A Permutation Disalignment Index-Based Complex Network Approach to Evaluate Longitudinal Changes in Brain-Electrical Connectivity*Entropy* **2017**, *19*(10), 548; doi:10.3390/e19100548 (registering DOI) - 17 October 2017**Abstract **

In the study of neurological disorders, Electroencephalographic (EEG) signal processing can provide valuable information because abnormalities in the interaction between neuron circuits may reflect on macroscopic abnormalities in the electrical potentials that can be detected on the scalp. A Mild Cognitive Impairment (MCI)

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In the study of neurological disorders, Electroencephalographic (EEG) signal processing can provide valuable information because abnormalities in the interaction between neuron circuits may reflect on macroscopic abnormalities in the electrical potentials that can be detected on the scalp. A Mild Cognitive Impairment (MCI) condition, when caused by a disorder degenerating into dementia, affects the brain connectivity. Motivated by the promising results achieved through the recently developed descriptor of coupling strength between EEG signals, the Permutation Disalignment Index (PDI), the present paper introduces a novel PDI-based complex network model to evaluate the longitudinal variations in brain-electrical connectivity. A group of 33 amnestic MCI subjects was enrolled and followed-up with over four months. The results were compared to MoCA (Montreal Cognitive Assessment) tests, which scores the cognitive abilities of the patient. A significant negative correlation could be observed between MoCA variation and the characteristic path length ($\lambda $ ) variation ($r=-0.56,p=0.0006$ ), whereas a significant positive correlation could be observed between MoCA variation and the variation of clustering coefficient (CC, $r=0.58,p=0.0004$ ), global efficiency (GE, $r=0.57,p=0.0005$ ) and small worldness (SW, $r=0.57,p=0.0005$ ). Cognitive decline thus seems to reflect an underlying cortical “disconnection” phenomenon: worsened subjects indeed showed an increased $\lambda $ and decreased CC, GE and SW. The PDI-based connectivity model, proposed in the present work, could be a novel tool for the objective quantification of longitudinal brain-electrical connectivity changes in MCI subjects.
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Compressed Secret Key Agreement:Maximizing Multivariate Mutual Information per Bit*Entropy* **2017**, *19*(10), 545; doi:10.3390/e19100545 - 14 October 2017**Abstract **

The multiterminal secret key agreement problem by public discussion is formulated with an additional source compression step where, prior to the public discussion phase, users independently compress their private sources to filter out strongly correlated components in order to generate a common secret

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The multiterminal secret key agreement problem by public discussion is formulated with an additional source compression step where, prior to the public discussion phase, users independently compress their private sources to filter out strongly correlated components in order to generate a common secret key. The objective is to maximize the achievable key rate as a function of the joint entropy of the compressed sources. Since the maximum achievable key rate captures the total amount of information mutual to the compressed sources, an optimal compression scheme essentially maximizes the multivariate mutual information per bit of randomness of the private sources, and can therefore be viewed more generally as a dimension reduction technique. Single-letter lower and upper bounds on the maximum achievable key rate are derived for the general source model, and an explicit polynomial-time computable formula is obtained for the pairwise independent network model. In particular, the converse results and the upper bounds are obtained from those of the related secret key agreement problem with rate-limited discussion. A precise duality is shown for the two-user case with one-way discussion, and such duality is extended to obtain the desired converse results in the multi-user case. In addition to posing new challenges in information processing and dimension reduction, the compressed secret key agreement problem helps shed new light on resolving the difficult problem of secret key agreement with rate-limited discussion by offering a more structured achieving scheme and some simpler conjectures to prove.
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Equilibration in the Nosé–Hoover Isokinetic Ensemble: Effect of Inter-Particle Interactions*Entropy* **2017**, *19*(10), 544; doi:10.3390/e19100544 - 14 October 2017**Abstract **

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We investigate the stationary and dynamic properties of the celebrated Nosé–Hoover dynamics of many-body interacting Hamiltonian systems, with an emphasis on the effect of inter-particle interactions. To this end, we consider a model system with both short- and long-range interactions. The Nosé–Hoover dynamics

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We investigate the stationary and dynamic properties of the celebrated Nosé–Hoover dynamics of many-body interacting Hamiltonian systems, with an emphasis on the effect of inter-particle interactions. To this end, we consider a model system with both short- and long-range interactions. The Nosé–Hoover dynamics aim to generate the canonical equilibrium distribution of a system at a desired temperature by employing a set of time-reversible, deterministic equations of motion. A signature of canonical equilibrium is a single-particle momentum distribution that is Gaussian. We find that the equilibrium properties of the system within the Nosé–Hoover dynamics coincides with that within the canonical ensemble. Moreover, starting from out-of-equilibrium initial conditions, the average kinetic energy of the system relaxes to its target value over a *size-independent* timescale. However, quite surprisingly, our results indicate that under the same conditions and with only long-range interactions present in the system, the momentum distribution relaxes to its Gaussian form in equilibrium over a scale that *diverges with the system size*. On adding short-range interactions, the relaxation is found to occur over a timescale that has a much weaker dependence on system size. This system-size dependence of the timescale vanishes when only short-range interactions are present in the system. An implication of such an ultra-slow relaxation when only long-range interactions are present in the system is that macroscopic observables other than the average kinetic energy when estimated in the Nosé–Hoover dynamics may take an unusually long time to relax to its canonical equilibrium value. Our work underlines the crucial role that interactions play in deciding the equivalence between Nosé–Hoover and canonical equilibrium.
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Investigating the Thermodynamic Performances of TO-Based Metamaterial Tunable Cells with an Entropy Generation Approach*Entropy* **2017**, *19*(10), 538; doi:10.3390/e19100538 - 13 October 2017**Abstract **

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Active control of heat flux can be realized with transformation optics (TO) thermal metamaterials. Recently, a new class of metamaterial tunable cells has been proposed, aiming to significantly reduce the difficulty of fabrication and to flexibly switch functions by employing several cells assembled

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Active control of heat flux can be realized with transformation optics (TO) thermal metamaterials. Recently, a new class of metamaterial tunable cells has been proposed, aiming to significantly reduce the difficulty of fabrication and to flexibly switch functions by employing several cells assembled on related positions following the TO design. However, owing to the integration and rotation of materials in tunable cells, they might lead to extra thermal losses as compared with the previous continuum design. This paper focuses on investigating the thermodynamic properties of tunable cells under related design parameters. The universal expression for the local entropy generation rate in such metamaterial systems is obtained considering the influence of rotation. A series of contrast schemes are established to describe the thermodynamic process and thermal energy distributions from the viewpoint of entropy analysis. Moreover, effects of design parameters on thermal dissipations and system irreversibility are investigated. In conclusion, more thermal dissipations and stronger thermodynamic processes occur in a system with larger conductivity ratios and rotation angles. This paper presents a detailed description of the thermodynamic properties of metamaterial tunable cells and provides reference for selecting appropriate design parameters on related positions to fabricate more efficient and energy-economical switchable TO devices.
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Kovacs-Like Memory Effect in Athermal Systems: Linear Response Analysis*Entropy* **2017**, *19*(10), 539; doi:10.3390/e19100539 - 13 October 2017**Abstract **

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We analyze the emergence of Kovacs-like memory effects in athermal systems within the linear response regime. This is done by starting from both the master equation for the probability distribution and the equations for the physically-relevant moments. The general results are applied to

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We analyze the emergence of Kovacs-like memory effects in athermal systems within the linear response regime. This is done by starting from both the master equation for the probability distribution and the equations for the physically-relevant moments. The general results are applied to a general class of models with conserved momentum and non-conserved energy. Our theoretical predictions, obtained within the first Sonine approximation, show an excellent agreement with the numerical results. Furthermore, we prove that the observed non-monotonic relaxation is consistent with the monotonic decay of the non-equilibrium entropy.
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Stationary Wavelet Singular Entropy and Kernel Extreme Learning for Bearing Multi-Fault Diagnosis*Entropy* **2017**, *19*(10), 541; doi:10.3390/e19100541 - 13 October 2017**Abstract **

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The behavioural diagnostics of bearings play an essential role in the management of several rotation machine systems. However, current diagnostic methods do not deliver satisfactory results with respect to failures in variable speed rotational phenomena. In this paper, we consider the Shannon entropy

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The behavioural diagnostics of bearings play an essential role in the management of several rotation machine systems. However, current diagnostic methods do not deliver satisfactory results with respect to failures in variable speed rotational phenomena. In this paper, we consider the Shannon entropy as an important fault signature pattern. To compute the entropy, we propose combining stationary wavelet transform and singular value decomposition. The resulting feature extraction method, that we call stationary wavelet singular entropy (SWSE), aims to improve the accuracy of the diagnostics of bearing failure by finding a small number of high-quality fault signature patterns. The features extracted by the SWSE are then passed on to a kernel extreme learning machine (KELM) classifier. The proposed SWSE-KELM algorithm is evaluated using two bearing vibration signal databases obtained from Case Western Reserve University. We compare our SWSE feature extraction method to other well-known methods in the literature such as stationary wavelet packet singular entropy (SWPSE) and decimated wavelet packet singular entropy (DWPSE). The experimental results show that the SWSE-KELM consistently outperforms both the SWPSE-KELM and DWPSE-KELM methods. Further, our SWSE method requires fewer features than the other two evaluated methods, which makes our SWSE-KELM algorithm simpler and faster.
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Backtracking and Mixing Rate of Diffusion on Uncorrelated Temporal Networks*Entropy* **2017**, *19*(10), 542; doi:10.3390/e19100542 - 13 October 2017**Abstract **

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We consider the problem of diffusion on temporal networks, where the dynamics of each edge is modelled by an independent renewal process. Despite the apparent simplicity of the model, the trajectories of a random walker exhibit non-trivial properties. Here, we quantify the walker’s

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We consider the problem of diffusion on temporal networks, where the dynamics of each edge is modelled by an independent renewal process. Despite the apparent simplicity of the model, the trajectories of a random walker exhibit non-trivial properties. Here, we quantify the walker’s tendency to backtrack at each step (return where he/she comes from), as well as the resulting effect on the mixing rate of the process. As we show through empirical data, non-Poisson dynamics may significantly slow down diffusion due to backtracking, by a mechanism intrinsically different from the standard bus paradox and related temporal mechanisms. We conclude by discussing the implications of our work for the interpretation of results generated by null models of temporal networks.
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Complexity-Entropy Maps as a Tool for the Characterization of the Clinical Electrophysiological Evolution of Patients under Pharmacological Treatment with Psychotropic Drugs*Entropy* **2017**, *19*(10), 540; doi:10.3390/e19100540 - 13 October 2017**Abstract **

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In the clinical electrophysiological practice, reading and comparing electroencephalographic (EEG) recordings are sometimes insufficient and take too much time. Tools coming from the information theory or nonlinear systems theory such as entropy and complexity have been presented as an alternative to address this

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In the clinical electrophysiological practice, reading and comparing electroencephalographic (EEG) recordings are sometimes insufficient and take too much time. Tools coming from the information theory or nonlinear systems theory such as entropy and complexity have been presented as an alternative to address this problem. In this work, we introduce a novel method—the permutation Lempel–Ziv Complexity vs. Permutation Entropy map. We apply this method to the EEGs of two patients with specific diagnosed pathologies during respective follow up processes of pharmacological changes in order to detect alterations that are not evident with the usual inspection method. The method allows for comparing between different states of the patients’ treatment, with a healthy control group, given global information about the signal, supplementing the traditional method of visual inspection of EEG.
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Is Cetacean Intelligence Special? New Perspectives on the Debate*Entropy* **2017**, *19*(10), 543; doi:10.3390/e19100543 - 13 October 2017**Abstract **

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In recent years, the interpretation of our observations of animal behaviour, in particular that of cetaceans, has captured a substantial amount of attention in the scientific community. The traditional view that supports a special intellectual status for this mammalian order has fallen under

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In recent years, the interpretation of our observations of animal behaviour, in particular that of cetaceans, has captured a substantial amount of attention in the scientific community. The traditional view that supports a special intellectual status for this mammalian order has fallen under significant scrutiny, in large part due to problems of how to define and test the cognitive performance of animals. This paper presents evidence supporting complex cognition in cetaceans obtained using the recently developed intelligence and embodiment hypothesis. This hypothesis is based on evolutionary neuroscience and postulates the existence of a common information-processing principle associated with nervous systems that evolved naturally and serves as the foundation from which intelligence can emerge. This theoretical framework explaining animal intelligence in neural computational terms is supported using a new mathematical model. Two pathways leading to higher levels of intelligence in animals are identified, each reflecting a trade-off either in energetic requirements or the number of neurons used. A description of the evolutionary pathway that led to increased cognitive capacities in cetacean brains is detailed and evidence supporting complex cognition in cetaceans is presented. This paper also provides an interpretation of the adaptive function of cetacean neuronal traits.
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Bowen Lemma in the Countable Symbolic Space*Entropy* **2017**, *19*(10), 532; doi:10.3390/e19100532 - 11 October 2017**Abstract **

We consider the sets of quasi-regular points in the countable symbolic space. We measure the sizes of the sets by Billingsley-Hausdorff dimension defined by Gibbs measures. It is shown that the dimensions of those sets, always bounded from below by the convergence exponent

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We consider the sets of quasi-regular points in the countable symbolic space. We measure the sizes of the sets by Billingsley-Hausdorff dimension defined by Gibbs measures. It is shown that the dimensions of those sets, always bounded from below by the convergence exponent of the Gibbs measure, are given by a variational principle, which generalizes Li and Ma’s result and Bowen’s result.
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Contact Hamiltonian Dynamics: The Concept and Its Use*Entropy* **2017**, *19*(10), 535; doi:10.3390/e19100535 - 11 October 2017**Abstract **

We give a short survey on the concept of contact Hamiltonian dynamics and its use in several areas of physics, namely reversible and irreversible thermodynamics, statistical physics and classical mechanics. Some relevant examples are provided along the way. We conclude by giving insights

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We give a short survey on the concept of contact Hamiltonian dynamics and its use in several areas of physics, namely reversible and irreversible thermodynamics, statistical physics and classical mechanics. Some relevant examples are provided along the way. We conclude by giving insights into possible future directions.
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Hydrodynamics of a Granular Gas in a Heterogeneous Environment*Entropy* **2017**, *19*(10), 536; doi:10.3390/e19100536 - 11 October 2017**Abstract **

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We analyze the transport properties of a low density ensemble of identical macroscopic particles immersed in an active fluid. The particles are modeled as inelastic hard spheres (granular gas). The non-homogeneous active fluid is modeled by means of a non-uniform stochastic thermostat. The

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We analyze the transport properties of a low density ensemble of identical macroscopic particles immersed in an active fluid. The particles are modeled as inelastic hard spheres (granular gas). The non-homogeneous active fluid is modeled by means of a non-uniform stochastic thermostat. The theoretical results are validated with a numerical solution of the corresponding the kinetic equation (direct simulation Monte Carlo method). We show a steady flow in the system that is accurately described by Navier-Stokes (NS) hydrodynamics, even for high inelasticity. Surprisingly, we find that the deviations from NS hydrodynamics for this flow are stronger as the inelasticity decreases. The active fluid action is modeled here with a non-uniform fluctuating volume force. This is a relevant result given that hydrodynamics of particles in complex environments, such as biological crowded environments, is still a question under intense debate.
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Informative Nature and Nonlinearity of Lagged Poincaré Plots Indices in Analysis of Heart Rate Variability*Entropy* **2017**, *19*(10), 523; doi:10.3390/e19100523 - 10 October 2017**Abstract **

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Lagged Poincaré plots have been successful in characterizing abnormal cardiac function. However, the current research practices do not favour any specific lag of Poincaré plots, thus complicating the comparison of results of different researchers in their analysis of heart rate of healthy subjects

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Lagged Poincaré plots have been successful in characterizing abnormal cardiac function. However, the current research practices do not favour any specific lag of Poincaré plots, thus complicating the comparison of results of different researchers in their analysis of heart rate of healthy subjects and patients. We researched the informative nature of lagged Poincaré plots in different states of the autonomic nervous system. It was tested in three models: different age groups, groups with different balance of autonomous regulation, and in hypertensive patients. Correlation analysis shows that for lag *l* = 6, *SD1/SD2* has weak (*r* = 0.33) correlation with linear parameters of heart rate variability (HRV). For *l* more than 6 it displays even less correlation with linear parameters, but the changes in *SD1/SD2* become statistically insignificant. Secondly, surrogate data tests show that the real *SD1/SD2* is statistically different from its surrogate value and the conclusion could be made that the heart rhythm has nonlinear properties. Thirdly, the three models showed that for different functional states of the autonomic nervous system (ANS), *SD1/SD2* ratio varied only for lags *l* = 5 and 6. All of this allow to us to give cautious recommendation to use *SD1/SD2* with lags 5 and 6 as a nonlinear characteristic of HRV. The received data could be used as the basis for continuing the research in standardisation of nonlinear analytic methods.
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An Approximated Box Height for Differential-Box-Counting Method to Estimate Fractal Dimensions of Gray-Scale Images*Entropy* **2017**, *19*(10), 534; doi:10.3390/e19100534 - 10 October 2017**Abstract **

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The Fractal Dimension (FD) of an image defines the roughness using a real number which is highly associated with the human perception of surface roughness. It has been applied successfully for many computer vision applications such as texture analysis, segmentation and classification. Several

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The Fractal Dimension (FD) of an image defines the roughness using a real number which is highly associated with the human perception of surface roughness. It has been applied successfully for many computer vision applications such as texture analysis, segmentation and classification. Several techniques can be found in literature to estimate FD. One such technique is Differential Box Counting (DBC). Its performance is influenced by many parameters. In particular, the box height is directly related to the gray-level variations over image grid, which badly affects the performance of DBC. In this work, a new method for estimating box height is proposed without changing the other parameters of DBC. The proposed box height has been determined empirically and depends only on the image size. All the experiments have been performed on simulated Fractal Brownian Motion (FBM) Database and Brodatz Database. It has been proved experimentally that the proposed box height allow to improve the performance of DBC, Shifting DBC, Improved DBC and Improved Triangle DBC, which are closer to actual FD values of the simulated FBM images.
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Cross Entropy Method Based Hybridization of Dynamic Group Optimization Algorithm*Entropy* **2017**, *19*(10), 533; doi:10.3390/e19100533 - 9 October 2017**Abstract **

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Recently, a new algorithm named dynamic group optimization (DGO) has been proposed, which lends itself strongly to exploration and exploitation. Although DGO has demonstrated its efficacy in comparison to other classical optimization algorithms, DGO has two computational drawbacks. The first one is related

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Recently, a new algorithm named dynamic group optimization (DGO) has been proposed, which lends itself strongly to exploration and exploitation. Although DGO has demonstrated its efficacy in comparison to other classical optimization algorithms, DGO has two computational drawbacks. The first one is related to the two mutation operators of DGO, where they may decrease the diversity of the population, limiting the search ability. The second one is the homogeneity of the updated population information which is selected only from the companions in the same group. It may result in premature convergence and deteriorate the mutation operators. In order to deal with these two problems in this paper, a new hybridized algorithm is proposed, which combines the dynamic group optimization algorithm with the cross entropy method. The cross entropy method takes advantage of sampling the problem space by generating candidate solutions using the distribution, then it updates the distribution based on the better candidate solution discovered. The cross entropy operator does not only enlarge the promising search area, but it also guarantees that the new solution is taken from all the surrounding useful information into consideration. The proposed algorithm is tested on 23 up-to-date benchmark functions; the experimental results verify that the proposed algorithm over the other contemporary population-based swarming algorithms is more effective and efficient.
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Analysis of Entropy Generation in Flow of Methanol-Based Nanofluid in a Sinusoidal Wavy Channel*Entropy* **2017**, *19*(10), 490; doi:10.3390/e19100490 - 8 October 2017**Abstract **

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The entropy generation due to heat transfer and fluid friction in mixed convective peristaltic flow of methanol-Al_{2}O_{3} nano fluid is examined. Maxwell’s thermal conductivity model is used in analysis. Velocity and temperature profiles are utilized in the computation of the

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The entropy generation due to heat transfer and fluid friction in mixed convective peristaltic flow of methanol-Al_{2}O_{3} nano fluid is examined. Maxwell’s thermal conductivity model is used in analysis. Velocity and temperature profiles are utilized in the computation of the entropy generation number. The effects of involved physical parameters on velocity, temperature, entropy generation number, and Bejan number are discussed and explained graphically.
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Bivariate Partial Information Decomposition: The Optimization Perspective*Entropy* **2017**, *19*(10), 530; doi:10.3390/e19100530 - 7 October 2017**Abstract **

Bertschinger, Rauh, Olbrich, Jost, and Ay (Entropy, 2014) have proposed a definition of a decomposition of the mutual information $MI(\mathbf{X}:\mathbf{Y},\mathbf{Z})$ into shared, synergistic, and unique information by way of solving a convex optimization problem. In

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Bertschinger, Rauh, Olbrich, Jost, and Ay (Entropy, 2014) have proposed a definition of a decomposition of the mutual information $MI(\mathbf{X}:\mathbf{Y},\mathbf{Z})$ into shared, synergistic, and unique information by way of solving a convex optimization problem. In this paper, we discuss the solution of their Convex Program from theoretical and practical points of view.
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Multivariate Dependence beyond Shannon Information*Entropy* **2017**, *19*(10), 531; doi:10.3390/e19100531 - 7 October 2017**Abstract **

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Accurately determining dependency structure is critical to understanding a complex system’s organization. We recently showed that the transfer entropy fails in a key aspect of this—measuring information flow—due to its conflation of dyadic and polyadic relationships. We extend this observation to demonstrate that

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Accurately determining dependency structure is critical to understanding a complex system’s organization. We recently showed that the transfer entropy fails in a key aspect of this—measuring information flow—due to its conflation of dyadic and polyadic relationships. We extend this observation to demonstrate that Shannon information measures (entropy and mutual information, in their conditional and multivariate forms) can fail to accurately ascertain multivariate dependencies due to their conflation of qualitatively different relations among variables. This has broad implications, particularly when employing information to express the organization and mechanisms embedded in complex systems, including the burgeoning efforts to combine complex network theory with information theory. Here, we do not suggest that any aspect of information theory is wrong. Rather, the vast majority of its informational measures are simply inadequate for determining the meaningful relationships among variables within joint probability distributions. We close by demonstrating that such distributions exist across an arbitrary set of variables.
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Coarse-Graining and the Blackwell Order*Entropy* **2017**, *19*(10), 527; doi:10.3390/e19100527 - 6 October 2017**Abstract **

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Suppose we have a pair of information channels, ${\kappa}_{1},{\kappa}_{2}$ , with a common input. The Blackwell order is a partial order over channels that compares ${\kappa}_{1}$ and ${\kappa}_{2}$ by the maximal expected utility an agent can obtain

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Suppose we have a pair of information channels, ${\kappa}_{1},{\kappa}_{2}$ , with a common input. The Blackwell order is a partial order over channels that compares ${\kappa}_{1}$ and ${\kappa}_{2}$ by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently, ${\kappa}_{1}$ is said to be Blackwell-inferior to ${\kappa}_{2}$ if and only if ${\kappa}_{1}$ can be constructed by garbling the output of ${\kappa}_{2}$ . A related partial order stipulates that ${\kappa}_{2}$ is more capable than ${\kappa}_{1}$ if the mutual information between the input and output is larger for ${\kappa}_{2}$ than for ${\kappa}_{1}$ for any distribution over inputs. A Blackwell-inferior channel is necessarily less capable. However, examples are known where ${\kappa}_{1}$ is less capable than ${\kappa}_{2}$ but not Blackwell-inferior. We show that this may even happen when ${\kappa}_{1}$ is constructed by coarse-graining the inputs of ${\kappa}_{2}$ . Such a coarse-graining is a special kind of “pre-garbling” of the channel inputs. This example directly establishes that the expected value of the shared utility function for the coarse-grained channel is larger than it is for the non-coarse-grained channel. This contradicts the intuition that coarse-graining can only destroy information and lead to inferior channels. We also discuss our results in the context of information decompositions.
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Generalized Skew-Normal Negentropy and Its Application to Fish Condition Factor Time Series*Entropy* **2017**, *19*(10), 528; doi:10.3390/e19100528 - 6 October 2017**Abstract **

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The problem of measuring the disparity of a particular probability density function from a normal one has been addressed in several recent studies. The most used technique to deal with the problem has been exact expressions using information measures over particular distributions. In

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The problem of measuring the disparity of a particular probability density function from a normal one has been addressed in several recent studies. The most used technique to deal with the problem has been exact expressions using information measures over particular distributions. In this paper, we consider a class of asymmetric distributions with a normal kernel, called Generalized Skew-Normal (GSN) distributions. We measure the degrees of disparity of these distributions from the normal distribution by using exact expressions for the GSN negentropy in terms of cumulants. Specifically, we focus on skew-normal and modified skew-normal distributions. Then, we establish the Kullback–Leibler divergences between each GSN distribution and the normal one in terms of their negentropies to develop hypothesis testing for normality. Finally, we apply this result to condition factor time series of anchovies off northern Chile.
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