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In this work, we consider the design of power-constrained networked control systems (NCSs) and a differential entropy-based fault-detection mechanism. For the NCS design of the control loop, we consider faults in the plant gain and unstable plant pole locations, either due to natural causes or malicious intent. Since the power-constrained approach utilized in the NCS design is a stationary approach, we then discuss the finite-time approximation of the power constraints for the relevant control loop signals. The network under study is formed by two additive white Gaussian noise (AWGN) channels located on the direct and feedback paths of the closed control loop. The finite-time approximation of the controller output signal allows us to estimate its differential entropy, which is used in our proposed fault-detection mechanism. After fault detection, we propose a fault-identification mechanism that is capable of correctly discriminating faults. Finally, we discuss the extension of the contributions developed here to future research directions, such as fault recovery and control resilience. [ABSTRACT FROM AUTHOR]

The study of Shannon differential entropy function and its associated measures using quantile function are of recent interest among researchers. However, a bivariate extension of the same have not been considered so far. Recently, a new method of deriving bivariate quantile function and their properties in the context of reliability modeling is available in literature. Motivated by this, the present paper introduces and obtain some properties of quantile-based entropy function in the bivariate case. We also examine various properties of bivariate quantile entropy function for residual random variables. Finally, we illustrate the bivariate quantile-based residual entropy function using a suitable lifetime data. [ABSTRACT FROM AUTHOR]

We investigate the cumulative Tsallis entropy, an information measure recently introduced as a cumulative version of the classical Tsallis differential entropy, which is itself a generalization of the Boltzmann-Gibbs statistics. This functional is here considered as a perturbation of the expected mean residual life via some power weight function. This point of view leads to the introduction of the dual cumulative Tsallis entropy and of two families of coherent risk measures generalizing those built on mean residual life. We characterize the finiteness of the cumulative Tsallis entropy in terms of L p -spaces and show how they determine the underlying distribution. The range of the functional is exactly described under various constraints, with optimal bounds improving on all those previously available in the literature. Whereas the maximization of the Tsallis differential entropy gives rise to the classical q -Gaussian distribution which is a generalization of the Gaussian having a finite range or heavy tails, the maximization of the cumulative Tsallis entropy leads to an analogous perturbation of the Logistic distribution. [ABSTRACT FROM AUTHOR]

The recent introduction of geometric partition entropy offered an alternative to differential Shannon entropy for the quantification of uncertainty as estimated from a sample drawn from a one-dimensional bounded continuous probability distribution. In addition to being a fresh perspective for the basis of continuous information theory, this new approach provided several improvements over traditional entropy estimators including its effectiveness on sparse samples and a proper incorporation of the impact from extreme outliers. However, a complimentary relationship exists between the new geometric approach and the basic form of its frequency-based predecessor that is leveraged here to define an entropy measure with no bias toward the sample size. This stable normalized measure is named the Boltzmann–Shannon interaction entropy (BSIE)) as it is defined in terms of a standard divergence between the measure-based and frequency-based distributions that can be associated with the two historical figures. This parameter-free measure can be accurately estimated in a computationally efficient manner, and we illustrate its utility as a quality metric for subsampling in the context of nonlinear polynomial regression. [ABSTRACT FROM AUTHOR]

We investigate the cumulative Tsallis entropy, an information measure recently introduced as a cumulative version of the classical Tsallis differential entropy, which is itself a generalization of the Boltzmann-Gibbs statistics. This functional is here considered as a perturbation of the expected mean residual life via some power weight function. This point of view leads to the introduction of the dual cumulative Tsallis entropy and of two families of coherent risk measures generalizing those built on mean residual life. We characterize the finiteness of the cumulative Tsallis entropy in terms of L p -spaces and show how they determine the underlying distribution. The range of the functional is exactly described under various constraints, with optimal bounds improving on all those previously available in the literature. Whereas the maximization of the Tsallis differential entropy gives rise to the classical q -Gaussian distribution which is a generalization of the Gaussian having a finite range or heavy tails, the maximization of the cumulative Tsallis entropy leads to an analogous perturbation of the Logistic distribution. [ABSTRACT FROM AUTHOR]

Response times (RT) distributions are routinely used by psychologists and neuroscientists in the assessment and modeling of human behavior and cognition. The statistical properties of RT distributions are valuable in uncovering unobservable psychological mechanisms. A potentially important statistical aspect of RT distributions is their entropy. However, to date, no valid measure of entropy on RT distributions has been developed, mainly because available extensions of discrete entropy measures to continuous distributions were fraught with problems and inconsistencies. The present work takes advantage of the cumulative residual entropy (CRE) function—a well-known differential entropy measure that can circumvent those problems. Applications of the CRE to RT distributions are presented along with concrete examples and simulations. In addition, a novel measure of instantaneous CRE is developed that captures the rate of entropy reduction (or information gain) from a stimulus as a function of processing time. Taken together, the new measures of entropy in RT distributions proposed here allow for stronger statistical inferences, as well as motivated theoretical interpretations of psychological constructs such as mental effort and processing efficiency. [ABSTRACT FROM AUTHOR]

A fundamental factor in relevant applications is the predictability of the life cycle of a coherent system consisting of more than one component. In this context, we examine how entropy can be applied to evaluate the degree of predictability. In particular, in order to calculate the Tsallis entropy of the past life, we consider a scenario in which all components of the system fail at a given time t and use the system signature to calculate the Tsallis entropy of the past life. We examine a number of analytical results, e.g., expressions, thresholds and orders for the measure at issue in our study. The results may provide insights into the predictability of a coherent system's life cycle. [ABSTRACT FROM AUTHOR]

We calculate differential Shannon entropies derived from time-dependent coordinate-space and momentum-space probability densities. This is performed for a prototype system of a coupled electron–nuclear motion. Two situations are considered, where one is a Born–Oppenheimer adiabatic dynamics, and the other is a diabatic motion involving strong non-adiabatic transitions. The information about coordinate- and momentum-space dynamics derived from the total and single-particle entropies is discussed and interpreted with the help of analytical models. From the entropies, we derive mutual information, which is a measure for the electron–nuclear correlation. In the adiabatic case, it is found that such correlations are manifested differently in coordinate- and momentum space. For the diabatic dynamics, we show that it is possible to decompose the entropies into state-specific contributions. [ABSTRACT FROM AUTHOR]

In parallel with the concept of Rényi entropy for residual lifetime distributions, the Rényi entropy of inactivity time of lifetime distributions belonging to asymmetric distributions is a useful measure of independent interest. For a system that turns out to be inactive in time t, the past entropy is considered as an uncertainty measure for the past lifetime distribution. In this study, we consider a coherent system that includes n components and has the property that all the components of the system have failed at time t. To assess the predictability of the coherent system's lifetime, we use the system's signature to determine the Rényi entropy of its past lifetime. We study several analytical results, including expressions, bounds, and order properties for this measure. [ABSTRACT FROM AUTHOR]

For a given system observed at time t, the past entropy serves as an uncertainty measure about the past life-time of the distribution. We consider a coherent system in which there are n components that have all failed at time t. To assess the predictability of the life-time of such a system, we use the signature vector to determine the entropy of its past life-time. We explore various analytical results, including expressions, bounds, and order properties, for this measure. Our results provide valuable insight into the predictability of the coherent system's life-time, which may be useful in a number of practical applications. [ABSTRACT FROM AUTHOR]

In this article, we propose three nonparametric estimators for quantile-based Shannon differential entropy function for complete samples. Asymptotic properties of the estimators are established under suitable regularity conditions. Simulation study are carried out to compare the performance of the proposed estimators and with other important entropy estimators. The usefulness of the new estimators are also illustrated using real data sets. [ABSTRACT FROM AUTHOR]

Adult attention deficit/hyperactivity disorder (ADHD), schizophrenia (SCHZ), and bipolar disorder (BP) have common symptoms and differences, and the underlying neural mechanisms are still unclear. This article will thoroughly discuss the differences between ADHD, BP, and SCHZ (31 healthy control and 31 ADHD; 34 healthy control and 34 BP; 42 healthy control and 42 SCHZ) relative to healthy subjects in combination with three atlases (et al., the Brainnetome atlas, the Dosenbach atlas, the Power atlas) and seven entropies (et al., approximate entropy (ApEn), sample entropy (SaEn), permutation entropy (PeEn), fuzzy entropy (FuEn), differential entropy (DiffEn), range entropy (RaEn), and dispersion entropy (DispEn)), as well as the prominent significant brain regions, in the hope of giving information that is more suitable for analyzing different diseases' entropy. First, the reliability (et al., intraclass correlation coefficient [ICC]) of seven kinds of entropy is calculated and analyzed by using the MSC dataset (10 subjects and 100 sessions in total) and simulation data; then, seven types of entropy and multiscale entropy expanded based on seven kinds of entropy are used to explore the differences and brain regions of ADHD, BP, and SCHZ relative to healthy subjects; and finally, by verifying the classification performance of the seven information entropies on ADHD, BP, and SCHZ, the effectiveness of the seven entropy methods is evaluated through these three methods. The core brain regions that affect the classification are given, and DiffEn performed best on ADHD, SaEn for BP, and RaEn for SCHZ. [ABSTRACT FROM AUTHOR]

In this paper, we propose nonparametric kernel estimators for Shannon differential entropy function under length-biased sampling. Asymptotic properties of the estimators are established under suitable regularity conditions. A simulation study is accomplished to compare the performance of proposed estimators. The usefulness of the estimators are also examined using a real data. [ABSTRACT FROM AUTHOR]

The entropy power inequality (EPI) and the Brascamp-Lieb inequality (BLI) are fundamental inequalities concerning the differential entropies of linear transformations of random vectors. The EPI provides lower bounds for the differential entropy of linear transformations of random vectors with independent components. The BLI, on the other hand, provides upper bounds on the differential entropy of a random vector in terms of the differential entropies of some of its linear transformations. In this paper, we define a family of entropy functionals, which we show are subadditive. We then establish that Gaussians are extremal for these functionals by adapting a proof technique from Geng and Nair (2014). As a consequence, we obtain a new entropy inequality that generalizes both the BLI and EPI. By considering a variety of independence relations among the components of the random vectors appearing in these functionals, we also obtain families of inequalities that lie between the EPI and the BLI. [ABSTRACT FROM AUTHOR]

The flow of a fluid across a revolving disc has several technical and industrial uses. Examples of rotating disc flows include centrifugal pumps, viscometers, rotors, fans, turbines, and spinning discs. An important technology with implications for numerous treatments utilized in numerous sectors is the use of hybrid nanofluids (HNFs) to accelerate current advancements. Through investigation of ternary nanoparticle impacts on heat transfer (HT) and liquid movement, the thermal properties of tri-HNFs were to be ascertained in this study. Hall current, thermal radiation, and heat dissipation have all been studied in relation to the use of flow-describing equations. The ternary HNFs under research are composed of the nanomolecules aluminum oxide (Al2O3), copper oxide (CuO), silver (Ag), and water (H2O). For a number of significant physical characteristics, the physical situation is represented utilizing the boundary layer investigation, which produces partial differential equations (PDEs). The rheology of the movement is extended and computed in a revolving setting under the assumption that the movement is caused by a rotatingfloppy. Before the solution was found using the finite difference method, complicated generated PDEs were transformed into corresponding ODEs (Keller Box method). A rise in the implicated influencing factors has numerous notable physical impacts that have been seen and recorded. The Keller Box method (KBM) approach is also delivered for simulating the determination of nonlinear system problems faced in developing liquid and supplementary algebraic dynamics domains. The rate of entropy formation rises as the magnetic field parameter and radiation parameter increase. Entropy production rate decreases as the Brinkman number and Hall current parameter become more enriched. The thermal efficiency of ternary HNFs compared to conventional HNFs losses to a low of 4.8% and peaks to 5.2%. [ABSTRACT FROM AUTHOR]

Entropy is re-examined as a quantification of ignorance in the predictability of a one dimensional continuous phenomenon. Although traditional estimators for entropy have been widely utilized in this context, we show that both the thermodynamic and Shannon's theory of entropy are fundamentally discrete, and that the limiting process used to define differential entropy suffers from similar problems to those encountered in thermodynamics. In contrast, we consider a sampled data set to be observations of microstates (unmeasurable in thermodynamics and nonexistent in Shannon's discrete theory), meaning, in this context, it is the macrostates of the underlying phenomenon that are unknown. To obtain a particular coarse-grained model we define macrostates using quantiles of the sample and define an ignorance density distribution based on the distances between quantiles. The geometric partition entropy is then just the Shannon entropy of this finite distribution. Our measure is more consistent and informative than histogram-binning, especially when applied to complex distributions and those with extreme outliers or under limited sampling. Its computational efficiency and avoidance of negative values can also make it preferable to geometric estimators such as k-nearest neighbors. We suggest applications that are unique to this estimator and illustrate its general utility through an application to time series in the approximation of an ergodic symbolic dynamics from limited observations. [ABSTRACT FROM AUTHOR]

Considering both biological and non-biological polygonal shape organizations, in this paper we introduce a quantitative method which is able to determine informational entropy as spatial differences between heterogeneity of internal areas from simulation and experimental samples. According to these data (i.e., heterogeneity), we are able to establish levels of informational entropy using statistical insights of spatial orders using discrete and continuous values. Given a particular state of entropy, we establish levels of information as a novel approach which can unveil general principles of biological organization. Thirty-five geometric aggregates are tested (biological, non-biological, and polygonal simulations) in order to obtain the theoretical and experimental results of their spatial heterogeneity. Geometrical aggregates (meshes) include a spectrum of organizations ranging from cell meshes to ecological patterns. Experimental results for discrete entropy using a bin width of 0.5 show that a particular range of informational entropy (0.08 to 0.27 bits) is intrinsically associated with low rates of heterogeneity, which indicates a high degree of uncertainty in finding non-homogeneous configurations. In contrast, differential entropy (continuous) results reflect negative entropy within a particular range (−0.4 to −0.9) for all bin widths. We conclude that the differential entropy of geometrical organizations is an important source of neglected information in biological systems. [ABSTRACT FROM AUTHOR]

Currently, entropy is quite often used to describe complex systems in various fields. The questions of using differential entropy for network structures presented in the form of connected graphs with correlations are considered. It is well known that the entropy of a continuous random vector can be split into two components, the entropy of randomness and the entropy of self-organization. Along with the assessment of the entropy itself, other useful characteristics have been proposed for network structures, the entropy measure of the relationship between several subsystems and the entropy of the system at a separate vertex; these expand the possibilities of entropy modeling for the study of network structures and allow estimating the interconnection of different sections with each other and determining how entropy within such systems changes. Examples on model data are considered. [ABSTRACT FROM AUTHOR]

Emotion recognition based on electroencephalography (EEG) has attracted high interest in fields such as health care, user experience evaluation, and human–computer interaction (HCI), as it plays an important role in human daily life. Although various approaches have been proposed to detect emotion states in previous studies, there is still a need to further study the dynamic changes of EEG in different emotions to detect emotion states accurately. Entropy-based features have been proved to be effective in mining the complexity information in EEG in many areas. However, different entropy features vary in revealing the implicit information of EEG. To improve system reliability, in this paper, we propose a framework for EEG-based cross-subject emotion recognition using fused entropy features and a Bidirectional Long Short-term Memory (BiLSTM) network. Features including approximate entropy (AE), fuzzy entropy (FE), Rényi entropy (RE), differential entropy (DE), and multi-scale entropy (MSE) are first calculated to study dynamic emotional information. Then, we train a BiLSTM classifier with the inputs of entropy features to identify different emotions. Our results show that MSE of EEG is more efficient than other single-entropy features in recognizing emotions. The performance of BiLSTM is further improved with an accuracy of 70.05% using fused entropy features compared with that of single-type feature. [ABSTRACT FROM AUTHOR]

The notion of compressibility of a random measure is a rather general concept which find applications in many contexts from data compression, to signal quantization, and parameter estimation. While compressibility for discrete and continuous measures is generally well understood, the case of discrete-continuous measures is quite subtle. In this paper, we focus on a class of multi-dimensional random measures that have singularities on affine lower-dimensional subsets. We refer to this class of random variables as affinely singular. Affinely singular random vectors naturally arises when considering linear transformation of component-wise independent discrete-continuous random variables. To measure the compressibility of such distributions, we introduce the new notion of dimensional-rate bias (DRB) which is closely related to the entropy and differential entropy in discrete and continuous cases, respectively. Similar to entropy and differential entropy, DRB is useful in evaluating the mutual information between distributions of the aforementioned type. Besides the DRB, we also evaluate the the RID of these distributions. We further provide an upper-bound for the RID of multi-dimensional random measures that are obtained by Lipschitz functions of component-wise independent discrete-continuous random variables (X). The upper-bound is shown to be achievable when the Lipschitz function is $A \mathrm {X}$ , where $A$ satisfies ${\mathrm{ SPARK}}({A_{m\times n}}) = m+1$ (e.g., Vandermonde matrices). When considering discrete-domain moving-average processes with non-Gaussian excitation noise, the above results allow us to evaluate the block-average RID and DRB, as well as to determine a relationship between these parameters and other existing compressibility measures. [ABSTRACT FROM AUTHOR]

In this paper, the fractional cumulative entropy is considered to get its further properties and also its developments to dynamic cases. The measure is used to characterize a family of symmetric distributions and also another location family of distributions. The links between the fractional cumulative entropy and the classical differential entropy and some reliability quantities are also unveiled. In addition, the connection the measure has with the standard deviation is also found. We provide some examples to establish the variability property of this measure. [ABSTRACT FROM AUTHOR]

In contrast to their seemingly simple and shared structure of independence and stationarity, Lévy processes exhibit a wide variety of behaviors, from the self-similar Wiener process to piecewise-constant compound Poisson processes. Inspired by the recent paper of Ghourchian, Amini, and Gohari, we characterize their compressibility by studying the entropy of their double discretization (both in time and amplitude) in the regime of vanishing discretization steps. For a Lévy process with absolutely continuous marginals, this reduces to understanding the asymptotics of the differential entropy of its marginals at small times, for which we obtain a new local central limit theorem. We generalize known results for stable processes to the non-stable case, with a special focus on Lévy processes that are locally self-similar, and conceptualize a new compressibility hierarchy of Lévy processes, captured by their Blumenthal–Getoor index. [ABSTRACT FROM AUTHOR]

The thick brane scenario built on the f (T , B) teleparallel gravity theory was considered for the study of phase transitions, internal structures and new classes of solutions in a model. In this theory, T denotes the torsion scalar and B is a boundary term. An interesting result was observed when brane splitting occurs, i.e. internal structures in the model arise as a consequence of the appearance of new domain walls in the theory. In fact, this preliminary result influences the profile of the matter field (from kink to multikink) so that for appropriate values of the parameters k 1 , 2 , multiple phase transitions are identified. To perform this analysis, the Differential Configurational Entropy (DCE) that has the ability to predict the existence of phase transitions through critical points was used. Furthermore, the DCE is able to select the most stable solutions since it gives us details about the informational content to the field settings. [ABSTRACT FROM AUTHOR]

The entropy power inequality (EPI) for convolution of two independent random variables was first proposed by Shannon, C. E., 1948. However, in practice, there are many situations in which the involved random variables are not independent. In this article, considering additive noise channels, it is shown that, under some conditions, EPI holds for the case when the involved random variables are dependent. In order to achieve our main result, meanwhile a lower bound for the Fisher information of the output signal is obtained which is useful on its own. An example is also provided to illustrate our result. [ABSTRACT FROM AUTHOR]

Both variance and entropy are commonly used measures for uncertainty. There exists many cases where variance is infinite and entropy is finite. In this note, we derive an upper bound illustrating the relationship between variance and entropy of random variables having a special class of distributions. We also derive an upper bound for kth absolute central moment proportional to entropy power for a special class of distributions. [ABSTRACT FROM AUTHOR]

In 1994, Jim Massey proposed the guessing entropy as a measure of the difficulty that an attacker has to guess a secret used in a cryptographic system, and established a well-known inequality between entropy and guessing entropy. Over 15 years before, in an unpublished work, he also established a well-known inequality for the entropy of an integer-valued random variable of given variance. In this paper, we establish a link between the two works by Massey in the more general framework of the relationship between discrete (absolute) entropy and continuous (differential) entropy. Two approaches are given in which the discrete entropy (or Rényi entropy) of an integer-valued variable can be upper bounded using the differential (Rényi) entropy of some suitably chosen continuous random variable. As an application, lower bounds on guessing entropy and guessing moments are derived in terms of entropy or Rényi entropy (without side information) and conditional entropy or Arimoto conditional entropy (when side information is available). [ABSTRACT FROM AUTHOR]

We revisit the problem of the estimation of the differential entropy $H(f)$ of a random vector $X$ in $R^{d}$ with density $f$ , assuming that $H(f)$ exists and is finite. In this note, we study the consistency of the popular nearest neighbor estimate $H_{n}$ of Kozachenko and Leonenko. Without any smoothness condition we show that the estimate is consistent ($E\{|H_{n} - H(f)|\} \to 0$ as $n \to \infty $) if and only if ${\mathbb E}\{ \log (\| X \| + 1)\} < \infty $. Furthermore, if $X$ has compact support, then $H_{n} \to H(f)$ almost surely. [ABSTRACT FROM AUTHOR]

The aim of several combustion experiments is the determination of the rate coefficients of important elementary reactions. The experimental conditions are usually selected on the basis of local sensitivity analysis. Shock tube and tubular flow reactor experiments are often designed in such a way that only one reaction step is important at the investigated conditions. Sheen and Manion (J. Phys. Chem. A, 118 (2014) 4929–4941) suggested a method for the design of shock tube experiments based on differential entropy. Their method was modified and extended in this work. In the extended method, both the experimental and residual errors of the measurements are considered at the calculation of the posterior uncertainty of the determined rate parameters, the differential entropy matrix is calculated in an analytical way and the net information flux value is calculated for each suggested experimental point. In an iterative procedure, all investigated experimental points with negative net information flux values are discarded and the remaining experimental conditions are recommended for the measurements. The most valuable candidate experimental points can be determined based on the net information flux values. The method was used for the selection of experimental conditions for the determination of the rate coefficient of reaction NO2 + H = NO + OH at conditions similar to the tubular flow reactor experiments of Alzueta et al. (Energy Fuels, 15 (2001) 724–729). In these experiments, the oxidation of methanol was investigated with and without NO addition. Our method suggested a range of temperature, equivalence ratio and initial NO concentration, where the experimental data carry the most information on the rate coefficient of this elementary reaction. [ABSTRACT FROM AUTHOR]

Research has shown multiscale entropy, brain signal behavior across time scales, to reliably increase at lower time scales with time-on-task fatigue. However, multiscale entropy has not been examined in short vigilance tasks (i.e., ≤ 10 min). Addressing this gap, we examine multiscale entropy during a 10-minute Psychomotor Vigilance Test (PVT). Thirty-four participants provided neural data while completing the PVT. We compared the first 2 min of the task to the 7th and 8th minutes to avoid end-spurt effects. Results suggested increased multiscale entropy at lower time scales later compared to earlier in the task, suggesting multiscale entropy is a strong marker of time-on-task fatigue onset during short vigils. Separate analyses for Fast and Slow performers reveal differential entropy patterns, particularly over visual cortices. Here, observed brain-behavior linkage between entropy and reaction time for slow performers suggests that entropy assays over sensory cortices might have predictive value for fatigue onset or shifts from on- to off-task states. • We present novel findings describing differential multiscale entropy changes across time in a brief psychomotor vigilance test. • One brain-behavior linkage, with slow (poorly motivated) performers exhibiting strong positive relationships between reaction times and entropy, together with different entropy patterns than fast (motivated) performers at occipital cortices, suggests entropy assays at this location may be predictive of fatigue onset and/or a shift from on- to off-task states in future work and in computational modeling of fatigue onset. [ABSTRACT FROM AUTHOR]

In this article we study weighted sums of $n$ i.i.d. Gamma($\alpha $) random variables with nonnegative weights. We show that for $n \geq 1/\alpha $ the sum with equal coefficients maximizes differential entropy when variance is fixed. As a consequence, we prove that among nonnegative quadratic forms in $n$ independent standard Gaussian random variables, a diagonal form with equal coefficients maximizes differential entropy, under a fixed variance. This provides a sharp lower bound for the relative entropy between a nonnegative quadratic form and a Gaussian random variable. Bounds on capacities of transmission channels subjects to $n$ independent additive gamma noises are also derived. [ABSTRACT FROM AUTHOR]

In this article, we propose a method for emotion classification based on multilayer convolutional neural network (MCNN) and combining differential entropy (DE) and brain network. First, we use continuous wavelet transform (CWT) to get the time–frequency representation (TFR) of electroencephalogram (EEG) signals on each channel and extract rich information from different frequency bands for subsequent analysis. Brain networks are then constructed in multiple bands to characterize the spatial connections hidden in the multichannel EEG signals. Based on brain networks, we further develop core brain networks through a set of key nodes determined by DE. These core brain networks are associated with brain activities and differ markedly between different emotional states. The final designed MCNN model takes DE features and core brain networks as inputs for emotion recognition. We evaluate our method on the SJTU emotion EEG dataset, and the average accuracy of 15 subjects achieves 91.45%. Utilizing the complementary features of DE and brain network, the proposed method provides an efficient framework for accurate emotion recognition from EEG signals. [ABSTRACT FROM AUTHOR]

To better understand the effects of strong coupling and QCD at high temperature in QGP, by using holographic model, we investigate the dissociation effect of bottomonium under the higher-order curvature corrections to the supergravity action corresponding to the corrections of large N expansion of boundary CFT in the side of field theory. The results show that effective potential is not a good physical quantity to estimate the dissociation strength of bottomonium in the case of finite wave number and considering the higher-order curvature corrections. Therefore, we calculate the quasinormal spectra (QNMs) and the differential configuration entropy (DCE). It is found that the dissociation effect is stronger for the stronger coupling. [ABSTRACT FROM AUTHOR]

• Default noninformative priors for the dirential entropy of the Weibull model are developed. • The developed noninformative priors are shown to be dir from the noninformative priors of the Weibull parameters. • The Bayes estimates are computed using random-walk Metropolis-Hastings algorithms. • Bayes estimates are illustrated by simulation experiments. • An application to the failure times from heat exchanger data. This article addresses the estimation of differential entropy of the Weibull distribution based on different non-informative prior distributions. We first derive non-informative priors using formal rules, such as Jeffreys prior and maximal data information prior based on Fisher information and entropy, respectively. We also develop reference prior and probability matching prior for the differential entropy. Next, we investigate the effects of these priors in the Bayes estimates of the differential entropy of the Weibull distribution based on complete sample. The Bayes estimates are computed using random-walk Metropolis-Hastings algorithms. We use Monte Carlo simulation to evaluate the performance of the Bayes estimates of differential entropy in terms of bias and root mean-squared error, while highest posterior density credible intervals are assessed in terms of average width and coverage probabilities. Finally, a real data set has been analyzed for illustrative purposes. [ABSTRACT FROM AUTHOR]

This paper studies convergence of empirical measures smoothed by a Gaussian kernel. Specifically, consider approximating $P\ast \mathcal {N}_\sigma $ , for $\mathcal {N}_\sigma \triangleq \mathcal {N}(0,\sigma ^{2} \mathrm {I}_{d})$ , by $\hat {P}_{n}\ast \mathcal {N}_\sigma $ under different statistical distances, where $\hat {P}_{n}$ is the empirical measure. We examine the convergence in terms of the Wasserstein distance, total variation (TV), Kullback-Leibler (KL) divergence, and $\chi ^{2}$ -divergence. We show that the approximation error under the TV distance and 1-Wasserstein distance ($\mathsf {W}_{1}$) converges at the rate $e^{O(d)} n^{-1/2}$ in remarkable contrast to a (typical) $n^{-\frac {1}{d}}$ rate for unsmoothed $\mathsf {W}_{1}$ (and $d\ge 3$). Similarly, for the KL divergence, squared 2-Wasserstein distance ($\mathsf {W}_{2}^{2}$), and $\chi ^{2}$ -divergence, the convergence rate is $e^{O(d)} n^{-1}$ , but only if $P$ achieves finite input-output $\chi ^{2}$ mutual information across the additive white Gaussian noise (AWGN) channel. If the latter condition is not met, the rate changes to $\omega \left ({n^{-1}}\right)$ for the KL divergence and $\mathsf {W}_{2}^{2}$ , while the $\chi ^{2}$ -divergence becomes infinite – a curious dichotomy. As an application we consider estimating the differential entropy $h(S+Z)$ , where $S\sim P$ and $Z\sim \mathcal {N}_\sigma $ are independent $d$ -dimensional random variables. The distribution $P$ is unknown and belongs to some nonparametric class, but n independently and identically distributed (i.i.d) samples from it are available. Despite the regularizing effect of noise, we first show that any good estimator (within an additive gap) for this problem must have a sample complexity that is exponential in d. We then leverage the above empirical approximation results to show that the absolute-error risk of the plug-in estimator converges as $e^{O(d)} n^{-1/2}$ , thus attaining the parametric rate in n. This establishes the plug-in estimator as minimax rate-optimal for the considered problem, with sharp dependence of the convergence rate both in n and d. We provide numerical results comparing the performance of the plug-in estimator to that of general-purpose (unstructured) differential entropy estimators (based on kernel density estimation (KDE) or k nearest neighbors (kNN) techniques) applied to samples of $S+Z$. These results reveal a significant empirical superiority of the plug-in to state-of-the-art KDE and kNN methods. As a motivating utilization of the plug-in approach, we estimate information flows in deep neural networks and discuss Tishby’s Information Bottleneck and the compression conjecture, among others. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the performance of energy detection-based spectrum sensing over $\mathcal {F}$ composite fading channels. To this end, an analytical expression for the average detection probability is first derived. This expression is then extended to account for collaborative spectrum sensing, square-law selection diversity reception, and noise power uncertainty. The corresponding receiver operating characteristics (ROC) are analyzed for different conditions of the average signal-to-noise ratio (SNR), noise power uncertainty, time-bandwidth product, multipath fading, shadowing, number of diversity branches, and number of collaborating users. It is shown that the energy detection performance is sensitive to the severity of the multipath fading and the amount of shadowing, whereby even small variations in either of these physical phenomena can significantly impact the detection probability. As a figure of merit to evaluate the detection performance, the area under the ROC curve (AUC) is derived and evaluated for different multipath fading and shadowing conditions. Closed-form expressions for the differential entropy and cross entropy are also formulated and assessed for different average SNR, multipath fading, and shadowing conditions. Then, the relationship between the differential entropy of $\mathcal {F}$ composite fading channels and the corresponding ROC/AUC is examined where it is shown that the average number of bits required for encoding a signal becomes small (i.e., low differential entropy) when the detection probability is high or when the AUC is large. The difference between composite fading and traditional small-scale fading is emphasized by comparing the cross entropy for Rayleigh and Nakagami- $m$ fading. A validation of the analytical results is provided through a careful comparison with the results of some simulations. [ABSTRACT FROM AUTHOR]

The distributed remote source coding (the so-called CEO) problem is studied in the case where the underlying source, not necessarily Gaussian, has finite differential entropy and the observation noise is Gaussian. The main result is a new lower bound for the sum-rate-distortion function under arbitrary distortion measures. When specialized to the case of mean-squared error, it is shown that the bound exactly mirrors a corresponding upper bound, except that the upper bound has the source power (variance), whereas the lower bound has the source entropy power. Bounds exhibiting this pleasing duality of power and entropy power have been well known for direct and centralized source coding since Shannon’s work. While the bounds hold generally, their value is most pronounced when interpreted as a function of the number of agents in the CEO problem. [ABSTRACT FROM AUTHOR]

We propose an alternative physical mechanism to explain the observed accelerated expansion of the Universe based on the configuration entropy of the cosmic web and its evolution. We show that the sheets, filaments, and clusters in the cosmic web act as sinks whereas the voids act as the sources of information. The differential entropy of the cosmic velocity field increases with time and also acts as a source of entropy. The growth of non-linear structures and the emergence of the cosmic web may lead to a situation where the overall dissipation rate of information at the sinks is about to dominate the generation rate of information from the sources. Consequently, the Universe either requires a dispersal of the overdense non-linear structures or an accelerated expansion of the underdense voids to prevent a violation of the second law of thermodynamics. The dispersal of the sheets, filaments, and clusters is not a viable option due to the attractive nature of gravity but the repulsive and outward peculiar gravitational acceleration at the voids makes it easier to stretch them at an accelerated rate. We argue that this accelerated expansion of the voids inside the cosmic web may mimic the behaviour of dark energy. [ABSTRACT FROM AUTHOR]

Abstract This study considers adjoint based a posteriori estimation of the error in a quantity of interest computed from numerical solutions based on multistage implicit–explicit (IMEX) time integration schemes and an entropy-viscosity formulation for damped hyperbolic partial differential equations (PDEs). Hyperbolic systems are challenging to solve numerically due to the need to stabilize systems with discontinuous or nearly discontinuous solutions in an attempt to limit non-physical oscillations and provide accurate solutions. The goal of this effort is to provide error estimates based on adjoint operators, variational analysis and computable residuals. The error estimates quantify the total error as well as different contributions to the error arising from the time integration schemes and choices of numerical parameters in the numerical method. [ABSTRACT FROM AUTHOR]

Abstract In this work, I propose a new method for the computation of informational entropy from Recurrence Plots when the analyzed time series are categorical in nature. In such cases, there is typically a simplification in choosing the parameters of the analysis, in the sense that no embedding in multidimensional space is usually assumed and that recurrence is restricted to exact matching (equivalence) of the numerically coded categories. However, such a simplified parameterization brings about some notable changes in the appearance of the obtained Recurrence Plots, which has consequences for the extraction of the standard dynamical measures. Specifically, a categorical Recurrence Plot is often composed of rectangular structures rather than line structures (diagonal and horizontal/vertical), over which the recurrence quantification measures were originally proposed. Starting from this observation, I consider alternative computational procedures to extract a non-biased measure of entropy for the categorical case, showing the viability of such a choice with simulated data Highlights • In categorical recurrence plots rectangular blocks are the prominent structures. • The extraction of several recurrence measures could be biased by this fact. • We propose to change the method of computation of informational entropy. • Our proposed method seems to be an unbiased estimate of entropy in this case. [ABSTRACT FROM AUTHOR]

We deal with the regularizing effect that, in scalar conservation laws in one space dimension, the nonlinearity of the flux function f has on the entropy solution. More precisely, if the set { w : f ″ (w) ≠ 0 } is dense, the regularity of the solution can be expressed in terms of BV Φ spaces, where Φ depends on the nonlinearity of f. If moreover the set { w : f ″ (w) = 0 } is finite, under the additional polynomial degeneracy condition at the inflection points, we prove that f ′ ∘ u (t) ∈ BV loc (ℝ) for every t > 0 and that this can be improved to SBV loc (ℝ) regularity except an at most countable set of singular times. Finally, we present some examples that show the sharpness of these results and counterexamples to related questions, namely regularity in the kinetic formulation and a property of the fractional BV spaces. [ABSTRACT FROM AUTHOR]

The present work investigates entropy production due to coupled natural convection/radiation heat transfer phenomenon in an inclined rectangular enclosure, isothermally heated from the bottom side and isothermally cooled from the other sides. The discrete-ordinate method is used in modeling the radiative transport equation while the statistical narrow band correlated-k model is adopted to deduce the radiative properties of the medium. The influence of pertinent parameters such as aspect ratio, inclination angle and walls emissivities on entropy generation is studied. It is found that the volumetric entropy generation is reduced when increasing the inclination angle of the enclosure. Moreover, it is shown that the minimum entropy production due to radiation heat transfer in participating media occurs at aspect ratio equal to unity. [ABSTRACT FROM AUTHOR]

Electromagnetic transitions involving nucleons and their resonances are here presented in the fermionic sector of the AdS/QCD soft-wall model, using the differential configurational entropy (DCE) as a measure of information entropy. The DCE is employed to derive adjustable parameters that define the minimal and nonminimal couplings in the nuclear interaction with a gauge vector field, in the nucleonic γ N → N ⁎ (1535) transition. The values obtained comply with phenomenological data with good accuracy. [ABSTRACT FROM AUTHOR]

This paper addresses the correspondence between linear inequalities for Shannon entropy and differential entropy for sums of independent group-valued random variables. We show that any balanced (with the sum of coefficients being zero) linear inequality for Shannon entropy holds if and only if its differential entropy counterpart also holds; moreover, any linear inequality for differential entropy must be balanced. In particular, our result shows that recently proved differential entropy inequalities by Kontoyiannis and Madiman can be deduced from their discrete counterparts due to Tao in a unified manner. Generalizations to certain abelian groups are also obtained. Our proof of extending inequalities for Shannon entropy to differential entropy relies on a result of Rényi which relates the Shannon entropy of a finely discretized random variable to its differential entropy and also helps in establishing that the entropy of the sum of quantized random variables is asymptotically equal to that of the quantized sum; the converse uses the asymptotics of the differential entropy of convolutions with weak additive noise. [ABSTRACT FROM PUBLISHER]

In a previous paper, we characterized the Devaney chaos, frequent hypercyclicity and dense distributional chaos of composition operators induced by continuous self-maps on the real line. The present paper further investigates the disjoint hypercyclicity and topological entropy of these operators. It is shown that the composition operator is -transitive if and only if it is Cesàro-hypercyclic, if and only if it is supercyclic, if and only if it has the specification property on the whole space. Furthermore, sufficient and necessary conditions for a pair of composition operators to be disjoint hypercyclic (disjoint mixing, respectively) are obtained. Finally, sufficient conditions for the composition operator to admit infinite topological entropy are provided. [ABSTRACT FROM AUTHOR]

Recent studies found that many channels are affected by additive noise that is impulsive in nature and is best explained by heavy-tailed symmetric alpha-stable distributions. Dealing with impulsive noise environments comes with an added complexity with respect to the standard Gaussian environment: the alpha-stable probability density functions do not possess closed-form expressions except in few special cases. Furthermore, they have an infinite second moment and the “nice” Hilbert space structure of the space of random variables having a finite second moment is lost along with its tools and methodologies. This is indeed the case in estimation theory, where classical tools to quantify the performance of an estimator are tightly related to the assumption of having finite variance variables. In alpha-stable environments, expressions, such as the mean square error and the Cramer–Rao bound, are hence problematic. In this paper, we tackle the parameter-estimation problem in the impulsive noise environments and develop novel tools that are tailored to the alpha-stable and heavy-tailed noise environments, tools that coincide with the standard ones adopted in the Gaussian setup, namely, a generalized “power” measure and a generalized Fisher information. We generalize known information inequalities commonly used in the Gaussian context: the de Bruijn identity, the Fisher information inequality, the isoperimetric inequality for entropies and the Cramer–Rao bound. Additionally, we derive upper bounds on the differential entropy of independent sums having a stable component. Intermediately, the new power measure is used to shed some light on the additive alpha-stable noise channel capacity in a setup that generalizes the linear average power constrained additive white Gaussian noise channel. Our theoretical findings are paralleled with numerical evaluations of various quantities and bounds using developed MATLAB packages. [ABSTRACT FROM PUBLISHER]

We compare two versions of Stampacchia Lemma and we give an application to the regularity of entropy solutions to some anisotropic boundary value problems. [ABSTRACT FROM AUTHOR]

The entropy of a random variable is well-known to equal the exponential growth rate of the volumes of its typical sets. In this paper, we show that for any log-concave random variable X , the sequence of the \lfloor n\theta \rfloor ^{\text {th}} intrinsic volumes of the typical sets of X in dimensions n \geq 1 grows exponentially with a well-defined rate. We denote this rate by h_{X}(\theta ) , and call it the \theta ^{\text {th}} intrinsic entropy of X . We show that h_{X}(\theta ) is a continuous function of \theta $ over the range [0, 1], thereby providing a smooth interpolation between the values 0 and $h(X)$ at the endpoints 0 and 1, respectively. [ABSTRACT FROM PUBLISHER]

We show a direct relationship between the variance and the differential entropy for subclasses of symmetric and asymmetric unimodal distributions by providing an upper bound on variance in terms of entropy power. Combining this bound with the well-known entropy power lower bound on variance, we prove that the variance of the appropriate subclasses of unimodal distributions can be bounded below and above by the scaled entropy power. As the differential entropy decreases, the variance is sandwiched between two exponentially decreasing functions in the differential entropy. This establishes that for the subclasses of unimodal distributions, the differential entropy can be used as a surrogate for concentration of the distribution. [ABSTRACT FROM PUBLISHER]

We provide a new derivation of the log likelihood spectral distance measure for signal processing using the logarithm of the ratio of entropy rate powers. Using this interpretation, we show that the log likelihood ratio is equivalent to the difference of two differential entropies, and further that it can be written as the difference of two mutual informations. These latter two expressions allow the analysis of signals via the log likelihood ratio to be extended beyond spectral matching to the study of their statistical quantities of differential entropy and mutual information. Examples from speech coding are presented to illustrate the utility of these new results. These new expressions allow the log likelihood ratio to be of interest in applications beyond those of just spectral matching for speech. [ABSTRACT FROM AUTHOR]