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Entropy
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Recent Progress in Uncertainty Measures
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Recent Progress in Uncertainty Measures

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Information Theory, Probability and Statistics".

Deadline for manuscript submissions: closed (17 March 2026) | Viewed by 2752

Special Issue Editors

Dr. Xinyang Deng

E-Mail Website
Guest Editor
School of Electronics and Information, Northwestern Polytechnical University, Xi’an 710072, China
Interests: uncertain information fusion; belief function theory
Prof. Dr. Yong Deng

grade E-Mail Website
Guest Editor
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China
Interests: uncertainty measure; Shannon entropy; Tsallis entropy; Renyi entropy; Deng entropy; evidence theory; fuzzy sets; fractal; complex network; time series
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Quantifying uncertainty is one of the crucial issues in information or intelligence systems, in which two types of uncertainties, stochastic uncertainty and epistemic uncertainty, together constitute the total uncertainty. The question of how to measure epistemic and total uncertainty in still an unsolved problem in many uncertainty reasoning theories, for example, fuzzy set theory, rough set theory, possibility theory, belief function theory, imprecise probability theory, and so forth, although there have been many advances in the research of uncertainty measures in these theories ,with unremitting efforts by related research communities. Especially, entropy-like definitions have been extensively involved in a large number of uncertainty measures, but axiomatic properties are not justified strictly in most of those. Justifications for the uncertainty measures, whether theoretical derivations in mathematics or essential requirements from applications, are required, similar as what has been performed for Shannon entropy, Renyi entropy, and Tsallis entropy.

This Special Issue aims to be a forum for the presentation of theoretical breakthroughs, improvement in methods, and progress in applications in the research of uncertainty measures in various uncertainty reasoning theories. In particular, studies on stochastic uncertainty based on Shannon entropy, Renyi entropy, Tsallis entropy, etc., fall within the scope of this Special Issue.

Dr. Xinyang Deng
Prof. Dr. Yong Deng
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • epistemic uncertainty
  • entropy and extropy
  • information theory
  • Deng entropy
  • plausibility entropy
  • fuzzy entropy
  • uncertainty reasoning
  • uncertain information fusion

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Published Papers (4 papers)

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Research

20 pages, 1793 KB  
Article
Nonparametric Tests for Exponentiality Against IFRA Alternatives Based on Cumulative Extropy Measures
by Anfal A. Alqefari
Entropy 2026, 28(2), 208; https://doi.org/10.3390/e28020208 - 11 Feb 2026
Viewed by 374
Abstract
This paper develops two nonparametric test statistics for testing exponentiality against alternatives in the increasing failure rate average (IFRA) class. The proposed procedures are constructed using information-theoretic functionals, namely the cumulative residual extropy and the cumulative past extropy of the first-order statistic. Exploiting [...] Read more.
This paper develops two nonparametric test statistics for testing exponentiality against alternatives in the increasing failure rate average (IFRA) class. The proposed procedures are constructed using information-theoretic functionals, namely the cumulative residual extropy and the cumulative past extropy of the first-order statistic. Exploiting fundamental properties of IFRA distributions, we derive explicit inequality relations that motivate the test statistics and establish their asymptotic normality under mild regularity conditions. To facilitate practical implementation, scale-invariant versions of the proposed tests are introduced, ensuring that their limiting distributions do not depend on unknown scale parameters. A comprehensive Monte Carlo simulation study demonstrates that the proposed tests possess strong power properties and frequently outperform several established competitors, particularly for moderate to large sample sizes. The applicability and effectiveness of the methodology are further illustrated through analyses of real lifetime datasets arising in reliability studies. The proposed tests are shown to be particularly effective for moderate sample sizes and provide a competitive alternative to existing IFRA-based procedures. Full article
(This article belongs to the Special Issue Recent Progress in Uncertainty Measures)
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32 pages, 4335 KB  
Article
Restricted Network Reconstruction from Time Series via Dempster–Shafer Evidence Theory
by Cai Zhang, Yishu Xian, Xiao Yuan, Meizhu Li and Qi Zhang
Entropy 2026, 28(2), 148; https://doi.org/10.3390/e28020148 - 28 Jan 2026
Viewed by 577
Abstract
As a fundamental mathematical model for complex systems, complex networks describe interactions among social, infrastructural, and biological systems. However, the complete connection structure is often unobservable, making topology reconstruction from limited data—such as time series of unit states—a crucial challenge. To address network [...] Read more.
As a fundamental mathematical model for complex systems, complex networks describe interactions among social, infrastructural, and biological systems. However, the complete connection structure is often unobservable, making topology reconstruction from limited data—such as time series of unit states—a crucial challenge. To address network reconstruction under sparse local observations, this paper proposes a novel framework that integrates epidemic dynamics with Dempster–Shafer (DS) evidence theory. The core of our method lies in a two-level belief fusion process: (1) Intra-node fusion, which aggregates multiple independent SIR simulation results from a single seed node to generate robust local evidence represented as Basic Probability Assignments (BPAs), effectively quantifying uncertainty; (2) Inter-node fusion, which orthogonally combines BPAs from multiple seed nodes using DS theory to synthesize a globally consistent network topology. This dual-fusion design enables the framework to handle uncertainty and conflict inherent in sparse, stochastic observations. Extensive experiments demonstrate the effectiveness and robustness of the proposed approach. It achieves stable and high reconstruction accuracy on both a synthetic 16-node benchmark network and the real-world Zachary’s Karate Club network. Furthermore, the method scales successfully to four large-scale real-world networks, attaining an average accuracy of 0.85, thereby confirming its practical applicability across networks of different scales and densities. Full article
(This article belongs to the Special Issue Recent Progress in Uncertainty Measures)
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23 pages, 491 KB  
Article
Properties of Residual Cumulative Sharma–Taneja–Mittal Model and Its Extensions in Reliability Theory with Applications to Human Health Analysis and Mixed Coherent Mechanisms
by Mohamed Said Mohamed and Hanan H. Sakr
Entropy 2026, 28(1), 32; https://doi.org/10.3390/e28010032 - 26 Dec 2025
Viewed by 421
Abstract
The entropy measure of residual cumulative Sharma–Taneja–Mittal is an alternative measure of uncertainty for residual cumulative entropy. This study investigates further theoretical properties and develops nonparametric estimation procedures for the proposed measure. The performance of the estimator is evaluated through simulation experiments, and [...] Read more.
The entropy measure of residual cumulative Sharma–Taneja–Mittal is an alternative measure of uncertainty for residual cumulative entropy. This study investigates further theoretical properties and develops nonparametric estimation procedures for the proposed measure. The performance of the estimator is evaluated through simulation experiments, and its practical relevance is illustrated using a real-world dataset on malignant tumor cases. Moreover, we investigate the properties of its dynamic version, including stochastic comparisons and its connections with the hazard rate function, mean residual function, and equilibrium random variables. Moreover, we introduce an alternative version of dynamic residual cumulative Sharma–Taneja–Mittal entropy and examine its monotonic properties. Additionally, we discuss this alternative version and its conditional form in the circumstances of record values. We introduce this alternative expression for the residual lifespan of upper record quantities in general distributions, characterizing it as a measure of upper record quantities derived from a distribution of uniform. Since Sharma–Taneja–Mittal entropy measures uncertainty, we also investigate its use in determining the entropy of the lifespan of mixed and coherent mechanisms, in which the lives of its constituent components are identically distributed and independent. Full article
(This article belongs to the Special Issue Recent Progress in Uncertainty Measures)
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28 pages, 1211 KB  
Article
Information-Theoretic Reliability Analysis of Consecutive r-out-of-n:G Systems via Residual Extropy
by Anfal A. Alqefari, Ghadah Alomani, Faten Alrewely and Mohamed Kayid
Entropy 2025, 27(11), 1090; https://doi.org/10.3390/e27111090 - 22 Oct 2025
Viewed by 701
Abstract
This paper develops an information-theoretic reliability inference framework for consecutive r-out-of-n:G systems by employing the concept of residual extropy, a dual measure to entropy. Explicit analytical representations are established in tractable cases, while novel bounds are derived for more complex [...] Read more.
This paper develops an information-theoretic reliability inference framework for consecutive r-out-of-n:G systems by employing the concept of residual extropy, a dual measure to entropy. Explicit analytical representations are established in tractable cases, while novel bounds are derived for more complex lifetime models, providing effective tools when closed-form expressions are unavailable. Preservation properties under classical stochastic orders and aging notions are examined, together with monotonicity and characterization results that offer deeper insights into system uncertainty. A conditional formulation, in which all components are assumed operational at a given time, is also investigated, yielding new theoretical findings. From an inferential perspective, we propose a maximum likelihood estimator of residual extropy under exponential lifetimes, supported by simulation studies and real-world reliability data. These contributions highlight residual extropy as a powerful information-theoretic tool for modeling, estimation, and decision-making in multicomponent reliability systems, thereby aligning with the objectives of statistical inference through entropy-like measures. Full article
(This article belongs to the Special Issue Recent Progress in Uncertainty Measures)
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