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OPERATIONS OF FUZZY SETS 97 Table 4.2 Representation of hedges in fuzzy logic Hedge Mathematical expression Graphical representation A little ½ A ðxÞŠ 1:3 Slightly ½ A ðxÞŠ 1:7 Very ½ A ðxÞŠ 2 Extremely ½ A ðxÞŠ 3 Very very ½ A ðxÞŠ 4 More or less pffiffiffiffiffiffiffiffiffiffiffi A ðxÞ Somewhat pffiffiffiffiffiffiffiffiffiffiffi A ðxÞ Indeed 2½ A ðxÞŠ 2 if 0 4 A 4 0:5 1 2½1 A ðxÞŠ 2 if 0:5 < A 4 1 Mathematical and graphical representation of hedges are summarised in Table 4.2. 4.4 Operations of fuzzy sets The classical set theory developed in the late 19th century by Georg Cantor describes how crisp sets can interact. These interactions are called operations.
98 FUZZY EXPERT SYSTEMS Figure 4.6 Operations on classical sets We look at four of them: complement, containment, intersection and union. These operations are presented graphically in Figure 4.6. Let us compare operations of classical and fuzzy sets. Complement . Crisp sets: Who does not belong to the set? . Fuzzy sets: How much do elements not belong to the set? The complement of a set is an opposite of this set. For example, if we have the set of tall men, its complement is the set of NOT tall men. When we remove the tall men set from the universe of discourse, we obtain the complement. If A is the fuzzy set, its complement :A can be found as follows: :A ðxÞ ¼1 A ðxÞ ð4:12Þ For example, if we have a fuzzy set of tall men, we can easily obtain the fuzzy set of NOT tall men: tall men ¼ð0=180; 0:25=182:5; 0:5=185; 0:75=187:5; 1=190Þ NOT tall men ¼ð1=180; 0:75=182:5; 0:5=185; 0:25=187:5; 0=190Þ Containment . Crisp sets: Which sets belong to which other sets? . Fuzzy sets: Which sets belong to other sets?
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MICHAEL NEGNEVITSKY Artificial Inte
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We work with leading authors to dev
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Pearson Education Limited Edinburgh
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Contents Preface Preface to the sec
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CONTENTS ix 7 Evolutionary computat
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Preface ‘The only way not to succ
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PREFACE xiii In Chapter 3, we prese
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Prefacetothesecondedition The main
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Introduction to knowledgebased inte
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INTELLIGENT MACHINES 3 Figure 1.1 T
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THE HISTORY OF ARTIFICIAL INTELLIGE
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SUMMARY 17 Although fuzzy systems a
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SUMMARY 19 Table 1.1 Period A summa
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QUESTIONS FOR REVIEW 21 or fuzzy se
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REFERENCES 23 Kosko, B. (1997). Fuz
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Rule-based expert systems 2 In whic
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RULES AS A KNOWLEDGE REPRESENTATION
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THE MAIN PLAYERS IN THE EXPERT SYST
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STRUCTURE OF A RULE-BASED EXPERT SY
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FUNDAMENTAL CHARACTERISTICS OF AN E
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FORWARD AND BACKWARD CHAINING INFER
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MEDIA ADVISOR: A DEMONSTRATION RULE
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MEDIA ADVISOR: A DEMONSTRATION RULE
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MEDIA: A DEMONSTRATION RULE-BASED E
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Page 68 and 69: CONFLICT RESOLUTION 49 . Fire the m
Page 70 and 71: SUMMARY 51 . Dealing with incomplet
Page 72 and 73: QUESTIONS FOR REVIEW 53 . Expert sy
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Page 76 and 77: BASIC PROBABILITY THEORY 57 Table 3
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Page 94 and 95: CERTAINTY FACTORS THEORY AND EVIDEN
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Page 102 and 103: SUMMARY 83 team also could assume t
Page 104 and 105: REFERENCES 85 . Both Bayesian reaso
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Page 108 and 109: FUZZY SETS 89 Range of logical valu
Page 110 and 111: FUZZY SETS 91 Figure 4.2 Crisp (a)
Page 112 and 113: FUZZY SETS 93 Figure 4.3 Crisp (a)
Page 114 and 115: LINGUISTIC VARIABLES AND HEDGES 95
Page 118 and 119: OPERATIONS OF FUZZY SETS 99 Similar
Page 120 and 121: OPERATIONS OF FUZZY SETS 101 Figure
Page 122 and 123: FUZZY RULES 103 Example: NOT (tall
Page 124 and 125: FUZZY RULES 105 Figure 4.9 Monotoni
Page 126 and 127: FUZZY INFERENCE 107 determined by f
Page 128 and 129: FUZZY INFERENCE 109 However, the OR
Page 130 and 131: FUZZY INFERENCE 111 Step 4: Defuzzi
Page 132 and 133: FUZZY INFERENCE 113 Figure 4.15 The
Page 134 and 135: BUILDING A FUZZY EXPERT SYSTEM 115
Page 144 and 145: SUMMARY 125 4.8 Summary In this cha
Page 146 and 147: BIBLIOGRAPHY 127 9 What is clipping
Page 148: BIBLIOGRAPHY 129 Zadeh, L.A. (1975)
Page 151 and 152: 132 FRAME-BASED EXPERT SYSTEMS Figu
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Page 161 and 162: 142 FRAME-BASED EXPERT SYSTEMS such
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148 FRAME-BASED EXPERT SYSTEMS Figu
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150 FRAME-BASED EXPERT SYSTEMS In a
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156 FRAME-BASED EXPERT SYSTEMS illu
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158 FRAME-BASED EXPERT SYSTEMS Area
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160 FRAME-BASED EXPERT SYSTEMS ) On
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162 FRAME-BASED EXPERT SYSTEMS The
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164 FRAME-BASED EXPERT SYSTEMS Bibl
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166 ARTIFICIAL NEURAL NETWORKS What
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168 ARTIFICIAL NEURAL NETWORKS Tabl
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170 ARTIFICIAL NEURAL NETWORKS Figu
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172 ARTIFICIAL NEURAL NETWORKS Step
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174 ARTIFICIAL NEURAL NETWORKS In F
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176 ARTIFICIAL NEURAL NETWORKS Why
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178 ARTIFICIAL NEURAL NETWORKS To p
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180 ARTIFICIAL NEURAL NETWORKS (b)
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182 ARTIFICIAL NEURAL NETWORKS Then
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184 ARTIFICIAL NEURAL NETWORKS The
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192 ARTIFICIAL NEURAL NETWORKS In o
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194 ARTIFICIAL NEURAL NETWORKS wher
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198 ARTIFICIAL NEURAL NETWORKS In t
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200 ARTIFICIAL NEURAL NETWORKS Ther
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204 ARTIFICIAL NEURAL NETWORKS Here
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206 ARTIFICIAL NEURAL NETWORKS The
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214 ARTIFICIAL NEURAL NETWORKS phys
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216 ARTIFICIAL NEURAL NETWORKS 10 D
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Evolutionary computation 7 In which
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SIMULATION OF NATURAL EVOLUTION 221
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GENETIC ALGORITHMS 223 Figure 7.2 A
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GENETIC ALGORITHMS 225 Figure 7.3 T
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GENETIC ALGORITHMS 227 Figure 7.5 T
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GENETIC ALGORITHMS 229 When necessa
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GENETIC ALGORITHMS 231 A surface of
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WHY GENETIC ALGORITHMS WORK 233 m x
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MAINTENANCE SCHEDULING WITH GENETIC
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EVOLUTION STRATEGIES 243 Define a s
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GENETIC PROGRAMMING 245 Which metho
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GENETIC PROGRAMMING 247 Table 7.3 T
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GENETIC PROGRAMMING 249 Figure 7.15
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QUESTIONS FOR REVIEW 255 . Evolutio
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BIBLIOGRAPHY 257 Whitley, L.D. (199
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260 HYBRID INTELLIGENT SYSTEMS repr
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262 HYBRID INTELLIGENT SYSTEMS enti
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264 HYBRID INTELLIGENT SYSTEMS Figu
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266 HYBRID INTELLIGENT SYSTEMS Now
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278 HYBRID INTELLIGENT SYSTEMS sets
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280 HYBRID INTELLIGENT SYSTEMS How
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282 HYBRID INTELLIGENT SYSTEMS In t
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292 HYBRID INTELLIGENT SYSTEMS word
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294 HYBRID INTELLIGENT SYSTEMS Step
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296 HYBRID INTELLIGENT SYSTEMS Step
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298 HYBRID INTELLIGENT SYSTEMS 4 De
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Knowledge engineering and data mini
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INTRODUCTION, OR WHAT IS KNOWLEDGE
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WILL AN EXPERT SYSTEM WORK FOR MY P
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WILL A FUZZY EXPERT SYSTEM WORK FOR
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WILL A NEURAL NETWORK WORK FOR MY P
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WILL GENETIC ALGORITHMS WORK FOR MY
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WILL A HYBRID INTELLIGENT SYSTEM WO
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DATA MINING AND KNOWLEDGE DISCOVERY
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SUMMARY 361 9.8 Summary In this cha
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REFERENCES 363 7 Why are fuzzy syst
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Glossary The glossary entries are c
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GLOSSARY 367 Back-propagation see B
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GLOSSARY 369 Clipping A common meth
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GLOSSARY 371 amounts of data in ord
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GLOSSARY 373 Evolution strategy A n
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GLOSSARY 375 Fuzzy rule A condition
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GLOSSARY 377 Heuristic search A sea
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GLOSSARY 379 Knowledge base A basic
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GLOSSARY 381 Method A procedure ass
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GLOSSARY 383 Output layer The last
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GLOSSARY 385 Roulette wheel selecti
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GLOSSARY 387 the network differs fr
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GLOSSARY 389 determines the strengt
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392 AI TOOLS AND VENDORS EXSYS, Inc
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394 AI TOOLS AND VENDORS M.4 A powe
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396 AI TOOLS AND VENDORS CynapSys,
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398 AI TOOLS AND VENDORS 215 Parkwa
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400 AI TOOLS AND VENDORS text files
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402 AI TOOLS AND VENDORS TransferTe
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404 AI TOOLS AND VENDORS La Jolla,
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406 AI TOOLS AND VENDORS Fax: +1 (3
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408 INDEX belongs-to 137-8 Berkeley
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410 INDEX fuzzification 107 fuzzifi
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412 INDEX Mamdani, E. 20, 106 Mamda
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INDEX 415 unsupervised learning 200
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