Fundamentals of fuzzy sets:
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Boston [u.a.]
Kluwer Acad. Publ.
2000
|
| Series: | The handbooks of fuzzy sets series
7 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 647 S. graph. Darst. |
| ISBN: | 079237732X |
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MARC
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| 245 | 1 | 0 | |a Fundamentals of fuzzy sets |c ed. by Didier Dubois ... |
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Record in the Search Index
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| adam_text | 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 21
1.2 The Historical Emergence of Fuzzy Sets 24
1.2.1 Fuzzy ism 25
1.2.2 Philosophical Background 26
1.2.3 From Logic to Fuzzy Logics 31
1.2.4 From Sets to Fuzzy Sets 36
1.3 Basic Notions of Fuzzy Set Theory 42
1.3.1 Representations of a Fuzzy Set 42
1.3.2 Scalar Characteristics of a Fuzzy Set 47
1.3.3 Extension Principles 50
1.3.4 Basic Connectives 53
1.3.5 Set Theoretic Comparisons Between Fuzzy Sets 58
1.3.6 Fuzzy Sets on Structured Referentials 66
1.4 Notions Derived from Fuzzy Sets 70
1.4.1 Fuzzy Relations 70
vi
1.4.2 Possibility Measures and Other Fuzzy Set Based
Functions 77
1.5 Generalisations and Variants of Fuzzy Sets 80
1.5.1 L FuzzySets 81
1.5.2 Fuzzy Sets as Ordering Relations 82
1.5.3 Toll Sets 84
1.5.4 Interval Valued Fuzzy Sets 86
1.5.5 Type 2 Fuzzy Sets 88
1.5.6 Probabilistic Extensions of Fuzzy Sets 89
1.5.7 Level 2 Fuzzy Sets 90
1.5.8 Fuzzy Rough Sets and Rough Fuzzy Sets 91
1.6 Semantics and Measurement of Fuzzy Sets 93
1.6.1 What Membership Grades May Mean 95
1.6.2 Measuring Membership Grades 97
1.6.3 The Semantic Meaningfulness of Fuzzy Logic 100
1.6.4 Membership Grades: Truth Values or Uncertainty
Degrees 102
1.6.5 Towards Membership Function Measurement 104
1.7 Conclusion 106
References 106
2
Fuzzy Set Theoretic Operators and Quantifiers 125
Janos Fodor and Ronald R. Yager
2.1 Introduction 125
2.2 Complementation 127
2.2.1 Representation of Negations 129
2.2.2 Other Important Results 129
2.3 Intersection and Union 130
2.3.1 Triangular Norms and Conorms 131
2.3.2 The Special Role of Minimum and Maximum 134
2.3.3 Continuous Archimedean t Norms and t Conorms 135
2.3.4 Parametered Families of t Norms and t Conorms 141
2.3.5 Complementation Defined from Intersection and Union 145
2.4 Inclusion and Difference 146
2.4.1 Fuzzy Implications 147
2.4.2 Fuzzy Implications Defined by t Norms, t Conorms and
Negations 148
2.4.3 Negations Defined by Implications 153
2.4.4 Axioms for Fuzzy Inclusions 154
2.4.5 Difference of Fuzzy Sets 156
2.5 Equivalence 158
2.6 Uninorms 159
2.6.1 Important Classes of Uninorms 160
2.7 Mean Aggregation Operators 162
2.8 Ordered Weighted Averaging Operators 165
2.9 Quantifiers 172
2.10 Linguistic Quantifiers and OWA Operators 173
vii
2.11 Weighted Unions and Intersections 179
2.12 Prioritized Fuzzy Operations 181
2.13 Other Aggregation Operators on Fuzzy Sets 184
2.13.1 Symmetric Sums 184
2.13.2 Weakt Norms 185
2.13.3 Compensatory Operators 186
References 187
3
Measurement of Membership Functions: Theoretical and
Empirical Work 195
Taner Bilgic and I.Burhan Turksen
3.1 Introduction and Preview 195
3.2 Interpretations of Grade of Membership 197
3.2.1 The Likelihood View 198
3.2.2 Random Set View 200
3.2.3 Similarity View 201
3.2.4 View from Utility Theory 202
3.2.5 View from Measurement Theory 203
3.3 Elicitation Methods 211
3.3.1 Polling 211
3.3.2 Direct Rating 212
3.3.3 Reverse Rating 213
3.3.4 Interval Estimation 213
3.3.5 Membership Exemplification 214
3.3.6 Pairwise Comparison 214
3.3.7 Fuzzy Clustering Methods 215
3.3.8 Neural Fuzzy Techniques 216
3.3.9 General Remarks 216
3.4 Summary 218
References 220
Appendix: Ordered Algebraic Structures and their Representations 228
PART II FUZZY RELATIONS
4
An Introduction to Fuzzy Relations 233
Sergei Ovchinnikov
4.1 Introduction 233
4.2 Basic Concepts 235
4.3 Coverings and Proximity Relations 238
4.4 Similarity Relations and Fuzzy Partitions 241
4.5 Fuzzy Orderings 246
4.6 Representation Theorems 254
References 258
viii
5
Fuzzy Equivalence Relations: Advanced Material 261
Dionis Boixader, Joan Jacas and Jordi Recasens
5.1 Introduction 261
5.2 How to Build Fuzzy Equivalence Relations 263
5.3 Fuzzy Equivalence Relations and Generalized Metrics 267
5.4 The Generators Set: Granularity, Observability and Approximation 270
5.5 Dimension and Basis — Their Calculation 279
References 288
6
Analytical Solution Methods for Fuzzy Relational Equations 291
Bernard De Baets
6.1 Introduction 291
6.2 Images and Compositions 293
6.2.1 Relational Calculus and Boolean Equations 293
6.2.2 Fuzzy Relational Calculus 294
6.3 Types of Inverse Problems 296
6.4 Sup G Equations 297
6.4.1 The Equation %(a,x) = b 297
6.4.2 Greatest Solution — Solvability Conditions 299
6.4.3 Complete Solution Set 301
6.4.4 Systems of Sup t Equations 307
6.4.5 Fuzzy Relational Equations 308
6.5 Left Inf 3 Equations 314
6.5.1 The Equation 3(x,b) = a 314
6.5.2 Greatest Solution — Solvability Conditions 316
6.5.3 Complete Solution Set 317
6.5.4 Systems of Left Inf 3 Equations 319
6.5.5 Fuzzy Relational Equations 320
6.6 Right nf 3 Equations 321
6.6.1 The Equation 3(a,x) = b 321
6.6.2 Smallest Solution — Solvability Conditions 323
6.6.3 Complete Solution Set 324
6.6.4 Systems of Right lnf 5 Equations 326
6.6.5 Fuzzy Relational Equations 327
6.7 Approximate Solution Methods 329
6.8 Further Reading 330
6.8.1 Various Generalizations 330
6.8.2 Miscellaneous Problems 330
6.8.3 Implementations 332
6.8.4 Applications 332
References 333
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 343
7.2 Some Misunderstandings Between Fuzzy Sets and Probability 346
7.2.1 Membership Function and Probability Measure 346
7.2.2 Fuzzy Relative Cardinality and Conditional Probability 349
7.2.3 Fuzzy Sets Can Be Cast in Random Set Theory 350
7.2.4. Membership Functions as Likelihood Functions 351
7.3 Possibility Theory 353
7.3.1 The Meaning of Possibility 354
7.3.2 Possibility Distributions 356
7.3.3 Information Content of a Possibility Distribution 358
7.3.4 Possibility and Necessity of Events 360
7.3.5 Joint Possibility, Separability and Non Interactive
Variables 364
7.3.6 Certainty and Possibility Qualification and the Extension
Problem 367
7.3.7 Conditional Possibility and Possibilistic Independence 368
7.3.8 Combination Rules in Possibility Theory 376
7.4 Quantitative Possibility Theory as a Bridge Between Probability
and Fuzzy Sets 378
7.4.1 Possibility Theory and Bayesian Statistics 378
7.4.2 Upper and Lower Probabilities 380
7.4.3 Possibility Distributions as Special Cases of Random
Sets and Belief Functions 381
7.4.4 Possibility Probability Transformations 383
7.4.5 Possibility Theory and the Calculus of Likelihoods 389
7.4.6 Probabilistic Interpretations of Fuzzy Set Operations 390
7.4.7 Possibility Degrees as Infinitesimal Probabilities 391
7.5 Towards Operational Semantics of Possibility Distributions and
Fuzzy Sets 393
7.5.1 Frequentist Possibility 393
7.5.2 Uncertainty Measures and Scoring Rules 394
7.5.3 Betting Possibilities 395
7.5.4 Possibility as Similarity 396
7.5.5 Possibility as Preference and Graded Feasibility 397
7.5.6 Refinements of Qualitative Possibility Theory 401
7.6 Possibility and Necessity of Fuzzy Events: A Tool for Decision
Under Uncertainty 402
7.6.1 Possibility and Necessity of Fuzzy Events 402
7.6.2 Sugeno Integrals 405
7.6.3 Quantitative Possibility and Choquet Integrals 406
7.6.4 Decision Theoretic Foundations of Possibility Theory 408
7.7 Conclusion 413
Mathematical Appendix 414
References 423
X
8
Measures of Uncertainty and Information 439
George J. Klir
8.1 Introduction 439
8.2 Measures of Nonspecificity 440
8.2.1 Classical Set Theory 440
8.2.2 Fuzzy Set Theory 443
8.2.3 Possibility Theory 444
8.2.4 Evidence Theory 446
8.3 Entropy Like Measures 447
8.3.1 Probability Theory 447
8.3.2 Evidence Theory 449
8.3.3 Possibility Theory 451
8.4 Measures of Fuzziness 452
8.4.1 Fuzzy Set Theory 452
8.4.2 Fuzzified Evidence Theory 453
8.5 Conclusions 454
References 454
9
Quantifying Different Facets of Fuzzy Uncertainty 459
Nikhil R. Pal and James C. Bezdek
9.1 Introduction 459
9.2 Different Facets of Fuzzy Uncertainty 461
9.3 Measuring Fuzziness 462
9.3.1 Postulates of Measures of Fuzziness 462
9.3.2 Various Measures of Fuzziness 464
9.4 Generalized Measure of Fuzziness 473
9.4.1 Higher Order Measures of Fuzziness 473
9.4.2 Weighted Fuzziness 474
9.5 Measuring Non Specificity 475
9.6 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 483
10.2 Fuzzy Quantities and Intervals 486
10.2.1 Definitions 486
10.2.2 Characteristics of a Fuzzy Interval 492
10.2.3 Noninteractive Fuzzy Variables 497
10.3 Basic Principles of Fuzzy Interval Analysis 498
10.3.1 The Extension Principle 498
xi
10.3.2 Functions on Non Interactive Fuzzy Variables: Basic
Results 501
10.3.3 Application to Usual Operations 505
10.3.4 Proper and Improper Representations of Functions 509
10.4 Practical Computing with Non Interactive Fuzzy Intervals 511
10.4.1 Parameterized Representations of a Fuzzy Interval 511
10.4.2 Exact Calculation of the Four Arithmetic Operations 514
10.4.3 Approximate Parametric Calculation of Functions of
Fuzzy Intervals 516
10.4.4 Approximate Calculation of Functions of Fuzzy Intervals
Using Level Cuts 519
10.5 Alternative Fuzzy Interval Calculi 521
10.5.1 Fuzzy Interval Calculations with Linked Variables 521
10.5.2 Additions of Fuzzy Intervals in the Sense of a Triangular
Norm 524
10.5.3 Multidimensional Fuzzy Quantities 530
10.5.4 Fuzzy Equations and the Optimistic Calculus of Fuzzy
Intervals 534
10.6 Comparison of Fuzzy Quantities 539
10.6.1 Positioning a Number with Respect to a Fuzzy Quantity 540
10.6.2 Ranking Fuzzy Intervals via Defuzzification 541
10.6.3 Goal Driven Ranking Methods 542
10.6.4 Fuzzy Ordering Relations Induced by Fuzzy Intervals 544
10.6.5 Fuzzy Dominance Indices and Linguistic Methods 553
10.6.6 Criteria for Ranking Fuzzy Intervals 554
10.7 Conclusion: Applications of Fuzzy Numbers and Intervals 558
References 561
11
Metric Topology of Fuzzy Numbers and Fuzzy Analysis 583
Phil Diamond and Peter Kloeden
11.1 Introduction 583
11.2 Calculus of Compact Convex Subsets in 5ln 585
11.2.1 Subsets and Algebraic Operations 585
11.2.2 The Hausdorff Metric 586
11.2.3 Compact Subsets of 9ln 587
11.2.4 Support Functions 588
11.2.5 LP Metrics 590
11.2.6 Continuity and Measurability 592
11.2.7 Differentiation 594
11.2.8 Integration 596
11.2.9 Bibliographical Notes 599
11.3 The Space £n 600
11.3.1 Definitions and Basic Properties 600
11.3.2 Useful Subsets of £ n and Sn 603
11.3.3 Bibliographical Notes 604
11.4 Metrics on £n 605
11.4.1 Definitions and Basic Properties 605
11.4.2 Completeness 607
11.4.3 Separability 608
xii
11.4.4 Convergence Relationships 608
11.4.5 Bibliographical Notes 609
11.5 Compactness Criteria 609
11.5.1 Introduction 609
11.5.2 Compact Subsets in (£ , dp) 611
11.5.3 Bibliographical Notes 613
11.6 Fuzzy Set Valued Mappings of Real Variables 613
11.6.1 Continuity and Measurability 613
11.6.2 Differentiation 615
11.6.3 Integration 621
11.6.4 Bibliographical Notes 624
11.7 Interpolation and Approximation 625
11.7.1 Interpolation and Splines 625
11.7.2 Bernstein Approximation 628
11.7.3 Bibliographical Notes 629
11.8 Fuzzy Differential Equations 630
11.8.1 Introduction 630
11.8.2 Existence and Uniqueness of Solutions 632
11.8.3 Reinterpreting Fuzzy DEs 632
11.8.4 Bibliographical Notes 637
11.9 Conclusion 637
References 637
Index 643
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| id | DE-604.BV013126288 |
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| indexdate | 2024-07-09T18:39:30Z |
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| isbn | 079237732X |
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| spelling | Fundamentals of fuzzy sets ed. by Didier Dubois ... Boston [u.a.] Kluwer Acad. Publ. 2000 XXI, 647 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier The handbooks of fuzzy sets series 7 Ensembles flous Fuzzy sets gtt Fuzzy sets Fuzzy-Menge (DE-588)4061868-7 gnd rswk-swf Fuzzy-Menge (DE-588)4061868-7 s DE-604 Dubois, Didier Sonstige oth The handbooks of fuzzy sets series 7 (DE-604)BV012607710 7 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008943043&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4061868-7 |
| title | Fundamentals of fuzzy sets |
| title_auth | Fundamentals of fuzzy sets |
| title_exact_search | Fundamentals of fuzzy sets |
| title_full | Fundamentals of fuzzy sets ed. by Didier Dubois ... |
| title_fullStr | Fundamentals of fuzzy sets ed. by Didier Dubois ... |
| title_full_unstemmed | Fundamentals of fuzzy sets ed. by Didier Dubois ... |
| title_short | Fundamentals of fuzzy sets |
| title_sort | fundamentals of fuzzy sets |
| topic | Ensembles flous Fuzzy sets gtt Fuzzy sets Fuzzy-Menge (DE-588)4061868-7 gnd |
| topic_facet | Ensembles flous Fuzzy sets Fuzzy-Menge |
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