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BACKGROUND AND OBJECTIVE: Ultrasound information entropy imaging is an emerging quantitative ultrasound technique for characterizing local tissue scatterer concentrations and arrangements. However, the commonly used ultrasound Shannon entropy imaging based on histogram-derived discrete probability estimation suffers from the drawbacks of histogram settings dependence and unknown estimator performance. In this paper, we introduced the information-theoretic cumulative residual entropy (CRE) defined in a continuous distribution of cumulative distribution functions as a new entropy measure of ultrasound backscatter envelope uncertainty or complexity, and proposed ultrasound CRE imaging for tissue characterization. METHODS: We theoretically analyzed the CRE for Rayleigh and Nakagami distributions and proposed a normalized CRE for characterizing scatterer distribution patterns. We proposed a method based on an empirical cumulative distribution function estimator and a trapezoidal numerical integration for estimating the normalized CRE from ultrasound backscatter envelope signals. We presented an ultrasound normalized CRE imaging scheme based on the normalized CRE estimator and the parallel computation technique. We also conducted theoretical analysis of the differential entropy which is an extension of the Shannon entropy to a continuous distribution, and introduced a method for ultrasound differential entropy estimation and imaging. Monte-Carlo simulation experiments were performed to evaluate the estimation accuracy of the normalized CRE and differential entropy estimators. Phantom simulation and clinical experiments were conducted to evaluate the performance of the proposed normalized CRE imaging in characterizing scatterer concentrations and hepatic steatosis (n = 204), respectively. RESULTS: The theoretical normalized CRE for the Rayleigh distribution was π/4, corresponding to the case where there were ≥10 randomly distributed scatterers within the resolution cell of an ultrasound transducer. The theoretical normalized CRE for the Nakagami distribution decreased as the Nakagami parameter m increased, corresponding to that the ultrasound backscattered statistics varied from pre-Rayleigh to Rayleigh and to post-Rayleigh distributions. Monte-Carlo simulation experiments showed that the proposed normalized CRE and differential entropy estimators can produce a satisfying estimation accuracy even when the size of the test samples is small. Phantom simulation experiments showed that the proposed normalized CRE and differential entropy imaging can characterize scatterer concentrations. Clinical experiments showed that the proposed ultrasound normalized CRE imaging is capable to quantitatively characterize hepatic steatosis, outperforming ultrasound differential entropy imaging and being comparable to ultrasound Shannon entropy and Nakagami imaging. CONCLUSION: This study sheds light on the theory and methodology of ultrasound normalized CRE. The proposed ultrasound normalized CRE can serve as a new, flexible quantitative ultrasound envelope statistics parameter. The proposed ultrasound normalized CRE imaging may find applications in quantified characterization of biological tissues. Our code will be made available publicly at https://github.com/zhouzhuhuang.

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We calculate the average differential entropy of a q-component Gaussian mixture in Rn. For simplicity, all components have covariance matrix σ21, while the means {Wi}i=1q are i.i.d. Gaussian vectors with zero mean and covariance s21. We obtain a series expansion in µ=s2/σ2 for the average differential entropy up to order O(µ2), and we provide a recipe to calculate higher-order terms. Our result provides an analytic approximation with a quantifiable order of magnitude for the error, which is not achieved in previous literature.

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Objective.Electroencephalogram (EEG) signals offer invaluable insights into the complexities of emotion generation within the brain. Yet, the variability in EEG signals across individuals presents a formidable obstacle for empirical implementations. Our research addresses these challenges innovatively, focusing on the commonalities within distinct subjects' EEG data.Approach.We introduce a novel approach named Contrastive Learning Graph Convolutional Network (CLGCN). This method captures the distinctive features and crucial channel nodes related to individuals' emotional states. Specifically, CLGCN merges the dual benefits of CL's synchronous multisubject data learning and the GCN's proficiency in deciphering brain connectivity matrices. Understanding multifaceted brain functions and their information interchange processes is realized as CLGCN generates a standardized brain network learning matrix during a dataset's learning process.Main results.Our model underwent rigorous testing on the Database for Emotion Analysis using Physiological Signals (DEAP) and SEED datasets. In the five-fold cross-validation used for dependent subject experimental setting, it achieved an accuracy of 97.13% on the DEAP dataset and surpassed 99% on the SEED and SEED_IV datasets. In the incremental learning experiments with the SEED dataset, merely 5% of the data was sufficient to fine-tune the model, resulting in an accuracy of 92.8% for the new subject. These findings validate the model's efficacy.Significance.This work combines CL with GCN, improving the accuracy of decoding emotional states from EEG signals and offering valuable insights into uncovering the underlying mechanisms of emotional processes in the brain.

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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.

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Complex biological systems often undergo sudden qualitative changes during their dynamic evolution. These critical transitions are typically characterized by a catastrophic progression of the system. Identifying the critical point is critical to uncovering the underlying mechanisms of complex biological systems. However, the system may exhibit minimal changes in its state until the critical point is reached, and in the face of high throughput and strong noise data, traditional biomarkers may not be effective in distinguishing the critical state. In this study, we propose a novel approach, mutual information weighted entropy (MIWE), which uses mutual information between genes to build networks and identifies critical states by quantifying molecular dynamic differences at each stage through weighted differential entropy. The method is applied to one numerical simulation dataset and four real datasets, including bulk and single-cell expression datasets. The critical states of the system can be recognized and the robustness of MIWE method is verified by numerical simulation under the influence of different noises. Moreover, we identify two key transcription factors (TFs), CREB1 and CREB3, that regulate downstream signaling genes to coordinate cell fate commitment. The dark genes in the single-cell expression datasets are mined to reveal the potential pathway regulation mechanism.

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To address the problem of low accuracy in multi-classification recognition of motor imagery electroencephalogram(EEG)signals,a recognition method is proposed based on differential entropy and convolutional neural network for 4-class classification of motor imagery.EEG signals are extracted into 4 frequency bands(Alpha,Beta,Theta,and Gamma)through the filter,followed by the computation of differential entropy for each frequency band.According to the spatial characteristics of brain electrodes,the data structure is reconstructed into three-dimensional EEG signal feature cube which is input into convolutional neural network for 4-class classification.The method achieves an accuracy of 95.88%on the BCI Competition IV-2a public dataset.Additionally,a 4-class classification motor imagery dataset is established in the laboratory for the same processing,and an accuracy of 94.50%is obtained.The test results demonstrate that the proposed method exhibits superior recognition performance.

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In this paper, we present the derivation of Jeffreys divergence, generalized Fisher divergence, and the corresponding De Bruijn identities for space-time random field. First, we establish the connection between Jeffreys divergence and generalized Fisher information of a single space-time random field with respect to time and space variables. Furthermore, we obtain the Jeffreys divergence between two space-time random fields obtained by different parameters under the same Fokker-Planck equations. Then, the identities between the partial derivatives of the Jeffreys divergence with respect to space-time variables and the generalized Fisher divergence are found, also known as the De Bruijn identities. Later, at the end of the paper, we present three examples of the Fokker-Planck equations on space-time random fields, identify their density functions, and derive the Jeffreys divergence, generalized Fisher information, generalized Fisher divergence, and their corresponding De Bruijn identities.

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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.

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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.

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Driving fatigue detection based on EEG signals is a research hotspot in applying brain-computer interfaces. EEG signal is complex, unstable, and nonlinear. Most existing methods rarely analyze the data characteristics from multiple dimensions, so it takes work to analyze the data comprehensively. To analyze EEG signals more comprehensively, this paper evaluates a feature extraction strategy of EEG data based on differential entropy (DE). This method combines the characteristics of different frequency bands, extracts the frequency domain characteristics of EEG, and retains the spatial information between channels. This paper proposes a multi-feature fusion network (T-A-MFFNet) based on the time domain and attention network. The model is composed of a time domain network (TNet), channel attention network (CANet), spatial attention network (SANet), and multi-feature fusion network(MFFNet) based on a squeeze network. T-A-MFFNet aims to learn more valuable features from the input data to achieve good classification results. Specifically, the TNet network extracts high-level time series information from EEG data. CANet and SANet are used to fuse channel and spatial features. They use MFFNet to merge multi-dimensional features and realize classification. The validity of the model is verified on the SEED-VIG dataset. The experimental results show that the accuracy of the proposed method reaches 85.65 %, which is superior to the current popular model. The proposed method can learn more valuable information from EEG signals to improve the ability to identify fatigue status and promote the development of the research field of driving fatigue detection based on EEG signals.

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The Gaussian law reigns supreme in the information theory of analog random variables. This paper showcases a number of information theoretic results which find elegant counterparts for Cauchy distributions. New concepts such as that of equivalent pairs of probability measures and the strength of real-valued random variables are introduced here and shown to be of particular relevance to Cauchy distributions.

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The evaluation of the information entropy content in the data analysis is an effective role in the assessment of fatigue damage. Due to the connection between the generalized half-normal distribution and fatigue extension, the objective inference for the differential entropy of the generalized half-normal distribution is considered in this paper. The Bayesian estimates and associated credible intervals are discussed based on different non-informative priors including Jeffery, reference, probability matching, and maximal data information priors for the differential entropy measure. The Metropolis-Hastings samplers data sets are used to estimate the posterior densities and then compute the Bayesian estimates. For comparison purposes, the maximum likelihood estimators and asymptotic confidence intervals of the differential entropy are derived. An intensive simulation study is conducted to evaluate the performance of the proposed statistical inference methods. Two real data sets are analyzed by the proposed methodology for illustrative purposes as well. Finally, non-informative priors for the original parameters of generalized half-normal distribution based on the direct and transformation of the entropy measure are also proposed and compared.

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Moisture adsorption and desorption isotherms of gelatin extracted from whitefish skin powder (FSGP) at different temperatures across a wide range of water activity were determined along with their thermodynamic properties. Nine mathematical models were utilized for fitting the experimental data and simulating the adsorption and desorption behavior. The thermodynamic properties were determined and fitted to the experimental data. The results showed that Peleg and GAB models were the best fit for FSGP. The energies involved in the adsorption and desorption process of FSGP indicated a stronger dependence on equilibrium moisture content (Xe). When Xe decreased, there was a consistent trend of increasing thermodynamic properties. Both the moisture adsorption and desorption behaviors of FSGP were, therefore, non-spontaneous processes. Linear correlations between the changes in enthalpy and entropy for adsorption and desorption were observed, indicating the presence of enthalpy-entropy compensation for FSGP. For preserving FSGP quality, it should be stored with Xw ≤ 8 (gw/gdm, d.b.) at temperatures below 53 °C and an RH of 50% to avoid it becoming rubbery. These findings are crucial for providing insight into the optimal drying and storage conditions.

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Objective. Auditory attention decoding (AAD) determines which speaker the listener is focusing on by analyzing his/her EEG. Convolutional neural network (CNN) was adopted to extract spectro-spatial-feature (SSF) from short-time-interval of EEG to detect auditory spatial attention without stimuli. However, the following factors are not considered in SSF-CNN scheme. (a) Single-band frequency analysis cannot represent the EEG pattern precisely. (b) The power cannot represent the EEG feature related to the dynamic patterns of the attended auditory stimulus. (c) The temporal feature of EEG representing the relationship between EEG and attended stimulus is not extracted. To solve these problems, SSF-CNN scheme was modified.Approach. (a) Multiple-frequency bands, but not a single alpha frequency band, of EEG, were analyzed to represent the EEG pattern more precisely. (b) Differential entropy, but not power, was extracted from each frequency band to represent the disorder degree of EEG, which was related to the dynamic patterns of the attended auditory stimulus. (c) CNN and convolutional-long-short-term-memory (ConvLSTM) were combined to extract spectro-spatial-temporal features from the 3D descriptor sequence constructed based on the topographical activity maps of multiple-frequency bands.Main results. Experimental results on KUL, DTU, and PKU with 0.1 s, 1 s, 2 s, and 5 s decision windows demonstrated that: (a) The proposed model outperformed SSF-CNN and state-of-the-art AAD models. Specifically, when the auditory stimulus was unavailable, AAD accuracy could be enhanced by at least3.25%,3.96%and5.08%on KUL, DTU, and PKU, respectively, compared with the baselines. And, on KUL, the longer decision window corresponded to lower enhancement, while on both DTU and PKU, the longer decision window corresponded to higher enhancement, except for two cases when decision window length was 2 s on PKU or 5 s on DTU. (b) Each modification contributed to the performance enhancement.Significance. DE feature, multi-band frequency analysis, and ConvLSTM-based temporal analysis help to enhance AAD accuracy.

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Emotion recognition is one of the most important issues in human-computer interaction (HCI), neuroscience, and psychology fields. It is generally accepted that emotion recognition with neural data such as electroencephalography (EEG) signals, functional magnetic resonance imaging (fMRI), and near-infrared spectroscopy (NIRS) is better than other emotion detection methods such as speech, mimics, body language, facial expressions, etc., in terms of reliability and accuracy. In particular, EEG signals are bioelectrical signals that are frequently used because of the many advantages they offer in the field of emotion recognition. This study proposes an improved approach for EEG-based emotion recognition on a publicly available newly published dataset, VREED. Differential entropy (DE) features were extracted from four wavebands (theta 4-8 Hz, alpha 8-13 Hz, beta 13-30 Hz, and gamma 30-49 Hz) to classify two emotional states (positive/negative). Five classifiers, namely Support Vector Machine (SVM), k-Nearest Neighbor (kNN), Naïve Bayesian (NB), Decision Tree (DT), and Logistic Regression (LR) were employed with DE features for the automated classification of two emotional states. In this work, we obtained the best average accuracy of 76.22% ± 2.06 with the SVM classifier in the classification of two states. Moreover, we observed from the results that the highest average accuracy score was produced with the gamma band, as previously reported in studies in EEG-based emotion recognition.

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This paper studies the properties of the derivatives of differential entropy H(Xt) in Costa's entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for m≥1, (-1)m+1(dm/dtm)H(Xt)≥0, while McKean conjectured a stronger statement, whereby (-1)m+1(dm/dtm)H(Xt)≥(-1)m+1(dm/dtm)H(XGt). Here, we study the higher dimensional analogues of these conjectures. In particular, we study the veracity of the following two statements: C1(m,n):(-1)m+1(dm/dtm)H(Xt)≥0, where n denotes that Xt is a random vector taking values in Rn, and similarly, C2(m,n):(-1)m+1(dm/dtm)H(Xt)≥(-1)m+1(dm/dtm)H(XGt)≥0. In this paper, we prove some new multivariate cases: C1(3,i),i=2,3,4. Motivated by our results, we further propose a weaker version of McKean's conjecture C3(m,n):(-1)m+1(dm/dtm)H(Xt)≥(-1)m+11n(dm/dtm)H(XGt), which is implied by C2(m,n) and implies C1(m,n). We prove some multivariate cases of this conjecture under the log-concave condition: C3(3,i),i=2,3,4 and C3(4,2). A systematic procedure to prove Cl(m,n) is proposed based on symbolic computation and semidefinite programming, and all the new results mentioned above are explicitly and strictly proved using this procedure.

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Human emotions are variant with time, non-stationary, complex in nature, and are invoked as a result of human reactions during our daily lives. Continuously detecting human emotions from one-dimensional EEG signals is an arduous task. This paper proposes an advanced signal processing mechanism for emotion detection from EEG signals using continuous wavelet transform. The space and time components of the raw EEG signals are converted into 2D spectrograms followed by feature extraction. A hybrid spatio-temporal deep neural network is implemented to extract rich features. A differential-based entropy feature selection technique adaptively differentiates features based on entropy, based on low and high information regions. Bag of Deep Features (BoDF) is applied to create clusters of similar features and computes the features vocabularies for reduction of feature dimensionality. Extensive experiments are performed on the SEED dataset, which shows the significance of the proposed method compared to state-of-the-art methods. Specifically, the proposed model achieved 96.7%, 96.2%, 95.8%, and 95.3% accuracy with the SJTU SEED dataset, for SVM, ensemble, tree, and KNN classifiers, respectively.

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Adsorption isotherms provide insight into the thermodynamic properties governed by food storage conditions. Adsorption isotherms of purple corn of the Canteño variety were evaluated at 18, 25, and 30 °C, for the equilibrium relative humidity (ERH) range between 0.065 and 0.95. The equilibrium moisture (Xe) was determined by the continuous weight-change method. Seven mathematical models of isotherms were modeled, using the coefficient of determination R2, mean absolute error (MAE), and estimated standard error (ESE) as the convergence criterion. Thermodynamic parameters such as isosteric heat (qst), Gibbs Free Energy (ΔG), differential entropy (ΔS), activation energy (Ea), and compliance with the isokinetic law were evaluated. It was observed that the adsorption isotherms presented cross-linking around 75% ERH and 17% Xe, suggesting adequate storage conditions at these values. The GAB and Halsey models reported better fit (R2 > 97%, MAE < 10%, ESE < 0.014 and random residual dispersion). The reduction of Xe from 17 to 7%, increases qst, from 7.7022 to 0.0165 kJ/g, while ΔG decreases considerably with the increase in Xe, presenting non-spontaneous endergonic behavior, and linear relationship with ΔS, evidencing compliance with the isokinetic theory, governed by qst. Ea showed that more energy is required to remove water molecules from the upper layers bound to the monolayer, evaluated using CGAB. The models predicted the storage conditions, and the thermodynamic parameters show the structural stability of the purple corn grains of the Canteño variety during storage.

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In several applications, the assumption of normality is often violated in data with some level of skewness, so skewness affects the mean's estimation. The class of skew-normal distributions is considered, given their flexibility for modeling data with asymmetry parameter. In this paper, we considered two location parameter (µ) estimation methods in the skew-normal setting, where the coefficient of variation and the skewness parameter are known. Specifically, the least square estimator (LSE) and the best unbiased estimator (BUE) for µ are considered. The properties for BUE (which dominates LSE) using classic theorems of information theory are explored, which provides a way to measure the uncertainty of location parameter estimations. Specifically, inequalities based on convexity property enable obtaining lower and upper bounds for differential entropy and Fisher information. Some simulations illustrate the behavior of differential entropy and Fisher information bounds.

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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.