr">24</a>,<a href="#B25-entropy-15-00144" class="html-bibr">25</a>].</p> Full article ">Figure 2
<p>Variation of viscosity ratio with volume fraction based on the data of [<a href="#B26-entropy-15-00144" class="html-bibr">26</a>,<a href="#B27-entropy-15-00144" class="html-bibr">27</a>,<a href="#B28-entropy-15-00144" class="html-bibr">28</a>].</p> Full article ">Figure 3
<p>Entropy generation rate ratio in microchannels.</p> Full article ">Figure 4
<p>Entropy generation rate ratio in minichannels.</p> Full article ">Figure 5
<p>Entropy generation in microchannels.</p> Full article ">Figure 6
<p>Entropy generation in minichannels.</p> Full article ">Figure 7
<p>Bejan number (Be) for microchannels.</p> Full article ">Figure 8
<p>Bejan number (Be) number for minichannels.</p> Full article ">
579 KiB  
Review
The Relation between Granger Causality and Directed Information Theory: A Review
by Pierre-Olivier Amblard and Olivier J. J. Michel
Entropy 2013, 15(1), 113-143; https://doi.org/10.3390/e15010113 - 28 Dec 2012
Cited by 94 | Viewed by 12443
Abstract
This report reviews the conceptual and theoretical links between Granger causality and directed information theory. We begin with a short historical tour of Granger causality, concentrating on its closeness to information theory. The definitions of Granger causality based on prediction are recalled, and [...] Read more.
This report reviews the conceptual and theoretical links between Granger causality and directed information theory. We begin with a short historical tour of Granger causality, concentrating on its closeness to information theory. The definitions of Granger causality based on prediction are recalled, and the importance of the observation set is discussed. We present the definitions based on conditional independence. The notion of instantaneous coupling is included in the definitions. The concept of Granger causality graphs is discussed. We present directed information theory from the perspective of studies of causal influences between stochastic processes. Causal conditioning appears to be the cornerstone for the relation between information theory and Granger causality. In the bivariate case, the fundamental measure is the directed information, which decomposes as the sum of the transfer entropies and a term quantifying instantaneous coupling. We show the decomposition of the mutual information into the sums of the transfer entropies and the instantaneous coupling measure, a relation known for the linear Gaussian case. We study the multivariate case, showing that the useful decomposition is blurred by instantaneous coupling. The links are further developed by studying how measures based on directed information theory naturally emerge from Granger causality inference frameworks as hypothesis testing. Full article
(This article belongs to the Special Issue Transfer Entropy)
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Figure 1
<p>Illustration of the problem of information flow in networks of stochastic processes. Each node of the network is associated to a signal. Edges between nodes stand for dependence (shared information) between the signals. The dependence can be directed or not. This framework can be applied to different situations such as solar physics, neuroscience or the study of turbulence in fluids, as illustrated by the three examples depicted here.</p> Full article ">
350 KiB  
Review
Machine Learning with Squared-Loss Mutual Information
by Masashi Sugiyama
Entropy 2013, 15(1), 80-112; https://doi.org/10.3390/e15010080 - 27 Dec 2012
Cited by 31 | Viewed by 10719
Abstract
Mutual information (MI) is useful for detecting statistical independence between random variables, and it has been successfully applied to solving various machine learning problems. Recently, an alternative to MI called squared-loss MI (SMI) was introduced. While ordinary MI is the Kullback–Leibler divergence from [...] Read more.
Mutual information (MI) is useful for detecting statistical independence between random variables, and it has been successfully applied to solving various machine learning problems. Recently, an alternative to MI called squared-loss MI (SMI) was introduced. While ordinary MI is the Kullback–Leibler divergence from the joint distribution to the product of the marginal distributions, SMI is its Pearson divergence variant. Because both the divergences belong to the ƒ-divergence family, they share similar theoretical properties. However, a notable advantage of SMI is that it can be approximated from data in a computationally more efficient and numerically more stable way than ordinary MI. In this article, we review recent development in SMI approximation based on direct density-ratio estimation and SMI-based machine learning techniques such as independence testing, dimensionality reduction, canonical dependency analysis, independent component analysis, object matching, clustering, and causal inference. Full article
(This article belongs to the Special Issue Estimating Information-Theoretic Quantities from Data)
1612 KiB  
Article
Ordered Regions within a Nonlinear Time Series Solution of a Lorenz Form of the Townsend Equations for a Boundary-Layer Flow
by LaVar King Isaacson
Entropy 2013, 15(1), 53-79; https://doi.org/10.3390/e15010053 - 24 Dec 2012
Cited by 5 | Viewed by 5340
Abstract
A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to a laminar three-dimensional boundary-layer flow. These equations are cast into a Lorenz-type system of equations. The initial system of Lorenz equations yields the generation of masked output [...] Read more.
A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to a laminar three-dimensional boundary-layer flow. These equations are cast into a Lorenz-type system of equations. The initial system of Lorenz equations yields the generation of masked output signals containing internal ordered regions. The self-synchronizing property of the Lorenz system of equations is then exploited by considering the initial Lorenz system as a transmitter system providing chaotic masked information signals to a series of identical Lorenz receiver systems. The output signal from each successive receiver system indicates the growing recovery of ordered regions in the chaotic output signal. Finally, the three-dimensional graph of the output velocity wave vector signal from the fourth receiver system and the spectral entropy rates for the output axial velocity wave vector indicate the presence of ordered regions which are characterized as axially-directed spiral vortices. Full article
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<p>Shown is a schematic diagram of the subsonic flow downstream of a normal shock wave. The x-y plane boundary-layer velocity profile is indicated.</p> Full article ">Figure 2
<p>The three-dimensional flow model and the coordinate system for the boundary layer flow environment. Note the y-location for the z-y plane flat plate boundary-layer velocity profile.</p> Full article ">Figure 3
<p>The time dependent internal feedback parameter, <span class="html-italic">F</span>, for the initial Lorenz system is shown as a function of the time step. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 4
<p>The axial velocity wave vector output, <span class="html-italic">a<sub>x1</sub></span> from the initial Lorenz system is shown as a function of the time step. Parameters: <span class="html-italic">M<sub>1</sub></span>= 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 5
<p>The phase plane, <span class="html-italic">a<sub>x1</sub></span> − <span class="html-italic">a<sub>y1</sub></span> for the output of the initial Lorenz system is shown. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 6
<p>The phase plane, <span class="html-italic">a<sub>z1</sub></span> − <span class="html-italic">a<sub>y1</sub></span> for the output of the initial Lorenz system is shown. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, y<sub>xz</sub> = 0.00417, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 7
<p>The three-dimensional representation of the fluctuating velocity wave vector trajectories produced by the initial Lorenz system is shown. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 8
<p>This figure presents the phase plane, <span class="html-italic">a<sub>x2</sub></span> − <span class="html-italic">a<sub>y2</sub></span> for the output of the first receiver system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 9
<p>This figure presents the phase plane, <span class="html-italic">a<sub>z2</sub></span> − <span class="html-italic">a<sub>y2</sub></span> for the output of the first receiver system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 10
<p>This figure presents a three-dimensional representation of the fluctuating velocity wave vector trajectories produced through the first receiver system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 11
<p>This figure presents the phase plane, <span class="html-italic">a<sub>x4</sub></span> - <span class="html-italic">a<sub>y4</sub></span> for the output of the third receiver system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0. 003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 12
<p>This figure presents the phase plane, <span class="html-italic">a<sub>z4</sub></span> − <span class="html-italic">a<sub>y4</sub></span> for the output of the third receiver system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 13
<p>This figure presents a three-dimensional representation of the fluctuating velocity wave vector trajectories produced through the third receiver system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 14
<p>This figure shows the phase plane, <span class="html-italic">a<sub>z5</sub></span> − <span class="html-italic">a<sub>y5</sub></span> for the output of the fourth receiver system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 15
<p>This figure presents the phase plane, <span class="html-italic">a<sub>z5</sub></span> − <span class="html-italic">a<sub>y5</sub></span> for the output of the fourth receiver system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 16
<p>This figure presents a three-dimensional representation of the fluctuating velocity wave vector trajectories produced through the fourth receiver system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, <span class="html-italic">j</span> = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 17
<p>The spectral entropy rate presented as a function of the segment number for the axial velocity wave vector, <span class="html-italic">a<sub>x1</sub></span> output from the initial transmitter system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, j = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 18
<p>The spectral entropy rate is shown as a function of the segment number for the axial velocity wave vector, <span class="html-italic">a<sub>x4</sub></span> output from the third receiver system system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, j = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">Figure 19
<p>The spectral entropy rate presented as a function of the segment number for the axial velocity wave vector, <span class="html-italic">a<sub>x5</sub></span> output from the fourth receiver system. Parameters: <span class="html-italic">M<sub>1</sub></span> = 1.44, <span class="html-italic">x</span> = 0.08, <span class="html-italic">z</span> = 0.003, j = 16 (<span class="html-italic">η</span> = 3.00).</p> Full article ">
1113 KiB  
Article
Function Based Fault Detection for Uncertain Multivariate Nonlinear Non-Gaussian Stochastic Systems Using Entropy Optimization Principle
by Liping Yin and Li Zhou
Entropy 2013, 15(1), 32-52; https://doi.org/10.3390/e15010032 - 21 Dec 2012
Cited by 7 | Viewed by 6067
Abstract
In this paper, the fault detection in uncertain multivariate nonlinear non-Gaussian stochastic systems is further investigated. Entropy is introduced to characterize the stochastic behavior of the detection errors, and the entropy optimization principle is established for the fault detection problem. The principle is [...] Read more.
In this paper, the fault detection in uncertain multivariate nonlinear non-Gaussian stochastic systems is further investigated. Entropy is introduced to characterize the stochastic behavior of the detection errors, and the entropy optimization principle is established for the fault detection problem. The principle is to maximize the entropies of the stochastic detection errors in the presence of faults and to minimize the entropies of the detection errors in the presence of disturbances. In order to calculate the entropies, the formulations of the joint probability density functions (JPDFs) of the stochastic errors are presented in terms of the known JPDFs of both the disturbances and the faults. By using the novel performance indexes and the formulations for the entropies of the detection errors, new fault detection design methods are provided for the considered multivariate nonlinear non-Gaussian plants. Finally, a simulation example is given to illustrate the efficiency of the proposed fault detection algorithm. Full article
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Figure 1
<p>Function based fault detection.</p> Full article ">Figure 2
<p>Flow chart of function based fault detection.</p> Full article ">Figure 3
<p>The response of the fault <span class="html-italic">δ</span>.</p> Full article ">Figure 4
<p>The response of the input disturbance <math display="inline"> <msub> <mi>ω</mi> <mi>k</mi> </msub> </math>.</p> Full article ">Figure 5
<p>The response of the output disturbance <math display="inline"> <msub> <mi>v</mi> <mi>k</mi> </msub> </math>.</p> Full article ">Figure 6
<p>The PDF of the fault signal <span class="html-italic">δ</span>.</p> Full article ">Figure 7
<p>The PDF of the input disturbance <math display="inline"> <msub> <mi>ω</mi> <mi>k</mi> </msub> </math>.</p> Full article ">Figure 8
<p>The PDF of the output disturbance <math display="inline"> <msub> <mi>v</mi> <mi>k</mi> </msub> </math>.</p> Full article ">Figure 9
<p>The residual value when fault occurs.</p> Full article ">Figure 10
<p>The residual value when no fault occurs.</p> Full article ">Figure 11
<p>3-D mesh of the system output <span class="html-italic">y</span>.</p> Full article ">Figure 12
<p>The optimization performance index <span class="html-italic">J</span>.</p> Full article ">
8814 KiB  
Article
Quantitative Analysis of Dynamic Behaviours of Rural Areas at Provincial Level Using Public Data of Gross Domestic Product
by Yi Chen, Guangfeng Zhang, Yiyang Li, Yi Ding, Bin Zheng and Qiang Miao
Entropy 2013, 15(1), 10-31; https://doi.org/10.3390/e15010010 - 20 Dec 2012
Cited by 9 | Viewed by 6858
Abstract
A spatial approach that incorporates three economic components and one environmental factor has been developed to evaluate the dynamic behaviours of the rural areas at a provincial level. An artificial fish swarm algorithm with variable population size (AFSAVP) is proposed for the spatial [...] Read more.
A spatial approach that incorporates three economic components and one environmental factor has been developed to evaluate the dynamic behaviours of the rural areas at a provincial level. An artificial fish swarm algorithm with variable population size (AFSAVP) is proposed for the spatial problem. A functional region affecting index θ is employed as a fitness function for the AFSAVP driven optimisation, in which a gross domestic product (GDP) based method is utilised to estimate the CO2 emission of all provinces. A simulation for the administrative provinces of China has been implemented, and the results have shown that the modelling method based on GDP data can assess the spatial dynamic behaviours and can be taken as an operational tool for the policy planners. Full article
(This article belongs to the Special Issue Entropy and Urban Sprawl)
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<p>The state distance between the <math display="inline"> <msup> <mi>i</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msup> </math> and <math display="inline"> <msup> <mi>j</mi> <mrow> <mi>t</mi> <mi>h</mi> </mrow> </msup> </math> individual.</p> Full article ">Figure 2
<p>The variable population size fish swarm algorithm workflow.</p> Full article ">Figure 3
<p>Framework of quantitative analysis of dynamic behaviours.</p> Full article ">Figure 4
<p><math display="inline"> <mrow> <mi>C</mi> <msub> <mi>O</mi> <mn>2</mn> </msub> </mrow> </math> emission estimation of province <span class="html-italic">j</span>.</p> Full article ">Figure 5
<p>Multi-state representation via power functions</p> Full article ">Figure 6
<p>Functional distance measurement of Sichuan province.</p> Full article ">Figure 7
<p>Θ fitness plot, 100 generations.</p> Full article ">Figure 8
<p>Θ fitness plot, 1000 generations.</p> Full article ">Figure 9
<p>Θ fitness plot, 10,000 generations.</p> Full article ">Figure 10
<p>Θ fitness plot, 100,000 generations.</p> Full article ">Figure 11
<p>Fitness phase portrait of the dynamic behaviours, 100,000 generations.</p> Full article ">
408 KiB  
Concept Paper
Urban Ecosystem Health Assessment and Its Application in Management: A Multi-Scale Perspective
by Meirong Su, Zhifeng Yang, Bin Chen, Gengyuan Liu, Yan Zhang, Lixiao Zhang, Linyu Xu and Yanwei Zhao
Entropy 2013, 15(1), 1-9; https://doi.org/10.3390/e15010001 - 20 Dec 2012
Cited by 13 | Viewed by 10762
Abstract
Urban ecosystem health assessments can be applied extensively in urban management to evaluate the status quo of the urban ecosystem, identify the limiting factors, identify key problems, optimize the scheme and guide ecological regulation. Regarding the multi-layer roles of urban ecosystems, urban ecosystem [...] Read more.
Urban ecosystem health assessments can be applied extensively in urban management to evaluate the status quo of the urban ecosystem, identify the limiting factors, identify key problems, optimize the scheme and guide ecological regulation. Regarding the multi-layer roles of urban ecosystems, urban ecosystem health should be assessed at different scales with each assessment providing a specific reference to urban management from its own viewpoint. Therefore, a novel framework of multi-scale urban ecosystem health assessment is established on global, national, regional and local scales. A demonstration of the framework is shown by using a case study in Guangzhou City, China, where urban ecosystem health assessment is conducted in the order of global, national, regional, and local scales, from macro to micro, and rough to detailed analysis. The new multi-scale framework can be utilized to generate a more comprehensive understanding of urban ecosystem health, more accurate orientation of urban development, and more feasible regulation and management programs when compared with the traditional urban ecosystem health assessment focusing at the local scale. Full article
(This article belongs to the Special Issue Entropy and Urban Sprawl)
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<p>Factors of concern for urban ecosystem health. (UEH is used in this figure to represent urban ecosystem health).</p> Full article ">Figure 2
<p>Multi-layer roles of urban ecosystems and respective concerns.</p> Full article ">Figure 3
<p>Framework of multi-scale urban ecosystem health assessment and its implications in management.</p> Full article ">
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