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Symmetry
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Uncertainty Theory: Symmetry and Applications
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Uncertainty Theory: Symmetry and Applications

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (18 September 2022) | Viewed by 20602

Special Issue Editors

Dr. Tingqing Ye

E-Mail Website
Guest Editor
School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China
Interests: uncertainty theory; mathematics
Dr. Waichon Lio

E-Mail Website
Guest Editor
School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China
Interests: uncertainty theory; mathematics
Prof. Dr. Baoding Liu

E-Mail Website
Guest Editor
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Interests: uncertainty theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Uncertainty theory is a branch of mathematics concerned with the analysis of belief degree. Recently, the study of uncertainty theory is in a period of rapid development, involving fields including biology, medical and biomedical sciences, finance, economics, social sciences, environmental sciences, engineering, and industry. The aim of this Special Issue is to attract leading researchers in these areas in order to include new high-quality results involving their symmetry properties, both from a theoretical and an applied point of view. All articles related to uncertainty theory are invited to be submitted for this Special Issue.

The topics of interest for this Special Issue include but are not limited to:

  1. uncertain statistics
  2. uncertain programming
  3. uncertain risk and reliability analysis
  4. uncertain logic
  5. uncertain set
  6. uncertain inference
  7. uncertain process
  8. uncertain calculus
  9. uncertain differential equation
  10. uncertain finance

Dr. Tingqing Ye
Dr. Waichon Lio
Prof. Dr. Baoding Liu
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 100 words) can be sent to the Editorial Office for announcement on this website.

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. Symmetry 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 2400 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

  • uncertainty theory
  • uncertain statistics
  • uncertain programming
  • uncertain process
  • uncertain differential equation
  • uncertain finance
  • symmetry in expert systems

Published Papers (13 papers)

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12 pages, 288 KiB  
Article
Uncertain Stochastic Optimal Control with Jump and Its Application in a Portfolio Game
by Chengyu Wu, Lu Yang and Chengke Zhang
Symmetry 2022, 14(9), 1885; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14091885 - 09 Sep 2022
Viewed by 1090
Abstract
This article describes a class of jump-uncertain stochastic control systems, and derives an Itô–Liu formula with jump. We characterize an optimal control law, that satisfies the Hamilton–Jacobi–Bellman equation with jump. Then, this paper deduces the optimal portfolio game under uncertain stochastic financial markets [...] Read more.
This article describes a class of jump-uncertain stochastic control systems, and derives an Itô–Liu formula with jump. We characterize an optimal control law, that satisfies the Hamilton–Jacobi–Bellman equation with jump. Then, this paper deduces the optimal portfolio game under uncertain stochastic financial markets with jump. The information of players is symmetrical. The financial market is constituted of a risk-free asset and a risky asset whose price process is subjected to the jump-uncertain stochastic Black–Scholes model. The game is formulated by two utility maximization problems, each investor tries to maximize his relative utility, which is the weighted average of terminal wealth difference between his terminal wealth and that of his competitor. Finally, the explicit expressions of equilibrium investment strategies and value functions for the constant absolute risk-averse and constant relative risk-averse utility function are derived by using the dynamic programming principle. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
13 pages, 270 KiB  
Article
Symmetry of Sampling Problem Based on Epistemic Uncertainty and Ellsberg Urn
by Waichon Lio and Rui Kang
Symmetry 2022, 14(9), 1790; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14091790 - 29 Aug 2022
Viewed by 909
Abstract
A general sampling problem can be described by an Ellsberg urn, which is a mathematical model that assumes that balls are randomly drawn from an urn with an uncertain numbers of colored balls. This means that the Ellsberg urn is essentially an intricate [...] Read more.
A general sampling problem can be described by an Ellsberg urn, which is a mathematical model that assumes that balls are randomly drawn from an urn with an uncertain numbers of colored balls. This means that the Ellsberg urn is essentially an intricate model with simultaneous randomness and epistemic uncertainty, and this is the core problem discussed in this paper. Since practical sampling is usually processed in an intricate environment, the solution for an equivalent mathematical problem is necessary. Suppose an Ellsberg urn contains three unknown numbers of colored balls (i.e., a two-degrees-of-freedom Ellsberg urn), and three balls are randomly drawn from the urn. Compared to the published papers, this paper first constructs a chance space with two-dimensional uncertainty space and three-dimensional probability space to rigorously calculate the color distributions for those drawn balls by uncertainty theory, probability theory, and chance theory. Moreover, it is interesting to find that all cases of the drawn balls are symmetric in such a specific situation of a sample problem with epistemic uncertainty. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
18 pages, 784 KiB  
Article
Uncertainty Theory-Based Resilience Analysis for LEO Satellite Communication Systems
by Ji Ma, Rui Kang, Ruiying Li, Qingyuan Zhang, Liang Liu and Xuewang Wang
Symmetry 2022, 14(8), 1568; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081568 - 29 Jul 2022
Cited by 1 | Viewed by 1445
Abstract
Resilience, the ability of a system to withstand disruptions and quickly return to a normal state, is essential for a low-Earth-orbit satellite communication system (LEO-SCS), as its large number of satellites, may encounter various disturbances. As a developing system with asymmetrical and insufficient [...] Read more.
Resilience, the ability of a system to withstand disruptions and quickly return to a normal state, is essential for a low-Earth-orbit satellite communication system (LEO-SCS), as its large number of satellites, may encounter various disturbances. As a developing system with asymmetrical and insufficient data, it is with both aleatory and epistemic uncertainties, which renders existing resilience measures inapplicable. This paper utilizes uncertainty theory to define belief instantaneous availability, based on which a new resilience measure for the LEO-SCS is proposed. As the resilience is mainly determined by the dynamic and uncertain characteristics of an LEO-SCS, an uncertain satellite network evolution model is developed to describe its operating patterns, and an evaluation method is proposed to estimate its resilience. To illustrate our proposed uncertain theory-based resilience evaluation method, an LEO-SCS case with 48 satellites is studied, and its resilience values under different backup strategies are compared to assist in system design decisions. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
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12 pages, 301 KiB  
Article
Types of Maintenance Based on Uncertain Data Envelope Analysis
by Mengqiao Lu, Shuyu Li and Meilin Wen
Symmetry 2022, 14(7), 1429; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071429 - 12 Jul 2022
Cited by 1 | Viewed by 1078
Abstract
Nowadays, one of the main challenges of marine maintenance is how to select an optimum maintenance strategy for each component of the complex ship machinery system. The uncertainty of the parameters is one of the main difficulties encountered. For example, engineers and experts [...] Read more.
Nowadays, one of the main challenges of marine maintenance is how to select an optimum maintenance strategy for each component of the complex ship machinery system. The uncertainty of the parameters is one of the main difficulties encountered. For example, engineers and experts are always questioning the credibility and integrity of the data collected, and historical maintenance records may also be lost during maintenance. The above data asymmetry will also lead to parameter uncertainty, which directly affects the accuracy of the prediction results. This paper proposes a method for determining maintenance types of ship equipment based on uncertainty theory and the Data Envelopment Analysis (DEA) model. Firstly, this paper constructs an uncertain maintenance optimization model (UMOM) based on the classic Data Envelopment Analysis model. Then, we converted the UMOM into an equivalent deterministic model for easy calculation by the uncertainty theory. Finally, a case study is given to verify this model. The results will conclude that the UMOM can meet the need for a reasonable classification of maintenance types of mechanical equipment in a marine system and provide valuable information for systemic management and storage. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
11 pages, 274 KiB  
Article
A New Uncertain Interest Rate Model with Application to Hibor
by Yang Liu, Huiting Jing and Tingqing Ye
Symmetry 2022, 14(7), 1344; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14071344 - 29 Jun 2022
Viewed by 1008
Abstract
This paper proposes a new interest rate model by using uncertain mean-reverting differential equation. Based on the model, the pricing formulas of the zero-coupon bond, the interest rate ceiling and interest rate floor are derived respectively according to Yao-Chen formula. The symmetry appears [...] Read more.
This paper proposes a new interest rate model by using uncertain mean-reverting differential equation. Based on the model, the pricing formulas of the zero-coupon bond, the interest rate ceiling and interest rate floor are derived respectively according to Yao-Chen formula. The symmetry appears in mathematical formulations of the interest rate ceiling and interest rate floor pricing formula. Furthermore, the model is applied to depict Hong Kong interbank offered rate (Hibor). Finally the parameter estimation by the method of moments and hypothesis test is completed. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
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22 pages, 3894 KiB  
Article
System Resilience Evaluation and Optimization Considering Epistemic Uncertainty
by Qiang Dong, Ruiying Li and Rui Kang
Symmetry 2022, 14(6), 1182; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14061182 - 08 Jun 2022
Cited by 3 | Viewed by 1447
Abstract
Epistemic uncertainties, caused by data asymmetry and deficiencies, exist in resilience evaluation. Especially in the system design process, it is difficult to obtain enough data for system resilience evaluation and improvement. Mathematics methods, such as evidence theory and Bayesian theory, have been used [...] Read more.
Epistemic uncertainties, caused by data asymmetry and deficiencies, exist in resilience evaluation. Especially in the system design process, it is difficult to obtain enough data for system resilience evaluation and improvement. Mathematics methods, such as evidence theory and Bayesian theory, have been used in the resilience evaluation for systems with epistemic uncertainty. However, these methods are based on subjective information and may lead to an interval expansion problem in the calculation. Therefore, the problem of how to quantify epistemic uncertainty in the resilience evaluation is not well solved. In this paper, we propose a new resilience measure based on uncertainty theory, a new branch of mathematics that is viewed as appropriate for modeling epistemic uncertainty. In our method, resilience is defined as an uncertainty measure that is the belief degree of a system’s behavior after disruptions that can achieve the predetermined goal. Then, a resilience evaluation method is provided based on the operation law in uncertainty theory. To design a resilient system, an uncertain programming model is given, and a genetic algorithm is applied to find an optimal design to develop a resilient system with the minimal cost. Finally, road networks are used as a case study. The results show that our method can effectively reduce cost and ensure network resilience. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
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15 pages, 442 KiB  
Article
Uncertain Hypergraphs: A Conceptual Framework and Some Topological Characteristics Indexes
by Jin Peng, Bo Zhang and Kiki Ariyanti Sugeng
Symmetry 2022, 14(2), 330; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020330 - 05 Feb 2022
Cited by 8 | Viewed by 1416
Abstract
In practical applications of hypergraph theory, we are usually surrounded by the state of indeterminacy. This paper employs uncertainty theory to address indeterministic information. We initially put forward the idea of an uncertain hypergraph by combining hypergraph theory with uncertainty theory in order [...] Read more.
In practical applications of hypergraph theory, we are usually surrounded by the state of indeterminacy. This paper employs uncertainty theory to address indeterministic information. We initially put forward the idea of an uncertain hypergraph by combining hypergraph theory with uncertainty theory in order to provide a useful tool to deal with a variety of uncertain complex systems and to create a new interdisciplinary research field. The main focus of this paper is to propose a conceptual framework of uncertain hypergraphs and to study the operations of uncertain hypergraphs. Moreover, some topological indexes are proposed to describe the characteristics of the structures of uncertain hypergraph. Additionally, some further research directions are discussed. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
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17 pages, 2294 KiB  
Article
Human Decision Time in Uncertain Binary Choice
by Lunhu Hu, Xing Pan, Song Ding and Rui Kang
Symmetry 2022, 14(2), 201; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020201 - 20 Jan 2022
Cited by 1 | Viewed by 2254
Abstract
Decision time, also known as choice reaction time, has been frequently discussed in the field of psychology. The Hick–Hyman Law (HHL) has been a fundamental model that has revealed the quantitative relationship between the mean choice reaction time of human and the information [...] Read more.
Decision time, also known as choice reaction time, has been frequently discussed in the field of psychology. The Hick–Hyman Law (HHL) has been a fundamental model that has revealed the quantitative relationship between the mean choice reaction time of human and the information entropy of stimuli. However, the HHL is only focused on rule-based behavior in which rules for selecting response according to stimulus are certain and neglects to model the knowledge-based behavior in which choices are uncertain and influenced by human belief. In this article, we explored the decision time related to one basic knowledge-based behavior—uncertain binary choice, where selections of response are determined by human belief degrees but not by stimuli uncertainties. Two experiments were conducted: one for verifying the HHL and the other for uncertain binary choice. The former (experiment) demonstrated the effectiveness of the HHL, and the latter one indicated that there is an exponential relationship existing between decision time and entropy of belief degree in uncertain binary choice. Moreover, data obtained from both experiments showed that the disturbance term of decision time should not be seen as probabilistic as existing studies have assumed, which highlighted the necessity and advantage of uncertain regression analysis. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
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12 pages, 343 KiB  
Article
Symmetry Analysis of the Uncertain Alternative Box-Cox Regression Model
by Liang Fang, Zaiying Zhou and Yiping Hong
Symmetry 2022, 14(1), 22; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14010022 - 24 Dec 2021
Cited by 4 | Viewed by 2136
Abstract
The asymmetry of residuals about the origin is a severe issue in estimating a Box-Cox transformed model. In the framework of uncertainty theory, there are such theoretical issues regarding the least-squares estimation (LSE) and maximum likelihood estimation (MLE) of the linear models after [...] Read more.
The asymmetry of residuals about the origin is a severe issue in estimating a Box-Cox transformed model. In the framework of uncertainty theory, there are such theoretical issues regarding the least-squares estimation (LSE) and maximum likelihood estimation (MLE) of the linear models after the Box-Cox transformation on the response variables. Heretofore, only weighting methods for least-squares analysis have been available. This article proposes an uncertain alternative Box-Cox model to alleviate the asymmetry of residuals and avoid λ tending to negative infinity for uncertain LSE or uncertain MLE. Such symmetry of residuals about the origin is reasonable in applications of experts’ experimental data. The parameter estimation method was given via a theorem, and the performance of our model was supported via numerical simulations. According to the numerical simulations, our proposed ‘alternative Box-Cox model’ can overcome the problems of a grossly underestimated lambda and the asymmetry of residuals. The estimated residuals neither deviated from zero nor changed unevenly, in clear contrast to the LSE and MLE for the uncertain Box-Cox model downward biased residuals. Thus, though the LSE and MLE are not applicable on the uncertain Box-Cox model, they fit the uncertain alternative Box-Cox model. Compared with the uncertain Box-Cox model, the issue of a systematically underestimated λ is not likely to occur in our uncertain alternative Box-Cox model. Both the LSE and MLE can be used directly without constructing a weighted estimation method, offering better performance in the asymmetry of residuals. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
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8 pages, 230 KiB  
Article
A New Formula for Calculating Uncertainty Distribution of Function of Uncertain Variables
by Yuxing Jia, Yuer Lv and Zhigang Wang
Symmetry 2021, 13(12), 2429; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122429 - 15 Dec 2021
Viewed by 1612
Abstract
As a mathematical tool to rationally handle degrees of belief in human beings, uncertainty theory has been widely applied in the research and development of various domains, including science and engineering. As a fundamental part of uncertainty theory, uncertainty distribution is the key [...] Read more.
As a mathematical tool to rationally handle degrees of belief in human beings, uncertainty theory has been widely applied in the research and development of various domains, including science and engineering. As a fundamental part of uncertainty theory, uncertainty distribution is the key approach in the characterization of an uncertain variable. This paper shows a new formula to calculate the uncertainty distribution of strictly monotone function of uncertain variables, which breaks the habitual thinking that only the former formula can be used. In particular, the new formula is symmetrical to the former formula, which shows that when it is too intricate to deal with a problem using the former formula, the problem can be observed from another perspective by using the new formula. New ideas may be obtained from the combination of uncertainty theory and symmetry. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
10 pages, 772 KiB  
Article
Analysis and Prediction for Confirmed COVID-19 Cases in Czech Republic with Uncertain Logistic Growth Model
by Chunxiao Ding and Wenjian Liu
Symmetry 2021, 13(12), 2264; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122264 - 28 Nov 2021
Cited by 3 | Viewed by 1208
Abstract
This paper presents an uncertain logistic growth model to analyse and predict the evolution of the cumulative number of COVID-19 infection in Czech Republic. Some fundamental knowledge about the uncertain regression analysis are reviewed firstly. Stochastic regression analysis is invalid to model cumulative [...] Read more.
This paper presents an uncertain logistic growth model to analyse and predict the evolution of the cumulative number of COVID-19 infection in Czech Republic. Some fundamental knowledge about the uncertain regression analysis are reviewed firstly. Stochastic regression analysis is invalid to model cumulative number of confirmed COVID-19 cases in Czech Republic, by considering the disturbance term as random variables, because that the normality test and the identical distribution test of residuals are not passed, and the residual plot does not look like a null plot in the sense of probability theory. In this case, the uncertain logistic growth model is applied by characterizing the disturbance term as uncertain variables. Then parameter estimation, residual analysis, the forecast value and confidence interval are studied. Additionally, the uncertain hypothesis test is proposed to evaluate the appropriateness of the fitted logistic growth model and estimated disturbance term. The analysis and prediction for the cumulative number of COVID-19 infection in Czech Republic can propose theoretical support for the disease control and prevention. Due to the symmetry and similarity of epidemic transmission, other regions of COVID-19 infections, or other diseases can be disposed in a similar theory and method. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
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28 pages, 8724 KiB  
Article
Multi-Objective UAV Trajectory Planning in Uncertain Environment
by Aoyu Zheng, Bingjie Li, Mingfa Zheng and Haitao Zhong
Symmetry 2021, 13(11), 2160; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112160 - 11 Nov 2021
Cited by 2 | Viewed by 1476
Abstract
UAV trajectory planning is one of the research focuses in artificial intelligence and UAV technology. The asymmetric information, however, will lead to the uncertainty of the UAV trajectory planning; the probability theory as the most commonly used method to solve the trajectory planning [...] Read more.
UAV trajectory planning is one of the research focuses in artificial intelligence and UAV technology. The asymmetric information, however, will lead to the uncertainty of the UAV trajectory planning; the probability theory as the most commonly used method to solve the trajectory planning problem in uncertain environment will lead to unrealistic conclusions under the condition of lacking samples, while the uncertainty theory based on uncertain measures is an efficient method to solve such problems. Firstly, the uncertainties in trajectory planning are sufficiently considered in this paper; the fuel consumption, concealment and threat degree with uncertain variables are taken as the objective functions; the constraints are analyzed according to the maneuverability; and the uncertain multi-objective trajectory planning (UMOTP) model is established. After that, this paper takes both the long-term benefits and its stability into account, and then, the expected-value and standard-deviation efficient trajectory model is established. What is more, this paper solves the Pareto front of the trajectory planning, satisfying various preferences, which avoids the defects of the trajectory obtained by traditional model only applicable to a certain specific situation. In order to obtain a better solution set, this paper proposes an improved backbones particle swarm optimization algorithm based on PSO and NSGA-II, which overcomes the shortcomings of the traditional algorithm such as premature convergence and poor robustness, and the efficiency of the algorithm is tested. Finally, the algorithm is applied to the UMOTP problem; then, the optimal trajectory set is obtained, and the effectiveness and reliability of the model is verified. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
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17 pages, 2315 KiB  
Article
Modeling RL Electrical Circuit by Multifactor Uncertain Differential Equation
by Yang Liu and Lujun Zhou
Symmetry 2021, 13(11), 2103; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112103 - 05 Nov 2021
Cited by 3 | Viewed by 1397
Abstract
The symmetry principle of circuit system shows that we can equate a complex structure in the circuit network to a simple circuit. Hence, this paper only considers a simple series RL circuit and first presents an uncertain RL circuit model based on multifactor [...] Read more.
The symmetry principle of circuit system shows that we can equate a complex structure in the circuit network to a simple circuit. Hence, this paper only considers a simple series RL circuit and first presents an uncertain RL circuit model based on multifactor uncertain differential equation by considering the external noise and internal noise in an actual electrical circuit system. Then, the solution of uncertain RL circuit equation and the inverse uncertainty distribution of solution are derived. Some applications of solution for uncertain RL circuit equation are also investigated. Finally, the method of moments is used to estimate the unknown parameters in uncertain RL circuit equation. Full article
(This article belongs to the Special Issue Uncertainty Theory: Symmetry and Applications)
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