Possibility theory with applications to data analysis:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Taunton
Research Studies Press [u.a.]
1998
|
| Series: | Control Systems Centre <Manchester>: UMIST Control Systems Centre series
5 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVII, 261 S. Ill., graph. Darst. |
| ISBN: | 086380229X 0471981613 |
Staff View
MARC
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| 245 | 1 | 0 | |a Possibility theory with applications to data analysis |c Olaf Wolkenhauer |
| 264 | 1 | |a Taunton |b Research Studies Press [u.a.] |c 1998 | |
| 300 | |a XXVII, 261 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Control Systems Centre <Manchester>: UMIST Control Systems Centre series |v 5 | |
| 650 | 4 | |a Fuzzy sets | |
| 650 | 4 | |a Possibility | |
| 650 | 4 | |a Probabilities | |
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Record in the Search Index
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| adam_text | Contents
List of figures xvii
Notation xxi
1 Introduction 1
1.1 Motivation , , 1
1.2 Methodology 3
1.3 Basic Ideas Examples 6
1.4 Fuzzy Systems Toolbox 8
1.5 Structure of the Book 12
2 Uncertainties in Control Systems 15
2.1 Condition Monitoring 15
2.2 Introductory Example 15
2.2.1 Model of a Heating System 16
2.2.2 Reliability Evaluation 17
2.2.3 Vagueness 17
2.2.4 Ambiguity 18
2.2.5 Variability 19
2.2.6 Imprecision 20
2.3 Summary 22
3 Uncertainty Techniques 25
3.1 Sets 30
3.1.1 Fuzzy Sets 31
3.1.2 Level set or a cut Representation 32
3.1.3 Semantics 35
3.2 Probabilities 36
3.3 Imprecise Probabilities 39
3.3.1 Random Closed Intervals 41
3.3.2 Multi Valued Mappings: Summary and Outlook .... 43
I xiii
I
xiv Contents
3.4 Uncertainties in Systems Analysis 46
3.4.1 Ordinary Relations 48
3.4.2 The Loss of Certainty 50
3.4.3 Metric Spaces 52
3.4.4 The Triangle Inequality and Triangular norms 54
3.4.5 Fuzzy Relations 57
3.5 Uncertainty Techniques: Summary and Conclusions 61
4 Possibility Theory 65
4.1 Extension Principle 66
4.2 Algebraic Operations with Fuzzy Quantities 69
4.3 Fuzzy Events 70
4.4 Assessing Fuzzy Sets 72
4.5 Confidence Measures 74
4.5.1 Possibility Distributions 74
4.5.2 Possibility and Necessity Measures 75
4.5.3 Confidence Measures as Probability Bounds 77
4.6 Probability Possibility Transformations 82
4.7 Upper and Lower Distribution Functions 85
4.8 Ranking Fuzzy Quantities 86
4.9 Mean Value of a Fuzzy Number 88
4.10 Fuzzy Measures 90
4.10.1 Fuzzy Integrals 91
4.10.2 Dynamic Reliability Evaluation 94
4.10.3 Fuzzy Dynamic Systems 97
4.11 Possibilities : Summary and Conclusions 99
5 Possibilistic Change Detection 101
5.1 Knowledge Representations 102
5.1.1 Boolean Logic 102
5.1.2 Approximate Reasoning 103
5.1.3 Conditioning 103
5.1.4 Gradual Rules 105
5.2 Comparing Distribution Functions 106
5.2.1 Problem Definition 106
5.2.2 Probability Distribution Estimates 108
5.2.3 The Feature Space C 108
5.2.4 From Probabilities to Possibilities 109
5.2.5 Fuzzy Restrictions on Space Y 110
5.2.6 The Detection of Changes in Signals 110
5.2.7 Properties of the Algorithm 111
Contents xv
5.2.8 Comparing Distribution Functions: Conclusions and Dis¬
cussion 112
5.3 Control Charts 113
5.3.1 Statistical Change Detection 113
5.3.2 Fuzzy Alarm Limits 115
5.3.3 Possibilistic Change Detection: Discussion 117
5.3.4 Possibilistic Control Charts 118
5.3.5 Possibilistic Change Detection: Conclusions and Discus¬
sion 128
6 Fuzzy Data Analysis 131
6.1 From Random Sets to Fuzzy Data 132
6.1.1 Random Sets 133
6.1.2 Random Sets and Set Valued Statistics 140
6.1.3 Level Set Valued Statistics 142
6.1.4 From Random Sets to Fuzzy Sets 142
6.1.5 Set Valued Statistics and Probabilistic Sets 145
6.1.6 Random Sets and Confidence Measures 147
6.1.7 Examples of Specific Random Sets 149
6.1.8 Context Mappings (Set Valuedisation) 151
6.1.9 Random Sets: Discussion and Conclusions 152
6.2 Fuzzy Data: Random Fuzzy Sets 153
6.2.1 Local Uncertainty Models for Time Series Data 154
6.2.2 Uncertain Datum 154
6.2.3 Changeability 157
6.2.4 Uncertain Trend 157
6.2.5 Local Uncertainty Models: Discussion 158
6.3 Collections of Fuzzy Observations 159
6.4 Comparing Time Series 161
6.4.1 Fuzzy Correlation 162
6.4.2 Co Similarity 164
6.5 Fuzzy Data Analysis: Conclusions and Discussion 174
7 Conclusions and Perspectives 179
A Probability Theory 187
A.I Probability Spaces 188
A.2 Random (Point) Variables 189
A.3 Lebesgue Theory 190
A.4 Probabilistic Independence 192
A.5 Stochastic Processes 193
xvi Contents
B Evidence Theory 195
B.I Belief Functions 196
B.2 Examples for Belief Assignments 197
B.3 Example: The Breakable Sensor 199
C Fuzzy Systems 201
C.I Fuzzy Logic 201
C.2 Projection and Extension 202
C.3 Composition of Fuzzy Relations 203
C.4 Fuzzy IF THEN Rules 204
C.5 Fuzzy Inference Rules for Approximate Reasoning 205
C.6 Possibilistic Reasoning 209
D Selected Topics 211
D.I Sets and Functions 211
D.2 The Expectation Operator 213
D.3 Li? Representation of Fuzzy Sets 218
D.4 Context Mappings 219
D.5 L Fuzzy Sets 220
D.6 Rough Set Approximation 221
D.7 Stochastic Fuzzy Sets 223
D.7.1 Random Fuzzy Sets 223
D.7.2 Probabilistic Sets 224
D.7.3 Fuzzy Random Variables 225
E Summary of Mappings 227
F Glossary 235
|
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| discipline | Mathematik Mess-/Steuerungs-/Regelungs-/Automatisierungstechnik / Mechatronik |
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| id | DE-604.BV012055504 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:20:52Z |
| institution | BVB |
| isbn | 086380229X 0471981613 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008159231 |
| oclc_num | 37688294 |
| open_access_boolean | |
| owner | DE-703 DE-188 |
| owner_facet | DE-703 DE-188 |
| physical | XXVII, 261 S. Ill., graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Research Studies Press [u.a.] |
| record_format | marc |
| series | Control Systems Centre <Manchester>: UMIST Control Systems Centre series |
| series2 | Control Systems Centre <Manchester>: UMIST Control Systems Centre series |
| spelling | Wolkenhauer, Olaf 1966- Verfasser (DE-588)133515931 aut Possibility theory with applications to data analysis Olaf Wolkenhauer Taunton Research Studies Press [u.a.] 1998 XXVII, 261 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Control Systems Centre <Manchester>: UMIST Control Systems Centre series 5 Fuzzy sets Possibility Probabilities Fuzzy-Logik (DE-588)4341284-1 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Fuzzy-Logik (DE-588)4341284-1 s DE-604 Control Systems Centre <Manchester>: UMIST Control Systems Centre series 5 (DE-604)BV009763376 5 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008159231&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Wolkenhauer, Olaf 1966- Possibility theory with applications to data analysis Control Systems Centre <Manchester>: UMIST Control Systems Centre series Fuzzy sets Possibility Probabilities Fuzzy-Logik (DE-588)4341284-1 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4341284-1 (DE-588)4079013-7 |
| title | Possibility theory with applications to data analysis |
| title_auth | Possibility theory with applications to data analysis |
| title_exact_search | Possibility theory with applications to data analysis |
| title_full | Possibility theory with applications to data analysis Olaf Wolkenhauer |
| title_fullStr | Possibility theory with applications to data analysis Olaf Wolkenhauer |
| title_full_unstemmed | Possibility theory with applications to data analysis Olaf Wolkenhauer |
| title_short | Possibility theory with applications to data analysis |
| title_sort | possibility theory with applications to data analysis |
| topic | Fuzzy sets Possibility Probabilities Fuzzy-Logik (DE-588)4341284-1 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Fuzzy sets Possibility Probabilities Fuzzy-Logik Wahrscheinlichkeitstheorie |
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