FUNDAMENTALS OF FUZZY SETS

Transcription

1 FUNDAMENTALS OF FUZZY SETS edited by Didier Dubois and Henri Prade IRIT, CNRS & University of Toulouse III Foreword by LotfiA. Zadeh 14 Kluwer Academic Publishers Boston//London/Dordrecht

2 Contents Foreword by Lotfi A. Zadeh Preface Series Foreword Contributing Authors General Introduction 1 Didier Dubois, Henri Prade I Fuzzy Sets: From Basic Concepts to Applications 4 II The Role of Fuzzy Sets in Information Engineering 9 III Conclusion: The Legitimacy of Fuzzy Sets 13 References ^ 16 PART I FUZZY SETS 1 Fuzzy Sets: History and Basic Notions 21 Didier Dubois, W. Ostasiewicz and Henri Prade 1.1 Introduction The Historical Emergence of Fuzzy Sets Fuzzy-ism Philosophical Background From Logic to Fuzzy Logics From Sets to Fuzzy Sets Basic Notions of Fuzzy Set Theory Representations of a Fuzzy Set Scalar Characteristics of a Fuzzy Set Extension Principles Basic Connectives Set-Theoretic Comparisons Between Fuzzy Sets 58 \ Fuzzy Sets on Structured Referentials Notions Derived from Fuzzy Sets Fuzzy Relations 70"

3 VI Possibility Measures and Other Fuzzy Set-Based Functions Generalisations and Variants of Fuzzy Sets L-FuzzySets Fuzzy Sets as Ordering Relations 82 f Toll Sets Interval-Valued Fuzzy Sets Type 2 Fuzzy Sets Probabilistic Extensions of Fuzzy Sets Level 2 Fuzzy Sets Fuzzy Rough Sets and Rough Fuzzy Sets Semantics and Measurement of Fuzzy Sets What Membership Grades May Mean Measuring Membership Grades The Semantic Meaningfulness of Fuzzy Logic Membership Grades: Truth Values or Uncertainty Degrees Towards Membership Function Measurement Conclusion 106 References Fuzzy Set-Theoretic Operators and Quantifiers 125 Janos Fodor and Ronald R. Yager 2.1 Introduction Complementation Representation of Negations Other Important Results Intersection and Union Triangulär Norms and Conorms The Special Role of Minimum and Maximum Continuous Archimedean t-norms and t-conorms Parametered Families of t-norms and t-conorms Complementation Defined from Intersection and Union Inclusion and Difference Fuzzy Implications Fuzzy Implications Defined by t-norms, t-conorms and Negations Negations Defined by Implications Axioms for Fuzzy Inclusions 154 > Difference of Fuzzy Sets Equivalence 158 x 2.6 Uninorms Important Classes of Uninorms Mean Aggregation Operators Ordered Weighted Averaging Operators Quantifiers Linguistic Quantifiers and OWA Operators 173

4 2.11 Weighted Unions and Intersections Prioritized Fuzzy Operations Other Aggregation Operators on Fuzzy Sets Symmetrie Sums Weakt-Norms Compensatory Operators 186 References Measurement of Membership Functions: Theoretical and Empirical Work 195 Taner Bilgic and I. Burhan Türksen 3.1 Introduction and Preview Interpretations of Grade of Membership The Likelihood View Random Set View Similarity View View from Utility Theory View from Measurement Theory Elicitation Methods Polling Direct Rating Reverse Rating Interval Estimation Membership Exemplification Pairwise Comparison 214 Y Fuzzy Clustering Methods Neural-Fuzzy Techniques General Remarks Summary 218 References 220 Appendix: Ordered Algebraic Structures and their Representations 228 Vll PART II FUZZY RELATION? 4 An Introduction to Fuzzy Relations 233 Sergei Ovchinnikov 4.1 Introduction Basic Concepte Coverings and Proximity Relations 238 )/ 4.4 Similarity Relations and Fuzzy Partitions Fuzzy Orderings Representation Theorems 254 References 258

5 5 Fuzzy Equivalence Relations: Advanced Material 261 Dionis Boixader, Joan Jacas and Jordi Recasens 5.1 Introduction How to Build Fuzzy Equivalence Relations Fuzzy Equivalence Relations and Generalized Metrics The Generators Set: Granularity, Observability and Approximation Dimension and Basis Their Calculation 279 References Analytical Solution Methods for Fuzzy Relational Equations 291 Bernard De Baets 6.1 Introduction Images and Compositions Relational Calculus and Boolean Equations Fuzzy Relational Calculus Types of Inverse Problems Sup-^G Equations The Equation ^(a,x) = b Greatest Solution Solvability Conditions Complete Solution Set Systems of Sup- G Equations Fuzzy Relational Equations Left Inf-5 Equations The Equation 3(x,b) = a Greatest Solution Solvability Conditions Complete Solution Set Systems of Left Inf-3 Equations Fuzzy Relational Equations Right Inf-,9 Equations The Equation H(a,x) = b Smallest Solution Solvability Conditions Complete Solution Set Systems of Right Inf-5 Equations Fuzzy Relational Equations Approximate Solution Methods Further Reading Various Generalizations Miscellaneous Problems Implementations Applications 332 References 333

6 ix PART III UNCERTAINTY 7 Possibility Theory, Probability and Fuzzy Sets: Misunderstandings, Bridges and Gaps 343 Didier Dubois, Hung T. Nguyen and Henri Prade 7.1 Introduction Some Misunderstandings Between Fuzzy Sets and Probability Membership Function and Probability Measure Fuzzy Relative Cardinality and Conditional Probability Fuzzy Sets Can Be Cast in Random Set Theory Membership Functions as Likelihood Functions Possibility Theory The Meaning of Possibility Possibility Distributions Information Content of a Possibility Distribution Possibility and Necessity of Events Joint Possibility, Separability and Non-Interactive Variables Certainty and Possibility Qualification and the Extension Problem Conditional Possibility and Possibilistic Independence Combination Rules in Possibility Theory Quantitative Possibility Theory as a Bridge Between Probability and Fuzzy Sets Possibility Theory and Bayesian Statistics Upper and Lower Probabilities Possibility Distributions as Special Cases of Random Sets and Belief Functions Possibility-Probability Transformations Possibility Theory and the Calculus of Likelihoods Probabilistic Interpretations of Fuzzy Set Operations Possibility Degrees as Infinitesimal Probabilities Towards Operational Semantics of Possibility Distributions and Fuzzy Sets Frequentist Possibility Uncertainty Measures and Scoring Rules Betting Possibilities " Possibility as Similarity Possibility as Preference and Graded Feasibility Refinements of Qualitative Possibility Theory Possibility and Necessity of Fuzzy Events: A Tool for Decision Under Uncertainty ^ Possibility and Necessity of Fuzzy Events Sugeno Integrals Quantitative Possibility and Choquet Integrals Decision-Theoretic Foundations of Possibility Theory Conclusion 413 Mathematical Appendix 414 References 423

7 8 Measures of Uncertainty and Information 439 George J. Klir 8.1 Introduction Measures of Nonspecificity Classical Set Theory Fuzzy Set Theory "* Possibility Theory Evidence Theory Entropy-Like Measures Probability Theory Evidence Theory Possibility Theory Measures of Fuzziness Fuzzy Set Theory Fuzzified Evidence Theory Conclusions 454 References Quantifying Different Facets of Fuzzy Uncertainty 459 Nikhil R. Pal and James C. Bezdek 9.1 Introduction Different Facets of Fuzzy Uncertainty Measuring Fuzziness Postulates of Measures of Fuzziness Various Measures of Fuzziness Generalized Measure of Fuzziness Higher Order Measures of Fuzziness Weighted Fuzziness Measuring Non-Specificity Conclusions 477 References 478 PART IV FUZZY SETS ON THE REAL LINE 10 Fuzzy Interval Analysis 483 Didier Dubois, Etienne Kerre, Radko Mesiar and Henri Prade 10.1 Introduction Fuzzy Quantities and Intervals Definitions Characteristics of a Fuzzy Interval Noninteractive Fuzzy Variables Basic Principles of Fuzzy Interval Analysis The Extension Principle 498

8 Functions on Non-Interactive Fuzzy Variables: Basic Results Application to Usual Operations Proper and Improper Representations of Functions Practical Computing with Non-Interactive Fuzzy Intervals Parameterized Representations of a Fuzzy Interval Exact Calculation of the Four Arithmetic Operations Approximate Parametric Calculation of Functions of Fuzzy Intervals Approximate Calculation of Functions of Fuzzy Intervals Using Level-Cuts Alternative Fuzzy Interval Calculi Fuzzy Interval Calculations with Linked Variables Additions of Fuzzy Intervals in the Sense of a Triangulär Norm Multidimensional Fuzzy Quantities Fuzzy Equations and the Optimistic Calculus of Fuzzy Intervals Comparison of Fuzzy Quantities 539 X Positioning a Number with Respect to a Fuzzy Quantity Ranking Fuzzy Intervals via Defuzzification Goal-Driven Ranking Methods Fuzzy Ordering Relations Induced by Fuzzy Intervals Fuzzy Dominance Indices and Linguistic Methods Criteria for Ranking Fuzzy Intervals Conclusion: Applications of Fuzzy Numbers and Intervals 558 References Metrie Topology of Fuzzy Numbers and Fuzzy Analysis 583 Phil Diamond and Peter Kloeden 11.1 Introduction Calculus of Compact Convex Subsets in 3l n Subsets and Algebraic Operations The Hausdorff Metrie Compact Subsets of 9l n Support Functions LP-Metrics Continuity and Measurability Differentiation Integration Bibliographical Notes The Space n Definitions and Basic Properties Useful Subsets of > n and n Bibliographical Notes Metricsont?" Definitions and Basic Properties Completeness Separability 608

9 Xll Convergence Relationships Bibliographical Notes Compactness Criteria Introduction Compact Subsets in (c? n, d p ) Bibliographical Notes Fuzzy Set Valued Mappings of Real Variables Continuity and Measurability Differentiation Integration Bibliographical Notes Interpolation and Approximation Interpolation and Splines Bernstein Approximation Bibliographical Notes Fuzzy Differential Equations Introduction Existence and Uniqueness of Solutions Reinterpreting Fuzzy DEs Bibliographical Notes Conclusion 637 References 637 Index 643