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Fuzzy number

About: Fuzzy number is a research topic. Over the lifetime, 35606 publications have been published within this topic receiving 972544 citations.


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Journal ArticleDOI
TL;DR: The concept of picture fuzzy sets (PFS), which are direct extensions of the fuzzy sets and the intuitonistic fuzzy sets, are introduced and the basic preliminaries of PFS theory are presented.
Abstract: In this paper, we introduce the concept of picture fuzzy sets (PFS), which are direct extensions of the fuzzy sets and the intuitonistic fuzzy sets. Then some operations on PFS with some properties are considered. The following sections are devoted to the Zadeh Extension Principle, picture fuzzy relations and picture fuzzy soft sets. Here, the basic preliminaries of PFS theory are presented.

528 citations

Journal ArticleDOI
TL;DR: This paper will give a characteristic of a field by a fuzzy ideal, as fuzzy invariant subgroups, fuzzy ideals, and to prove some fundamental properties of fuzzy algebra.

526 citations

Journal ArticleDOI
01 Mar 1998
TL;DR: This paper introduces the fuzzy association rules of the form, 'If X is A then Y is B', to deal with quantitative attributes, using the fuzzy set concept, to find association rules more understandable to human.
Abstract: Data mining is the discovery of previously unknown, potentially useful and hidden knowledge in databases. In this paper, we concentrate on the discovery of association rules. Many algorithms have been proposed to find association rules in databases with binary attributes. We introduce the fuzzy association rules of the form, 'If X is A then Y is B', to deal with quantitative attributes. X, Y are set of attributes and A, B are fuzzy sets which describe X and Y respectively. Using the fuzzy set concept, the discovered rules are more understandable to human. Moreover, fuzzy sets handle numerical values better than existing methods because fuzzy sets soften the effect of sharp boundaries.

524 citations

Book
27 Nov 2000
TL;DR: This chapter discusses the development of model-free Logic Control for Fuzzy Systems, as well as some of the techniques used in the model-based approach to Logic Control.
Abstract: FUZZY SET THEORY Classical Set Theory Fuzzy Set Theory Interval Arithmetic Operations on Fuzzy Sets FUZZY LOGIC THEORY Classical Logic Theory The Boolean Algebra Multi-Valued Logic Fuzzy Logic and Approximate Reasoning Fuzzy Relations Fuzzy Logic Rule Base FUZZY SYSTEM MODELING Modeling of the Static Fuzzy Systems Stability Analysis of Discrete-Time Dynamic Fuzzy Systems Modeling of Continuous-Time Dynamic Fuzzy Systems Stability Analysis of Continuous-Time Fuzzy Systems Controllability Analysis of Continuous-Time Dynamic Fuzzy Systems Analysis of Nonlinear Continuous-Time Dynamic Fuzzy Systems FUZZY CONTROL SYSTEMS Classical Programmable Logic Control Fuzzy Logic Control I: A General Model-Free Approach Fuzzy Logic Control II: A General Model-Based Approach FUZZY PID CONTROLLERS Conventional PID Controllers Fuzzy PID Controllers Fuzzy PID Controllers: Stability Analysis ADAPTIVE FUZZY CONTROL Fundamental Adaptive Fuzzy Control Concept Gain Scheduling Fuzzy Self-Tuning Regulator Model Reference Adaptive Fuzzy Systems Dual Control Sub-Optimal Fuzzy Control APPLICATIONS IN FUZZY CONTROL Health Monitoring Fuzzy Diagnostic Systems Fuzzy Control of Image sharpness for Auto-focus Cameras Fuzzy Control for Servo Mechanic Systems Fuzzy PID Controllers for Servo Mechanic Systems Fuzzy Controllers for Robotic Manipulator Note: Each chapter also contains Problems and References

523 citations

Book
25 Nov 2005
TL;DR: This chapter discusses the Hartley Measure, a measure of uncertainty based on the Shannon Entropy model, which was developed in the second half of the 1990s to help clarify the role of uncertainty in evidence-based decision-making.
Abstract: Preface. Acknowledgments. 1 Introduction. 1.1. Uncertainty and Its Significance. 1.2. Uncertainty-Based Information. 1.3. Generalized Information Theory. 1.4. Relevant Terminology and Notation. 1.5. An Outline of the Book. Notes. Exercises. 2 Classical Possibility-Based Uncertainty Theory. 2.1. Possibility and Necessity Functions. 2.2. Hartley Measure of Uncertainty for Finite Sets. 2.2.1. Simple Derivation of the Hartley Measure. 2.2.2. Uniqueness of the Hartley Measure. 2.2.3. Basic Properties of the Hartley Measure. 2.2.4. Examples. 2.3. Hartley-Like Measure of Uncertainty for Infinite Sets. 2.3.1. Definition. 2.3.2. Required Properties. 2.3.3. Examples. Notes. Exercises. 3 Classical Probability-Based Uncertainty Theory. 3.1. Probability Functions. 3.1.1. Functions on Finite Sets. 3.1.2. Functions on Infinite Sets. 3.1.3. Bayes' Theorem. 3.2. Shannon Measure of Uncertainty for Finite Sets. 3.2.1. Simple Derivation of the Shannon Entropy. 3.2.2. Uniqueness of the Shannon Entropy. 3.2.3. Basic Properties of the Shannon Entropy. 3.2.4. Examples. 3.3. Shannon-Like Measure of Uncertainty for Infinite Sets. Notes. Exercises. 4 Generalized Measures and Imprecise Probabilities. 4.1. Monotone Measures. 4.2. Choquet Capacities. 4.2.1. Mobius Representation. 4.3. Imprecise Probabilities: General Principles. 4.3.1. Lower and Upper Probabilities. 4.3.2. Alternating Choquet Capacities. 4.3.3. Interaction Representation. 4.3.4. Mobius Representation. 4.3.5. Joint and Marginal Imprecise Probabilities. 4.3.6. Conditional Imprecise Probabilities. 4.3.7. Noninteraction of Imprecise Probabilities. 4.4. Arguments for Imprecise Probabilities. 4.5. Choquet Integral. 4.6. Unifying Features of Imprecise Probabilities. Notes. Exercises. 5 Special Theories of Imprecise Probabilities. 5.1. An Overview. 5.2. Graded Possibilities. 5.2.1. Mobius Representation. 5.2.2. Ordering of Possibility Profiles. 5.2.3. Joint and Marginal Possibilities. 5.2.4. Conditional Possibilities. 5.2.5. Possibilities on Infinite Sets. 5.2.6. Some Interpretations of Graded Possibilities. 5.3. Sugeno l-Measures. 5.3.1. Mobius Representation. 5.4. Belief and Plausibility Measures. 5.4.1. Joint and Marginal Bodies of Evidence. 5.4.2. Rules of Combination. 5.4.3. Special Classes of Bodies of Evidence. 5.5. Reachable Interval-Valued Probability Distributions. 5.5.1. Joint and Marginal Interval-Valued Probability Distributions. 5.6. Other Types of Monotone Measures. Notes. Exercises. 6 Measures of Uncertainty and Information. 6.1. General Discussion. 6.2. Generalized Hartley Measure for Graded Possibilities. 6.2.1. Joint and Marginal U-Uncertainties. 6.2.2. Conditional U-Uncertainty. 6.2.3. Axiomatic Requirements for the U-Uncertainty. 6.2.4. U-Uncertainty for Infinite Sets. 6.3. Generalized Hartley Measure in Dempster-Shafer Theory. 6.3.1. Joint and Marginal Generalized Hartley Measures. 6.3.2. Monotonicity of the Generalized Hartley Measure. 6.3.3. Conditional Generalized Hartley Measures. 6.4. Generalized Hartley Measure for Convex Sets of Probability Distributions. 6.5. Generalized Shannon Measure in Dempster-Shafer Theory. 6.6. Aggregate Uncertainty in Dempster-Shafer Theory. 6.6.1. General Algorithm for Computing the Aggregate Uncertainty. 6.6.2. Computing the Aggregated Uncertainty in Possibility Theory. 6.7. Aggregate Uncertainty for Convex Sets of Probability Distributions. 6.8. Disaggregated Total Uncertainty. 6.9. Generalized Shannon Entropy. 6.10. Alternative View of Disaggregated Total Uncertainty. 6.11. Unifying Features of Uncertainty Measures. Notes. Exercises. 7 Fuzzy Set Theory. 7.1. An Overview. 7.2. Basic Concepts of Standard Fuzzy Sets. 7.3. Operations on Standard Fuzzy Sets. 7.3.1. Complementation Operations. 7.3.2. Intersection and Union Operations. 7.3.3. Combinations of Basic Operations. 7.3.4. Other Operations. 7.4. Fuzzy Numbers and Intervals. 7.4.1. Standard Fuzzy Arithmetic. 7.4.2. Constrained Fuzzy Arithmetic. 7.5. Fuzzy Relations. 7.5.1. Projections and Cylindric Extensions. 7.5.2. Compositions, Joins, and Inverses. 7.6. Fuzzy Logic. 7.6.1. Fuzzy Propositions. 7.6.2. Approximate Reasoning. 7.7. Fuzzy Systems. 7.7.1. Granulation. 7.7.2. Types of Fuzzy Systems. 7.7.3. Defuzzification. 7.8. Nonstandard Fuzzy Sets. 7.9. Constructing Fuzzy Sets and Operations. Notes. Exercises. 8 Fuzzification of Uncertainty Theories. 8.1. Aspects of Fuzzification. 8.2. Measures of Fuzziness. 8.3. Fuzzy-Set Interpretation of Possibility Theory. 8.4. Probabilities of Fuzzy Events. 8.5. Fuzzification of Reachable Interval-Valued Probability Distributions. 8.6. Other Fuzzification Efforts. Notes. Exercises. 9 Methodological Issues. 9.1. An Overview. 9.2. Principle of Minimum Uncertainty. 9.2.1. Simplification Problems. 9.2.2. Conflict-Resolution Problems. 9.3. Principle of Maximum Uncertainty. 9.3.1. Principle of Maximum Entropy. 9.3.2. Principle of Maximum Nonspecificity. 9.3.3. Principle of Maximum Uncertainty in GIT. 9.4. Principle of Requisite Generalization. 9.5. Principle of Uncertainty Invariance. 9.5.1. Computationally Simple Approximations. 9.5.2. Probability-Possibility Transformations. 9.5.3. Approximations of Belief Functions by Necessity Functions. 9.5.4. Transformations Between l-Measures and Possibility Measures. 9.5.5. Approximations of Graded Possibilities by Crisp Possibilities. Notes. Exercises. 10 Conclusions. 10.1. Summary and Assessment of Results in Generalized Information Theory. 10.2. Main Issues of Current Interest. 10.3. Long-Term Research Areas. 10.4. Significance of GIT. Notes. Appendix A Uniqueness of the U-Uncertainty. Appendix B Uniqueness of Generalized Hartley Measure in the Dempster-Shafer Theory. Appendix C Correctness of Algorithm 6.1. Appendix D Proper Range of Generalized Shannon Entropy. Appendix E Maximum of GSa in Section 6.9. Appendix F Glossary of Key Concepts. Appendix G Glossary of Symbols. Bibliography. Subject Index. Name Index.

522 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023202
2022446
2021696
2020649
2019653
2018733