Entropy Measures Research Articles

Probabilistic hesitant fuzzy multiple criteria decision-making with triangular norm based similarity and entropy measures

Existing probabilistic hesitant fuzzy set (PHFS) measures are constructed using two information measures: hesitancy and unwrapped probabilities. We argue that unifying these semantic terms in PHFS information theory is not logical. We introduce a new class of information measures for PHFSs, which address the logical wrapping of hesitant fuzzy sets (HFS) and probability. We propose several similarity measures for these sets that use the Triangular norm operator. We consider the relationship between measures of entropy and similarity and represent the axiomatic definition of PHFS entropy measures. Finally, we use case studies to demonstrate applications of these information measures. We describe two multiple-criteria decision-making algorithms. The last step is devoted to PHFS ranking procedures: one based on the score function of alternatives and the other based on the relative closeness of alternatives. This contribution describes new information measures and uses case studies to illustrate how they can be applied to decision-making processes.

Markov decision processes with risk-sensitive criteria: an overview

AbstractThe paper provides an overview of the theory and applications of risk-sensitive Markov decision processes. The term ’risk-sensitive’ refers here to the use of the Optimized Certainty Equivalent as a means to measure expectation and risk. This comprises the well-known entropic risk measure and Conditional Value-at-Risk. We restrict our considerations to stationary problems with an infinite time horizon. Conditions are given under which optimal policies exist and solution procedures are explained. We present both the theory when the Optimized Certainty Equivalent is applied recursively as well as the case where it is applied to the cumulated reward. Discounted as well as non-discounted models are reviewed.

Prognostic value of different discretization parameters in 18fluorodeoxyglucose positron emission tomography radiomics of oropharyngeal squamous cell carcinoma.

We aim to interrogate the role of positron emission tomography (PET) image discretization parameters on the prognostic value of radiomic features in patients with oropharyngeal cancer. A prospective clinical trial (NCT01908504) enrolled patients with oropharyngeal squamous cell carcinoma (; mixed HPV status) undergoing definitive radiotherapy and evaluated intra-treatment 18fluorodeoxyglucose PET as a potential imaging biomarker of early metabolic response. The primary tumor volume was manually segmented by a radiation oncologist on PET/CT images acquired two weeks into treatment (20Gy). From this, 54 radiomic texture features were extracted. Two image discretization techniques-fixed bin number (FBN) and fixed bin size (FBS)-were considered to evaluate systematic changes in the bin number ({32, 64, 128, 256} gray levels) and bin size ({0.10, 0.15, 0.22, 0.25} bin-widths). For each discretization-specific radiomic feature space, an LASSO-regularized logistic regression model was independently trained to predict residual and/or recurrent disease. The model training was based on Monte Carlo cross-validation with a 20% testing hold-out, 50 permutations, and minor-class up-sampling to account for imbalanced outcomes data. Performance differences among the discretization-specific models were quantified via receiver operating characteristic curve analysis. A final parameter-optimized logistic regression model was developed by incorporating different settings parameterizations into the same model. FBN outperformed FBS in predicting residual and/or recurrent disease. The four FBN models achieved AUC values of 0.63, 0.61, 0.65, and 0.62 for 32, 64, 128, and 256 gray levels, respectively. By contrast, the average AUC of the four FBS models was 0.53. The parameter-optimized model, comprising features joint entropy (FBN = 64) and information measure correlation 1 (FBN = 128), achieved an AUC of 0.70. Kaplan-Meier analyses identified these features to be associated with disease-free survival ( and , respectively; log-rank test). Our findings suggest that the prognostic value of individual radiomic features may depend on feature-specific discretization parameter settings.

A Unified Approach for the Calculation of Different Sample-Based Measures with the Single Sampling Method

This paper explores two sample-based methods for analysing multistable systems: basin stability and basin entropy. Both methods rely on many numerical integration trials conducted with diverse initial conditions. The collected data is categorised and used to compute metrics that characterise solution stability, phase space structure, and system dynamics predictability. Basin stability assesses the overall likelihood of reaching specific solutions, while the basin entropy measure aims to capture the structure of attraction basins and the complexity of their boundaries. Although these two metrics complement each other effectively, their original procedures for computation differ significantly. This paper introduces a universal approach and algorithm for calculating basin stability and entropy measures. The suitability of these procedures is demonstrated through the analysis of two non-linear systems.

On topological indices and entropy measures of beryllonitrene network via logarithmic regression model

Chemical graph theory, a subfield of graph theory, is used to investigate chemical substances and their characteristics. Chemical graph analysis sheds light on the connection, symmetry, and reactivity of molecules. It supports chemical property prediction, research of molecular reactions, drug development, and understanding of molecular networks. A crucial part of computational chemistry is chemical graph theory, which helps researchers analyze and manipulate chemical structures using graph algorithms and mathematical models. Beryllonitrene , a compound of interest due to its potential applications in various fields, is examined through the lens of graph theory and mathematical modeling. The study involves the calculation and interpretation of topological indices and graph entropy measures, which provide valuable insights into the structural and energetic properties of Beryllonitrene’s molecular graph. Logarithmic regression models are employed to establish correlations between these indices, entropy, and other relevant molecular attributes. The results contribute to a deeper understanding of Beryllonitrene’s complex characteristics, facilitating its potential applications in diverse scientific and technological domains. In this study, degree-based topological indices TI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{TI}$$\\end{document} are determined, as well as the entropy of graphs based on these TI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{TI}$$\\end{document}.

On characterization of entropy measure using logarithmic regression model for Copper(II) Fluoride.

The versatile uses of Copper(II) Fluoride (CuF2) are well known; these include its usage as a precursor in chemical synthesis as well as its contribution to the creation of sophisticated materials and electronics. There are interesting opportunities to study the interactions between these elements because of their unique crystal structure, which contains copper ions and fluoride anions. Its potential in optoelectronic devices and conductive qualities also make it a viable material for next-generation technologies. To better understand the structural properties of CuF2 and how they affect its entropy, we present new Zagreb indices in this study and use them to calculate entropy measures. We also build a regression model to clarify the relationship between the calculated indices and entropy levels. The findings of our investigation offer significant understanding regarding the ability of the suggested Zagreb indices to extract meaningful content and their correlation with entropy in the context of CuF2. This information is important for understanding CuF2 alloys and for exploring related complex materials.

Evidence of ordering in Cu-Ni alloys from experimental electronic entropy measurements

Evidence of ordering in Cu-Ni alloys from experimental electronic entropy measurements

DSD²: Can We Dodge Sparse Double Descent and Compress the Neural Network Worry-Free?

Neoteric works have shown that modern deep learning models can exhibit a sparse double descent phenomenon. Indeed, as the sparsity of the model increases, the test performance first worsens since the model is overfitting the training data; then, the overfitting reduces, leading to an improvement in performance, and finally, the model begins to forget critical information, resulting in underfitting. Such a behavior prevents using traditional early stop criteria. In this work, we have three key contributions. First, we propose a learning framework that avoids such a phenomenon and improves generalization. Second, we introduce an entropy measure providing more insights into the insurgence of this phenomenon and enabling the use of traditional stop criteria. Third, we provide a comprehensive quantitative analysis of contingent factors such as re-initialization methods, model width and depth, and dataset noise. The contributions are supported by empirical evidence in typical setups. Our code is available at https://github.com/VGCQ/DSD2.

Exploring the Entropy-Based Classification of Time Series Using Visibility Graphs from Chaotic Maps

The classification of time series using machine learning (ML) analysis and entropy-based features is an urgent task for the study of nonlinear signals in the fields of finance, biology and medicine, including EEG analysis and Brain–Computer Interfacing. As several entropy measures exist, the problem is assessing the effectiveness of entropies used as features for the ML classification of nonlinear dynamics of time series. We propose a method, called global efficiency (GEFMCC), for assessing the effectiveness of entropy features using several chaotic mappings. GEFMCC is a fitness function for optimizing the type and parameters of entropies for time series classification problems. We analyze fuzzy entropy (FuzzyEn) and neural network entropy (NNetEn) for four discrete mappings, the logistic map, the sine map, the Planck map, and the two-memristor-based map, with a base length time series of 300 elements. FuzzyEn has greater GEFMCC in the classification task compared to NNetEn. However, NNetEn classification efficiency is higher than FuzzyEn for some local areas of the time series dynamics. The results of using horizontal visibility graphs (HVG) instead of the raw time series demonstrate the GEFMCC decrease after HVG time series transformation. However, the GEFMCC increases after applying the HVG for some local areas of time series dynamics. The scientific community can use the results to explore the efficiency of the entropy-based classification of time series in “The Entropy Universe”. An implementation of the algorithms in Python is presented.

Decoding the News Media Diet of Disinformation Spreaders.

In the digital era, information consumption is predominantly channeled through online news media and disseminated on social media platforms. Understanding the complex dynamics of the news media environment and users' habits within the digital ecosystem is a challenging task that requires, at the same time, large databases and accurate methodological approaches. This study contributes to this expanding research landscape by employing network science methodologies and entropic measures to analyze the behavioral patterns of social media users sharing news pieces and dig into the diverse news consumption habits within different online social media user groups. Our analyses reveal that users are more inclined to share news classified as fake when they have previously posted conspiracy or junk science content and vice versa, creating a series of "misinformation hot streaks". To better understand these dynamics, we used three different measures of entropy to gain insights into the news media habits of each user, finding that the patterns of news consumption significantly differ among users when focusing on disinformation spreaders as opposed to accounts sharing reliable or low-risk content. Thanks to these entropic measures, we quantify the variety and the regularity of the news media diet, finding that those disseminating unreliable content exhibit a more varied and, at the same time, a more regular choice of web-domains. This quantitative insight into the nuances of news consumption behaviors exhibited by disinformation spreaders holds the potential to significantly inform the strategic formulation of more robust and adaptive social media moderation policies.

Assessing Motor Variability during Squat: The Reliability of Inertial Devices in Resistance Training.

Movement control can be an indicator of how challenging a task is for the athlete, and can provide useful information to improve training efficiency and prevent injuries. This study was carried out to determine whether inertial measurement units (IMU) can provide reliable information on motion variability during strength exercises, focusing on the squat. Sixty-six healthy, strength-trained young adults completed a two-day protocol, where the variability in the squat movement was analyzed at two different loads (30% and 70% of one repetition maximum) using inertial measurement units and a force platform. The time series from IMUs and force platforms were analyzed using linear (standard deviation) and non-linear (detrended fluctuation analysis, sample entropy and fuzzy entropy) measures. Reliability was analyzed for both IMU and force platform using the intraclass correlation coefficient and the standard error of measurement. Standard deviation, detrended fluctuation analysis, sample entropy, and fuzzy entropy from the IMUs time series showed moderate to good reliability values (ICC: 0.50-0.85) and an acceptable error. The study concludes that IMUs are reliable tools for analyzing movement variability in strength exercises, providing accessible options for performance monitoring and training optimization. These findings have implications for the design of more effective strength training programs, emphasizing the importance of movement control in enhancing athletic performance and reducing injury risks.

HAG-NET: Hiding Data and Adversarial Attacking with Generative Adversarial Network.

Recent studies on watermarking techniques based on image carriers have demonstrated new approaches that combine adversarial perturbations against steganalysis with embedding distortions. However, while these methods successfully counter convolutional neural network-based steganalysis, they do not adequately protect the data of the carrier itself. Recognizing the high sensitivity of Deep Neural Networks (DNNs) to small perturbations, we propose HAG-NET, a method based on image carriers, which is jointly trained by the encoder, decoder, and attacker. In this paper, the encoder generates Adversarial Steganographic Examples (ASEs) that are adversarial to the target classification network, thereby providing protection for the carrier data. Additionally, the decoder can recover secret data from ASEs. The experimental results demonstrate that ASEs produced by HAG-NET achieve an average success rate of over 99% on both the MNIST and CIFAR-10 datasets. ASEs generated with the attacker exhibit greater robustness in terms of attack ability, with an average increase of about 3.32%. Furthermore, our method, when compared with other generative stego examples under similar perturbation strength, contains significantly more information according to image information entropy measurements.

Measures of Distance and Entropy Based on the Fermatean Fuzzy-Type Soft Sets Approach

The definition of Fermatean fuzzy soft sets and some of its features are introduced in this study. A Fermatean fuzzy soft set is a parameterized family of Fermatean fuzzy sets and a generalization of intuitionistic and Pythagorean fuzzy soft sets. This paper presents a definition of the Fermatean fuzzy soft entropy. Also acquired are the formulae for standard distance measures such as Hamming and Euclidean distance. Other formulas have also been proposed for calculating the entropy and distance measurements of FFSSs. Even if the entropy and distance measures are defined for other set extensions, they cannot be applied directly to Fermatean fuzzy soft sets. It can be used to determine the uncertainty associated with a Fermatean fuzzy soft set, discover similarities between any two Fermatean fuzzy soft sets using the proposed distance measures, and compare it to other existing structures in the literature. Fermatean fuzzy soft set applications in decision-making and pattern recognition difficulties are also examined. Finally, comparison studies with other known equations are performed.

A family of robust chaotic S-unimodal maps based on the Gaussian function

This research paper introduces a family of one-dimensional S-unimodal maps based on the Gaussian function, designed to exhibit robust chaos across a wide range of parameters. These maps are developed to display robust chaos by avoiding multiple fixed points that are primarily responsible for the coexisting attractors in 1D maps. The parameter space analysis reveals that chaotic behaviour is sustained across the entire parameter space, except for a very narrow region. The study employs a comprehensive computational approach, including quantitative measures such as sample entropy, Lyapunov exponent, and invariant measures. The uniformly higher values of sample entropy, uniform positive values of the Lyapunov exponent, and the existence of invariant measures in a region of parameter space confirm the presence of robust chaos in these maps. Such a promising class of robust chaotic maps may be potentially used in diverse fields such as chaos-based cryptography, pseudo-random number generation, communication systems, and more.

Entanglement Trajectory and its Boundary

In this article, we present a novel approach to investigating entanglement in the context of quantum computing. Our methodology involves analyzing reduced density matrices at different stages of a quantum algorithm's execution and representing the dominant eigenvalue and von Neumann entropy on a graph, creating an "entanglement trajectory." To establish the trajectory's boundaries, we employ random matrix theory. Through the examination of examples such as quantum adiabatic computation, the Grover algorithm, and the Shor algorithm, we demonstrate that the entanglement trajectory remains within the established boundaries, exhibiting unique characteristics for each example. Moreover, we show that these boundaries and features can be extended to trajectories defined by alternative entropy measures. The entanglement trajectory serves as an invariant property of a quantum system, maintaining consistency across varying situations and definitions of entanglement. Numerical simulations accompanying this research are available via open access.

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem—and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.

Predictive models on potential energies of zeolite ZK-5 using bond weighted information entropy measures

Zeolites are hydrated aluminosilicate mineral characterised by a uniform pore structure. Zeolite ZK-5 or KFI framework was synthesized due to it's important physicochemical characteristics. So far, computational material development has improved efficiency by identifying novel zeolites with desirable properties including system equilibrium and stability. Topological indices, which characterise KFI zolite's complex structure, can help researchers develop new techniques for studying the complex structure. Shannon's entropy modelling help us prototype and understand our zeolites element changes and complexity characteristics. Modelling and understanding these changes and relationships can help us improve our zeolite in the field of the microporous material by QSAR/QSPR studies. We have also studied accuracy of the cubic regression model between the derived degree type entropies of KFI zeolite and it's potential energy. Therefore, for investigating the property of large-shaped KFI zeolite, the anticipated cubic correlation analysis reveals a significant correlation with electrostatic, dispersion energies.

Entropy for q-rung linear diophantine fuzzy hypersoft set with its application in MADM

A notable advancement in fuzzy set theory is the q-rung linear diophantine fuzzy set. The soft set theory was expanded into the hypersoft set theory. By combining both the q-rung linear diophantine fuzzy set and hypersoft set, this study describes the notion of q-rung linear diophantine fuzzy hypersoft set that can handle multi sub-attributed q-rung linear diophantine fuzzy situations in the real world. Furthermore, some of its algebraic operations such as union, intersection and complement are described in this study. In addtion, the entropy measure of the q-rung linear diophantine fuzzy hypersoft set is established as it is helpful in determining the degree of fuzziness of q-rung linear diophantine fuzzy hypersoft sets. A multi-attribute decision making algorithm based on suggested entropy is presented in this study along with a numerical example of selecting a suitable wastewater treatment technology to demonstrate the effectiveness of the proposed algorithm in real-life situations. A comparative study was undertaken that describes the validity, robustness and superiority of the proposed algorithm and notions by discussing the advantages and drawbacks of existing theories and algorithms. Overall, this study describes a novel fuzzy extension that prevails over the existing ones and contributes to the real world with a valid real-life multi-attribute decision making algorithm that can cover many real-world problems that are unable to be addressed by the existing methodology.

On the application of compound optimal design strategy in progressively type-II censored life-testing experiments

Progressive censoring schemes allow removal of censoring units at the time of each failure. This flexibility of removing items during the experiment makes progressive censoring very effective but it also increases the number of possible censoring schemes. So the problem of choosing optimal progressive censoring schemes has gained a lot of attention in the literature. In this article, we propose a new criterion based on cumulative residual entropy measure for the determination of optimal progressive type II censoring scheme. The proposed criterion represents an overall information of the progressive type II censored life-testing experiment and we obtain optimal schemes by maximizing this measure. Also, we develop optimal design by maximizing the information subject to a given cost constraint. Finally, we perform a compound optimal design strategy that simultaneously optimizes our proposed criterion and the total cost of the experiment. Our proposed design does not depend on any asymptotic results. Sensitivity analysis for the constraint design and COD are studied for the mis-specification of the input parametrs. A real data set is analyzed for the purpose of illustration.

Dynamic nonlinear simplified neutrosophic sets for multiple-attribute group decision making

In this paper, the concept of a dynamic nonlinear simplified neutrosophic set (DNSNS) is proposed for describing the real-time changing expert preference information. Furthermore, the DNSNS aggregation model and decision algorithm are provided to solve the actual multiple-attribute group decision making (MAGDM) problems. The basic notions, the similarity measure, the entropy measure, and the index of distance of DNSNS are presented first. Secondly, the univariate time series of DNSNS are projected into dynamic nonlinear simplified neutrosophic curves in three-dimensional space. The areas of the surface enclosed by the curves represent the variance among the DNSNSs. Thus, the DNSNS aggregation model is established correctly without preprocessing the original data. Afterward, the aggregation algorithm extended from the plant growth simulation algorithm (PGSA) is proposed for calculating the optimal aggregation preference curve and constructing the collective matrix. Additionally, a novel corresponding decision algorithm based on TOPSIS and projection theory is proposed for obtaining the overall ranking of alternatives in the actual MAGDM problem. Finally, a typical example is presented to illustrate the feasibility and effectiveness of the proposed model and algorithm.