Cross-entropy Measures Research Articles

Parametric picture fuzzy cross-entropy measures based on [formula omitted]-Choquet integral for building material recognition

This research paper introduces a novel picture fuzzy cross-entropy measure that utilizes Lin’s divergence as a basis for comparing two picture fuzzy sets. To improve the flexibility and applicability of this new cross-entropy measure, a family of parametric cross-entropy measures is defined. By adjusting the parameters in applications, the influence of the degree of positive membership, negative membership, and neutral membership on the picture fuzzy cross-entropy can be observed, allowing for a more tailored analysis. Additionally, we investigate the relationship between power weighted means and the d-Choquet integral, which serves as an extension of the ordinary Choquet integral. Using this knowledge, picture fuzzy cross-entropy measures based on the d-Choquet integral are introduced to yield more sensitive results, particularly when interactions between criteria exist in specific problem domains. This consideration of criterion interactions is often absent in existing cross-entropy measures. Moreover, an algorithm is presented for solving pattern recognition problems and applied to a building material recognition problem sourced from existing literature. Then, we proposed another algorithm and use it to investigate a novel material classification problem. Through these applications, the effectiveness of the proposed cross-entropy measures in pattern recognition is demonstrated. The paper conducts a comparative analysis between existing methods and the proposed approaches, followed by a sensitivity analysis. This analysis involves manipulating the parameters derived from both the parametric cross-entropy measures and the d-Choquet integrals to assess their respective impacts and sensitivities. Finally, the results regarding the classification problem are examined with performance metrics such as accuracy, precision, recall, and F1 score.

A consensus-based single valued neutrosophic model for selection of educational vendors under metaverse with extended reality

Selecting the right educational vendors in the Metaverse, particularly those utilizing extended reality (XR), is crucial for creating an engaging and immersive learning environment. Careful vendor selection ensures the delivery of high-quality XR educational content, considering factors such as seamless integration with different XR devices and addressing associated challenges, which can affect the comfort and well-being of learners.This paper proposes a decision-making model to address the complex task of selecting educational vendors within the Metaverse with a focus on extended reality (XR). Recognizing the challenges inherent in such selections, this work presents a consensus-based measurement of alternatives and ranking according to compromise solution (MARCOS) model, incorporating single-valued neutrosophic (SVN) information and SVN Dombi-Archimedean weighted aggregation operators. Attribute weights crucial for decision-making are determined through an optimization tool employing cross-entropy measures with partial weight information. The evaluation encompasses nine criteria, including cost, security, privacy, pedagogical alignment, configuration, maintenance, level of control, output tools, degree of immersion, availability of XR learning contents, and time. Four leading vendors including Victory XR, ARuVR, zSpace, and Varjo Technologies are assessed, with zSpace emerging as the most favorable alternative with the highest overall utility value of 0.6889. The paper further conducts sensitivity analysis and performance comparisons to establish the robustness of the ranking results, providing a comprehensive methodology for educational stakeholders addressing the evolving dynamics of the Metaverse.

Multi-attribute decision making based on VIKOR with probabilistic linguistic term sets: An application to the risk evaluation of foreign direct investment.

The multiple global environments have triggered changes in the international environment, leading to a sharp decline of foreign direct investment (FDI) compared to pre-pandemic level. To evaluate the investment risk of FDI and make optimal investment decision becomes the most important issue for investors. This paper focuses on the evaluation of investment risk for FDI. First, an index system for risk evaluation of FDI is constructed. Then, we introduce the probabilistic linguistic entropy and cross entropy measures, based on which, a programming model is developed to identify the objective attribute weights. A composite weight derivation method, which takes both the objective attribute weights and the subjective attribute weights into account, is further introduced. In view of attributes' uncertainty and fuzziness and the conflicting characteristics of some attributes, the VIKOR (the Serbian name: VlseKriterijumska Optimizacija I Kompromisno Resenje, means multi-criteria optimization and compromise solution) method is used to evaluate the risk of FDI under the probabilistic linguistic environment. Furthermore, a case study is presented to illustrate the proposed method. The comparative analysis and some further discussions verify the validity of the proposed method for the FDI risk evaluation.

A novel uncertain information modeling method based on cosine similarity and cross entropy under spherical uncertain linguistic fuzzy set

Multi-attribute group decision-making (MAGDM) is one of the research hotspots in human cognitive and decision-making theory. However, there are still challenges to the existing MAGDM methods in modeling uncertain linguistics of decision-makers’ (DMs’) cognitive information and objectively obtaining weights. Therefore, this paper aims to develop a new MAGDM method considering incomplete known weight information under spherical uncertain linguistic sets (SULSs) to model uncertain information in MAGDM problems. The method mainly includes the following aspects. Firstly, a new concept, which enables an intuitive evaluation of neutral membership and hesitancy degrees at the linguistic evaluation, has been is first developed for capturing the more uncertain information. Secondly, the cosine similarity measure (CSM) and cross-entropy measure (CEM) are widely used to measure ambiguous information because of their robustness of measurement results. The CSM and CEM are extended to SULSs to calculate the DMs’ and attributes weights quantitively, respectively. Thirdly, in terms of effective integration of fuzzy information to obtain more accurate decision results, the Hamy mean (HM) and dual Hamy mean (DHM) operators are valued due to their consideration of the interrelationships between inputs. Two extension operators, named spherical fuzzy uncertain linguistic weight HM and DHM, are proposed to integrate spherical fuzzy uncertain linguistic information in the third stage. In the experiment, a decision case is presented to illustrate the applicability of the proposed method, and results show the effectiveness, flexibility and advantages of the proposed method are demonstrated by numerical examples and comparative analysis.

A heterogeneous picture fuzzy SWARA-MARCOS evaluation framework based on a novel cross-entropy measure

Multi-criteria decision-making (MCDM) entails a heterogeneous decision-making problem, which poses challenges for decision-makers (DMs) in generating an optimal solution. To address this, we have proposed a heterogeneous evaluation framework. First, a novel picture fuzzy cross-entropy measure was defined with the simultaneous consideration of uncertainty and hesitancy of picture fuzzy information, overcoming the shortcomings of the existing cross-entropy measure in relation to its validity and properties. Next, an optimisation-model for determining the objective weights of criteria was constructed based on the proposed closeness measurement and the step-wise weight assessment ratio analysis (SWARA) method. This model was constituted from both objective and subjective perspectives under the circumstance of completely unknown criterion information. Additionally, the conventional Measurement of Alternatives and Ranking according to Compromise Solution (MARCOS) method was extended to the picture fuzzy environment, and a normalisation method was developed to transform heterogeneous judgments, including linguistic terms, interval numbers, and picture fuzzy numbers, into a unified representation form. A heterogeneous picture fuzzy SWARA-MARCOS evaluation framework was then established and used to solve a military equipment supplier selection problem. The results demonstrated the validity and feasibility of the proposed evaluation framework, while sensitivity, comparative, and complexity analyses demonstrated the robustness and superiority of it.

A novel CE-PT-MABAC method for T-spherical uncertain linguistic multiple attribute group decision-making

A T-spherical uncertain linguistic set (TSULS) is not only an expanded form of the T-spherical fuzzy set and the uncertain linguistic set but can also integrate the quantitative judging ideas and qualitative assessing information of decision-makers. For the description of complex and uncertain assessment data, TSULS is a powerful tool for the precise description and reliable processing of information data. However, the existing multi-attribute border approximation area comparison (MABAC) method has not been studied in TSULS. Thus, the goal of this paper is to extend and improve the MABAC method to tackle group decision-making problems with completely unknown weight information in the TSUL context. First, the cross-entropy measure and the interactive operation laws for the TSUL numbers are defined, respectively. Then, the two interactive aggregation operators for TSUL numbers are developed, namely T-spherical uncertain linguistic interactive weighted averaging and T-spherical uncertain linguistic interactive weighted geometric operators. Their effective properties and some special cases are also investigated. Subsequently, a new TSULMAGDM model considering the DM’s behavioral preference and psychology is built by integrating the interactive aggregation operators, the cross-entropy measure, prospect theory, and the MABAC method. To explore the effectiveness and practicability of the proposed model, an illustrative example of Sustainable Waste Clothing Recycling Partner selection is presented, and the results show that the optimal solution is h3. Finally, the reliable, valid, and generalized nature of the method is further verified through sensitivity analysis and comparative studies with existing methods.

Evaluation of Propulsion Technologies for Sustainable Road Freight Distribution Using a Dual Probabilistic Linguistic Group Decision-Making Approach

For the first time, this study addresses the problem of evaluating propulsion technologies for sustainable road freight distribution from the multicriteria group decision-making perspective. It aims to help logistics companies involved in global supply chains significantly reduce GHG emissions from freight distribution activities. To evaluate sustainable propulsion technologies, this article introduces a novel group decision-making approach with DPL term sets, which takes into account the simultaneous occurrence of stochastic and nonstochastic uncertainty. Since the existing methods do not consider the dependency among multiple numbers of criteria, the evaluation results become insignificant. To overcome this, we employ the Archimedean operations on DPL elements to develop DPL weighted Archimedean Maclaurin symmetric mean operator. For measuring the weights of criteria, an optimization model using a cross-entropy measure of DPL elements is developed. A case study for one of the major logistics companies in Serbia is presented to exhibit the relevance of the suggested approach, as well as offer decision-making guidelines in real-world settings. The research findings show that hybrid electric trucks are the most advantageous propulsion technology solution to reduce GHG emissions from distribution activities. The superiority of the proposed approach is demonstrated through the comparative investigation.

Stable and Fast Deep Mutual Information Maximization Based on Wasserstein Distance.

Deep learning is one of the most exciting and promising techniques in the field of artificial intelligence (AI), which drives AI applications to be more intelligent and comprehensive. However, existing deep learning techniques usually require a large amount of expensive labeled data, which limit the application and development of deep learning techniques, and thus it is imperative to study unsupervised machine learning. The learning of deep representations by mutual information estimation and maximization (Deep InfoMax or DIM) method has achieved unprecedented results in the field of unsupervised learning. However, in the DIM method, to restrict the encoder to learn more normalized feature representations, an adversarial network learning method is used to make the encoder output consistent with a priori positively distributed data. As we know, the model training of the adversarial network learning method is difficult to converge, because there is a logarithmic function in the loss function of the cross-entropy measure, and the gradient of the model parameters is susceptible to the "gradient explosion" or "gradient disappearance" phenomena, which makes the training of the DIM method extremely unstable. In this regard, we propose a Wasserstein distance-based DIM method to solve the stability problem of model training, and our method is called the WDIM. Subsequently, the training stability of the WDIM method and the classification ability of unsupervised learning are verified on the CIFAR10, CIFAR100, and STL10 datasets. The experiments show that our proposed WDIM method is more stable to parameter updates, has faster model convergence, and at the same time, has almost the same accuracy as the DIM method on the classification task of unsupervised learning. Finally, we also propose a reflection of future research for the WDIM method, aiming to provide a research idea and direction for solving the image classification task with unsupervised learning.

An Entropy-Based Approach to Measuring Diet Adherence.

The aim of this study is to put forward an approach designed to calculate and sum up discrepancies between the actual food acquisition or intake and any standard or norm. Based on secondary analysis of cross-sectional data from the Mexican National Survey of Household Income and Expenditure, our proposed method to produce classes of entropy-based Diet Adherence Indices generates a Food Basket Adherence Index (FBAI) for Mexico City (2129 households). Findings suggest that it is possible to measure and decompose diet adherence using a cross entropy measure. Using food expenditure data and a normative food basket for Mexico City results, we show households' deviations from the suggested norm for different food groups. The average FBAI was 0.44, far above the minimum score of 0 which would indicate full adherence to the normative food basket. Our measure has a distribution wide enough to detect meaningful changes and distinguish between groups with known differences, providing important new insights on the linkages between home food environments and income distribution, and food insecurity and household distribution.

Spoofing Cross-Entropy Measure in Boson Sampling.

Cross-entropy (XE) measure is a widely used benchmark to demonstrate quantum computational advantage from sampling problems, such as random circuit sampling using superconducting qubits and boson sampling (BS). We present a heuristic classical algorithm that attains a better XE than the current BS experiments in a verifiable regime and is likely to attain a better XE score than the near-future BS experiments in a reasonable running time. The key idea behind the algorithm is that there exist distributions that correlate with the ideal BS probability distribution and that can be efficiently computed. The correlation and the computability of the distribution enable us to postselect heavy outcomes of the ideal probability distribution without computing the ideal probability, which essentially leads to a large XE. Our method scores a better XE than the recent Gaussian BS experiments when implemented at intermediate, verifiable system sizes. Much like current state-of-the-art experiments, we cannot verify that our spoofer works for quantum-advantage-size systems. However, we demonstrate that our approach works for much larger system sizes in fermion sampling, where we can efficiently compute output probabilities. Finally, we provide analytic evidence that the classical algorithm is likely to spoof noisy BS efficiently.

Time-sequential hesitant fuzzy entropy, cross-entropy and correlation coefficient and their application to decision making

In decision making problems, the hesitance of the decision maker to the same decision object is reflected not only in the length of multiple possible memberships under, but also in the time dimension. The newly proposed time-sequential hesitant fuzzy set (TSHFS) can show the hesitance in decision making information from the view of time dimension, and in which the concept of hesitance of hesitant fuzzy sets is improved from a constructive perspective. However, it is difficult to further explore the application prospects of TSHFS in decision making as the current measurement methods, such as entropy, cross-entropy, and correlation coefficient, could not be leveraged directly in the TSHFS environment. For that entropy, cross-entropy, and correlation coefficient are important metric components of fuzzy sets and in related scenarios, we propose corresponding methods for TSHFS in this work. Meanwhile, to make full use of the importance of attribute information in the decision matrix and make the decision results have better discrimination ability, hesitant fuzzy diffusion decision model is proposed. The proposed entropy measures are constructed on trigonometric functions, the new cross-entropy measure and correlation coefficients are derived from classical counterparts, which enrich ways to deal with problems under the TSHFS environment, especially referring to decision making. In hesitant fuzzy diffusion decision model, the prior ordered value is defined as the final decision result to measure relationships among alternatives. The effectiveness and advantages of proposed entropy measures, cross-entropy measure, correlation coefficients, and the proposed decision model are demonstrated through the application of decision making.

Transformer's frequency response analysis results interpretation using a novel cross entropy based methodology

Transformer defects can be identified by the FRA (frequency response analysis) that is a promising diagnostic technique. Despite the standardization in FRA measuring technique, its results interpretation is yet a research area. Because different faults types can be identified in various frequency bounds of the FRA signatures, it is necessary to identify the possible relationships between specific failures and frequency ranges in this contribution. For this purpose, a real transformer is used to conduct the essential tests, which include both healthy and faulted circumstances (axial displacement (AD), radial deformation (RD), and short-circuits (SC)). To identify efficient characteristics from the produced frequency response traces and improve interpretation accuracy of such traces, a new hyperbolic fuzzy cross entropy (FCE) measure is demonstrated and then utilized for the aim of discrimination and classification of transformer winding defects in pre-defined frequency ranges. After normalizing FRA results of the transformer under healthy and various fault circumstances the lower bounds from such responses have been extracted and then utilized to construct the desired form of the fuzzy sets of healthy and faulted circumstances. Then, a new hyperbolic FCE measure-based discrimination and classification of winding faults methodology is offered on the basis of highest and lowest FCE measure values. The highest FCE measure value between the fuzzy sets of healthy and faulted circumstances such as AD, RD and SC is designated to confirm the occurrence of winding faults in a suitable frequency range. The suggested methodology ensures smart interpretation of FRA signature and accurate classification of winding faults as it can effectively discriminate both healthy and faulted circumstances in the desired frequency ranges. The proposed approaches' performance is tested and compared by applying the experimental data after feature extraction.

Emergency response scheme selection with T-spherical hesitant probabilistic fuzzy TODIM-TPZSG approach

This paper aims to investigate an innovative framework to handle emergency response scheme selection (ERSS) issues by integrating TODIM and TPZSG (two-person zero-sum game) methods under novel T-spherical hesitant probabilistic fuzzy set (T-SHPFS) environments. First, T-SHPFS is defined as an extension of the existing tools, which can depict the complex assessment information including several possible values of the various membership functions’ degrees and the associated statistical uncertainty information. Concomitantly, T-SHPFS’s normalization method, comparison laws, operation rules, cross-entropy measure and Hausdorff distance are explored. Then, an objective attribute weight determining model is constructed, considering the credibility of T-SHPF evaluations and the divergence degrees between attribute assessments simultaneously. Next, an integrated TODIM-TPZSG decision-making approach is developed to select the most desirable emergency response scheme. Finally, an illustrative example concerning the selection of the best medical waste disposal method during the COVID-19 epidemic is conducted to verify the effectiveness of the proposed TODIM-TPZSG method. Sensitivity analysis and comparisons between the TODIM-TPZSG and other representative methods are also provided to demonstrate the superiorities of the proposed method. The results reveal that the developed T-SHPFSs give DMs more assessment freedom; the proposed TODIM-TPZSG approach considers the decision makers’ psychological behaviors; the ranking results of the proposed method can reflect the specific divergence degrees among the alternatives; and the needed computation burden and computational complexity are low and less affected by the number of alternatives and criteria than most current ERSS methods.

Some uncertainty measures for probabilistic hesitant fuzzy information

The probabilistic hesitant fuzzy sets (PHFSs), which have received a lot of attention and have been widely used in many domains, are effective in describing hesitant evaluations. Despite significant progress, entropy and cross-entropy, as the famous uncertainty measures of probabilistic hesitant fuzzy information, have not yet been thoroughly investigated. However, these uncertainty measures are quite important and useful in the context of multi-attribute decision-making (MADM). In this contribution, we revisit the concept of uncertainty measures for probabilistic hesitant fuzzy information by comprehensively considering their fuzziness and hesitancy, and proposed some novel entropy and cross-entropy measures for them. Specifically, the hybrid entropy measures are designed as convex combinations of the fuzzy entropy factor and the hesitant entropy factor, and in the same line of thought, the hybrid cross-entropy measures are also designed as convex combinations of the fuzzy cross-entropy factor and the hesitant cross-entropy factor. It is shown that the hybrid entropy measures are simple in structure, clear in physical meaning, and strong in discriminating capability, while the hybrid cross-entropy measures can avoid the design flaws of existing cross-entropy measures and employ natural symmetry. It is further shown that the hybrid entropy and cross-entropy measures can be designed in pairs and are mutually supportive, which suggests that they are internally unified as the uncertainty measures of probabilistic hesitant fuzzy information. In the end, based on the proposed uncertainty measures, an MADM model is developed within the TOPSIS framework to evaluate a company’s strategic initiatives under probabilistic hesitant fuzzy environment. Numerical results show that these uncertainty measures can support group decision-making effectively and efficiently with probabilistic hesitant fuzzy information.

Entropy of credibility distribution for intuitionistic fuzzy variable

<abstract> <p>This paper handles the new information entropy measure and divergence measure associated with intuitionistic fuzzy variables (IFVs). Based on credibility distribution and credibility measure of intuitionistic fuzzy variable, the credibility entropy formulas of discrete and continuous IFVs are proposed and some of their properties are investigated. The cross-entropy of intuitionistic fuzzy variable and its relationship with credibility entropy are then discussed. Finally, some numerical examples are given to illustrate the practicability of the presented credibility entropy and cross-entropy of intuitionistic fuzzy variable. Also, we make some comparative analysis on the credibility cross-entropy measure and some existing distance measures of IFVs in the pattern recognition problem.</p> </abstract>

A new hybrid entropy-based decision support method and its application to online shopping selection

The current study proposes a new method called the hybrid entropy-based decision support method (named HEBM) to address multi-criteria decision making (MCDM) problems with uncertainty. In the proposed HEBM, the decision makers (DMs)’ imprecise evaluations are characterized with intuitionistic multiplicative values; weights of the criteria are generated by an improved method with hybrid entropy and cross-entropy measures; and alternatives rankings are determined by defined closeness coefficients. Simulation experiments verify the validity of the improved criteria weights generation method with hybrid entropy and cross-entropy measures. Compared with existing MCDM methods, the proposed HEBM has the following advantages: (1) it avoids the information loss and has higher accuracy; (2) the weights of the criteria derived can directly characterize the DMs’ preferences on the criteria; and (3) quantitative and qualitative information are both analyzed. Simulation experiments and comparative analyses are conducted to demonstrate the effectiveness and superiority of the proposed method. A case study of online shopping selection problems is presented to illustrate the applicability of the proposed method.

Rényi Cross-Entropy Measures for Common Distributions and Processes with Memory

Two Rényi-type generalizations of the Shannon cross-entropy, the Rényi cross-entropy and the Natural Rényi cross-entropy, were recently used as loss functions for the improved design of deep learning generative adversarial networks. In this work, we derive the Rényi and Natural Rényi differential cross-entropy measures in closed form for a wide class of common continuous distributions belonging to the exponential family, and we tabulate the results for ease of reference. We also summarise the Rényi-type cross-entropy rates between stationary Gaussian processes and between finite-alphabet time-invariant Markov sources.

Probabilistic linguistic decision-making based on the hybrid entropy and cross-entropy measures

Probabilistic linguistic decision-making based on the hybrid entropy and cross-entropy measures

T-Spherical fuzzy ORESTE method based on cross-entropy measures and its application in multiple attribute decision-making

T-Spherical fuzzy ORESTE method based on cross-entropy measures and its application in multiple attribute decision-making

Maximal overlap discrete wavelet packet transforms-based bipolar neutrosophic cross entropy measure for identification of rotor defects

This study propounds a novel methodology for automatic identification of rotor defects severity, when the machine is operated at constant speed, through maximal overlap discrete wavelet packet transforms (MODWPT) and proposed cross entropy measures of bipolar neutrosophic sets, single valued neutrosophic sets and fuzzy sets respectively. After the 3rd level of decomposition of raw vibration signals into eight frequency bands (called as WPT3 bands) through MODWPT, the energy reading in each WPT3 band is monitored. Thereafter, the lower (here called as truth membership degrees) and upper energy bounds from the training and testing samples of various rotor defect conditions at each WPT3 band are extracted and then normalized to lie in [0,1]. Finally, the energy intervals are constructed and thereafter converted into the desired forms of fuzzy sets, bipolar neutrosophic sets and single valued neutrosophic sets consecutively. The minimum cross entropy measure values between the fuzzy sets (Method 1), single valued neutrosophic sets (Method 2) and bipolar neutrosophic sets (Method 3) of training and testing samples of rotor defect conditions are computed and utilized to identify defect conditions such as angular/parallel misalignment of 10 Mils, unbalance, rub and defect free respectively. The novelty of the proposed rotor defect identification methodology lies in the fact that all the proposed variants of bipolar neutrosophic cross entropy measure have been found skilled enough to identify the desired rotor defects under study. On another hand, the existing cosine similarity measure [1] does not possess the necessary capability of identifying all rotor defects, especially, the rub defect condition. Furthermore, like the existing cosine similarity measure, our proposed variants of bipolar neutrosophic cross entropy measures have been found equally compatible for fairly furnishing the consistent and feasible results under intuitive analysis.