Entropy Measures Research Articles

Optimizing the measurement of sample entropy in resting-state fMRI data.

The complexity of brain signals may hold clues to understand brain-based disorders. Sample entropy, an index that captures the predictability of a signal, is a promising tool to measure signal complexity. However, measurement of sample entropy from fMRI signals has its challenges, and numerous questions regarding preprocessing and parameter selection require research to advance the potential impact of this method. For one example, entropy may be highly sensitive to the effects of motion, yet standard approaches to addressing motion (e.g., scrubbing) may be unsuitable for entropy measurement. For another, the parameters used to calculate entropy need to be defined by the properties of data being analyzed, an issue that has frequently been ignored in fMRI research. The current work sought to rigorously address these issues and to create methods that could be used to advance this field. We developed and tested a novel windowing approach to select and concatenate (ignoring connecting volumes) low-motion windows in fMRI data to reduce the impact of motion on sample entropy estimates. We created utilities (implementing autoregressive models and a grid search function) to facilitate selection of the matching length m parameter and the error tolerance r parameter. We developed an approach to apply these methods at every grayordinate of the brain, creating a whole-brain dense entropy map. These methods and tools have been integrated into a publicly available R package ("powseR"). We demonstrate these methods using data from the ABCD study. After applying the windowing procedure to allow sample entropy calculation on the lowest-motion windows from runs 1 and 2 (combined) and those from runs 3 and 4 (combined), we identified the optimal m and r parameters for these data. To confirm the impact of the windowing procedure, we compared entropy values and their relationship with motion when entropy was calculated using the full set of data vs. those calculated using the windowing procedure. We then assessed reproducibility of sample entropy calculations using the windowed procedure by calculating the intraclass correlation between the earlier and later entropy measurements at every grayordinate. When applying these optimized methods to the ABCD data (from the subset of individuals who had enough windows of continuous "usable" volumes), we found that the novel windowing procedure successfully mitigated the large inverse correlation between entropy values and head motion seen when using a standard approach. Furthermore, using the windowed approach, entropy values calculated early in the scan (runs 1 and 2) are largely reproducible when measured later in the scan (runs 3 and 4), although there is some regional variability in reproducibility. We developed an optimized approach to measuring sample entropy that addresses concerns about motion and that can be applied across datasets through user-identified adaptations that allow the method to be tailored to the dataset at hand. We offer preliminary results regarding reproducibility. We also include recommendations for fMRI data acquisition to optimize sample entropy measurement and considerations for the field.

A New Logic, a New Information Measure, and a New Information-Based Approach to Interpreting Quantum Mechanics.

The new logic of partitions is dual to the usual Boolean logic of subsets (usually presented only in the special case of the logic of propositions) in the sense that partitions and subsets are category-theoretic duals. The new information measure of logical entropy is the normalized quantitative version of partitions. The new approach to interpreting quantum mechanics (QM) is showing that the mathematics (not the physics) of QM is the linearized Hilbert space version of the mathematics of partitions. Or, putting it the other way around, the math of partitions is a skeletal version of the math of QM. The key concepts throughout this progression from logic, to logical information, to quantum theory are distinctions versus indistinctions, definiteness versus indefiniteness, or distinguishability versus indistinguishability. The distinctions of a partition are the ordered pairs of elements from the underlying set that are in different blocks of the partition and logical entropy is defined (initially) as the normalized number of distinctions. The cognate notions of definiteness and distinguishability run throughout the math of QM, e.g., in the key non-classical notion of superposition (=ontic indefiniteness) and in the Feynman rules for adding amplitudes (indistinguishable alternatives) versus adding probabilities (distinguishable alternatives).

Information entropy facilitates (not impedes) lexical processing during language comprehension.

It is well known that contextual predictability facilitates word identification, but it is less clear whether the uncertainty associated with the current context (i.e., its lexical entropy) influences sentence processing. On the one hand, high entropy contexts may lead to interference due to greater number of lexical competitors. On the other hand, predicting multiple lexical competitors may facilitate processing through the preactivation of shared semantic features. In this study, we examined whether entropy measured at the trial level (i.e., for each participant, for each item) corresponds to facilitatory or inhibitory effects. Trial-level entropy captures each individual's knowledge about specific contexts and is therefore a more valid and sensitive measure of entropy (relative to the commonly employed item-level entropy). Participants (N = 112) completed two experimental sessions (with counterbalanced orders) that were separated by a 3- to 14-day interval. In one session, they produced up to 10 completions for sentence fragments (N = 647). In another session, they read the same sentences including a target word (whose entropy value was calculated based on the produced completions) while reading times were measured. We observed a facilitatory (not inhibitory) effect of trial-level entropy on lexical processing over and above item-level measures of lexical predictability (including cloze probability, surprisal, and semantic constraint). Extra analyses revealed that greater semantic overlap between the target and the produced responses facilitated target processing. Thus, the results lend support to theories of lexical prediction maintaining that prediction involves broad activation of semantic features rather than activation of full lexical forms.

Uncertainty measurement for incomplete lattice-valued information system

We analyze the properties and characteristics of the information structure in incomplete lattice-valued information system (ILIS), we redefine the information structure and the dependence and information distance between the two information structures. In addition, in order to evaluate the uncertainty of ILIS, the concepts of granular measure and entropy measure are expounded, including information granulation, information quantity, rough entropy and information entropy. Finally, we carry out numerical experiments to verify the feasibility of the method, and carry out effective statistical analysis. These results are conducive to the establishment of granular computing framework in ILIS.

On K-Banhatti, Revan Indices and Entropy Measures of MgO(111) Nanosheets via Linear Regression

The structure and topology of chemical compounds can be determined using chemical graph theory. Using topological indices, we may uncover much about connectivity, complexity, and other important aspects of molecules. Numerous research investigations have been conducted on the K-Banhatti indices and entropy measurements in various fields, including the study of natural polymers, nanotubes, and catalysts. At the same time, the Shannon entropy of a graph is widely used in network science. It is employed in evaluating several networks, including social networks, neural networks, and transportation systems. The Shannon entropy enables the analysis of a network’s topology and structure, facilitating the identification of significant nodes or structures that substantially impact network operation and stability. In the past decade, there has been a considerable focus on investigating a range of nanostructures, such as nanosheets and nanoparticles, in both experimental and theoretical domains. As a very effective catalyst and inert substrate, the MgO nanostructure has received a lot of interest. The primary objective of this research is to study different indices and employ them to look at entropy measures of magnesium oxide(111) nanosheets over a wide range of p values, including p=1,2,3,…,j. Additionally, we conducted a linear regression analysis to establish the correlation between indices and entropies.

Dynamic Weighted Cumulative Residual Entropy Estimators for Laplace Distribution: Bayesian Approach

Objectives: To develop Bayesian estimators of dynamic weighted cumulative residual entropy (DWCRE) for Laplace distribution and to investigate posterior risks using various priors and loss functions. Methods: Weighted entropy measure of information is provided by a probabilistic experiment whose basic events are described by their objective probabilities and some qualitative (objective or subjective) weights. In this paper, we have used priors (Jeffrey’s, Hartigan, Uniform and Gumble Type II) and several loss functions. Findings: Bayesian estimators and associated posterior risks for Laplace distribution have been derived for different priors and loss functions. Monte Carlo Simulation study and graphical analyses have also been presented along with the conclusion. Through the comprehensive simulation study in the paper, it has been observed that Hartigan prior is better than other priors in terms of the posterior risk whereas Uniform prior has always higher posterior risk. Novelty: The introduction of new Bayesian estimators and their posterior risks for dynamic weighted cumulative residual entropy (DWCRE) of Laplace distribution. Keywords: Bayesian estimators, Laplace distribution, Fisher information matrix, Loss functions, Priors

Fractal dimensions and machine learning for detection of Parkinson’s disease in resting-state electroencephalography

Parkinson’s disease (PD) is an incurable neurological disorder that degenerates the cerebrospinal nervous system and hinders motor functions. Electroencephalography (EEG) signal analysis can provide reliable information regarding PD conditions. However, EEG is a complex, multichannel, and nonlinear signal with noise that problematizes identifying PD symptoms. A few studies have employed fractal dimension (FD) to extract distinguishing PD features from EEG signals. However, no exploratory study exists, as per our knowledge, on the efficiency of the different FD measures. We aim to conduct a comparative analysis of the various FDs that, as feature extraction measures, can discriminate PD patients who are ON and OFF medication from healthy controls using ML architecture. This study has implemented and analyzed several techniques for segmentation, feature extraction, and ML models. The results show that k-nearest neighbors (KNN) classifier with Higuchi FD and 90% overlap for segmented window delivers the highest accuracies, yielding a mean accuracy of 99.65±0.15%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$99.65\\pm 0.15\\%$$\\end{document} for PD patients ON medication and 99.45±0.18%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$99.45\\pm 0.18\\%$$\\end{document} for PD patients OFF medication, respectively. The model accurately identifies the signs of the disease in resting-state EEG with almost equivalent accuracy in both OFF and ON medication patients. To enhance the interpretability in our study, we leveraged XGB’s feature importance to generate brain topographic plots. This integration of explainable AI (XAI) enhanced the transparency and comprehensibility of our model’s classifications. Additionally, a comparison between the performance of FD and a few entropy measures has also been drawn to validate the significance of FD as a superior feature extraction measure. This study contributes to the body of knowledge with an architectural pipeline for detecting PD in resting-state EEG while emphasizing fractal dimension as an effective way of extracting salient features from EEG signals.

The confined helium atom: An information–theoretic approach

AbstractIn this article, we study the helium atom confined in a spherical impenetrable cavity by using informational measures. We use the Ritz variational method to obtain the energies and wave functions of the confined helium atom as a function of the cavity radius . As trial wave functions we use one uncorrelated function and five explicitly correlated basis sets in Hylleraas coordinates with different degrees of electronic correlation. We computed the Shannon entropy, Fisher information, Kullback–Leibler entropy, Tsallis entropy, disequilibrium and Fisher–Shannon complexity, as a function of . We found that these entropic measures are sensitive to electronic correlation and can be used to measure it. As expected these entropic measures are less sensitive to electron correlation in the strong confinement regime ( a.u.).

Graph Entropy for Adjacently and Non-Adjacently Configured Pentagonal Structure of Carbon Nanocones

aims: Carbon nanocones possess exceptional properties compared to other nanomaterials, such as nanotubes, graphene etc. Hence they can be used as an alternative to other nanostructures. This work helps in determining various properties of the nanocone structure through topological descriptors. In this paper, various degree based topological descriptors along with its entropy measures are evaluated for adjacently and non-adjacently configured pentagonal structure of carbon nanocones. background: Carbon nanoparticles are gaining much importance in the contemporary world. Among which carbon nanocones have wide range of acceptance in the field of nanotechnology due to its effective properties and applications. Nanocones, also known as nanohorns are carbon networks which are planar in structure and has majority of hexagonal faces along with some nonhexagonal faces, which are most commonly pentagons. Nanocones which include pentagons in their structure can be referred to as adjacently or non- adjacently configured pentagonal structure of nanocones. Various topological descriptors for few nanocone classes were derived by researchers earlier, but not for the class of nanocones in this paper. Through this work, we try to vanish the research gap in this field. objective: : The degree based descriptors and corresponding graph entropies for the adjacently and non-adjacently configured pentagonal structure of nanocones were determined, which further can be applied in quantitative structure – activity – property relationship studies. The concept of graph entropy is to assign a probability function to the edges in the chemical graph using the topological descriptor and it helps to characterise the complexity of graphs. method: We have employed the degree counting method and edge partition based on the vertices and edges of the adjacently and non-adjacently configured pentagonal nanocone structures to obtain the edge partitions and then using the corresponding mathematical expression, the degree based descriptors and its corresponding entropies were determined. result: Result: The analytically closed formulas to compute the degree based topological descriptors and graph entropies for any generation of the class of nanocone structures were obtained. conclusion: In this work, the degree based descriptors and the corresponding graph entropies for the adjacently and non-adjacently configured pentagonal structure of carbon nanocones are determined applying the degree counting method and edge partition based on the vertices and edges. Also a graphical comparative study was also done with the help of obtained results. other: end.

Scale based entropy measures and deep learning methods for analyzing the dynamical characteristics of cardiorespiratory control system in COVID-19 subjects during and after recovery

COVID-19, known as Coronavirus Disease 2019 primarily targets the respiratory system and can impact the cardiovascular system, leading to a range of cardiorespiratory complications. The current forefront in analyzing the dynamical characteristics of physiological systems and aiding clinical decision-making involves the integration of entropy-based complexity techniques with artificial intelligence. Entropy-based measures offer promising prospects for identifying disturbances in cardiorespiratory control system (CRCS) among COVID-19 patients by assessing the oxygen saturation variability (OSV) signals. In this investigation, we employ scale-based entropy (SBE) methods, including multiscale entropy (MSE), multiscale permutation entropy (MPE), and multiscale fuzzy entropy (MFE), to characterize the dynamical characteristics of OSV signals. These measurements serve as features for the application of traditional machine learning (ML) and deep learning (DL) approaches in the context of classifying OSV signals from COVID-19 patients during their illness and subsequent recovery. We use the Beurer PO-80 pulse oximeter which non-invasively acquired OSV and pulse rate data from COVID-19 infected patients during the active infection phase and after a two-month recovery period. The dataset comprises of 88 recordings collected from 44 subjects(26 men and 18 women), both during their COVID-19 illness and two months post-recovery. Prior to analysis, data preprocessing is performed to remove artifacts and outliers. The application of SBE measures to OSV signals unveils a reduction in signal complexity during the course of COVID-19. Leveraging these SBE measures as feature sets, we employ two DL techniques, namely the radial basis function network (RBFN) and RBFN with dynamic delay algorithm (RBFNDDA), for the classification of OSV data collected during and after COVID-19 recovery. To evaluate the classification performance, we employ standard metrics such as sensitivity, specificity, false positive rate (FPR), and the area under the receiver operator characteristic curve (AUC). Among the three scale-based entropy measures, MFE outperformed MSE and MPE by achieving the highest classification performance using RBFN with 13 best features having sensitivity (0.84), FPR (0.30), specificity (0.70) and AUC (0.77). The outcomes of our study demonstrate that SBE measures combined with DL methods offer a valuable approach for categorizing OSV signals obtained during and after COVID-19, ultimately aiding in the detection of CRCS dysfunction.

A hyperspectral band selection strategy based on adjacency clustering and local structural correlation

ABSTRACT A popular strategy to reduce the spectral size of HyperSpectral Images (HSI) is to suppress all bands whose information content is largely duplicated in surviving bands. This is commonly termed Band Selection (BS). In this manuscript, a novel, unsupervised BS technique is proposed, consisting of three main steps: 1) Adjacent structural-similarity-based band partitioning, 2) for each partition, correlation-based, cluster-representative band selection, and 3) bad cluster removal using an Entropy measure. Initially, the adjacent structural similarity is measured, using Structural SIMilarity Index (SSIM); the next step is adjacency-partitioned cluster formation using a derivative-based approach. Then, a representative band with a high correlation is selected for each cluster. Finally, according to the Entropy measure, the cluster representatives are ranked to obtain the optimal subset of bands. The proposed Adjacency Clustering and Local Structural Correlation (ACLSC)-based BS is compared with other state-of-art approaches in terms of classification performance on benchmark HSI datasets. Overall Accuracy (OA), Average Accuracy (AA), and Kappa ( κ ) are computed on three datasets, namely, Indian Pines, Salinas, and Pavia University, to validate the proposed algorithm. Validation results support the use of adjacency clustering and incorporation of spatial-structural features for BS purposes.

Analyzing household income inequality in Pakistan: a subgroup decomposition of generalized entropy measures

Analyzing household income inequality in Pakistan: a subgroup decomposition of generalized entropy measures

Retracted: On Entropy Measures and Eccentricity-Based Descriptors of Polyamidoamine (PAMAM) Dendrimers

Retracted: On Entropy Measures and Eccentricity-Based Descriptors of Polyamidoamine (PAMAM) Dendrimers

Retracted: Correlation Coefficient and Entropy Measures Based on Complex Dual Type-2 Hesitant Fuzzy Sets and Their Applications

Retracted: Correlation Coefficient and Entropy Measures Based on Complex Dual Type-2 Hesitant Fuzzy Sets and Their Applications

Probabilistic Hesitant Fuzzy Evidence Theory and Its Application in Capability Evaluation of a Satellite Communication System.

Evaluating the capabilities of a satellite communication system (SCS) is challenging due to its complexity and ambiguity. It is difficult to accurately analyze uncertain situations, making it difficult for experts to determine appropriate evaluation values. To address this problem, this paper proposes an innovative approach by extending the Dempster-Shafer evidence theory (DST) to the probabilistic hesitant fuzzy evidence theory (PHFET). The proposed approach introduces the concept of probabilistic hesitant fuzzy basic probability assignment (PHFBPA) to measure the degree of support for propositions, along with a combination rule and decision approach. Two methods are developed to generate PHFBPA based on multi-classifier and distance techniques, respectively. In order to improve the consistency of evidence, discounting factors are proposed using an entropy measure and the Jousselme distance of PHFBPA. In addition, a model for evaluating the degree of satisfaction of SCS capability requirements based on PHFET is presented. Experimental classification and evaluation of SCS capability requirements are performed to demonstrate the effectiveness and stability of the PHFET method. By employing the DST framework and probabilistic hesitant fuzzy sets, PHFET provides a compelling solution for handling ambiguous data in multi-source information fusion, thereby improving the evaluation of SCS capabilities.

Exploring the Hidden Complexity: Entropy Analysis in Pulse Oximetry of Female Athletes.

This study examines the relationship between physiological complexity, as measured by Approximate Entropy (ApEn) and Sample Entropy (SampEn), and fitness levels in female athletes. Our focus is on their association with maximal oxygen consumption (VO2,max). Our findings reveal a complex relationship between entropy metrics and fitness levels, indicating that higher fitness typically, though not invariably, correlates with greater entropy in physiological time series data; however, this is not consistent for all individuals. For Heart Rate (HR), entropy measures suggest stable patterns across fitness categories, while pulse oximetry (SpO2) data shows greater variability. For instance, the medium fitness group displayed an ApEn(HR) = 0.57±0.13 with a coefficient of variation (CV) of 22.17 and ApEn(SpO2) = 0.96±0.49 with a CV of 46.08%, compared to the excellent fitness group with ApEn(HR) = 0.60±0.09 with a CV of 15.19% and ApEn(SpO2) =0.85±0.42 with a CV of 49.46%, suggesting broader physiological responses among more fit individuals. The larger standard deviations and CVs for SpO2 entropy may indicate the body's proficient oxygen utilization at higher levels of physical demand. Our findings advocate for combining entropy metrics with wearable sensor technology for improved biomedical analysis and personalized healthcare.

A Novel Trigonometric Entropy Measure Based on the Complex Proportional Assessment Technique for Pythagorean Fuzzy Sets

The extension of intuitionistic fuzzy sets (IFS) to Pythagorean fuzzy sets (PFS) is a significant advancement, addressing the inherent limitations of IFS. This study introduces a novel entropy measure specifically designed for Pythagorean fuzzy sets, establishing its axiomatic definition and presenting key properties. Decision making guided by entropy is advantageous, as it effectively mitigates ambiguity with increasing entropy values. Furthermore, a numerical example is provided to facilitate a comparative assessment of our newly introduced entropy measure in contrast to existing PFS entropy measures. The validation of our findings is achieved through the application of the COPRAS method, which determines decision outcomes based on a multitude of influencing factors. Notably, the determination of weights in this method is underpinned by the utilization of our innovative entropy measure.

Measures of Distance in Quantum Mechanics

Combining gravity with quantum theory is still a work in progress. On the one hand, classical gravity is the geometry of space-time determined by the energy–momentum tensor of matter and the resulting nonlinear equations; on the other hand, the mathematical description of a quantum system is Hilbert space with linear equations describing evolution. In this paper, various measures in Hilbert space will be presented. In general, distance measures in Hilbert space can be divided into measures determined by energy and measures determined by entropy. Entropy measures determine quasi-distance because they do not satisfy all the axioms defining distance. Finding a general rule to determine such a measure unambiguously seems to be fundamental.

Topological characterization of cove-edged graphene nanoribbons with applications to NMR spectroscopies

Graphene nanoribbons (GNRs) have recently garnered significant attention due to their potential applications in transistors, photovoltaics, and quantum electronic devices. Cove-edged graphene nanoribbons (CGNRs) are particularly intriguing due to their unique electromagnetic properties. These nanoribbons, characterized by asymmetric edges composed of alternating hexagons, exhibit exceptional electronic properties. In this study, we employed a variety of degree and degree-sum-based topological descriptors (TDs) to calculate the topological entropy measures of CGNRs. The explicit mathematical formulae of the TDs for the CGNRs were derived using the edge partition method. Additionally, we determined the spectral diameters, delocalization energies, resonance energies, and HOMO-LUMO gaps to assess these compounds' thermodynamic and kinetic stabilities. We also utilized NEWGRAPH and MATLAB software to predict the NMR patterns for the CGNRs. The shallow HOMO-LUMO gaps in these calculations suggest that CGNRs are promising candidates for topological spintronics applications.

Confidence Levels Measurement of Mobile Phone Selection Using a Multiattribute Decision-Making Approach with Unknown Attribute Weight Information Based on T-Spherical Fuzzy Aggregation Operators

Advancement in mobile phone (MP) technology has revolutionized the lifestyle. In recent years, we observed that MP technology had been involved in almost all aspects of life, such as communication purposes, e-commerce, mobile baking, and social media connectivity. So, it becomes a hot research topic to select the best MP that fulfills the desired feathers requirement. In this paper, the expert’s familiarity with the examined objects is factored into the initial judgments under the T-spherical fuzzy sets (T-SFSs) environment. The T-SFS is the extension of the picture fuzzy (PF) set (PFS), which gives wider scope for finding the most precise options than existing fuzzy frameworks. The multiattribute decision-making (MADM) is a common and valuable method for aggregating information. For MADM, various aggregation operators (AOs) have been created over the years. The article introduces the newly proposed approach T-spherical fuzzy (T-SF) confidence level weighted averaging T−SFWAc and T-SF confidence level weighted geometric T−SFWGc. Also, some desired properties of AOs are discussed, and the T-SF entropy measure is introduced for selecting the weight criteria. A MADM framework is introduced, on the behalf of proposed operators. The proposed MADM framework is applied to solve the real-life example of consumers’ preferences to show effectiveness and practicality. Lastly, the developed framework is set side by side with other prevailing approaches to demonstrate the superiority and significance of other existing AOs.