Axiomatic Fuzzy Set Theory and Its Applications:
Saved in:
| Format: | Electronic eBook |
|---|---|
| Language: | English |
| Published: |
Berlin
Springer-Verlag GmbH
2009
|
| Edition: | 1. Aufl. |
| Series: | Studies in Fuzziness and Soft Computing
244 |
| Subjects: | |
| Online Access: | BTU01 FHN01 FHR01 Volltext Inhaltsverzeichnis |
| Physical Description: | 1 Online-Ressource (500 S.) 235 mm x 155 mm |
| ISBN: | 9783642004025 |
| DOI: | 10.1007/978-3-642-00402-5 |
Staff View
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Record in the Search Index
| _version_ | 1804139102635819008 |
|---|---|
| adam_text | IMAGE 1
CONTENTS
PART I: REQUIRED PRELIMINARY MATHEMATICAL KNOWLEDGE 1 FUNDAMENTALS 3
1.1 SETS, RELATIONS AND MAPS 3
1.1.1 SETS 3
1.1.2 RELATIONS 7
1.1.3 MAPS 9
1.1.4 COUNTABLE SETS 14
1.1.5 PARTIALLY ORDERED SETS, DIRECTED SETS AND NETS 16 1.1.6 MAXIMAL
CONDITIONS AND MINIMAL CONDITIONS 19 1.2 TOPOLOGICAL SPACES 21
1.2.1 NEIGHBORHOOD SYSTEMS AND TOPOLOGIES 21
1.2.2 LIMIT POINTS CLOSURE OF A SET AND CLOSED SETS 23 1.2.3 INTERIOR
AND BOUNDARY 26
1.2.4 BASES COUNTABILITY AXIOMS SEPARABILITY 27
1.2.5 SUBSPACE SEPARATION AND CONNECTED SETS 30
1.2.6 CONVERGENCE AND HAUSDORFF SPACES 32
1.2.7 VARIOUS SPECIAL TYPES OF TOPOLOGICAL SPACES 35 1.3 METRIZATION 36
1.3.1 CONTINUOUS FUNCTIONS 37
1.3.2 METRIC AND PSEUDO-METRIC SPACES 38
1.3.3 SOME PROPERTIES OF METRIZABLE AND PSEUDO METRIZABLE SPACE 39
1.4 MEASURES 44
1.4.1 ALGEBRAS AND SIGMA-ALGEBRAS 45
1.4.2 MEASURES 47
1.5 PROBABILITY 50
1.5.1 PROBABILITY SPACE AND PROBABILITY MEASURE 51 1.5.2 SOME USEFUL
PROPERTIES OF PROBABILITY 52
1.6 COMBINATORIC SYSTEMS 56
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/992310261
DIGITALISIERT DURCH
IMAGE 2
XIV CONTENTS
EXERCISES 58
REFERENCES 60
2 LATTICES 61
2.1 LATTICES 61
2.2 DISTRIBUTIVE LATTICES 67
2.3 BOOLEAN ALGEBRA 72
2.4 COMPLETELY DISTRIBUTIVE LATTICES 77
2.5 TOPOLOGICAL MOLECULAR LATTICES 86
EXERCISES 92
REFERENCES 94
PART II: METHODOLOGY AND MATHEMATICAL FRAMEWORK OF AFS THEORY
3 BOOLEAN MATRICES AND BINARY RELATIONS 97
EXERCISES 109
OPEN PROBLEMS 110
REFERENCES 110
4 AFS LOGIC, AFS STRUCTURE AND COHERENCE MEMBERSHIP FUNCTIONS I LL
4.1 AFS FUZZY LOGIC I LL
4.1.1 EL ALGEBRA 113
4.1.2 SIMPLE CONCEPTS AND COMPLEX CONCEPTS 123
4.1.3 AFS STRUCTURE OF DATA 128
4.1.4 COHERENCE MEMBERSHIP FUNCTIONS OF THE AFS FUZZY LOGIC AND THE AFS
STRUCTURE OF DATA 130 4.2 COHERENCE MEMBERSHIP FUNCTIONS VIA AFS LOGIC
AND PROBABILITY 134
4.2.1 COHERENCE MEMBERSHIP FUNCTIONS BASED ON THE PROBABILITY MEASURES
135
4.2.2 COHERENCE MEMBERSHIP FUNCTIONS BASED ON WEIGHT FUNCTIONS AND
PROBABILITY DISTRIBUTIONS 136
4.3 COHERENCE MEMBERSHIP FUNCTIONS AND THE PROBABILITY MEASURES OF FUZZY
EVENTS 142
4.3.1 THE PROBABILITY MEASURES OF FUZZY EVENTS 142 4.3.2 COHERENCE
MEMBERSHIP FUNCTIONS BASED ON PROBABILITY MEASURE OF THE FUZZY EVENTS
145
4.4 COHERENCE MEMBERSHIP FUNCTIONS OF MULTI-NORMAL DISTRIBUTIONS AND
GAUSSION WEIGHTS 146
4.5 EXPERIMENTAL STUDIES 151
4.5.1 PROBABILITY DISTRIBUTION OF IRIS PLANT DATA 151
4.5.2 CONSISTENCY OF THE MEMBERSHIP FUNCTIONS ON THE OBSERVED DATA AND
THE WHOLE SPACE 153
IMAGE 3
CONTENTS XV
4.5.3 UNIVERSALITY OF THE LAWS DISCOVERED ON THE OBSERVED DATA 155
4.5.4 STABILITY ANALYSIS OF AFS FUZZY LOGIC ON THE DIFFERENT OBSERVED
DATA SETS DRAWN FROM THE SAME PROBABILITY SPACE 158
EXERCISES 162
OPEN PROBLEMS 163
REFERENCES 163
5 AFS ALGEBRAS AND THEIR REPRESENTATIONS OF MEMBERSHIP DEGREES 167
5.1 AFS ALGEBRA 167
5.1.1 EI N ALGEBRAS 167
5.1.2 E*L N ALGEBRAS 169
5.2 AFS ALGEBRA REPRESENTATION OF MEMBERSHIP DEGREES 174 5.3 NORM OF AFS
ALGEBRA AND MEMBERSHIP FUNCTIONS OF FUZZY CONCEPTS 178
5.4 FURTHER ALGEBRAIC PROPERTIES OF AFS ALGEBRA 183
5.4.1 ORDER OF EL ALGEBRA 184
5.4.2 ALGEBRAIC STRUCTURES OF EL ALGEBRA 190
5.4.3 ALGEBRAIC STRUCTURES OF EI N ALGEBRA 197
5.4.4 ALGEBRAIC STRUCTURES OF E*I N ALGEBRA 206
5.5 COMBINATORIC PROPERTIES OF THE AFS STRUCTURE 214
EXERCISES 219
OPEN PROBLEMS 222
REFERENCES 223
PART III: APPLICATIONS OF AFS THEORY
6 AFS FUZZY ROUGH SETS 227
6.1 ROUGH SETS AND FUZZY ROUGH SETS 227
6.1.1 ROUGH SETS AND THEIR EQUIVALENT DEFINITION 228 6.1.2 FUZZY ROUGH
SETS 230
6.2 FUZZY ROUGH SETS UNDER FRAMEWORK OF AFS THEORY 235 6.2.1 AFS
STRUCTURE OF INFORMATION SYSTEMS 235
6.2.2 REPRESENTATIONS OF FUZZY CONCEPTS IN THE INFORMATION SYSTEMS 236
6.2.3 DEFINITIONS OF AFS FUZZY ROUGH SETS 241
6.2.4 SOME PROPERTIES OF AFS FUZZY ROUGH SETS 245 6.2.5 ALGORITHM OF
APPROXIMATING UPPER AND LOWER APPROXIMATIONS OF FUZZY CONCEPTS 249
6.2.6 EXPERIMENTS 253
6.3 COMPARISONS WITH FUZZY ROUGH SETS AND OTHER CONSTRUCTS OF ROUGH SETS
257
6.3.1 COMPARISONS WITH FUZZY ROUGH SETS 257
IMAGE 4
XVI CONTENTS
6.3.2 COMPARISONS WITH OTHER CONSTRUCTS OF ROUGH SETS 260
EXERCISES 262
OPEN PROBLEMS 263
REFERENCES 265
7 AFS TOPOLOGY AND ITS APPLICATIONS 269
7.1 TOPOLOGY ON AFS STRUCTURES AND TOPOLOGICAL MOLECULAR LATTICE ON *EI
N ALGEBRAS 269
7.2 TOPOLOGY ON AFS STRUCTURES AND TOPOLOGICAL MOLECULAR LATTICE ON EI N
ALGEBRAS 279
7.3 FUZZY SIMILARITY RELATIONS BASED ON TOPOLOGICAL MOLECULAR LATTICES
282
7.4 FUZZY CLUSTERING ALGORITHMS BASED ON TOPOLOGICAL MOLECULAR LATTICES
285
7.5 EMPIRICAL STUDIES 287
7.5.1 EMPIRICAL EXAMPLES OF TAIWAN AIRFREIGHT FORWARDER 287
7.5.2 EXPERIMENTAL STUDIES ON THE IRIS DATA SET 290 EXERCISES 293
OPEN PROBLEMS 294
APPENDIX A 295
APPENDIX B 296
REFERENCES 299
8 AFS FORMAL CONCEPT AND AFS FUZZY FORMAL CONCEPT ANALYSIS 303
8.1 CONCEPT LATTICES AND AFS ALGEBRAS 304
8.2 SOME AFS ALGEBRAIC PROPERTIES OF FORMAL CONCEPT LATTICES 312 8.3
CONCEPT ANALYSIS VIA ROUGH SET AND AFS ALGEBRA 320 8.3.1 MONOTONE
CONCEPT 321
8.3.2 AFS FORMAL CONCEPT 323
8.3.3 ROUGH SET THEORY APPROACH TO CONCEPT APPROXIMATION 330
8.4 AFS FUZZY FORMAL CONCEPT ANALYSIS 336
EXERCISES 343
OPEN PROBLEMS 345
REFERENCES 346
9 AFS FUZZY CLUSTERING ANALYSIS 351
9.1 ELEMENTARY FUZZY CLUSTERING METHOD VIA AFS FUZZY LOGIC 354
9.2 APPLICATIONS OF THE ELEMENTARY FUZZY CLUSTERING FOR MANAGEMENT
STRATEGIC ANALYSIS 370
IMAGE 5
CONTENTS XVII
9.2.1 IMPROVEMENTS OF THE ELEMENTARY FUZZY CLUSTERING METHOD 373
9.2.2 EXPERIMENTAL STUDY OF THE LIANG S ALGORITHM 375 9.2.3 APPLICATIONS
OF THE IMPROVED ALGORITHM 377
9.3 FEATURE SELECTION, CONCEPT CATEGORIZATION AND CHARACTERISTIC
DESCRIPTION VIA AFS THEORY 381
9.3.1 FEATURE SELECTION 382
9.3.2 PRINCIPAL CONCEPT SELECTION AND CONCEPT CATEGORIZATION 384
9.3.3 CHARACTERISTIC DESCRIPTION OF SAMPLES 390
9.3.4 AFS FUZZY CLUSTERING ALGORITHM BASED ON THE I/A: - A NEAREST
NEIGHBORS 391
9.3.5 AFS FUZZY CLUSTERING ALGORITHM BASED ON PRINCIPAL CONCEPTS AND THE
CATEGORIES OF CONCEPTS 393
9.4 EXPERIMENTAL EVALUATION OF THE ALGORITHMS 395
9.4.1 EXPERIMENTAL STUDIES OF AFS FUZZY CLUSTERING ALGORITHM BASED ON
PRINCIPAL CONCEPTS AND THE CATEGORIES OF CONCEPTS 395
9.4.2 EXPERIMENTAL RESULTS OF AFS FUZZY CLUSTERING ALGORITHM BASED ON
THE 1/K - A NEAREST NEIGHBORS 405
EXERCISES 416
OPEN PROBLEMS 417
REFERENCES 417
10 AFS FUZZY CLASSIFIERS 423
10.1 CLASSIFIER DESIGN BASED ON AFS FUZZY LOGIC 424
10.1.1 CLASSIFIER DESIGN BASED ON AFS FUZZY LOGIC 425 10.1.2
EXPERIMENTAL RESULTS 427
10.2 AFS CLASSIFIER DESIGN BASED ON ENTROPY ANALYSIS 434 10.2.1 FUZZY
ENTROPY 435
10.2.2 CLASSIFIER DESIGN 437
10.2.3 EXPERIMENTAL STUDIES 439
10.2.4 EXPERIMENT STUDIES OF PARAMETERS 446
10.2.5 STABILITY AND UNIVERSALITY OF THE COHERENCE MEMBERSHIP FUNCTIONS
450
10.2.6 COMPARATIVE ANALYSIS 453
10.3 AFS FUZZY CLASSIFIER BASED ON FUZZY DECISION TREES 454 10.3.1
GENERATION OF FUZZY RULES FROM AFS DECISION TREES 456
10.3.2 CONSTRUCTION OF AFS DECISION TREES 459
10.3.3 RULE EXTRACTION AND PRUNING 461
10.3.4 DETERMINING THE OPTIMAL THRESHOLD 6 464
IMAGE 6
XVIII CONTENTS
10.3.5 INFERENCE OF DECISION ASSIGNMENT AND ASSOCIATED CONFIDENCE
DEGREES 468
10.3.6 EXPERIMENTAL STUDIES 469
10.3.7 STRUCTURE COMPLEXITY 477
EXERCISES 479
OPEN PROBLEMS 479
APPENDIX A 480
REFERENCES 485
BIBLIOGRAPHY 491
INDEX 507
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| indexdate | 2024-07-09T21:38:45Z |
| institution | BVB |
| isbn | 9783642004025 |
| language | English |
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| series | Studies in Fuzziness and Soft Computing |
| series2 | Studies in Fuzziness and Soft Computing |
| spelling | Axiomatic Fuzzy Set Theory and Its Applications Xiaodong Liu ; Witold Pedrycz 1. Aufl. Berlin Springer-Verlag GmbH 2009 1 Online-Ressource (500 S.) 235 mm x 155 mm txt rdacontent c rdamedia cr rdacarrier Studies in Fuzziness and Soft Computing 244 Fuzzy-Menge (DE-588)4061868-7 gnd rswk-swf Axiomatik (DE-588)4004038-0 gnd rswk-swf Fuzzy-Menge (DE-588)4061868-7 s Axiomatik (DE-588)4004038-0 s DE-604 Liu, Xiaodong Sonstige oth Pedrycz, Witold 1953- Sonstige (DE-588)122838203 oth Erscheint auch als Druck-Ausgabe, Hardcover 978-3-642-00401-8 Studies in Fuzziness and Soft Computing 244 (DE-604)BV021858135 244 https://doi.org/10.1007/978-3-642-00402-5 Verlag Volltext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017545078&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Axiomatic Fuzzy Set Theory and Its Applications Studies in Fuzziness and Soft Computing Fuzzy-Menge (DE-588)4061868-7 gnd Axiomatik (DE-588)4004038-0 gnd |
| subject_GND | (DE-588)4061868-7 (DE-588)4004038-0 |
| title | Axiomatic Fuzzy Set Theory and Its Applications |
| title_auth | Axiomatic Fuzzy Set Theory and Its Applications |
| title_exact_search | Axiomatic Fuzzy Set Theory and Its Applications |
| title_full | Axiomatic Fuzzy Set Theory and Its Applications Xiaodong Liu ; Witold Pedrycz |
| title_fullStr | Axiomatic Fuzzy Set Theory and Its Applications Xiaodong Liu ; Witold Pedrycz |
| title_full_unstemmed | Axiomatic Fuzzy Set Theory and Its Applications Xiaodong Liu ; Witold Pedrycz |
| title_short | Axiomatic Fuzzy Set Theory and Its Applications |
| title_sort | axiomatic fuzzy set theory and its applications |
| topic | Fuzzy-Menge (DE-588)4061868-7 gnd Axiomatik (DE-588)4004038-0 gnd |
| topic_facet | Fuzzy-Menge Axiomatik |
| url | https://doi.org/10.1007/978-3-642-00402-5 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017545078&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV021858135 |
| work_keys_str_mv | AT liuxiaodong axiomaticfuzzysettheoryanditsapplications AT pedryczwitold axiomaticfuzzysettheoryanditsapplications |