NeuroFuzzy nd Soft Computing A n Ac Lnn n cn nnc Sig g g
[email protected].
[email protected] thu.edu.tw tw Computer Science Depament, Tsing Hua University Taiwan The MathWorks Natick Massachusetts USA
Ti S
[email protected] Depament of Computer and Information Science National Chiao Tung University Hsinchu Taiwan
Eiji Mii
[email protected] @joho.kan sai.co.jp p Information Systems Depament Kansai Paint Co. Ltd. Fushimi Chuo-ku Osaka Japan
Pretie H Upper dde River J 7
La Cgs CgsCaagga aa
Jang, Jyh-Shing Roger. Neurofuzzy d so computing: a computatonal aproach to ling d machine intelligence / yhShing Roger ang, ChueTsai Sun, Eiji Mizuti p. cm Includes biblioaphca eferences d ndex ISBN 01326163 1 So Computng 2 Neural netwoks (Computer science) science) 3 Fuzzy systems I Sun ChuenTsai. II. Mizutani, Eiji III. Title. 9629050 QA769S6336 1997 CIP 3dc20 Acqusitions Edior Tom Robbins Production Editor oseph Scordao Diector Prd Mfg David Riccardi Prducton Mager Bayi Mendoza Leon Cover Designer Bruce Kenselaar Copy Editor Patricia Daly Buyer Donna Sullivan Editoral Assistt Ncy Garcia
&
@997 by PrenticeHall PrenticeHall Inc Simon Schust /A Viacom Company Upr Saddle Rver, N 07458
&
All rghts reserved. No pa of this book may be reproduced, in any form or by y means, without rmission in wrting frm he publisher. The author d publishe of this book have used their st effos in peparing this this book These effos include the development esearch, d testng of the theores and pogams to determine their effectiveness The author d publishe make no waanty of y knd, expressed or implied, with egard to these programs or the documetaion contained in this book. The author d publishe publishe shall not liable liable in y event event for incidental or consuential damages in connection with or arising ou of, the fishing rfomance, or use of these programs. Prnted n the United States of Amerca 10 9 8 7 6 5 4 3 2
N 1313 PreniceHall Ineational (UK) Limited London PrenticeHall of Australia Pty. Limited, Sydney PrentceHall Cada Inc, Toronto PrenticeHal Hispoamercana, SA, Mexico PeniceHall of Inda Private Limted, New lhi PrenticeHall of ap, Inc Tokyo Simon Schuste Asa Pte Ltd, Singapoe Editora PrnticeHall do Basil, Ltda, Rio de eiro
&
La Cgs CgsCaagga aa
Jang, Jyh-Shing Roger. Neurofuzzy d so computing: a computatonal aproach to ling d machine intelligence / yhShing Roger ang, ChueTsai Sun, Eiji Mizuti p. cm Includes biblioaphca eferences d ndex ISBN 01326163 1 So Computng 2 Neural netwoks (Computer science) science) 3 Fuzzy systems I Sun ChuenTsai. II. Mizutani, Eiji III. Title. 9629050 QA769S6336 1997 CIP 3dc20 Acqusitions Edior Tom Robbins Production Editor oseph Scordao Diector Prd Mfg David Riccardi Prducton Mager Bayi Mendoza Leon Cover Designer Bruce Kenselaar Copy Editor Patricia Daly Buyer Donna Sullivan Editoral Assistt Ncy Garcia
&
@997 by PrenticeHall PrenticeHall Inc Simon Schust /A Viacom Company Upr Saddle Rver, N 07458
&
All rghts reserved. No pa of this book may be reproduced, in any form or by y means, without rmission in wrting frm he publisher. The author d publishe of this book have used their st effos in peparing this this book These effos include the development esearch, d testng of the theores and pogams to determine their effectiveness The author d publishe make no waanty of y knd, expressed or implied, with egard to these programs or the documetaion contained in this book. The author d publishe publishe shall not liable liable in y event event for incidental or consuential damages in connection with or arising ou of, the fishing rfomance, or use of these programs. Prnted n the United States of Amerca 10 9 8 7 6 5 4 3 2
N 1313 PreniceHall Ineational (UK) Limited London PrenticeHall of Australia Pty. Limited, Sydney PrentceHall Cada Inc, Toronto PrenticeHal Hispoamercana, SA, Mexico PeniceHall of Inda Private Limted, New lhi PrenticeHall of ap, Inc Tokyo Simon Schuste Asa Pte Ltd, Singapoe Editora PrnticeHall do Basil, Ltda, Rio de eiro
&
Contents
Preface
�e Organization
Oba te Eme Pom Acknowledgments How to Contact Us 1
Introduction to Neuro-Fzzy and So Computing 11 1.2
1.3
I 2
Introduction So Computing Constituents Constituents and Conventional Artifcial Intelligence 1.21 om Conventional AI to Computational Computat ional Intelligence 12.2 Neural Networks zzy Set Theory T heory 123 Evolutionary Computation 124 Neuro-zzy and So Computing Characteristics Characteris tics
Fzzy Set Theory Fzzy Sets Introdu ction 2 Introduction 2.2 Bic Defnitions and Terminology 2.3 Settheoretic Operations Formulation and Parameterization 24 MF Formulation
xix �
xi xxiii
xiv vi 1
1 3
6 6 7 7
11 13
13 4 21 24
iii
iv
Contnts
24 1 MFs of One Dimension 242 MFs of Two Dimensions 243 Derivatives of Parameterized MFs 25 More on zzy Union, Intersection, and Complement* 25 zzy Complement* 252 zzy Intersection and Union* 253 Parameterized Tnorm and Tconorm* 26 Summary xercises 3 uzzy Rules and uzzy Reasonng
Introduction 32 xtension Principle and zzy Relations 321 xtension Principle 32 2 zzy Relations 33 zzy IfThen Rules 331 Linguistic Variales 332 zzy Then Rules 3.4 zzy Reoning 341 Compositional Rule of Inference 342 zzy Reoning 35 Summary xercises
3.1
4 uzzy Inference Systems
41 Introduction 42 Mamdani zzy Models 421 Other Variants 4.3 Sugeno zzy Models 44 Tsumoto zzy Models 45 Other Considerations 45 1 Input Space Partitioning 452 zzy Modeling 46 Summary xercises
24 30 34 35 35
36 40
42 42 47
47 47 47 50
54 54 59
62 63 64 70 70 73 73 74
79 81 84 8 5
86 87 89 90
CNTENT
II
Regression and Optimization
5 LeastSquares Metods for System Identication
5 System Identication An Introduction 5 2 ics of Matrix Manipulation and Calculus 53 LetSquares Estimator 54 Geometric Interpretation of LSE 55 Recursive LetSquares Estimator 56 Recursive LSE for TimeVarying Systems* 57 Statistical Properties and the Maximum Likelihood Estimator* 58 LSE for Nonlinear Models 59 Summary Exercises 6 Derivativebased Optimization
61 Introduction 62 Descent Methods 621 Gradiented Methods 63 The Method of Steepest Descent 64 Newton's Methods 641 Clsical Newton's Method 642 Modied Newton ' s Methods* 643 QuiNewton Methods* 65 Step Size Determination 651 Initial racketing 652 Line Searchs 653 Termination Rules* 66 Conjugate Gradient Methods* 66 Conjugate Directions* 662 om Orthogonality to Conjugacy* 663 Conjugate Gradient Algorithms* 67 Analysis of Quadratic Ce 671 Descent Methods with Line Minimization 672 Steepest Descent Method without Line Minimization 68 Nonlinear Letsquares Prolems 681 GaussNewton Method 682 LevenergMarquardt Concepts 69 Incorporation of Stochtic Mechanisms
v
93 95
95 97
04 0 113 116 118 122 125 126 129
129 129 130 133 134 34 136 39 141 141 142 146 48 48 149
152 54 156 156 160 16 163 166
vi
Contnts
6.10 Summary Exercises
168 168
7 Derivativeee Optimzation
173
7 Introduction 7.2 Genetic Algorithms 7.3 Simulated Annealing 7.4 Random Search 75 Downhill Simplex Search 76 Summary Exercises
73 75 181 86 89
III
193
194
9
Neural Networks
8 Adaptive Networks
199
81 8.2 83 8.4
Introduction Architecture ackpropagation for Feedforward Networks Extended ackpropagation for Recurrent Networks 8.41 Synchronously Operated Networks P and RRL 8.4.2 Continuously Operated Networks Mon's Gain Formula* 85 Hyrid Learning Rule Comining Steepest Descent and LSE 8.5 OLine Learning atch Learning 85.2 OnLine Learning PatternyPattern Learning 8.5.3 Dierent Ways of Comining Steepest Descent nd LSE 8.6 Summary Exercises
9 Supervised Learnng Neural Networks
91 Introduction 9.2 Perceptrons 92.1 Architecture and Learning Rule 922 ExclusiveOR Prolem 9.3 Adaline 9.4 ackpropagation Multilayer Perceptrons 9.41 ackpropagation Learning Rule 942 Methods of Speeding Up MLP aining
99 200 205 210 212 215 219 220 222 222 223 223 226
226 227 227 229 230 233 234 236
vii
CNTENT
94.3 MLP's Approximation Power 9.5 Radial is nction Networks 95.1 Architectures and Learning Methods 9.5.2 nctional Equivalence to FIS 9.5.3 Interpolation and Approximation RFNs 95.4 Examples 96 Modular Networks 97 Summary Exercises 1 Learning from Reinforcement
238 238 238 241 242 244 246 250 25 258
258 10.1 Introduction 259 10.2 Failure Is the Surest Path to Success 259 10.2.1 Jackpot Journey 10.2.2 Credit Assignment Prolem 262 10.2.3 Evaluation nctions 263 103 Temporal Dierence Learning 264 10.3.1 TD Formulation 265 10.3.2 Expected Jackpot 266 268 10.3.3 Predicting Cumulative Outcomes 270 104 The Art of Dynamic Programming 10.4. 1 Formulation of Clsical Dynamic Programming 270 10.4.2 Incremental Dynamic Programming 272 10.5 Adaptive Heuristic Critic 273 273 10.5.1 Neuronlike Critic 105.2 An Adaptiv Neural Critic Algorithm 274 277 1053 Exploration and Action Selection 106 Qearning 278 278 106.1 ic Concept 279 10.6.2 Implementation 281 10.7 A Cost Path Prolem 0 7. 1 Expected Cost Path Prolem y TD methods 282 0.7 .2 Finding an Optimal Path in a Deterministic Minimum Cost Path Prolem 283 07.3 State Representations for Generalization 287 10.8 World Modeling* 288 10 .8.1 Modelfree and Modeled Learning* 288 289 0.8.2 Distal Teacher*
vii
Contnts
08.3 earnng Speed* 09 Other Network Congurations* 1091 DiideandConquer Methodology* 0.92 Recurent Networks* 10.10Reinforcement earnng by Eolutonay Computaton* 001 ucket rigade* 1002 Genetic Reinforcers* 0. 0.3 Immune Modeling* 10. 11Summary Exercises 1 1 Unsupervised Learnng and Oter Neural Networks
1 Introduction 2 Competitie earnng Networks 3 Kohonen SelfOganizing Networks 114 earnng Vector Quntization .5 Hebbian earning 11.6 Principal Component Networks .6 . Princpal Component Analysis 1162 Oja's Moded Hebbian Rule .7 he Hopeld Network 71 ContentAddressable Nature .7 .2 inary Hopeld Networks 7 .3 ContinuousValued Hopeld Networks 1 7.4 aelng Salesperson Problem 7.5 The oltzmann Machne 11.8 Summary Exercises
IV
Neuro-Fuzzy Modeling
12 ANFIS: Adaptive Neurouzzy Inference System
121 Introduction 12.2 ANFIS Archtecture 12.3 Hybrid earnng Algorthm 24 earning Methods that Crossfertilize ANFIS nd RFN 25 ANFIS 3 ' a Unersal Approximator*
289 290 290 291 292 292 292 293
293 294 31
30 302 305 308 310 312 312 316 316 317
318 32 324 326 327 328
333 335
335 336 340 341 342
CNTENT
ix
126 Simulation Examples 12.6 ractical Considerations 2 62 Example 1 Modeling a Two-Input Sinc nction 12 .6.3 Example 2 Modeling a Three-Input Nonlinear nction 12.6 4 Example 3 Online Identication in Control Systems 12 6.5 Example 4 redicting Chaotic Time Series 127 Extensions and Adanced Topics Exercises
345 345 346 348 351 353 360
13 Coactive Neurozzy Modeling: owards Generalized ANFIS
369
13.1 Introduction 132 amework 13.2 Toward Multiple Inputs/ Outputs Systems 322 Architectural Comparisons 13.3 Neuron nctions for Adaptie Networks 13.31 zzy Membership nctions ersus Receptie Field Units 33.2 Nonlinear Rule 13.33 Modied Sigmoidal and uncation Filter nctions 3.4 Neuro-zzy Spectrum 135 Analysis of Adaptie earning Capability 135 Conergence ed on the Steepest Descent Method Alone 135.2 Interpretability Spectrum 13.5 3 Eolution of Antecedents (MFs ) 35 4 Eolution of Consequents ( Rules) 135.5 Eoling Prtitions 13.6 Summary Exercises
369 370 370 370 372 373 376 380 382 385 385 386 387 389 390 393 395
V
Advanced NeurFuzzy Modeling
14 Classication and Regression ees
41 Introduction 142 Decision ees 4.3 CAR Algorithm for ee Induction 43.1 ee Growing 1432 ee Pruning 144 Using CART for Structure Identication in ANFIS
363
4 43
403 404 406 407 41 3
416
Contnts
145 Summary Exercises 15 Data Clustering Algoritms
42 421 423
5 Introduction 52 KMeans Clustering 15.3 zzy CMeans Clustering 154 Mountain Clustering Method 55 Subtractie Clustering 15.6 Summary Exercises
423 424 425 427 43 432 432
16 Rulebase Structure Identication
434
16 Introduction 16.2 Input Selection 163 Input Space Partitionng 164 Rulebe Organzaton 165 Focus Seted Rule Combination 16.6 Summary Exercses
VI
Neuro-zzy Control
17 Neurouzzy Control I
7.1 Introduction 17 2 Feedback Control Systems and Neuro-zzy Control: An Oeriew 172.1 Feedback Control Systems 172.2 Neuro-zzy Control 17.3 Expert Control: Mimicking An Expert 174 Inerse earnng 17.4 ndamentals 17.42 Ce Studies 175 Specalzed earnng 176 ackpropagation Through Time and Realime Recurrent earning 176.1 ndamentals 176.2 Ce Studies: the Inerted Pendulum System 77 Summary
434 435 436 441 446 447 448
45 453
453 454 454 458 458 460 460 463 465 469
469 470 476
CNTENT
Exercses 18 NeroFzzy Control II
181 Introducton 182 Renforcement earnng Control 182.1 Control Enronment 182.2 Neurozzy Renforcement Controllers 183 Gradent-ee Optmzaton 18.3. 1 GAs Codng and Genetc Operators 18.32 GAs Formulatng Objecte nctons 184 Gan Schedulng 18.41 ndamentals 18.42 Ce Studes 18.5 Feedback nearzaton and Sldng Control 18.6 Summary Exercses
VII
Advanced Applications
19 ANFIS Applications
xi 477
48
480 480 480 481 483 48 4
488 48 9
489 491 493 496 497
5 53
191 Introducton 192 Prnted Character Recognton 193 Inerse Knematcs Problems 194 Automoble MPG Predcton 195 Nonlnear System dentcaton 196 Channel Equalzaton 19.7 Adapte Nose Cancellaton
503
2 FzzyFiltered Neral Networks
535
201 Introducton 20.2 zzyFltered Neural Networks 203 Applcaton 1 Plma Spectrum Analyss 203. 1 Multlayer Perceptron Approach 20.3.2 zzy-Fltered Neural Network Approach 204 Applcaton 2: Hand-Wrtten Numeral Recognton 20.4 1 One Dmensonal zzy Flters 2042 Two Dmensonal zzy Flters
503
507
510 514
516 523 535 536 538 538 539 540 541 542
xii
Contnts
205 Genetc Algorthmbed zzy Flters 2051 A General Model 2052 Varatons and Dscusson 206 Summary 21 zzy Sets and Genetic Algoritms in Game laying
543 543
545 549 551
211 Introducton 212 Varants of Genetc Algorthms 213 Usng Genetc Algorthms n Game Playng 214 Smulaton Results of the Bc Model 215 Usng zzly Characterzed Features 216 Usng Polyplod GA n Game Playng 217 Summary
55 55 553 556 559 560 564
22 So Computing for Color Recipe rediction
568
221 222 223 224
Introducton Color Recpe Predcton Sngle MP approaches CANFS modelng for Color Recpe Predcton 2241 zzy Parttonngs 2242 CANFIS Archtectures 2243 Knowledgeembedded structures 2244 CANFIS Smulaton 225 Color Pant Manufacturng Intellgence 2251 Manufacturng Intellgence Archtecture 2252 Knowledge Be 2253 Mult-eltes Generator 2254 zzy Populaton Generator 2255 Ftness ncton 2256 Genetc Strateges 226 Expermental Ealuaton 227 Dscusson 228 Concludng Remarks and ture Drectons
568 569 569 57 572 573 576 576 577 579 580 581 581 582 584 587 589 591
A Hints to Selected xercises
595
B List of Internet Resources
598
C List of MALAB rograms
61
CNTENT
xiii
D List of Acronyms
64
Index
67
Foreword
Lotf Zadeh
Among my many PhD students some hae forged new tools n their work. Jyh Shng Roger Jang and Chuen-Tsa Sun fall nto ths category Neurozzy and So Computng makes sble ther mtery of the subject matter, ther nsightlness, and ther expostory skll Ther coauthor, Eji Mizutan, h made an mportant contrbution by brngng to the writng of the text hs extense experience in dealing wth realworld problems n an ndustral settng Neurozzy and So Computng s one of the rst texts to focus on so com putng a concept whch h drect bearng on machne ntelligence. n this connecton, a bt of hstory s n order The concept of so computng began to crystallze durng the pt seeral years and s rooted n some of my earlier work on so data analyss, fuzzy logc, and ntellgent systems. Today, close to four decades aer artical ntellgence A w born, t can nally be sad wth some justcaton that ntellgent systems are becomng a realty Why did t take so long for the era of intelligent systems to arre? In the rst place, the A communty had greatly underestmated the dculty of attanng the ambtous goals whch were on ts agenda The needed technol ges were not in place and the conceptual tools n AI ' s armamentarium mainly predcate logc and symbol manipulaton technques were not the right tools for buldng machnes whch could be called intellgent n a sense that matters in real world applicatons Today we hae the requsite hardware, soware, and sensor technologes at our dsposal for buldng ntellgent systems ut, perhaps more mportant, we are also n possesson of computatonal tools which are far more eecte in the conception and design of ntellgent systems than the predcatelogc-bed methods, whch form the core of tradtonal AI The tools n queston dere om a collecton of methodologes whch fall under the rubric of what h come to be known so computng SC In large meure, the employment of so computing techniques underles the rapd growth n the arety and sblty of consumer products and ndustral systems whch qualify to be sessed possessng a signcantly hgh
v
xvi
Forword by Zadh
MIQ machne ntellgence quotent. The essence of so computng is that unlike the tradtonal, hard computng so computng s amed at an accommodaton wth the pere mprecson of the real world Thus, the gudng princple of soft computng s to explot the tolerance for mprecson, uncertanty, and partal truth to achee tractablty, robustness low soluton cost, and better rapport wth realty. In the nal analysis, the role model for so computng is the human mind. So computng s not a sngle methodology. Rather, t s a partnershp. The prncpal partners at this juncture are zzy logc F, neurocomputng NC, and probablstc reonng PR, wth the latter subsumng genetc algorthms GA chaotc systems, belef networks, and parts of learnng theory. The potal contr buton of F s a methodology for computng wth words that of NC s system dentcaton, learnng, and adaptaton; that of PR is propagaton of belef; and that of G A s systematzed random search and optmzaton. In the man, F, NC, and PR are complementary rather than compette. For this reon, it is frequently adantageous to use F, NC, and PR n combnaton rather than exclusely, leadng to scalled hybrd intellgent systems. At this juncture, the most sble systems of ths type are neurzzy systems. We are also begnnng to see zzy-genetic, neurogenetc, and neurozzy-genetc systems Such systems are lkely to become ubiqutous in the not dstant ture. In comng years, the ubquty of intellgent systems s certan to hae a pr found mpact on the ways whch human-made systems are conceed, desgned manufactured, employed, and nteracted wth. Ths s the perspecte n whch the contents of Neurozzy and So Computing should be ewed. Takng a closer look at the contents of Neurozzy and So Computng, what should be noted s that today most of the applcatons of fuzzy logc nole what mght be called the calculus of zzy rules, or CFR for short. To a consderable de gree, CFR is self-contaned. rthermore, CFR s relately ey to mter because t s close to human ntuton. Taking adantage of ths, the authors focus ther attenton on CFR and mnmze the tme and eort needed to acqure sucient expertse n zzy logc to apply it to real-world problems. One of the central ssues in CFR s the nducton of rules om obseratons. In this context, neural network technques and genetc algorthms play potal roles, whch are dscussed n Neurozzy and So Computng, in consderable detal and wth a great deal of insght. In the applcaton of neural network technques, the man tool s that of gradient progammng. y contrt, in the application of genetic algorthms, smulated annealng, and random search methods, the existence of a gradent s not sumed. The complementarty of gradent programmng and gradent-ee methods prodes a bs for the concepton and design of neurogenetc systems. A notable contrbuton of Neurozzy and Soft Computng, s the exposition of ANFIS Adapte Neuro zzy Inference System a system deeloped by the authors whch s ndng numerous applcations n a arety of elds. ANFIS and ts ariants and relates in the realms of neural, neurofuzzy, and renforcement
i
Fowod
xvii
learnng systems represent a directon of bc mportance n the concepton and desgn of intellgent systems wth high MIQ. Neurozzy and So Computng is a thoroughly up-todate text wth a wealth of nformaton whch s well-organzed, clearly presented, and llustrated by many examples. It is required reading for anyone who is interested in acquiring a solid background n so computng a partnershp of methodologes whch play potal roles n the concepton, desgn, and application of ntellgent systems ot A Zadeh
Preface
Durng the pt few years, we hae wtnessed a rapd growth n the number and arety of applcatons of fuzzy logc and neural networks, rangng from consumer electroncs and ndustral process control to decson support systems and nancal tradng. Neurofuzzy modelng, together wth a new drng force from stochtc, gradent-free optmzaton technques such genetc algorthms and smulated an nealng, forms the consttuents of socalled soft computng, whch s amed at solng realworld decson-makng, modelng, and control problems These problems are usually mprecsely dened and requre human nterenton. Thus, neurozzy and so computng, wth ther ablty to ncorporate human knowledge and to adapt ther knowledge be a new optmzaton technques are lkely to play ncrengly mportant roles n the concepton and desgn of hybrd ntellgent systems. Ths book prodes the rst comprehense treatment of the consttuent method ologes underlyng neurfuzzy and so computng, an eolng branch wthn the scope of computatonal ntellgence that s drawng ncrengly more attenton t deelops. Its man features nclude fuzzy set theory, neural networks, data clus terng technques, and seeral stochtc optmzaton methods that do not requre gradent nformaton. n partcular, we put equal emphes on theoretcal pects of coered methodologes, wll emprcal obseratons and ercatons of arous applcatons n practce.
ADIENCE
Ths book s ntended for use a text n courses on computatonal ntellgence at ether the senor or rst-year graduate leel. It s also sutable for use a self-study gude by students and researchers who want to learn bc and adanced neurofuzzy and so computng wthn the framework of computatonal ntellgence Prerequ stes are mnmal; the reader s expected to hae bc knowledge of elementary calculus and lnear algebra xix
ac
ORGANIZAION
Chapter 1 ges an oerew of neurozzy and so computng ref historcal traces of releant techniques are descrbed to drect our rst step toward the neur zzy and so computng world. The remander of the book is organzed into the eight parts described next Part I Chapters 2 through 4 presents a detaled introducton to the theory and terminology of zzy discplnes, includng fuzzy sets, fuzzy rules, fuzzy reonng, and fuzzy inference systems. Part II Chapters 5 through 7 prodes an oerew of system dentcaton and optmzaton technques that proe to be eectie for neural-zzy and so comput ng Chapter 5 ntroduces let-squares methods n the context of system iden tcation Chapter 6 descrbes deratiebed nonlnear optmizaton technques, ncludng nonlnear let-squares methods These two chapters are netably math ematcally orented, and for the rst readng, many sections labeled wth an terisk * can be omtted Chapter 7 dscusses deriatee optmzaton technques, ncludng genetc algorithms, smulated annealng, the downhll Smplex method, and random search Part II Chapters 8 through 11 ntroduces a ariety of mportant neural net work paradgms found in the literature, ncludng adapte networks the most generalzed amework for model constructon, superised learning neural networks for data regression and clscaton, reinforcement learnng for infrequent and de layed ealuate sgnals, unsupersed learning neural networks for data clusterng, and some other networks that do not belong to any of the orementoned categores Part Chapters 12 and 13 explans how to buld ANFIS Adapte Neuro Fuzzy Inference stems and CANFIS Cotie NeuroFuzzy ference stems core neurzzy models that can incorporate human expertse well adapt themseles through repeated training Part V Chapters 14 through 16 coers structure identcaton technques for neural networks and fuzzy modelng, ncludng the CART Clscation and Re gresson e method, whch s qute popular n multarate analysis of statstcs; seeral data clusterng algorthms amed at batch-mode model bulding , and ecent rulebe formulaton and organzaton a tree parttonng of input space Part V Chapter 17 and 18 consders arous approaches to the desgn of neuro fuzzy controllers, ncludng expert control, nerse learnng, specalized learnng, backpropagation through time, real-time recurrent learnng, renforcement learnng, genetc algorthms, gan schedulng, and feedback lnearzaton n conjuncton with sldng mode control The lt part, Part VII Chapters 19 through 22, ges a arety of applcaton examples n dierent domans, such prnted character recogniton, nerse kine matcs problems n robotcs, adapte channel equalzaton, multarate nonlnear regresson, adaptie nose cancellaton, nonlnear system dentcaton, plma spec trum analysis, hand-wrtten numeral recognton, game playng, and color recpe predcton
i
rac
: : : Pa 2: Rrsion and Optimization
Pa 1: Fu Set Th
-------------------- 11 1
--------------
Ch
Ch.2: Fu Sets
4
Ch 7: DerivativeFree Optim
Ch Fuy Inference Systems
-- 4 :� 1-------------
_[
Ch 12: ANFIS
Ch 13: CANFIS
Ch 8: Adaptive
1, "
---- -------
---�------------ 1 eurFUZ Ctl
Ch 17: NeurFuy Control I
l
i
Ch.18:
Ch 16: Rulebase Stcture
------------------ Pa Advanc NeurFu Modeling
Figure 0.1
and LSE
Ch 6: DervativeBas
Ch 3: Fu Rules and Fu Reasoning
6
System
Ch 9: Supeis-Learning NNs Ch 10: Learning from Reinforcement
11
Ch 11: nsupeis-Learning and Other Networks
;
��
'
:
1 1
Ch 19: ANFIS Applications Ch 20: FuyFilter Neural Networks Ch 21: Fu GA in Game Playing
Ch So Computing for Color Priction
Pa 7: Advanc Applitions
P qs s mo s o s book
he prerequisite dependenies mon the individul hpters re shown in i ure 0.1. his dir rrnes hpters with respet to the level of dvnement so the reder h some exibility in studyin the whole boo Setions mred with n terisk * n be sipped for the rst redin
xxii
rac
FEATURES
he orienttion of the book is towrd methodoloies tht re likely to be of pr til use mny stepbystep exmples re inluded to omplement explntions in the text. Sine one piture is worth thousnds of words, this book ontins spe illy desined ures to visulize mny ide nd onepts possible, nd thus help reders understnd them t lne. Most sienti plots in the exmples were enerted by M© nd S© . or the reder's onveniene, these M prorms n be obtned by llin out the reply rd in this book or vi FP or WWW. Se the next setion for detils of how to obtin these M pro ms. Some of the exanples nd demonstrtions require the zzy Loi oolbox by he MthWorks In the ontt informtion is
¥
he MthWorks, In. 24 Prime Prk Wy Ntik, MA 0 17601500, USA Phone: 508 6477000 x: 508 6477001 mil: infoathwork . com WWW: http : //www mathwork . com Chpters 2 throuh 18 re eh followed by set of exerises some of them involve M prormin t&ks, whih n be expnded into suitble te projets. his serves to onrm nd reinfore understndin of the mteril pre sented in eh hpter, well to equip the reder with hndson prormmin experienes for prtil problem solvin. Hints to seleted exerises re in the ppendix t the end of this book. or instrutors who use this book text, the solution mnul is vilble om the publisher. A set of viewrphs tht ontins importnt illustrtions in the book is vilble for lsroom use. hese viewgrphs re diretly essible vi the book's home pe t http : //www . c . nthu edu . tw/- j ang/oft . htm
his is the ple where the reder n do other thins suh :
• Get the most updted informtion suh ddendum, errtum, et. bout the book.
• • • •
Get enhnements nd buxes of M prorms. Give omments d suestions. View the sttistis of omments nd suestions of other reders.
Link to the uthors' WWW home pes nd other Internet resoures. M and S are regstered trademarks of The Math Works, Inc.
xxiii
rac
A reference list is iven at the end of each chapter that contains references to the research literature. his enables readers to pursue individu topics in greater depth. Moreoer, neurfuzzy and so computin is a relatively new eld and con tinues to eole rapidly within the scope of computation and articial intelligence. hese references provide an entry point from the welldened core knowlede con tained in this book to another dimension of innovative and challenin research and applications. OBTAINING THE EXAMPLE PROGRAMS
For people without Internet access, the eiest way to et the ee MATLA pr grams used in this book is to ll the reply card bound in the book) and send it back to the MathWorks. he MathWorks will send you a oppy disk contnin the MATLA les, free of chare. Some of the MATLA prorams rely on the zzy Logic oolbox, which is not freely available.) For people with Internet access, the example MATLA programs are available electronicly in two ways by F le transfer protocol) or WWW worldwide web). he FP address is ftp . mathwork com
he les are at /pub/book/jang/*
For WWW access to the FP site, the UR univers resource locator) address is ftp//ft p . mathwork . com/pub/book/j ang/
ou can also access it ia the book's home pae at http//w w·c nthu . edu . tw/ -j ang/ oft . htm
A sample FP session is shown next, with what you should type in boldface unix> matworkscom Connected to ftp . mathwork com . 220 ftp FTP erver ( Verion 2 4 (2 Tue Aug 1 10 : 3 1 : 36 EDT 1 995 ready. Nme (ftp . mathwork . comlin anonymous 33 1 Guet login ok, end your complete email addre a paword Paword slin@lsilcom ( se yur ea adress ere) 230 Guet login ok, acce retrict ion apply . ftp> cd /ub/bo oks/jang 250 CW co and ucc eful . ftp> binary ( u ust se bnary transfer fr se ata es) 200 Type et to I ftp> romt (S yu n t nee t transfer ea e nteratvey)
xxiv
rac
Interactive mode off . ft> mget * (Get eve e n te dt) 200 PORT coand ucce ful . 150 Oening BI NARY mode data connection for addvec . m (6 91 byte . 226 Tranfer comlete . local addvec . m remote addvec . m 69 1 byt e rece ived in 0 01 1 econd (62 Kbyte/ 200 PORT coand ucceful . ft> bye 221 Goodbye .
ou may want to use a mirror site near you ft//unix . hena . ac . uk/mirror/matl ab ft//ft . ak . unikarlruhe . de/ub/matlab ft//novell . fe lk cvut . cz/ub/mirror/mathwork ft//ft . uaizu . ac . j /ub/vendor/mathwork
ACKNOWLEDGMENTS
his book ould not have been nished without the sistane of many individuls. First, we would like to aknowlede Professor Lot A Zadeh at the EECS eletril enineerin and omputer siene epartment of the University of Clifornia at erkeley his onstant support and enouraement w the major drivin fore for the birth of this book. We would lso like to thank him for oerin the title of this book. We appreiate the onstrutive and enourain omments of the manusript reviewers Siva Chittajallu Purdue University, Irena Naisetty the Ford Motor Company, Hao in, University of ex Medial ranh, and Mark J. Wierman Creihton University. We would like to thank the Prentie Hll editoril stespeilly om Robbins, who oered insihtful suestions and led us throuh the maze of details soiated with publishin a book. We re also greatly indebted to Phyllis Moran, who metiulously handled orrespondene and the reviewin proesses. We also wish to thank our proof editor, Patriia ly, who spent hours polishin our Enlish. We re deeply ratel to Mr. Issei Nhio, who desined the lovely over for the book. Individu aknowledments of eah author follow. Acknowedgments of JS Roger Jang his book w submitted for publiation while I w with he MathWorks, In., and published er I joined the eprtment of Computer Siene, Nationl sin Hua
rac
v
University Hsinchu Taiw. I'd like to thank The MathWorks Inc. for makin available a stimulatin d n workin environment. I am also ratel to the Department of Computer Science at Tsin Hua University for providin a scholaly environment for both teachin and resech. Most of all the book project w recommended by my Ph.. advisor Professor Lot Zadeh and initiated while I w a research sociate at the University of California at erkeley back in 992 I'd like to express my heartfelt ratitude to Professor Zadeh for his advice and continuous support. t but not let I am indebted to my wife Sue d my children Timmy and Annie. In particular without Sue's takin care of all the family matters this book would neer hae been possible Acknowledgments of ChuenTsai Sun First I greatly appreciate the support of Professor Lot Zadeh who h been en courain me to do research on so computin since my Ph.D. years. I am deeply grateful to many scholars in fuzzy systems neur networks and eolutionary com putin om whom I have learned oer the pt 10 years. I want to express my appreciation to Dr Chi un from the Lawrence Livermore National Labora tory who encouraed me to study zzy lterin techniques. I also want to thank my raduate students at Nation Chiao n Uniersity whose eaer quest for knowlede h been a continual challene and inspiration to me. Finally I want to thank my family for their constant support throuhout these years. Acknowledgments of Eiji Mizutani First I would like to express my sincere ratitude to Professor Stuart E. rey fus Department of Industrial Enineerin and Operations Research University of California at erkeley and Professor aid M. Auslander epartment of Mechan ical Enineerin Uniersity of California at erkeley for their insihtful comments on our manuscript and thei patience for replyin to my repeated inquiries throuh email. More important Professor reyfus expertly ided me in the labyrinth of scalled articial neural networks and tauht me much of what I know about their computational izmos. In pticular Chapter 10, "Learnin from Reinforcemen would neer have been completed without his quintessential mathematical uide. Professor Auslander supported me in a friendly manner listenin patiently to all my frustrations with school and cultural dierences. The research commencement with him ave me a ticket for enterin this research world. rthermore reected in this book e the early inuences of Professor Kazuo Inoue and Mr. Atusi Ichimura who directed my rst step toward the challenes of cuttin ede research in constructin articially intellient systems. I would like to take this opportunity to thank Mr. Kenichi Nishio Sony Cor poration for supportin me with warm friendship d Professor Hideyuki Tak ai Kyushu Institute of esin for ener sistance. I am also deeply indebted to Professor Mayoshi Tomizuka Professor Toshio
xxv
rac
kud, Professor Sieru Omtu, Professor Nkji Hond, en Miezkowski, Jk Hsmontr, vid Mrtin, eo i, Arun iliiri, Anes Conepion, Shron Smith, Zenji Nkano, ohru irym, Akio Kwmoto, Mihel L, nd Myuki Mu rkami. All their sistne retly smoothed the wy. Finlly, I wish to thnk Professor J.-S. oer Jn rst uthor) for his on dene in me that inspired ondene in myself. HOW TO CONTACT US
We'd like to her your omments, orretions of errors, nd suestions for future editions. Plee send them to J.S. oer Jn jang�c.nthu.edu.tw Chuensi Sun: un�ci.nctu.edu.tw eiji�joho.kanai.co.j or Eiji Miutni eiji�bioy2.me.berkele.edu
ou n lso send your omments vi the book's WWW home pe t htt//www . c . nthu . edu . tw/ -j ang/ oft . htm
Chapter 1 Introduction to Neuro-Fuzzy and Soft Computing
1.1
INTRODUCION
So computing SC , a inovtive pproah to ostrtig ompttioally
telliget systems, h jst ome ito the limelight. It is ow realized tht omplex re-world problems reqire itelliget systems tht ombie knowledge, tehniqes, ad methodologies from varios sores. These itelliget systems are spposed to possess hmlike expertise withi a spei domai, adpt themselves d ler to do better i hagig eviromets, d expli how they mke deisios or take tios. I ootig re-world omptig problems, it is freqetly advatgeos to se several omptig tehiqes syergistily rather th exlsively, reslt ig i ostrtio of omplemetary hybrid itelliget systems. The qitessee of desigig itelliget systems of this kid is neurofuzzy computing er etworks that reogize ptters ad dpt themselves to ope with hgig evi romets; fzzy iferee sstems tht iorporte hm kowledge ad perform ifereig ad deisio makig. The itegrtio of these two omplemetary p proahes, together with ertai derivtive-free optimiztio tehiqes, reslts i ovel disiplie alled neurofuzzy and so computing. As a prelde, we shall provide bird'seye view of relevat itelliget system approhes, og with bits of their history, ad disss the fetres of erfzzy ad so omptig. 1.2
SOFT COMPUI NG CONSTITUENS AND CONVENIONAL ARIFICIAL INTELLIGENCE S utn s an een ara t utn w araes te rearkabe abty f te uan n t reasn an ea n an envrnent f unertanty an resn (Lot A. Zadeh, 1992 12 1
Introduton to NeuroFuzzy and So Computng
2
TT I I I I I I I
I I I I I
; I• p ..
. . .. . . . .
Neual Charaer Rongnize
.. ·
.
.
C
I I I I I I I
. . . . . .": :: . . d
Fie 1 1 A ne arater ner an a knwee base erate n re snn t tree anwrtten araters tat fr a w "
Tble 11 S utn nsttuents te rst tree tes an nventna arta nteene
Stregth Lerig d dpttio Kowledge represettio vi zzy if-the rles Geeti lgorithm d Systemti radom serh simlted elig Symboli mipltio Covetiol AI
Methodology Nerl etwork zzy set theory
So omptig osists of severl omptig prdigms, ildig erl et works, fzzy set theory, approximte reoig, d derivtivefree optimiztio methods sh geeti lgorithms d simted elig. Eh of these o stitet methodologies h its own stregth, smmrized i Tble 1 .1 The semless itegrtio of these methodologies forms the ore of so omptig; the syergism llows so omptig to iorporte hm kowledge eetively, de with impreisio d ertity, d lear to dpt to kow or hgig e viromet for better performe. For lerig d dpttio, so omptig reqires extesive ompttio. I this sese, so omptig shres the sme har-
Se So Computng Consttuents and Conventonal Artal ntellgene
3
ateristis omptatioal itelligee. I geera, so omptig does ot perform mh symboi maiplatio, so we a view it a ew disiplie that omplemets ovetioal artiial itelligee (AI) approahes, ad vie versa. For istae, Figre 1 1 illstrates a sitatio i whih a er harater reogizer ad a kowledge be are sed together to determie the meaig of a had-writte word. The eral harater reogizer geerates two possibe aswers dog ad dag, sie the middle harater old be either a or a. the kowledge be provides a extra piee of iformatio that the give word is related to aimals, the the aswer dog is piked p orretly. Figre 12 is a list of ovetioal AI approahes ad eah of the so ompt ig ostitets i hroologi order. We disss the featres of ovetio AI i Setio 1.21, ad those of so omptig ostitets i Setios 12.2 throgh 1.24. I Setio 13, we smmarize the erzzy ad so omptig harateristis. 1.2.1 From Conventional AI to Computationa I nteligence Hmas sally employ natura anuaes i reoig ad drawig olsios. Covetioal AI researh foses o a attempt to mimi hma itelliget behav ior by expressig it i lagage forms or symboli rles. Covetio AI bily maiplates symbos o the smptio that sh behavior a be stored i sym bolially strtred kowledge bes. This is the salled ysa syb syste ytess [3, 5 Symboli systems provide a good bis for modelig hma ex perts i some arrow problem are if expliit kowledge is availabe Perhaps the most sessfl ovetioal AI prodt is the kowledgebed system or expert system (S); it is represeted i a shemati form i Figre 1.3 Covetio AI literatre reets earlier work o itelliget systems. May AI prersors deed AI i light of their ow philosophy; some represetative AI deitios are listed ext alg with a ope of S deitios.
• AI is the stdy of aents that exist i a eviromet ad pereive ad at. (S. Rsse ad P. Norvig) [6
• AI is the art of making ompters do smart thigs. (Waldrop) 9 • AI is a programmig stye, where programs operate on data aordig to rles i order to aomplish goals. (W. A. Taylor) [8]
• AI is the ativity of providig sh mahies ompters with the ability
to display behavior that wold be regarded itelliget if it were observed i hmas. (R. Meod) [4
• ES is a ompter program sig expert kowledge to attai high levels of performae i a narrow problem area. A. Waterma) [1 0
Introduton to NeuroFuy and So Computng
4
neninl 98
98
1 947 Cybeet
1956 Acia Integence 1 960 LIs la gua g e
Nerl newrks
1 943 McCh-Pitt nen me
1957 Peeptron 1 A dalin e
Madaine
968
1970s 978 mid Kno
Enge eng ( ex systems )
98
her
:
15
Fuy s et
________ L_______ __________ 1970 Genec 197 Bh o ck o a thm � gato a m 197 Fuy ntoe
1975 Cognltron N e nitron 1 1 9 Seoganiing ma 1 982 Hofied Net
1
1 1 9
:
:
. 1985 Fuy mng 1983 man mane ( TSK m ) 16 Bkpon
:
99
C
1
Neurfuz m e lng
1991 AN IS CAFIS
19
mid 1980 Afia ife Immne mng
1 9 G ene tic
1
pr ogra m min g
Fig A stra sket f s utn nsttuents an nventna AI araes
• ES is ritre of the hm expert, i the sese tht it kows lmost everythig bot lmost othig (A R. Mirzi) 2]
These deitios provide ospios AI mework lthogh they my be some wht ephemerl bese the oeptl frmework is metmorphosig rpidly The reder my well woder, H AI beome obsolete ledy? Cllig s utn nsttuents prts of moder AI ievitbly depeds o persol jdgmet. It is tre tht tody my books o moder AI desribe erl etworks d perhps other so omptig ompoets, see i [6, 1].
Se So Computng Consttuents and Conventonal Artal Intellgene
%
5
Inference Engine
Qio R
Expanation Faciity
User (Novice)
Knowledge Engineer
Human Expe
Host
Knowedge Acquisition
Computer ec .
Figure An epert system: one of the most successful conventional AI prod
ucts.
This means that the AI eld is steadily expanding; the bondary between AI and so ompting is beoming indistint and obviosly sessive generations of AI methodologies wil be growing more sophistiated. rther disssion of these phil sophial AI territories [7] is beyond the sope of this book. In pratie the symboi manipations limit the sitations to whih the onven tional AI theories an be applied bease knowedge aqisition and representation are by no means ey bt are ardos tks. More attention h been direted toward biologially inspired methodologies sh brain modeling evoltionary al gorithms and immne modeling; they simlate biologial mehanisms responsible for generating natral inteligene. These methodologies are somewhat orthogonal to onventional AI approahes and generally ompensate for the shortomings of symboliism. The long-term goal of AI researh is the reation and nderstanding of macine intelligence. om this perspetive so ompting shares the same ltimate goal with AI. Figre .4 is a shemati representation of an intelligent system that an sense its environment (pereive) and at on its pereption (reat). An ey extension of ES may aso rest in the same ideal omptationaly intelligent system soght by so ompting resehers. So ompting is apparently evolving nder AI inenes that sprang from cybernetics (the stdy of information and ontrol in hmans and mahines).
Introduton to Neuro-Fuzzy and So Computng
6
Pcptions
Snsing Dvcs (Vision) Natual nguage Pes
Actions
Mchanical Dvics
II Task II Gneato II II Knowldge II Handl II Data II Handle L
C
Machin Laing Infencing (Rsing) Planning
Fige An intelligent sY$tem.
1 2.2 Neura Networks The hman brain is a sore of natra inteligene and a try remarble parael ompter. The brain proesses inomplete information obtained by pereption at an inredibly rapid rate. Nerve ells ntion abot 0 times slower than eletroni irit gates bt hman brains proess visa and aditory information mh fter than modern ompters Inspired by biologia nervos systems many researhers espeiay brain model ers have been exploring artiia nera networks a novel nonagorithmi approah to information proessing They model the brain a ontinos-time nonnear dynami system in onnetionist arhitetres that are expeted to mimi bran mehanisms to simlate intelligent behavior Sh onnetionism replaes sym bolialy strtred representations with distribted representations in the form of weights between a msive set of interonneted nerons (or proessing nits) It does not need ritia deision ows in its agorithms. A variety of onnetionist approahes have been stdied some representative methodologies and their omptational apaities are disssed in sbseqent hap ters 12 3 Fuzzy Set Theory The hman brain interprets impreise and inomplete sensory information provided by pereptive organs zzy set theory provides a systemati als to deal with sh information lingistially and it performs nmeria omptation by sing lingisti labels stipated by membership ntions. Moreover a seletion of fzzy if-then rles forms the key omponent of a zzy inferene system (FIS) that an eetively model hman expertise in a spei appliation. Althogh the zzy inferene system h a strtred knowledge representation
Se Neuro-Fuzzy an d So Computng Caratersts
7
in the form of zzy if-then rles it laks the adaptability to deal with han ng external environments. Ths we inorporate neral network earning onepts in fzzy inferene systems reslting in neurozzy modeling a pivotal tehniqe in so ompting. We disss fzzy sets fzzy rles and zzy inferene systems in Chapters 2 3 and 4 Approahes to nerzzy modeling are desribed in Chapters 2 and 3. 1 .2 .4 Evolutionary Computation Natral inteligene is the prodt of milions of years of biolo al evoltion. Sim ating omplex biologial evotionary proesses may lead s to disover how evol tion propels living systems toward higher-level inteligene Greater attention is ths being paid to evoltionary ompting tehniqes sh geneti algorithms ( GAs) whih are bed on the evoltionary priniple of natral seletion. Immune model ing and Articial Life are simiar disiplines and are bed on the smption that hemial and physial laws may be able to expain living inteigene. In parti lar Artiial ife an inlsive paradigm attempts to reaize lifelike behavior by imitating the proesses that or in the development or mehanis of ife [ Heuristically informed searh tehniqes are employed in many AI appliations. When a searh spae is too large for an exhastive (blind brte-fore) searh and it is dilt to identify knowedge that an be applied to rede the searh spae we have no hoie bt to se other more eient searh tehniqes to nd less thanoptimm soltions. The GA is a andidate tehniqe for this prpose; it oers the apaity for poplation-bed systemati random searhes. Simulated annealing and random search are other andidates that explore the searh spae in a stohti manner. Those optimization methods are disssed in Chapter 7. 1.3
NEROFZZY AND SOF COMPING CHARACTER I STICS
With nerfzzy modeing a bakbone the harateristis of so ompting an be smmarized follows: Hma exetise So ompting tiizes hman expertise in the form of zzy
if-then rles well in onventional knowedge representations to solve pratial problems.
Biologically isied comtig models Inspired by bioloal neral networks
artiial neral networks are employed extensively in so ompting to deal with pereption pattern reognition and nonlinear regression and lsia tion problems.
New otimizatio techiqes So ompting applies innovative optimization methods arising from varios sores; they are geneti algorithms (inspired by
8
Introduton to Neuro-Fuzy and So Computng
C
the evoltion and seletion proess) simlated annealing motivated by ther modynamis) the random searh method and the downhill Simplex method These optimization methods do not reqire the gradient vetor of an obje tive fntion so they are more exible in dealing with omplex optimization problems. Nmeical comtatio Unlike symboli AI so ompting relies mainly on n
merial omptation. Inorporation of symboli tehiqes in so ompting is an ative researh area within this eld
New alicatio domais Bease of its nmerial omptation so ompt
ing h fond a nmber of new appliation domains besides that of AI ap proahes These appliation domains are mostly omptation intensive and inlde adaptive signal proessing adaptive ontrol nonlinear system identi ation nonlinear regession and pattern reognition.
Modelee leaig Neral networks and adaptive zzy inferene systems have
the ability to onstrt models sing only target system sample data. Detailed insight into the target system heps set p the initial model strtre bt it is not mandatory
Itesive "tatio Withot sming too mh bakgrond knowledge of
the problem being solved nerzzy and so ompting rely heaviy on high-speed nmber-rnhing omptation to nd res or reglarity in data sets This is a ommon featre of all are of omptational inteligene.
alt toleace Both neral networks ad fzzy inferene systems exhibit falt
toerane. The deletion of a neron in a neral network or a rle in a zzy inferene system does not neessarily destroy the system. Instead the sys tem ontines performing bease of its parallel and redndant arhitetre althogh performane qality gradally deteriorates.
Goal dive chaacteistics Nerfzzy and so ompting are goal driven;
the path leading om the rrent state to the soltion does not realy matter ong we are moving toward the goa in the long rn This is partilary tre when sed with derivativefree optimization shemes sh geneti algorithms simated anneaing and the random searh method. Domain spei knowledge helps redes the amont of omptation and searh time bt it is not a reqirement.
Realwold alicatios Most realworld problems are large sale and inevitaby
inorporate bilt-in nertainties; this preldes sing onventional approahes that reqire detailed desription of the problem being solved. So ompting is an integrated approah that an sally tiize spei tehniqes within sbtks to onstrt generally satifatory sotions to real-world probems.
EFEENCES
9
The eld of so ompting is evolving rapidly; new tehniqes and appliations are onstanty being proposed. We an see that a rm fondation for so ompting is being bilt throgh the oletive eorts of researhers in vios disiplines all over the world. The nderlying driving fore is to onstrt highly atomated inteigent mahines for a better ife tomorrow whih is already jst arond the orner.
REFERENCES
1 C GLangton editor. Artcal lf volume 6. AddisonWesley Reading MA, 1989 2 A. R Mirzai Artcal ntllgnc concpts and applcatons n ngnrng MIT ress Cambridge, MA, 1990 3] A Newell and H. A Simon Computer science empiric inquiry Symbols and search Communcatons of th A CM 19(3)113-126, 1976. Jr Raymond McLeod. Managmnt nformaton systms. Science Research Associates Chicago 1979 E Rich and K Knight Artcal ntllgnc McGrawHill New York, 2nd edition 1991. [6 S. Russell and Norvig. Artcal ntllgnc a mod approach rentice Hall, Upper Saddle River NJ 199. 7 Smolensky On the proper treatment o connectionism Bhavol and Bn cncs, 2(1), 1988 8 W A Taylor. Wat vry ngnrs should know about A MIT ress, Cambridge, MA, 1988 9 Mitchell M Wdrop. Manmad mnds th proms of artcal ntllgnc Walker, New York 1987 10 Dond A. Waterman gud to rt systms. AddisonWesley Reading MA 1986 11] atrick H. Winston Artcal ntllgnc. AddisonWesley Reading, MA 3rd edi tion 1992 12 Lotf A Zadeh. zzy logic neur networks and sot computing. Onepage course announcement o CS 29, Spring 1993 the University o Caliornia at Berkeley November 1992
Part I Fzzy Set Theory
Chapter Fuzzy Sets
J.-S. R. Jang
This hapter introdes the bi denitions notation and operations for zzy sets that wi be needed in the folowing hapters. Sine researh on fzzy sets and their appiations h been nderway for amost 30 years now it is impossible to over over al al pets of rrent rrent development developmentss in this t his eld. el d. Theref Therefore the t he aim ai m of this hapter is to provide a onise introdtion to and a smmary of the bi onepts entral to the stdy of fzzy sets. Detaied Det aied treatments of spei sbjets sbj ets an b e fond in the referene list at the end of this hapter. 2.1
INTRODCTION
A lsia lsi a set is a set with a risp bondary. For For example a sia set A of rea nmbers greater than 6 an be expressed
A = {x l x > 6 } , (2 1) where there is a ear nambigos bondary 6 sh that if x is greater greater than than this nmber then x belongs to te set A; otherwise x does not not beong to the set. Al
thogh lsial sets are sitable for varios appliations and have proven to be an important tool for mathematis and ompter siene they do not reet the na tre of hman onepts and and thoghts whih tend to be abstrat and impreise. impre ise. As an ilstration mathematialy we an express the set of tal persons a olletion of persons whose height is more than 6 ; this is the set denoted by Eqation (21), if we let A = tall person and x = height height . et this is an an nnatra and inad eqate way of representing or sa onept of tal person. person . For one thing thi ng the dihotomos natre of the lsia set wold lsify a person 6.001 tal a tall person bt not a person 5.999 5. 999 tall. This distintion distintion is intitively intitively nreona nreonable. ble. The aw omes from the sharp transition between inlsion and exlsion in a set. In ontrt to a lsia set a fzzy set, the name implies is a set withot a risp bondary. That is the transition from belong to a set to not belong to a set is grad grad and this smooth transition is haraterized by membership fnti fntions ons that give fzzy sets exibiity in modeling ommony sed lingisti expressions
y y
Sets Sets
C C
sh sh the water water is hot hot or the temperare is high As Zadeh pointed pointed ot in 1965 in his semnal paper entitled zzy Sets 11, sh impreisely dened sets or lses play an important roe in hman thinking partiarly in the domains of patern reognition ommiation of information and abstration. Note that the fzziess does not ome from the randomness of the onstitent members of the ses bt from the nertain and impreise natre of abstra hoghs ad onepts. Let s now set forth several bi denitions onerning fzzy ses. 2.2
BASIC BA SIC DEFIN ITIONS AND TERMI NOLOGY NOLOG Y
Let X be a spae of objets and x be a generi element of X A lsia set A X , i s dened dened a olletion olletion of elemen elements ts or objets x E X , sh that eah x an either belong or not belong to the set By denng a chaacteistic fnction for eah element x in X we an represent a lsial et by a set of ordered ordered pairs x , 0) or x , 1 ) whih whih indiates indiates x or x E , respetively. Unlike the aforementioned onventional set a zzy set 11 expresses the degree to whih an element element belongs to a set. Hene the harateristi fntion fntion of a zzy set is alowed to have have vales vales between 0 ad 1 whih denotes the degree of membership member ship of an element in a gven set.
Denition Fuzzy uzzy sets and membership membership nctions X
is a oletion of objets denoted generially by dened a set of ordered pairs = {(()) I
x then a fzzy set in X is
x E X},
2 2
where () is aled the membeshi fnction (or M for short) for the fzzy set The MF maps eah element of X to a membership grade (or membership vale) beween 0 and 1
o Obviosly the denition of a fzzy set is a simple extension of the denition of a sial set in whih the harateristi ntion is permitted to have any vales between 0 and 1 If the vale vale of the membership ntion x is restrited to either 0 or 1 then is reded to a lsial set and () ( ) is the harateristi fntion of For arity we shall also refer to lsial sets ordinary sets rsp sets nonfzzy sets or jst sets. Usally X is referred to he nivese of discose or simply the nivese, and it may onsist of disrete (ordered or nonordered) objets or ontinos spae. his an be laried by the followig examples.
Examle Fuzzy sets with a discrete nonordered universe
Se Bas Dentons and Termnolo
b MF o Coiuou Uiv
MF o Dic Uiv
"
1
"
0c C 06 :C 02
� 0.4
2 6 X Numb of Chd
ig a
years old "
1
0 cC 06 0C : 02
XAg
= sensible number of children in a family "; b
ATLAB e: mf _univ m)
B
1
about
Let X = {San aniso, Boston, Los Angeles} be the set of ities one may hoose to live in. The fzzy set C = desirable ity to live in may be desribed follows: C = {(San aniso, 0.9), (Boston, 08), (Los Angeles, 0.6)} Apparently the niverse of disorse X is disrete and it ontains nonordered objetsin this e, three big ities in the United States. As one an see, the foregoing membership grades isted above are qite sbjetive; yone an ome p with three dierent bt legitimate vaes to reet his or her preferene.
o
Examl Fuzzy sets with a discrete ordered universe
Let X = {, 1 2, 3, 4, 5, 6} be the set of nmbers of hildren a family may hoose to have. Then the fzzy set = sensible nmber of hildren in a family may be desribed folows
= {(, 0. ), (1, 0.3), (2, 07), (3 1), (4, 07), (5, 0.3), (6, 0.1)}
Here we have a disrete ordered niverse X; the MF for the fzzy set is shown in Figre 21(a) Again, the membership grades of this zzy set are obviosly sbjetive meres
o Examl Fuzzy sets with a continuous universe
Let X = R+ be the set of possible ages for hm beings Then the fzzy set abot 50 years od may be expressed
B
= {(, () X},
B
=
uy
where ()
ts
C
x
1 + 1 50 4
Ths s lustrated n Fgure 21(b)
o om the precedng exampes, t s obvous that the constructon of a zzy set depends on two thngs: the dentcaton of a sutable unverse of dscourse and the speccaton of an approprate membershp ncton The speccaton of membershp functons s subjective whch means that the membershp nctons speced for the same concept (say, sensble number of chdren n a fany ) by derent persons may vary consderaby Ths subjectvty comes fom ndvdua derences n percevng or expressng abstract concepts and h lttle to do wth randomness Therefore, the sbjectivity and oadomess of fuzzy sets s the prmary derence between the study of fuzzy sets and probablty theory, whch deas wth objectve treatment of random phenomena For smplcty of notaton, we now ntroduce aternatve way of denotng a zzy set A zzy set can be denoted follows: =
{ x
E E ()/ , f X s a collecton of dscrete objects
f X s a contnuous space (usualy the rea lne R). (23) The summaton and ntegraton sgs n Equaton (23) stand for the unon of (, ()) pars; they do not ndcate summaton or ntegaton Smlarly, / s only a marker and does not mply dvson ()/,
Exale Alteative epression
Usng the notaton of Equaton (2 3) , we can rewrte the fuzzy sets n Examples 21, 22, and 23 C 09/San ancsco + 08/Boston + 06/Los Angees, = 01/0 + 03/1 + 07/2 + 10/3 + 07/4 + 03/5 + 01/6,
d respectvely
B /R
1 1+
x
o In practce, when the unverse of dscourse X s a contnuous space (the real lne R or ts subset), we usually partton X nto severa fuzzy sets whose MFs cover X n a more or less unform manner These fuzzy sets, whch usually carry names
Se Bas Dentons and Temnolo
� 12 E C � 2
Youn
Od
dde Aged
' "
"
2 3 4 6 7
o
X=Age
9
ige Typical MFs of linguistic values young " middle aged and old "
ATLAB e linf m
that conform to adjectives appearing n our daily lingustic usage, such large medum, or smal, are called lnguistic vaues or inguistic labels Thus, the universe of discourse X is oen called the lnguistic varable Formal denitons of linguistc varables and lnguistic vaues are given n the next chapter; here we shall give only a simple example
=
Exle Linguistic variables and linguistic values
Suppose that X age Then we can dene fuzzy sets young, middle aged and old that are characterzed by MFs Jold x , Jmiddleaged x , and Jold x , re spectively Just a variable can sume various values, a linguistc variable Age can sume derent lngustc values, such young, mddle aged, d old in this ce age sumes the value of young, then we have the expression age is young, and so forth for the other values Typca MFs for these lnguistic values are displayed n Figure 22, where the unverse of discourse X is totally covered by the MFs and the transton from one MF to another is smooth and gradual
D A fuzzy set is uniquely specied by ts membership function To descrbe mem bershp functions more specically, we shall dene the nomenclature used n the literature (Unless otherwse specied, we shall sume that the universe of the zzy sets under discussion is the real line R or its subset) Deitio Support
The sot of a fuzzy set is the set of all ponts
x in X such that JA x > 0:
support() = {XI A(X) > o}
(24)
D Deitio ore
u y
The coe of a zzy set s the set of all points core() = {XIJA(X)
ts
C
x in X such that JA(X) = 1:
I }
(2 5)
o
Deitio Normality
A zzy set is omal ts core s nonempty I other words, we can always nd a pont x E X such that JA(X) = 1.
o Deitio ssover points
= =· . .
A cossove oit of a zzy set is a pont crossover()
x E X at which JA(X) = 05: 0 5}
{XIJA(X)
(26)
o
Deitio Fuzzy singleton
A zzy set whose support is a single pont in X with JA(X) = 1 is called a fzzy sigleto.
o
Figures 23(a) and 23(b) illustrate the cores, supports, and crossover points of the bell-shaped membership nction representng middle aged and of the zzy singleton chacterizng 45 years old Deitio cut stng -cut
The ct or level set of a zzy set is a crisp set dened by (27)
=
.
Stog ct or stog level set are dened smlarly:
{XIJA(X) > }
(28)
o
Usng the notaton for a leve set, we can express the support and core of a zzy set support() = , and core() , respectvely
=
c Basc Dntons and Tmnoo Membership Grades Mddle Aged
---------- --+
·· ··
''
Points - Suppo
( a)
Age
Membership Grades
Year Old ------------------ -
/
Age
Co and Sppo
(b) ig Ces suts and ssve ints f a te zzy set middle aed" and b te fuzzy sinletn yeas ld "
Ditio Cnvexity
zzy set is covx f and only f for any X , X2 E X and any E 0, 1 ' (2.9) JA (X + (1 ) X2 ) > min { JA ( x d , JA ( x2 ) } . lternatively, is conve f all ts -level sets are conve.
o crisp set C in R n is conve if and ony f for any two points X E C and X2 E C, their conve combination X + (1 ) X2 is stll in C, where 0 < < l
Hence the conveity of a ( crisp ) level set implies that s composed of a sngle lne segment ony. Note that the denition of convety of a fuzy set is not strict the common denition of conveity of a function. For comparison, the denition of conveity of a ncton f ( x ) is (210)
u y
(a Two Covx
Fz St
b A Nocovx
ts
C
Fz St
1 08 0.6
"
1 8 0.6 � 04 02
c
cO C C
C C
� 0 02
ig a w nvex membesi ntins; b a nnnvex membesi ntin (MTLB e: o m)
whch s a tghter condton than Equaton (9). Fgure .4 ustrates the concept of convety of fuzy sets; Fgure .4(a) shows two conve fuzzy sets [the e fuzzy set satses both Equatons (9) and (10, whe the rght one satses Equaton (9) ony] ; Fgure .4(b) s a nonconve fuzzy set. Dnition Fuzzy numbes
A fuzy number s a fuzzy set n the rea ne (R) that satses the condtons for normaty and convety.
Most (noncomposte) fuzy sets used n the terature satsfy the condtons for normaty and convety, so fuy numbers are the most bc type of fuzy sets. Dnition andwidts f nmal and nvex zzy sets
For a norma and conve fuzzy set, the bandwidth or width s dened the dstance between the two unque crossover ponts: (11)
Dnition Symmety
A fuzy set s s"mtic f ts MF s symmetrc around a cetan pont namey, A(C + x = A( x for a x E X
x = c,
c t-totc patons
D
Deitio en le en it lsed
A fuzzy set A is oe left if lm A = 1 and lim A = 0; oe ight ifim A 0 and lim A = 1; and closed iflim_ A = im A o
= =
D
For nstance, the fuzzy set young in Figure is open le; old s open right; and middle aged is closed 2.3
SEHEO REIC OPE RAI ON S
Union, intersecton, and complement are the most bc operatons on csical sets On the bis of these three operations, a number of identities can be established, lsted n Table 1 These denttes can be vered usng Venn dagrams Corresponding to the ordinary set operations of unon, ntersection, and com plement, fuzzy sets have similar operations, which were initially dened in Zadeh's semnal paper [11] Before ntroducng these three zzy set operatons, rst we shall dene the notion of contnment, which pays a central role in both ordinary and fuzzy sets This denition of contanment s, of course, a natural extension of the ce for ordnary sets Deitio Cntainment subset
zzy set A s cotaied n zzy set or, equivalently, A is a sbset of or s smaller than or equal to if and only f A (X) for all I symbols,
<
(.1)
D
Figure 5 ilustrates the concept of A C Deitio nin disjuntin
The io of two fuzzy sets A and s a zzy set C, wrtten C = A u or C = A OR , whose MF is reated to those of A and by
(.13)
D
As ponted out by Zadeh [11], a more intuitive but equivalent dention of union is the smallest fuzzy set containing both A d Aternatively, if is any zzy set that contans both A and , then t so contans A U The ntersecton of zzy sets can be dened alogously
D
uy
ts
C
Tble asi identities f lassial sets wee and C ae is ses
and C ae tei esndin mlements; is te univese; and is te emty set
Law of contradiction Law of the excluded mddle Idempotency Involuton Commutativty Assocativity Distributivity Absorpton Absorption of complement DeMorgan ' s laws
U , U , ( U U C U C) ( C C) C) ( U ( U C) U C) ( U ( C) U ( B) ( U B) U ( ( U U
=
U
=
U
U
=
Deitio ntesetin njuntin
The itesectio of two fuzzy sets and is a zzy set C, wrtten C or C AND , whose MF is related to those of and by
( 4)
o
As in the ce of the union, it is obvious that the intersecton of and s the largest fuzzy set which s contained in both and This reduces to the ordinary ntersection operation if both and are nonfuzzy Deitio Cmlement neatin
The comlemet of zzy set , denoted by (, NOT ), s dened
( 5)
c ttotc patons A Is Conain in B
58 :
ie The concept of C B (MTLB le: ubet . m) (a) Sets and
(b) St ant Aa
08 06 0. 0 0
08 06 0 0 0 (c) Set aA OR a
d St a ND a
08 0.6 0. 0 0
08 0.6 0. 0. 0
ie Operations on zz sets: a two zz sets and B b ; c UB;
d B (MTLB le: fuzetom)
Fgure 2.6 demonstrates these three bic operations: Figure 2.6 a llustrates two zzy sets and B Fgure 2.6 b is the complement of Figure 2.6 c is the union of and B and Figure 2.6 d s the ntersection of and B Equations 2.13 , 214 , and 2.15 perform exactly the corresponding o erations for ordinary sets if the alues of the membership nctons are restricted to either 0 or 1. However, t s understood that these functions are not the only
Fuzzy ts
C
possible generalzatons of the crsp set operatons For each of the aforementoned thre set operatons, several dierent cses of functons wth desrable properties have been proposed subsequently n the lterature; we wil introduce some of these nctions in Section 25 The approprateness of these functions can be checked via the denttes in Table 21 see Exercises 3 and 4) For distnction, the max [Equa tion 2.13), mn [Equation 2.14)], d the complement operator Equaton 215) wll be referred to the classical or stadad fzzy oeatos for intersecton, unon, and negation, respectvely, on zzy sets Next we dene other operations on zzy sets whch are also drect generaza tions of operations on ordnary sets Deitio artesian pduct and co-pduct
Let d B be zzy sets n and , respectvely The Catesia odct of space x wth the and B, denoted by x B, is a fuzzy set in the product membershp nction 216) Simlary, the Catesia co-odct + B is a fuzzy set with the membership function 217) Both x B and + B are characterized by twdmensonal MFs, which are explored in greater detai in Section 2.4.2 24
M F FORMU LAION AN D PARAM EERIZAION
As mentioned earlier, a zzy set is completely characterized by its MF Since most zy sets n use have a universe of discourse consstng of the rea line R , it would be impractical to lst all the pairs denng a membershp functon A more convenient and concise way to dene an MF is to express t a mathematcal formula, in Example 23 I this secton we describe the clses of parameterized functions commony used to dene MFs of one and two dimensons MFs of higher dmensions can be dened smilarly Moreover, we give the derivatives of some of the MFs with respectve to ther nputs and parameters These dervatves are important for ne-tunng a fuzzy inference system to acheve a desired input/output mapping; technques for ne-tuning fuzzy nference systems are discussed in detai n Chapter 4. 2. 4. 1 M Fs of One Di mension First we dene several clses of parameterized MFs of one dmensonthat s, MFs with a single input Deitio iangular MFs
M ormto d Prmeterzto
b,
A tiala MF is seced by three arameters { } follos
b,
triangle ( ) =
0 0
< << < < <
b
b
( 28 )
b b
By using min and max e have an alternative exression for the receding euaton
b,
triangle ( ) = max min
b
b,
0
( 29 )
The arameters { } (ith < < ) determine the coordnates of the three corners of the underlyng trangular M
gure 27 ( a) illustrates a triangul M dened by triangle ( 20 60 80) Ditio aezdal MFs
b,
A tazoidal MF is secied by four arameters { } follos
b,
traezoid ( ) =
0 < << < < < < 0 <
b
b
b
An alternatve concise exresson using min and max is traezoid ( ) = max min 0
b,
b,
b
( 220 )
( 22 )
The ameters { } (ith < < < ) determine the coordinates of the four corners of the underlyng traezoidal M
igure 2 7 (b) llustrates a traezoidal M dened by traezoid ( 0 20 60 9) Note that a traezoidal M ith arameter { } reduces to a triangul M hen is eual to Due to their simle formul and comutational eciency both triangular Ms and traezodal Ms have been used extensively esecally in realtime imlemen tations Hoever since the Ms are comosed of straght line segments they e not smooth at the corner onts secied by the ameters I the folloing e introduce other tyes of Ms dened by smooth and nonlnear nctons
b
b,
Fuz (a) Tangue MF
�!eC 08 Q C 04 C
: 0.6
� 0.4 C
� 0.2
0.2
20
00
d) Geneed Be MF
(c) Gaan MF
�� 08 Q 0 04 C 02
C
b) Tapez MF
�� 08 Q 0 C
00
ets
�!eC Q 0 04 C 02
(
20
40
0.8
00
Eaples of four classes of paraeterze MFs a tranle( 20 60 80) b trapezo(x 020609) aussan(x 0 20) bell(x 20 4 0) (TLB e: d f . m
Fig
Ditio Gaussan MFs
{c, } c c, c ; c
A Gassia MF s seced by to arameters gaussanx
=e
x
222)
A Gaussan M s determned cometey by and reresents the Ms center and determnes the Ms dth gure 27c) ots a Gaussan M dened by gaussan; 0 20)
Ditio Generalze bell MFs
A gaizd bll MF or bll MF) s seced by three ameters be; b = 2
a, c) + l x a c l
{a c}: b
223)
here the arameter b s usuay ostve ( b s negatve the shae of ths M becomes an usdedon be) Note that th M s a drect generazaton of
c M Fomton nd Pmtzton
27 (b) hangng b'
(a) hangng 'a
0.8
0.8
0 0. 0.2 -0
-5
0.6
\\
0.4
5
0.2
0
-5
0
5
b
(d) hangin g 'a' and
(c) hanging 'c'
0.8 0.6 0.4
0.2
0
-0
5
5
0
Fig 28 The eects of changing paraeters in bell MFs a changing para
c;
eter ; b changing paraeter b; c changing paraeter d changing and b siultaneousl but keeping their ratio constant ( e: allbell
the Cauchy distribution used in robabiity theory so it is aso referred to the Cachy MF.
Figure 27d) iustrates a generaized be MF dened by be 2 4 5) A desired generaized be MF can be obtained by a proer seection of the parmeter set { b Specicy we can adjust and to vary the center nd idth of the M and then use b to contro the soes at the crossover points Figure 9 shows the physica meanings of each parameter in a be MF Figure 8 further iustrates the eects of chnging each parmeter o obtain handson experience of these eects the reader is encouraged to run the e bel lanu hich is aiabe via FP and WWW see page xxiii) Another e bellani. gives more vivid visua eects by animating two be MFs hen their parameters re changing Because of their smoothness and concise notation Gaussian and be MFs are becoming increingy pour for secifying fuzzy sets Gaussian functions re we known in robabiity nd statistics and they possess usefu properties such iniance under mutipication the product of two Gaussians is a Gaussian ith a scaing factor) and Fourier transform the Fourier transform of a Gaussian is sti a
c}
c
Fuz
M
1.
Sp =
ets
C
-b/2s
�
c-s
c+s
c
2s
Fig Phsca eaning of paraeters n a geneize be MF
Gaussian). The bell MF h one more arameter than the Gaussian MF so it h one more degree of eedom to adjust the steeness at the crossover oints. Although the Gaussian MFs and bell MFs achieve smoothness they are unable to seci ymmetric MFs hich are mortant in ertain alicatons Next e dene the sigmoidal MF hi is either oen le or right. Asymmetric and close MFs can be synthesized using either the absolute dierence or the roduct of to sigmoidal nctions exlained next. Ditio Sigoia MFs
A sigmoidal MF is dened by
a,
sig; c) = here
+ ex
a c ]
2.24)
a controls the sloe at the crossover oint = c
a,
Deending on the sign of the arameter a sgmoidal MF is inherently oen right or le and thus is aroriate for reresenting concets such very lare or very negative. Sigmoidal nctions of this kind are emloyed idely the activation nction of articial neural netorks. Therefore for a neural netork to simulate the behavior of a fuzzy inference system more on this later chaters) the rst roblem e face is ho to synthesize a close MF through a sigmoidal nction. To simle ays for achieving this are shon n the folloing examle.
i
Examl lose an asetc MFs base on sigoia nctons
Figure 2.0a) shos to sigmodal functions = sig; ) and = sig; 2 ) ; a close and ymmetric MF can be obtained by takng their dierence shon in Fiure 2. 10b) . Fiure 2. 0c) shos an additional sigodal MF dened = sig; 2 ) ; another ay to form a close and ymmetric MF is to take ther roduct shon in Figure 2. 0 d) . The reader is encouraged to try the TLB le aailable via FTP and WWW see age iii) hch dislay the animaton of to composite MFs bed on sigoidal nctions.
M Fomaton and Paameterzaton a)
0.8 0.6 04 02 0 0
g(; ,5); 2 g(2 5)
5
1 08 0 04 02 0 -10
0
0
5
-5
g(,5) 3 g(; 2,5)
\
08 06 04 0.2 0
(b) l 2
(c)
I
5
5
5
0
5
0
(d) *3
08 0 04 02 0 -0
3
0
0
5
0
Fig 10 a Two sigoal nctons Yl an Y2 ; b a close MF obtane o
I Yl - Y21 ; c two sigoial nctions Yl an Y3 ; a close MF obtaine o Yl Y3 · (MTLB e: di ig m)
In the fooing e dene a much more genera tye of MF the eright MF. This tye of M, athough extremey exibe in seciing zzy sets is not used often in ractice because of ts unnecessary comexity. Ditio 3 Le-ght MF L-R MF
�j
l-ight MF or L-R MF is secied by three ameters { } ) =
, X < C. L >
2.2)
here ) and ) are monotonicay decreing functions dened on 0 ) ith O) = O) = 1 and im ) = im ) = O
Exl L-R MF
Fuz
(a
(C 8 f .
X
Fig
Ch
()
C .8 f .
�.CC
ts
Two L-R MFs: (MTLB e: difflr m)
�CC
X
a LR(x 65 60 10); b LR(x 25 10 40)
Let
Bed on the receding £(x) and () Figure 211 iustrates two Ms secied by (x 65 60 10) and (x 25 10 40)
The ist of MFs introduced in this section is by no means exhaustive other seciaized MFs can be created for specic aications if necessary In articuar any tye of continuous robabiity distribution functions can be used an MF here rovided that a set of ameters is given to secify the approriate meanings of the MF Severa other tyes of ameterized MFs such S Z and twosided Gaus sian MFs e examined in Exercises 6 7 8 and 10 respectivey 2.4.2 MFs of Two Dimensions Sometimes it is advantageous or necessary to use MFs with two inputs each in a dierent universe of discourse. MFs of this kind are generay referred to twodimensiona MFs where ordinry MFs MFs with one input) are referred to onedimensiona MFs. One natura way to extend onedimensiona MFs to twodimension ones is via cyindrica extension dened next Dtio linical etensons of one-diensional zz sets
A is a fuzzy set in then its cylidcal xtsio in ( A dened by cA = ()()
x
is a fuzzy set (226)
S M ormtion nd Pmeteriztio
(a)
B zy
yll E f
08
06 f 0.2 G
0.4
Fi a ase set ts clnrcal etenson ()
(TLB le
cylet m
U suly is referred to a bas st.)
The cocet of cylidrical extesio is uite straightforwd; it is illustrated i Figure 2 2 The oeratio of rojectio o the other had decrees the dimesio of a give multidimesioal) membershi ctio Ditio Projectons of zz sets
R
Let be a twodimesioal zzy set o x The the ojctio of ad e deed max )/ =
R x Ry
ad
R oto
y
=
max )/ x
resectively
R;
Figure 23a) shows the MF for fuzzy set Figure 23b) ad Figure 23c) are the roectios of oto ad resectively Geerally seakig MFs of two dimesios fall ito two categories comosite ad ocomosite a MF of two dimesios ca be exressed aalytic exressio of two MFs of oe dimesio the it is comosite; otherwise it is o comosite A examle is give ext
R
Examl oposte an noncoposte MFs
(a)
1
Fuz
A
b Projecto oto
Twomesoal MF
1
Ch
(c) Projecto oto
X
Y
1
0.5
0.5
05
o
0
o
y
S
y
X
y
X
X
Fi a Two-ensional set ; b projection of onto ;
an c projecton of onto . (MATLA e roj ect m Supose that zzy set
A ) i near 3 4) is dened by
y 3
) ex
2
4) .
Then this todimensiona M is comosite since it can be decomosed into to Gaussian Ms ) exp ex gausian; 3 2) gaussian ; 4 ) .
A
Note that e can vie the zzy set a to statements joined by the connective AND is near 3 AND is near 4 here the rst statement is dened by near 3 ) gaussian; 3 2) and the second statement is dened by Jnear 4 (Y )
=
gausian(y; 4, 1).
Thus the mutiication of these to Ms i used to interret the AND oeration of these to statements. On the other hand if this zzy set is dened by )
+ 3 4
(227)
then it is noncomosite.
As demonstrated in the receding exame a comosite todimension M is usuay the resut of to statements oined by the AND or OR connectives. Under
Sc MF Fomo d mterzo
()
(a) z = min(ap(x) ap(y»
05
05
10
10
10
10 10
0 0
x
d) z = max(l(x l(y
05
05
10
10
10
0
x
10 -10
10
x
(c) min((x) l(y
= max(ap(x) p(y»
10 0
x
Fig Two-diensional MFs dene b the mi and x operators a
= (trap(x) trap()); b = (trap(x) trap()) c = ( bell(x) bell()) ; = (bell(x) bell()) (MALA e: f2d )
ti onition te twoimeniona MF i dened te AND r OR aggregatin o it two contituent MF Cia AND an OR peratin n uzzy et are min and m ee Setin 23 teir eet n generating twoimenina MF are iutrated in te owing exampe
(
Exl oposite two-diensional MFs based on min and m operators
(
(
(
(
Let trap = trapezoi 6 2 2 6 an trap = trapezi 6 2 2 6 e tw trapezoida MF n an repetivey Aer appying te mi an operator we ave twoimeniona MF n x wn in Figure 214 a and Figure 24 an d repeat te ame pt exept tat te trapezia MF are repaed y e MF e = e 4, 3 ) an e = e 4 3 )
(
(
( (
(
(
(
(
When te min operatr i ued to aggregate neimenina MF te reuting twoimeniona MF an e viewed eiter te reut appying cica uzzy interetion Denitin 215 to te yinria extenin eac neimenina MF or a Carteian prdut o two neimenina uzzy et Denitin 21 Simiar interpretatin appy t te perator
(
(
zzy S
Ch
It i ovious tat the denition for onedimeniona MFs introdued in Se tion . have natura extensions to the e of twodimensiona MFs with some appropriate adjustments. For intane rossover points shoud e hange to rosover urves ut is a risp set in instead of and so on.) Moreover the onepts introdued in thi setion an e generaized eiy to form the onepts o ndimensiona MFs. 2. 4. 3 Deriaties of Parameterized M Fs To make a zzy ystem adaptive we need to know the derivatives of an MF with respet to its gument input) and pameters. This derivative information pays a entra roe in the earning or adaptation of a zzy system whih wi e disused in depth in usequent hapters. Here we ist these deritives for the Gaussian and e MFs the reader is enouraged to derive them independenty. For te Gaussi MF et y
=
gausian(x; (, c)
e
-� (
x
c
(
)
2
.8)
Then c
y y y c
9)
)
3
2.3)
y.
c
2.3)
I the preeing expressions the derivatives are arranged to inude y and thu
save omputation. For the e MF et
y e c =
l l
1+
Then y y 8y
y
Bc
2 c y y ) c if = c
y ( y) 2n
I cI
2.32)
233) 2.34)
y y if c
if = c
b y 1 y if c if x c.
=
Derition of tese formu are e Exerises 1 and 3
35 23)
35
ec More on Fy Union, Inersecon, and Complemen*
2.5
MORE ON FUZZY U NI ON, INTERSECTION, AN D COM PLEMENT*
Ti etion iue more avane topi onerning ompement, union, an interetion operation on fuzzy et. For a rttime reaing, ti etion may e omitte witout iontinuity. Trougout ti ook, etion or uetion marke wit a tar ( * ) an e kippe at rt reaing. Atoug te ia fuzzy et operator Equation 2.13, 2.14, an 2.15 poe more rigorou axiomati propertie own in ti etion, tey are not te ony way to ene reonae an onitent operation on zzy et. Ti etion examine oter viae enition of te zzy ompement, interetion, an union operator. 2. 5. 1 Fuzzy Complement* A zzy ompement operator i a ontinuou ntion N : , 1 meet te foowing iomati requirement:
� , 1 wi
N(O) = 1 an N() = ounary monotoniity. N() > N() f <
2.3
A ntion atifying tee requirement form te genera of zzy ompe ment. It i evient tat vioation of any of tee requirement wou a to ti ome ntion wi are unaeptae ompement operator. Speiay, a vioation of te ounary onition wou inue ntion tat o not onform to te orinary ompement for rip et. Te monotoni ereing requirement i eentia ine we intuitivey expet tat an inree in te memerip grae of a zzy et mut reut in a eree in te memerip grae of it ompement. Tee two requirement are te i requirement tat a fuzzy ompement op erator ou meet. Anoter optiona requirement impoe ivoltio on a zzy ompement: 2.38 N(N()) = invoution, wi guarantee tat te oue ompement of a zzy et i ti te et itef. Te foowing exampe of fuzzy ompement atify two of te i requirement in Equation 2.3 we te aforementione optiona one. Examl Sugeno s complement
One of fuzzy ompement i Sgo's comlmt 8, ene y N () = 1 + s '
239
were s i a parameter greater tan 1. For ea vaue of te parameter s we otain a partiuar fuzzy ompement operator, own in Figure 2.15a.
Fzzy es
Ch
(b Yaer's Complens
(a Sueno's Cplemens
02 05 X=a
0.5 X=a
Fig Sugeno s nd ger s complements. (MATLA e negation.m) Examl gers complement
Another of zzy ompement i ag's comlmt 1] ene y ) = 1 W) / W
24)
where w i a poitive parameter Figure 2 15) emontrate ti of n tion for vaiou vue of w. Note tat ue to te invoution requirement, ot Sugeno n ager' ompement are ymmetri aout te 45egree traigt ine onneting , ) an 1, 1)
Oviouy, tee iomati requirement for zzy ompement o not eter mine ) uniquey However, ) i equa to te ia zzy ompement) if the foowing requirement i ntroue 1] 241)
A
Ti requirement enure tat a ange in te memerip vaue in ou ave a orreponing eet on te memerip in Ti requirement togeter wit the i requirement ounary n monotoniity) for zzy ompement enta () = 1 wi automatiay atie te invoution requirement
A
25 2 Fuzzy Intersection and U nion* The interetion of two zzy et n B i peie in genera y a ntion , ] , wih aggregate two memerip gae foow T , 1] x , ]
�
A
242)
were i a inary operator for te ntion T. Ti of zzy nteretion operator, wi are uuay referre to Tnorm tringuar norm) operator, meet te foowing i requirenent
More on F Union nersecion, and Complemen ·
Dnition -norm
A -nom operator [3 i a twopae funtion T(· ·) atiing (OO) = , ( 1) = ( ) =
ounary) T(a, b) < T(c, d) if a < c and b < d monotoniity) ommutativity) ( ) = ( ) oiativity) . ( ( )) = T (( ) )
(43)
Te rt requirement impoe the orret generaization to rip et. Te e on requirement impie tat a eree in te memerip vaue in or B annot proue an inree in te memerip vaue in B Te tir requirement ini ate tat te operator i inierent to te orer of te fuzzy et to e omine. Finay, te fourt requirement aow u to take te interetion of any numer of et in any orer of pairwie grouping. Te foowing expe iutrate four of te mot equenty enountere Tnorm operator.
A
An
Examl Four -norm opertors
Four of te mot frequenty ue Tnorm operator are
/
Minimm Tmin(a ) = min, ) = Algbaic odct ( ) = Bondd odct ( ) = ( + 1)
{
° V
Dastic odct
Tdp(a, b)
=
(44)
if = 1
b,
ia
0,
if
=
1.
a, b < 1.
°
Wit te unertaning tat an are etween an 1 we an raw urfae pot of tee four Tnorm operator ntion of an ee te rt row of Figure 16 Te eon row of Figure 16 ow te orreponing urfae wen = A(X) = trapezoi; 3 8 1 17) an = B(Y) = trapezoi; 3 8 1 1 7) tee twoimenion MF an e viewe te Carteian prout of an B uner four ierent Tnorm operator. om Figure 16 it an e oerve tat
A
(45)
Ti an e verie matematiay.
Like zzy interetion, te zzy union operator i peie in genera y a ntion S [0 1 x [0 1 [0 1 In ymo,
�
(46)
Fzzy es
Ch
D lu
• 0 0 Frst w Four orm opetors Tmin(a b) Tp(a b) Tp(a b) nd Tdp(a b); scond w the cospondg srfces for a (MATLA e traz(x 3 8 1 17) nd b = traz(y 8 1 17)
Fi
tnorm.m
were a nary operator for te funon S T of zzy unon operator, wh are oen referre to Tonorm or Sorm) operor, tfy te foowng requrement Ditio conorm Snorm
A -coom or S-om) operator [3] twope nton S (· ) atfyng ounary) S( 1) = 1 S(O a) = Sea ) = a S(a b) < S(c ) f a < c n b < monoonty) ommuttvty) Sea b) = S( a) oatvty) Sea S b c)) = S(S(a b) c)
(47)
Te jutaton of tee requrement miar to tat of the requrement for Tnorm operator xaml For cono opertors
ec More on Fzzy Unon nersecon and Compement*
M
V_y 0 0
b i S
39
e S
Yy 0
Dti m
V_y 0 0
0
Fig
Fist ow Fou -conom opeatos Smi(a b) Sp(a b) Sp(a b) an Sp(a b) secon w te coesponing sufaces fo a MLB e: tapezoi(x 3 8 12 1) an b = tapezoi(y 3 8 12 1)
nm
Correponing to te four Tnom operator in te preiou exampe we ae te foowing ou Tonorm opeator. Maximm S(a b) = max (a b) = a v b Algbaic sm S(a b) a + b abo Bondd sm S(a b) = 1 (a + b)
Dastic sm
S(a , b) =
a b, 1,
if b = O i a = . i a b > .
(248)
Te rt row o Figure 21 ow te urfae pot of tee T onorm operator Te eon row emontrate te orreponing twoimeniona MF we a = A(X) = trapezoi( 3 8 12, 1) an b = (X) = trapezoi( 3, 8 12 1) tee MF are te Carteian oprout o an B uing tee ou Tonorm opeator It an ao e erie tat
A
(249)
Note tat tee eentia requirement for Tnom an Toorm operator an not uniquey etermine te ia uzzy inteetion an unionnamey te mi an max operator Stonger etition ae to e taken into oieatio to pinpoint te min an max opeator For a etaie teatment of ti uet ee [1
Fzzy es
Ch
heoem Genelized eogan s law
Tnorms ·, ·) and Tconorms , ·) are duals which support the generaization of
== =
DeMoga's law
, ) ) ))) , 250) , ) ) , )) ), where ·) is a zzy complement operator we use and for Tnorm and Tconorm operators, respectively, then the preceding equations can be rewrtten
))), = ))).
251)
Thus for a given Tnorm operator, we can always nd a corresponding Tconorm operator through the generalized DeMorgan's law, an vice versa In fact, the four Tnorm and Tconorm operators in Examples 212 and 213, respectivey, are du in the sense of the generalized DeMorgan's law The reader is encouraged to verfy this) 2.5.3 Parameterized -norm and conorm* Several aa"eteized -oms and dual Tconorms have ben proposed in the pt, such those of ager [9], Dubois and Prade [4] , Schweizer and Sklar [7] , and Sugeno [ or instance, Schweizer and Sklar's Tnorm operator can be expressed Tss ( b, p)
, ) It is observed that
=
[max{O, ( - P + b P
-
l )}] -
1 [max{, «1 ) + 1 ) 1) }]
252)
lim , , ) = , 253) lim , ) = min, ), which correspond to two of the more commonly used Tnorms for the zzy AND operation To give a general idea of how the parameter aects the Tnorm and Tconorm operators, Figure 21a) shows typical membership functions of zzy sets and B Figure 21b) and Figure 2 1c) are , ) and ) , r espectivey, with = 0 soid line) 1 dhed line), dotted line) nd 1 dhdotted line) Note that the bellshaped membership nctions of and B in Figure 21a) are dened ' follows 1 254) A(X) = bell; 5,2,75) 1
A
°
( ) = bell; 5, 1, 5)
=
=
A
1
1 For completeness, other types of parameterized Tnorms are given next
255)
ec More on Fzzy Un on, In ersecon , and Complemen*
f A B 0.5 15
0
5
0
5
10
5
5
10
5
5
0
5
o
Fig Scweizer nd Sklr s prmeterized -norms nd -conorms
A
p
p
memersip nctions for zz set and B; , , ) nd c , , ) wit = 0 solid line 1 dsed line dotted line nd 1 das-dotted line (MATLA e tnormm
p
Yag 9] For
q > 0,
{
, , , ,
q)q)
°
= 1 mi{, 1 ) + 1 )] }, = mi { , + ) }.
2.56)
Dbois ad Pad 4] For E 0, 1]
{
,,) = m{,,}, , , = + mi {, , 1 m{ , 1 , }. 257)
Hamach ] For > 0,
{
, , ) = + 1 ) + )], , , ) = + + 2)]/ 1 + )].
258)
Fzzy es
{
Ch
k [5 or s > 0,
, , s = og [1 + (s ) (s )/(s 1), , , s) = 1 log [ + (s )/(s 1)].
{
l S
259)
Sgo [ or > 1,
, , ) = m� { O , + )(a + 1) }, , , ) = m{, + }.
2.60)
Dombi [ or > 0,
{ 26
TD (a, b, A) = + 1 [ (a SD (, b, ) = + 1 [ ( -
_
1 >
(b - 1) >] /> '
_
(b.
_
2 61)
1) >] - /
SUMMARY
his chapter introduces the bic denitions nottion, and operations for fuzzy sts, incuding their mmbershi ncion represntaons, setthoretic operaions AND, R and N), various types of memership nctions, and advnced zzy st operaors such norms nd cnoms Most membership nctins are determined by domain eperts he human dtermined memership nctons, however, may not e precise enough for cerain plictins herefor, i always advisabe to apply optimization techniques t netune parametrized memership ncions for better performance I the discussion of neurozzy modeling n the subsequent chapters, we shall come across parameterizd membership nctons gain and use their derivatives for derivative bed optimization zzy sets ay the foundation for the entire zzy set thory and reated disci plines I the ne chpter, we shall introduce the use f zzy sets in zzy ifthen rules and zzy reoning.
EXERCISES 1.
Sometimes it is usefu to decompose an M into combination of its evel sets' Msj this is the soltio picil, which states
A(X) ma min[, A " )] ,
Aa
a
A
2.62)
Aa·
where is the cut (eve set) of zzy set and A a is the M of [Remember hat is crisp set and hus Aa can take ny values in
Aa
mma
43
Resolto Pcple
�
1
.
O 2
.. . .... . : .. 5
:
. .
o
.
.
10
Resolution principle (MATLA e: reolut m
{, }.] Figure 2.19 iutrate te onept of te reoution prinipe, were ea reange uner te MF repreent min[; A a )] for a pei etween an 1. a) Aapt Equation 262) to a zzy et wit a irete univere of ioure ) Put te MF of fuzzy et in Exampe 2.2 into the reoution format
°
A
Verify te ientitie in Tae 2.1 uing Venn iagram. Determine if te ia fuzzy operator [Equation 2 .13) , 2.14) , an 2. 15)] o for ea ientity in Tae 2.1. Expain wy y giving impe proof or ounterexampe epeat Exerie 3, uming tat te zzy union an interetion are ene ierenty
AUB(X) A(X) + B (X) A (X)B(X) A nB(X) A (X)B (X) .
A
a
2.63) 2.64)
Suppoe tat fuzzy et i erie y A ( ) = e , c). Sow tat te ia zzy ompement of i erie y A ) = e; , c).
A
a
Te S- wit two parameter an < i an Sape openrigt MF ene y 0,
SX ; , ) =
2' 2 ,
2( ) 1 _ 2( X ) 1,
for < for 1 < < 1 for 1 < < for <
2.65)
Fzzy es es
Ch
nton to mpement mpement t MF. MF . ) ) Pot P ot ntane ntane of t a) Wrte a MATLA nton MF w w varou vaue of parmeter ) Fn te t e rosover pont of S X ; l, r) ontnuou. ) Prove tt te ervtve S; l , r ) wt respet o X ontnuou. Te Z-MF wth two parameters n r (l < r ) a Zape opene MF ene y 2.66) Z ; l r) = 1 S; l r) where ; l, r) te SMF n the prevou ex exere. ere. epeat epeat ) troug ) of Exere 6 wth the ZMF
Te MF wt two parameter n shape MF ene va te S an ZMF ntroue earer, a foow , , )
{
S for < Z ; , + ) , for > ,
2.67)
)
were te enter an > 0) the pred on eh e of te MF. a Wrte a MATLA nton to mpement th MF. ) Po ntne of t MF wth vrou vaue of pareters Fn the roover roover ponts n wt of ; l , r
)
)
Te twsidd -MF an extenon of e MF ntroue prevouy; t ene wt four parameter , , , n
t t ; , , , ) =
0, S, , , 1, Z,,), 0
for < for < < for < < for < < for <
2.8)
mpemen th MF ) Pot ntne of of th a) Wrte MATLA nton oo mpemen varou vaue of prmeter. ) Fn the roover roover pont n n wt MF wth varou of ; ; , , , ) ) The twsidd assia MF s ene y for < Cl . for Cl < < Cl . for C2 < X . 2.69) Wrte MATLA nton nton to mpement mpement th MF MF ) Pot ntane of th MF wt vrou vaues of pareter. Fn the roover pont an wth of t M.
)
)
ec umma
Find the eve set and its width for a fzzy set dened by (x trapezoid(x c )
y
Derive the partia derivatives of a Gassian MF assian(x c with respect to its arment x and parameters and c and ths verify Eqa tions (229) to (231)
y
Derive the partia derivatives of a be MF be(x c with respect to its arment x and parameters and c and ths verify Eqations (233) to (236)
et the fzzy set be dened by a Gassi MF assian(x c how that width(/width( is a constant independent of the parameters c d
et the zzy set be dened by a eneraized be MF be(x c how that width width ( ( /width( /width( is a fnction nction of parame parameter ter ony
et the fzzy set be dened by a be MF be(x c Demonstrate that for a Q E [0 1) im width( = 2 b -o
his demonstrates that a be MF wi approach a characteristic fnction of a crisp set if 0. how that the th e enos eno s and Ya Yaer's er' s compement compement operators (Eampes 210 and 211) satisfy the invotion reqirement of Eqation (238) Verify that the for -norm and -conorm operators in Eampes 212 d th e sense of the eneraized DeMoran DeMoranss aw aw 213 are d to each other in the
Prove the ineqaities o Eqations 245 and (249) how that (a ( ( ) 0.
=
when 0 (b ( ( )
=
min( ) when
how that the foowin foowin operators on fzzy fzzy sets satisfy satisfy DeMorans DeMoran s aw: (a Dombi's -norm d -conorm with 1 (b Hamacher's norm and conorm with 1 (c ma and min with () enos compement
how that the twodimensiona MF dened by
is a composite MF bed on the onedimensiona eneraized be MF are ated ated by Dombis Dombi s norm operator operator
46
Fuzzy es es
C
3
U beu a a tmpat t wrt nw tat aw t man ua tunng t wng wng paramtrz MF MF (a) tranguar trangu ar MF MF () trap za MF (c) Gauan MF () gma MF () SMF ( Exrc 6) an () MF (a Exrc 8)
4
U bei (r sigi) a a tmpat t wrt nw tat w t anmatn t wng wng paramtrz MF : (a) tranguar MF () trapza MF (c) Gauan MF () SMF ( Exrc 6) an () MF ( Exrc Exrc 8) 8)
ATLAB
ATLAB
REFERENCES
11 R E. Bema ad M. Giertz. On the aaytic aaytic orma orma ism ism o the theory theory o uzzy sets Informaton cncs 5 1 9156 1973 2 J Dombi A genera genera cs oo uzy operators operators the De Morga cls o uzzy uzzy operators ad uzziess meures induced by uzzy operators zzy ts and ystms, 819 163 1982 3 D Dubois ad ad H rade rade zzy zzy sts and an d systms systm s thory and applcatons. applcatons. Academic press press New York, 1980 D Dubois Du bois ad a d H rade. New results about properties properties and sematics o uzzy set theoretic operators. In . Wang ad S K Chag editors fuzzy sts thory and applcatons to polcy analyss and nformaton systms pages 5975. enum, New York 1980. simutaeous so sociativi ciativityty o F(x, F(x ,y) ad x+yF(x,y).) . Aquatons 5] M. J ak On the simutaeous Math., 19 1 9226 1979. 6] H. Hamacher ber logische verknupugen unscher aussagen und deren zugehrge bewertungs unktionen. I R appl, G J Kir, ad Riccidi L editors Prgrss n cybtcs and systms rsarch I pages 276288 Hemisphere, New York 1975 7 B Schweizer ad A. Skl. Associative unctions ad abstract semigroups Publ. 981 1963. 1963. Math. Dbrcn, 10 6 981 8 M. Sugeno zy meures and uzzy itegras a survey In M M. Gupta, G N. Sidis ad B R Gaies, editors zzy automata and dcson procsss pages 89 102 NorthHoland New York, 1977 9] R Yager On a genera genera cls o zzy zz y connectives zzy ts and ystms, 23522 1980 10] R R Yager On the meure meure o uzziness ad negation, pt p t I membership in the unit interva Intatonal Joual of ManMachn tuds, 5221229 1979 11] L A Zadeh zzy sets. Informaton and Control, 8:338353 1965
Chapter 3 Fuzzy Rules and Fuzzy Reasoning
JS R Jang
3.1
INRODUCION
In ti capt w intuc t cncpt t xtnin pincip an uzzy a tin wic xpan t ntin an appicaiity uzzy t intuc pviuy n w pnt t nitin inguitic vaia an inguitic vau an x pain w t u tm in uzzy u wic a an cint t quantitativ ming w ntnc in a natua aticia anguag. By intpting zzy u apppiat uzzy atin w invtigat int cm zzy ning w innc pcu n t cncpt t cmpitina u innc a u t iv cncuin m a t zzy u an knwn act zzy u an uzzy ning a t ackn zzy innc ytm wic a t mt imptant ming t n zzy t ty. Ty av n uccuy appi t a wi ang a uc autmatic cnt xpt ytm pattn cgnitin tim i pictin an ata cicatin Inpt icuin aut uzzy innc ytm i pvi in Capt 4. 3.2
EXENSION PRINCIPLE AND FUZZY RELAIONS
W a tat y ving nitin an xamp t xtnin pincip an uzzy atin wic a t atina in zzy ning. 3.2. 1 Extension Principe extesio icile 4, ] i a ic cncpt zzy t ty tat pvi a gna pcu xtning cip main matmatica xpin t uzzy main i pcu gnaiz a cmmn pinttpint mapping a nctin (·) t a mapping twn zzy t M pcicay upp tat i
uz Rules and uzzy Reasoning
a nctin m X t an
C
A i a uzzy t n X n
Tn t xtnin pincip tat tat th imag zzy t f ( . ) can xp a zzy t B
A un th mapping
w Yi = f (X i ) = 1 , n. In th w h zzy t B can n thug th vau f() in X . . Xn . f(·) i a manytn mapping thn t xit X X2 E X X X2 uc that ( X ) = (X 2 ) = y* y* E . I ti c t mmhip ga B at y = y* i th maxmum t mmip ga at X = X an X = X2 inc f( ) = y * may rut m ithr X = X r X = X 2 . M gnay w av (y) = m - l ()
H
=
A
x=/ ( y)
A imp xamp w. xamle pplcton of the extenson prncple to zzy sets wth dscrete n verses
Lt
A 0.1/2 + 0.4/1 + 08/0 + 09/1 + 03/2
an
f ( ) = x2 3
Upn appying t xtnin pincip w hav
+
+
+ + + V
B 0.1/1 0.4/2 08/3 0.9/2 0.3/1 0.8/3 (0.4 0.9)/2 (01 0.3)/1 08/ 3 0.9/ 2 0.3/1
V
+ V + +
w pnt m Figu 31 iutat thi xamp. F a zzy t wit a cntinuu univr f icur X an anaogu prc u appi xame 3. pplcton of the extenson prncple to zzy sets wth contnos nverses
Lt an
() (; 1520.5)
{ ( f ( ) =
-
1) 2 - 1
i
>
o.
i < 0.
ec Exension Principle and uzzy Relaions
2 0 2
Y
49
0 2
Fige Extenson prncple on zzy sets wth dscrete nverses.
Figur 32(a) i th pot of y = ); igur 32(c) is () th M of A. Ar mpoyig th xtio pricip w obtai a zzy st B its MF is show i Figur 32(b) whr th pot of () is rotatd 90 dgrs for y viwig Sic () i a maytoo mappig for E 1 2] th m oprator is usd to obta th mmbrhip grads of B wh y E [0 1] . This causs discotiuitis of ( ) at y = 0 d 1. h drivatio of (y) is l Exrcis 1 at th d of this chaptr
ow w cosidr a mor gra situatio Suppos that is a mappig o dimioa product spac X x Xn to a sigl uivrs such that ( n ) = y ad thr is a fuzzy t Ai i ach Xi = 1 Sic ach mt i a iput vctor ( n ) occurs smltneosly this iplis a AD opratio Thrfor th mmbrship grad of fuzzy st B iducd by th mappig houd b th iimum of th mmbrship grads of th costitut zzy t Ai = 1 With this udrstadig w v a compt forma ditio of th xtsio pricipl Deitio Extenson prncple
Suppos that fuctio is a mappig om a dimsioa Cartsia product pac X x X2 x · Xn to a odimsioa uivrs such that y = ) ad suppo A . . . An ar zzy sts i X . . Xn rspcivly Th th xtio pricipl srts that th fuzzy st B iducd by th mappig is dd by (31)
o
zzy Rules nd zzy Resoning
0
3 2 1
2 1
Y = f(x)
°
>
Ch
B
° -1 .
-1 -2
2 -2
°
2
X
Mmbip Grds
A
GU 5 °·8
GO f:G 0.2 2
°
X
2
Fige Extension principle on zzy sets with continos niverses s ex plined in Exmple 3.2. he lower plot is zzy set ; the pper le is the nction y = f); nd the pper right is the zzy set B indced vi the extension principe. (MATLA extenio )
Th forgoig xtio prcpl sums that y = f . . . n) is a crisp c tio. I cs whr f is a zzy fucto or mor prcisly wh y = f l . . . n) is a fuzzy st charactrizd by a + 1 ) dimsioal M] th w ca mpoy th compositioal rul of frc itroducd i Sctio 34.1 pag 63 ) of th xt chaptr to d th iducd zzy st B.
3 2. 2 Fuzzy Reations Bary zzy rlatio 4 6] ar fuzzy sts i X x whch map ach mt i X x to a mmbrship grad btw 0 ad 1. I ptcuar uary zzy ratos ar fuzzy sts with odimsioa MF; biary zzy rato ar zzy sts wh twodimioa Ms ad o o Applcatos of zzy ratios clud ar uch fuzzy cotro ad dciso maig r w rstrict our attto to bary fuzzy rlatios a graizatio to ary ratos s straightforward Deitio inry zzy reltion
ec Exension Pncipe and uzzy Relaions
t X ad Y b two uivr of dicour h R = {( ( ) ( )
) I ( ) E X x Y}
(32)
i a bay fzzy elato i X x Y [ot that ( ) i i fact a twdimioa M itroducd i Sctio 242]
xamle ry reltos
t X = Y = R (th poitiv ra i) ad R = " i much gratr tha . h MF of th fuzzy ratio R ca b ubjctivy dd
{ ( ) + +
0,
. f Y > . if y
<
(3.3)
x.
I X = {3 4 5} ad Y = {3 4 5 6 7} th it i covit to xpr th fuzzy ratio R a elato matx R=
0 0111 0200 0273 0333 0 0 0091 0 167 0231 0 0 0077 0 143
o
(34)
whr th mt at row ad colum is qua to th mmbrhip grad btw th th mt of X ad th mt of Y
Othr commo xamp of biary fuzzy ratio ar foow
• • • •
i co to ( ad ar umbr) dpd o ( ad ar vt) ad oo ali ( ad ar pros objct ad o o) I i arg th i smal ( is a obsrvd radig d i a corrpodig
actio).
T
h t xprio I i A, th i B i ud rpatdy i a zzy ifrc ytm. W wi xpor fuzzy ratios of this id i th foowig chaptr. zzy ratio i dirt product pacs ca b combid through a compo itio opratio. Dirt compoitio opratios hav b uggtd for fuzzy ratio; th bt ow i th maxmi compoitio propod by Zadh [4. Deto x-m composto
uzzy Rules nd uzzy Reasoning
52
C
Z,
t Rl ad R2 b two zzy ratio dd o X x Y ad Y x rpctivly. Th max-mi comositio of Rl ad R2 i a zzy t dd by
Z} ,
35)
V t with th udrtadig that V ad t rprt m ad mi rpctivy.
36)
Rl o R2
=
{( ) m mi( ( ) ()) E X E Y E
or quivaty ( ) ma min[R l (x, y) , JR2 (y, z)] ) )]
o
Wh Rl ad R2 ar xprd rlaio matri th caculatio of RI O R2 i amot th am matrix multiplicatio xcpt that x ad + ar rplacd by ad rpctivy. or thi ro th mmi compoitio i ao cald th
t
V
max-mi odct.
Svra proprti commo to biy rlatio ad mmi compoitio ar giv xt whr R, 8, ad T ar biary rlatio o X x Y Y x ad x W, rpctivy.
Z,
R (8 0 T = (R 0 8) 0 T Aociativity R 0 (8 T = (R 8) U (R 0 T Ditributivity ovr uio Wak ditributivity ovr itrctio R o (8 n T � (R 0 8) n (R T 8 � T = R8 C RoT Mootoicity
Z
37)
Athough mmi compoitio i widly ud it i ot ily ubjctd to mathmatical aayi. To achiv gratr mathmatical tractability mproduct compoitio h b propod a atrativ to mmi compoitio. Deitio 34 x-prodct composto
Aumig th am otatio ud i th ditio of mmi compoitio w ca d max-odct comositio folow 3.8)
o Th foowig xamp dmotrat how to apply mmi ad mproduct compoitio ad how to itrprt th rutig zzy rlatio Rl 0 R2 . Examle 34 x-m d mx-pdct composto
ec Exension Pinciple and uzzy Relaions
t
R1 = "x i rvat to y R2 "y i rvat to z"
b two zzy ratio dd o X x Y ad Y x Z rpctivy whr X { 2 3} Y = { } ad Z { } Aum that R ad R2 ca b xprd th foowig ratio matric
a
R1
[
01 03 05 07 04 02 08 09 06 08 03 02
]
09 01 02 03 R2 05 06 07 02 ow w wat to d R1 R2, which ca b itrprtd a drivd fuzzy ratio "x i rlvat to z bd o R ad R2. For impicity uppo that w ar oy itrtd i th dgr of rvac btw 2 E X) ad E Z). I w adopt mmi compoitio th
a
(2
a
t
t
t
t
m04 09 02 02 08 05 09 07) m(04 02 05 07) 07 (by mmi compoitio)
O th othr had if w choo mproduct compoitio istad w hav m(04 x 09 02 x 02 08 x 05 09 x 07) m(036 004 040 063) 063 by mproduct compoitio) Figur 33 iutrat th compoitio of two zzy ratio whr th ratio btw mt 2 i X ad mt i Z i buit up via th four posib paths (oid i) coctig th two mt h dgr of rvac btw 2 ad i th mimum of th four path' trgth whi ach pah's trgth is th miimum (or product) of th trgth of it costitut is
a
a
Both th mmi ad mproduct compoitio of two ratio matrics ca b obtaid through th MATLA mxstr m th prviou xamp w ud m to itrprt OR ad * to itrprt AD whr * ca b ithr mi or product h MATLA mxstr.m availabl va F or WWW pag ) ca b ud to comput th max* compositio of two ratio matric gra w ca hav (coorm) (orm) compositio that itrprt OR ad AD uig coorm ad Torm oprator rpctivly hs xtdd maig for zzy OR ad AD ar discud i th xt sctio
4
uz Rules nd uzzy Reasoning
x
y
C
z
Fige Composto of y ltos
3.3
FU ZZY I FH EN RU LES
I thi ctio th ditio ad xampl of liguitic variabl ar giv rt.
Th w xplai two itrprtatio of fuzzy ifth rul ad how to obtai a zzy rlatio that rprt th maig of a giv zzy ru.
3. 3. 1 inguistic Variabes A w poitd out by Zadh [7] covioa tchqu for ytm aalyi ar triically uuitd for daig with humaitic ytm who bhavior i trogly iucd by huma judgmt prcptio ad motio Thi i a maiftatio of what might b cald th icile of icomatibiity "A th complxity of a ytm icr our abiity to mak prci ad yt igicat tatmt about it bhavior dimiih util a thrhold i rachd byod which prciio ad g icac bcom amo mutualy xcluiv characritic 7] w bcau of hi blif hat Zadh propod th cocpt of liguitic varabl 5] a altra tiv approach to modig huma thikig�a approach hat i a approximat mar rv to ummariz iformatio ad xpr it i trm of zzy t tad of crip umbr . W prt th formal ditio of liguiic variabl xt; th xamp that folow wil carify th ditio which may m cryptic at th rt radig Deitio gstc vrles d other relted termology
A ligistic vaiable i charactrizd by a quitupl ( ) X G, M) i whch i th am of th variabl; ) i th tem set of that i th t of it igistic es or ligistic tems; X i th uivr of dicour; G i a sytactic le which grat th trm ); ad M i a sematic le whch ociat with ach iguiic valu A it maig M{A) whr M{A) dot a zzy t i X
i
Ife Rl
Mdde g
10
0
40
Xa
70
9
Fig ypcl membershp ctos of the term set (g) (MATLA le v.m)
The followig example helps cari he precedig deiio. Exam gstc vrbles d lgstc vles I g is ierpreed a iguisic variable he is erm se ( g) could be
(g)
{ yog ot yog very yog ot very yog
mddle ged ot mddle ged old ot old very old more or less old ot very old ot very yog d ot very old }
39
where each erm i (ge) is characerized by a zzy se of a uiverse of discourse X = 0 100] show i Figure 34 Usually we use "age is youg o deoe he sigme of he iguisic value "youg o he liguisic variable g By cor whe g is ierpreed a umerical variable we use he expressio "age = 20 isead o si he umerical lue "20 o he umerical variable g The syacic rule refers o he way he liguisic lues i he erm se (ge) are geeraed. The semaic rule dees he membership fucio of each liguisic value of he erm se; Figure 3.4 displays some of he ypical membership cios.
om he precedig example we ca see ha he erm se cosiss of several i may tms (yog mddle ged old) modied by he gatio "o ador he hdgs (very more or less te extremely, ad so forh ad he lied by coctivs such d or ether, ad ether he sequel we shl rea he coecives he hedges ad he egaio operaors ha chage he meaig of heir operads i a specied coexidepede fhio. Ditio Cocetrto d dlto of lgstc vles
zzy Rles and zzy Reasoning
JA(')'
C
Le be a liguisic value characerized by a fuzzy se wih membership cio The is ierpreed a modied versio of he origial liguisic value expressed . = 3.10
k [JA(X)]k
I paricular he operaio of coctatio is deed
CO
)
3.11
•
3.12
=
while ha of diatio is expressed by DL
Coveioly we ae CO ad DL o be he resuls of applyig he hedges very ad more or less, respecively o he liguisic erm . owever oher cosise deiios for hese liguisic hedges are possible ad well jusied for various applicaios. Followig he deiios i he previous chaper we ca ierpre he egaio operaor OT ad he coecives AD ad O OT
[1[A(JA(XX))]/XJ(, X)]/X, [JA(X) V J(X)]/X,
AD B = B =
3.13
O B B =
JA(') J()
respecively where ad B are wo liguisic ues whose meaigs are deed by ad Through he use of CO ad DL . for he liguisic hedges very ad more or less, ogeher wih he ierpreaios of egaio ad he coecives AD ad O i Equaio 3 13 we are ow able o cosruc he meaig of a comosit igistic tm, such "o very youg ad o very old ad "youg bu o oo youg. Exam Costrctg Fs for composte lgstc terms
Le he meaigs of he liguisic erms yog ad old be deed by he followig membership cios 1 314 bell 20 2 0 = youg 1 + 2x0 4 1 3.15 bell 30 3 100 00 l 6 ' old 1+ where is he age of a give perso wih he ierl 0 100] he uiverse of discourse The we ca cosruc MFs for he followig composie liguisic erms
ec zzy fTen Rles
•
( / X
mre r less ld = DL
=
• = • = [ + � 1 - � r] / =
t yg d t ld
=
yg n
X
=
yg bt t t yg
•
yg n yg
extremely ld
=
CO N(C ON (C ON (old
=
({ ol c ) 2 ) 2
=
.
l [1 +
] 8/
x.
We ume ha he meig of he hedge t i he ame ha of very ad he meig of extremely i he ame ha of very very very Figure 3.5a how he MF for he primary iguiic erm yg ad ld Figure 3.5b how he MF for he compoie iguiic erm mre r less ld, t yg d t ld, yg bt t t yg, ad extremely ld
Aoher operaio ha reduce he fuzzie of a fuzzy e A i deed folow. Ditio Ctrst tesct
The operaio of cotast itsicatio o a liguiic ue i deed by
.
for 0 < (X ) < 0.5 for 0.5 < (X ) <
(3 6)
The cor ieier T icree he vue of (X ) which are above 0.5 ad dimiihe hoe whi are beow hi poi. Thu cor ieicaio h he eec of reducig he fuzzie of iguiic vaue The ivere operaor of cor ieier i cor dimiiher D which i expored i Exercie 9 Exam Ctrst teser
e be deed by
(X ) = riagle 1 3 9 which i a riagular MF wih he verex a 3 ad he be ocaed a = o = 9. Figure 36 ilurae he reul of applyig he cor ieier T
=
zzy R d Fzy Reg
C
(a) Pma Lg Vale
�� EC 4
0.8 0.6
: 0.2
��
o
10
5
70
�
9
1
X = age (b) oe Lg Vale
0.8
EC 4 :
0.6
Yong b No T Yo
0.2
X = age
Fig Fs for prmry d composte lgstc vles Exmple
3 6
(MATLA le comv m)
to severa times the solid lie is ; the dotted lie is T ; the dhed lie is T = T T ; ad the dhdot ie is T = T T T . Thus repeated applicatios of T reduces the fuzziess of a zzy set; i the extreme ce the zzy set becomes a crisp set with boudaries at the crossover poits.
Whe we dee MFs of liguistic values i a term set it is ituitively reo abe to have these MFs roughy satisfy the requiremet of orthogoality which is described ext. Ditio rthogolty
A term set = t , , t of a liguistic variable o the uiverse X is othogoa if it fuls the folowig property
L
= 1 \ E X
317
where the t are covex ad ormal zzy sets deed o X ad these fuzzy ses me up the term set
S Fuzzy Ife R ff of on Inenier
Fig xmple
3 7:
the eects of the cotrst teser
inensif m)
(MATLA le
For he MFs i a erm se o be iuiivey reoabe he orhogoaiy re quireme h o be folowed o some exe This is show i Figure 2.2 where he erm se coais hree iguisic erms "youg "middle aged ad "od A ideph exposiio of liguisic variables ad heir applicaios ca be foud i 8] . ex we sh discuss he use of liguisic variabes ad iguisic values i fuzzy ifhe rules 3. 3. 2 Fuzzy If-hen Rules A fzzy ifth aso ow fzzy , fzzy imicatio, or zzy coditioa statmt ) sumes he form
if x is he y is B,
3.18)
where ad B are iguisic vaues deed by fuzzy ses o uiverses of discourse X ad Y respecively Oe "x is is called he atcdt or mis, while "y is B is caed he cosqc or cocsio. Exampes of zzy ifhe rules are widespread i our daiy liguisic expressios such he followig
• • • •
I pressure is high he voume is small I he road is sippery he drivig is dagerous I a omao is red he i is ripe I he speed is high he apply he bre a lie
Before we ca empoy zzy ifhe rules o model ad aalyze a sysem rs we have o formaize wha is mea by he expressio "if x is he y is B, which is someimes abbreviaed B. essece he expressio describes a reaio
Fuzzy Rues nd Fzzy Reanng
C
bewee wo variables x ad Y his suggess ha a fuzzy ifhe rule be deed a biary zzy relaio R o he produc space X Y. Geerally speaig here are wo ways o ierpre he fuzzy rule B. I we ierpre B cod with B he
/ JA (X) JB ( Y)/(X, y) where is a Torm operaor ad B is used agai o represe he zzy relaio R O he oher had if B is ierpreed tais B he i ca R
=
A
-
B
=
A
B
x
*
=
be wrie four diere formul
• Maerial implicaio
R
=
B
B.
3.19
( B.
3.20
( B B.
3.21
=
• roposiioal calculus R
=
B
=
• Exeded proposiioal calculus R
=
B
=
• Geeralizaio of modus poes JR (X , y)
=
sup{c I JA (X) *
C
< JB ( Y ) and 0 < c < I} ,
3.22
B ad is a Torm operaor Alhough hese four formul are diere i appearace hey all reduce o he familiar ideiy B = B whe ad B are proposiios i he sese where R =
of wvalued logic Figure 3.7 illusraes hese wo ierpreaios of a zzy rule Bed o hese wo ierpreaios ad rious Torm ad Tcoorm operaors a umber of qualied mehods ca be formulaed o calculae he fuzzy relaio R = B. oe ha R ca be viewed a zzy se wih a wdimesioal MF
(X , y) wih
=
! ( (X) (Y))
=
! (, )
(X ) , = (Y ) , where he cio ! called he zzy imicatio fctio, performs he of rasformig he membership gades of x i ad Y i B io hose of x y) i B. Suppose ha we adop he rs ierpreaio coupled wih B" he mea ig of B. The four diere zzy relaios B resul om employig four of he mos commoly used Torm operaors =
S Fzzy Ifen Res
A
x
Fig wo terettos of zzy mplcto copled wth B b etls B
• • • •
R m =
B = J (X) J (y)/(X , y), or (, b) = b This relaio
which w proposed by Mamdi [3] resuls from usig he mi operaor for cojucio . Rp
=
B = J (X) J (y)/( X y) or (, b) = b roposed by
arse [2] his relaio is bed o usig he algebraic produc operaor for cojucio. R bp = B = J (X) J (y)/(X y) = 0 V (J (X) + J (y) )/(x, y) , or (, b) = 0 V ( + b 1) This formula employs he bouded produc operaor for cojucio. Rdp = B = J(X ) J (y)/(X , y), or
8
if b = if = 1 oherwise This formua uses he dric produc operaor for cojucio. The rs row of Figure 3.8 shows hese four zzy implicaio fucios [wih = J (X) ad b = J ( Y )] he secod row shows he correspodig zzy relaios R m , Rp , R bp , ad Rdp whe J (X) = be; 4 3 10) ad J (y) = bel; 4 3 10) Whe we adop he secod ierpreaio eails B, he meaig of B, agai here are a umber of fuzzy implicaio fucios ha are reoable cadidaes. The folowig four have bee proposed i he lieraure
�
•
R a = UB = 1 ( 1 J(x) + J (y))/(x,y), or fa (, b) = 1(1 + b)
•
Rmm
This is Zadeh's arihmeic rue which follows Equaio (3. 19) by usig he bouded sum operaor for U =
U(B) = ( J(x)) V (J (x) J (y))/(x, y) , or (, b)
=
(1 ) V ( b) This is Zadeh's mmi rule which follows Equaio (3.20) by usig mi for ad m for U
Fzy Res nd uzzy Reag
n
P
2
2
v
(d)
C
s P
Fig Fist ow: zzy impication nctions base on the inteetation
coupe with B "; secon ow: the coesponing zzy eations.
(MATLA e
fui )
• = B = ( (X)) V ( X), or ! (, b) (1 ) V b This is Booea zzy impicaio usig m for U
• = ( (X) (y)) /(x, y), where
-
< b =
b/
if < b if > b.
This is Gogue's fuzzy impicaio which folows Equaio (322) by usig he algebraic produc for he Torm operaor. Figure 3.9 shows hese four fuzzy implicaio cios wih = ( X) ad b ( Y )] ad he resuig fuzzy relaios , , , ad whe (X) = bell ; 4 3, 10) ad (y) = bel ; 4 3, 10). shoud be ep i mid ha he zzy implicaio fucios iroduced here are by o meas exhausive. eresed readers ca d oher feibe fuzzy impli caio cios i 1]
34
FUZZY REASON IN G
zzy reoig also ow approximae reoig is a iferece procedure ha derives coclusios om a se of fuzzy ifhe rules ad ow facs. Before iroducig zzy reoig we shal discuss he composiioal rue of iferece which plays a ey role i zzy reoig.
2
S Fzzy Reag
s
A Rul
e
s M Rul
Bl
F Im
s
F I
v
X_
Fig First w zzy impitio tios bsed o the iterettio etis B "; seod row: the orrespodig zzy retios.
M le
fui )
3.4. 1 Compositional Rule of Inference The cocept behid the compositioal rule of iferece proposed by Zadeh 7 should ot be totly ew to the reader; we employed the same idea to explai the mmi compositio of relatio matrices i Sectio 322 Moreover the extesio priciple i Sectio 32 1 is actually a special ce of the compositioal rule of iferece The compositioal rule of iferece is a geeralizatio of the followig famili otio Suppose that we have a curve = x) that regulates the relatio betwee x ad Whe we are give x = , the om = x) we ca ifer that y b () see Figure 310 a) A geeralizatio of the aforemetioed process would allow to be a iterval ad (x) to be a itervalvalued fuctio show i Figure 3 10b) To d the resultig iterval y = b correspodig to the iterval x = , we rst costruct a cylidrical extesio of ad the d its itersectio with the itervalalued curve The projectio of oto the is yields the iteval = b. Goig oe step further i our geeralizatio we sume that F is a fuzzy relatio o X Y ad is a fuzzy set of X show i Figures 311a) ad 311b) To d the resultig fuzzy set B, agai we costruct a cylidrical extesio () with be . The itersectio of () ad Figure 311c)] forms the aalog of the regio of itersectio i Figure 310b) By projectig () n oto the is we ifer y a fuzzy set B o the is show i Figure 311d) Specically let J, Jc , JB , ad JF be the MFs of , (), B, ad respectively where Jc is related to J through
Jc x, y)
=
J (X.
Fy Rl and Fy Reang
C
b
X=B
x
Fig Dervto of = b om =
pots = f(x) s rve b to.
The
a
B
x
a d = f(x)
a
d b re d b re tervs = f(x) s tervved
mi ] mi ] . By projecig () n F oo he is we have
=
m mi ] ]. This formula reduces o he mmi composiio see Deiio 3.3 i Secio 3.2) of wo relaio marices if boh A a uary zzy relaio ad F a biary zzy relaio have ie uiverses of discourse. Coveioally B is represeed B = A o F,
where deoes he composiio operaor. is ieresig o oe ha he exesio priciple iroduced i Secio 3.2.1 of he previous chaper is i fac a special ce of he composiioal rule of iferece. Specically if = f(x) i Figure 3.10 is a commo crisp oeoe or mayoe cio he he derivaio of he iduced zzy se B o Y is exacly wha is accomplished by he exesio priciple. Usig he composiioal rule of iferece we ca formalize a iferece proce dure upo a se of zzy ifhe rules. This iferece procedure geerally called approximae reoig or zzy reoig is he opic of he ex subsecio. 3. 4. 2 Fuzzy Reasoning The bic rule of iferece i radiioal wvalued logic is mods os, ac cordig o which we ca ifer he ruh of a proposiio B from he ruh of A ad he implicaio A B. For isace if A is ideied wih "he omao is red ad B wih "he omao is ripe he if i is rue ha "he omao is red i is also rue ha "he omao is ripe This cocep is illusraed follows
65
S o
(b)
a) Fy Reatio F o X ad Y
Cyd Eeso of
X
()
Mmm of a) ad
(b)
d) Pecto of
()
oto Y
Fig Compositio e of iferee (MATLA l cri . m
prmi 1 fac x i , prmi 2 rul if x i h y i B, coquc cocluio y i B.
owvr, i much of huma roig, modu po i mployd i a approx ima mar For xampl, if w hav h am implicaio rul "if h omao i rd, h i i rip ad w ow ha "h omao i mor or l rd, h w may ifr ha "h omao i mor or l rip Thi i wri prmi 1 fac x i ', prmi 2 rul if x i h y i B, coquc cocluio y i B',
whr ' i clo o ad B' i clo o B Wh , B, ', ad B' ar zzy of appropria uivr, h forgoig ifrc procdur i calld aoximat asoig or fzzy asoig; i i alo calld galizd mods os (GMP for hor , ic i h modu po a pcial c Uig h compoiio rul of ifrc iroducd i h prviou ubcio, w ca formula h ifrc procdur of zzy roig h followig diio
Ditio pproximte resoig zzy resoig
, ', ad B b fuzzy of X, X , ad Y, rpcivly Aum ha h zzy implicaio B i xprd a fuzzy rlaio o X x Y Th h zzy
66
uz Rues ad uz Reasog
C
B ducd by x ' d h zzy ru "f x h y B dd by JB' (y)
or, quvay,
maxz min[A' ( x) , JR (X , y)] [ x) (X ,y),
B' = ' = ' (
B).
(323) (324)
o
ow w c u h frc procdur of zzy og o drv cocuo, provdd ha h zzy mpcao B dd appropra bay fuzzy rao wha foow, w ha dcu h compuaoa pc of h zzy rog roducd h prcdg do, d h xd h dcuo o uao whch mup zzy ru wh mup cd ar vovd dcrbg a ym' bhavor owvr, w w rrc our codrao o Mamd' zzy mpcao co d h cca maxm compoo, bcau of hr wd appcaby d y aphc rprao
Si ngle Rule with Si ngle Antecedent
Th h mp c, d h formua avaab Equao (323) A rhr mpcao of h quao yd (y) [V ( ( X) (X) (y) W " (Y) ohr word, r w d h d of mach w h mmum of (X ) x ) h hadd aa h cd par of Fgur 312) h h MF of h rug B' qua o h MF of B cppd by w , how h hadd aa h coqu par of Fgur 312 uvy, w rpr a mur of dg of bf for h cd pa of a ru; h mur g propagad by h fh ru d h rug d of bf or MF for h coqu par (B' Fgur 312) houd b o gar h w . Single Rule with Multiple Antecedents
A zzy fh ru wh wo cd uuay wr � "f x ad y B h z C Th corrpodg probm for GMP xprd prm 1 fac) prm 2 (ru) coquc cocuo)
x ' d y B', f x d y B h z C, z C'
Th zzy ru prm 2 c b pu o h mpr fom x B C uvy, h fuzzy ru ca b rformd o a ray zzy rao
67
ec uzzy Reag min A
B
A'
B'
& - -
x
y
Fig Grphi iterettio of GP sig mdi s zzy mpitio d the mxmi ompositio.
bd o Mamdi ' fuzzy impicaio cio, foow (,B,C)
=
( B) C =
(X) (Y ) (z)(x,y,z)
Th ruig C' i xprd C'
=
( ' B') ( B
C)
Thu
(3 25)
W (Wl " W2 ) (z),
�
rig rgh
whr Wl ad W2 ar h maxima of h MF of n ' ad B n B', rpcivy gra, Wl do h dgs of comatibility bw ad ' imiary for W2 . Sic h acd par of h zzy ru i corucd by h cocv "ad, Wl " W2 i cad h ig stgth or dg of flllmt of h zzy ru, which rpr h dgr o which h cd par of h ru i aid. A graphic irpraio i how i Figur 313, whr h MF of h ruig C' i qua o h MF of C cippd by h rig rgh w , W = Wl W2 . Th graizaio o mor ha wo cd i raighforward. A araiv way of cacuaig C' i xpaid i h foowig horm hom Deompositio method for tig B'
C' (' B') ( B C) [' ( C)] n [B' (B =
C)
(3 26)
68
uzzy Rus ad uzzy Reasoig
C
min A
A'
B
B'
c
\ ----- ---WW W - ---
---------- -
x
c'
z
y
Fige pproximte resoig for mtipe teedets Poof J (z)
VZ [J (X) J (y) [ (X ) J (Y ) J (z) V { J (X ) J (Y ) J (z) V [J ( ) J (Y ) J (z)]} J (X) J (Y ) J (z) J ( ) } J (Y ) J (Y ) ·
(3.27) o
Th prcdig horm a ha h ruig coquc ' c b xprd h ircio of = ' ( ) d � = B' (B ), ach of which corrpod o h ifrrd zzy of a GMP probm for a ig zzy ru wih a ig cd
Mu ltiple Rules with Multiple Antecedents
Th irpraio of muip ru i uuay a h uio of h zzy r aio corrpodig o h zzy ru Thrfor, for a GMP probm wri x i ' d Y i B', prmi 1 fac) if x i d Y i B h z i , prmi 2 ru 1) if x i d Y i B h z i , prmi 3 ru 2) coquc cocuio) Z i ', w ca mpoy h zzy roig how i Figur 3.14 ifrc procdur o driv h ruig oupu zzy '. To vri hi ifrc procdur, R = X B ad R = X B • Sic h maxmi compoiio opraor i diribuiv ovr h opraor, i foow ha
'
U U
(' x B') (R R ) [(' x B') R] [(' = � ,
x
B') R ]
U
(3.28)
whr ad � ar h ifrrd zzy for ru 1 d 2 , rpcivy Fig ur 3.14 how gaphicay h opraio of zzy roig for muip ru wih muip cd
ec Fuzzy eaning min J
J
A 1 A'
_ _ _ _ . _ _ _ _ _ _ _ _ .
�_ _ _ J _ _ _ _ _ _ _ _
X J
y J
J
B 1 B'
C z
J
B' B2
x
;
y
z
J
z
Fig Fzzy resoig for mtipe es with mtipe teedets
Wh a giv fuzzy rul um h form "if x i or y i B h z i C, h rig rgh i giv h maximum of dgr of mach o h acd par for a giv codiio. Thi zzy rul i quival o h uio of h wo fuzzy rul "if x i h z i C ad "if y i B h z i C ummary, h proc of zzy roig or approxima roig ca b dividd io four p Dgs of comatibility Compar h ow fac wih h acd of fuzzy
rul o d h dgr of compaibiliy wih rpc o ach acd MF.
Fiig stgth Combi dgr of compaibiliy wih rpc o acd MF
i a rul uig fuzzy AD or OR opraor o form a rig rgh ha idica h dgr o which h acd par of h rul i aid.
Qalid idcd cosqt MFs Apply h rig rgh o h co
qu MF of a rul o gra a qualid coqu MF. Th qualid coqu MF rpr how h rig rgh g propagad ad ud i a zzy implicaio am)
Ovall ott MF Aggrga all h qualid coqu MF o obai a
ovrall oupu MF.
Th four p ar alo mployd i a fuzzy ifrc ym, which i iro ducd i Chapr 4.
70
3.5
u Rues ad u Reasoig
Ch
SUMMARY
Thi chapr iroduc h xio pricipl, zzy rlaio, zzy ifh rul, d zzy roig. Th xio pricipl provid a procdur for mappig bw fuzzy . zzy rlaio d hir compoiio rul ipula zzy ad hir combiaio d irpraio i a mulidimioal pac. By irprig zzy ifh rul fuzzy rlaio, variou chm of fuzzy roig bd o h cocp of h compoiioal rul of ifrc) ar commoly ud o driv cocluio fom a of fuzzy ifh rul. zzy ifh rul d zzy roig ar h bacbo of zzy ifrc ym, which ar h mo impor modlig ool bd o zzy hory. dph dicuio abou fuzzy ifrc ym i providd i h x chapr; hir adapiv vrio d corrpodig applicaio ar ivigad i Chap r 12 d 19.
EXERCISES
Apply h xio pricipl o driv h MF of h zzy B i Ex pl 32 Figur 3.2). Prov h idii i Equaio 3.7) for maxmi compoiio. Do h idii i Equaio 3.7) hold for ohr yp of compoiio? Carry ou h calculaio of Rl R2 i Expl 3.4, uig boh maxmi ad maxproduc compoiio. Doubl chc your rul uig h M l st .
5 pa Expl 3.6 wih h w mig of od d dd by h
followig rigular MF
Jyoung x)
=
gaussi x, 0, 20)
=
( "- )2
e- ( t )2 ,
old ) = gaui, 100, 30) = e Plo h MF for h liguiic valu i Exampl 3.6 uig M.
6 U h MF of od d sm i Exrci 5 o gra h MF for h followig
oprimary rm a) ot ver yog d ot ver od b) ver yog or ver od Plo h MF for h wo liguiic valu uig M.
7 crig h magiud abolu valu) of h paramr of a bll MF h
a c imilar o ha of h corac iir i Equaio 3.16)ha i,
71
EFEENCE
i icr h mmbrhip grad abov 05 ad dimiih ho blow 05 Explai why 8 Suppo ha h MF for fuzzy ad ar a rapzoidal MF rapzoid; , b c )
d a woidd MF , b, c, ) uaio 268)] , rpcivly Show ha T) =
Fid a opraor cotast dimiish DM ha i h ivr of cor iir T amly, for ay fuzzy , DM hould aify h followig DMT» = pa h plo i Figur 36 bu u h cor dimiihr iad 10 Vri ha Equaio 319) o 322) rduc o h famili idiy
u
wh ad ar propoiio i h of wovalu logic
11 pa h plo of h zzy rlaio i h cod row of Figur 38 ad 39,
umig ad ar dd by a) (X) = riagl, 5, 10, 1 5) ad (Y ) = riagl, 5, 10, 15) b) (X ) = rapzoid, 3, 8, 12, 17) ad (Y) = rapzoid, 3, 8, 12, 17)
Aohr fuzzy implicaio fucio bd o h irpraio "A ail B ca b xprd or, alraivly,
J X Y sgn[JB (Y) - JA (X ) ]/(x , y) ,
( b) = g (b ) =
{I
if b > , 0 hr ,
whr = (X ) ad b = (Y ) Plo = (, b) ad , ) = g [ (Y) (X ) ], umig h MF for ad (X) = bll, 4, 3, 10) d ( Y ) = bll, 4, 3, 10) , rpcivly U h idii i h prviou xrci o how ha h fuzzy rul "if i or Y i h i i quival o h uio of h wo fuzzy rul "if i h i udr maxmi compoiio ad "if Y i h i
e" e"
e"
REFERENCES
1 S kami M Mizumoto and K. Tanaka Some considerations on uzzy conditiona inerence. zzy ts and ystms 23-273, 1980
72
uzzy Rus ad uzzy Reasog
C
[2] P. M Larsen Industrial application o zzy logic control. Intational Joual of ManMachin tudis 12(1):3-10, 1980. [3] E. H Mmdani and S. Assilian. An experiment in linguistic synthesis with a uzzy logic controller. Intational Joual of Man-Machin tudis 7(1)1-13, 1975 [4] L A Zdeh zzy sets Infoation and Control 8338-353, 1965. [5] L A. Zdeh Quatitative zzy semantics Information cincs 3:159176, 1971 [6] L. A Zdeh. Similarity relations nd zzy ordering Infoation cincs, 3177206, 1971 [7] L. A. Zdeh Outlie o a new approh to the analysis o complex systems ad decision processes IEEE ansactions on ystms Man and Cybtics 3(1):284, Januy 1973 [8] L. A Zdeh The concept o a linguistic viable and its application to approximae reoning Parts 1, 2 nd 3. Infoation cinces 8:199249, 8301357, 9:4380, 195
Chapter Fuzzy Inference Systems
J
4.1
s
R. Jang
INTRODUCTION
Th fzzy ifc systm i a popuar compuig framwor bd o h cocp of zzy hory, fuzzy ifh ru, ad zzy roig. h foud uccfu appicaio i a wid variy of d, uch auomaic coro, daa cicaio, dciio aayi, xpr ym, im ri prdicio, roboic, ad par rcogiio Bcau of i muidicipiary aur, h zzy ifrc ym i ow by umrou ohr am, uch fzzy-l-basd systm, fzzy xt systm [2], fzzy modl [10, 9], fzzy associativ mmoy [], zzy logic cotoll [6, 4, 5], ad impy d ambiguouy zzy systm.
Th bic rucur of a zzy ifrc ym coi of hr cocpua
compo a l bas, which coai a cio of zzy ru; a databas or dictioay , which d h mmbrhip fucio ud i h zzy ru; ad a asoig mchaism, which prform h ifrc procdur uuay h zzy roig iroducd i Scio .42) upo h ru ad giv fac o driv a roab oupu o cocuio. o ha h bic zzy ifrc ym ca a ihr fuzzy ipu or crip ipu which ar viwd zzy igo , bu h oupu i produc ar a mo away zzy Somim i i cary o hav a crip oupu, pciay i a iuaio whr a fuzzy ifrc ym i ud a coror. Thrfor, w d a mhod of dfzzicatio o xrac a crip vau ha b rpr a fuzzy . A fuzzy ifrc ym wih a crip oupu i how i Figur 4.1, whr h dhd i idica a bic fuzzy ifrc ym wih zzy oupu ad h dfuzzicaio boc rv h purpo of raformig a oupu zzy io a crip ig vau. A xamp of a fuzzy ifrc ym wihou d zzicaio boc i h woru woipu ym of Figur .14. Th cio of h dfuzzicaio boc i xpaid i Scio 4.2. Wih crip ipu ad oupu, a fuzzy ifrc ym impm a oiar mappig from i ipu pac o oupu pac. Th mappig i accompihd by a umbr of fuzzy ifh ru, ach of which dcrib h oca bhavior of h
7
7
uzzy Inference ystms
C
ul • • •
•• •
ul
• • •
-B !fulfl Y � I
1 I
Fg 1 ok digrm for zzy iferee system.
mappig. I paricuar, h cd of a ru d a zzy rgio i h ipu pac, whi h coqu pci h oupu i h zzy rgio. wha foow, w ha r iroduc hr yp of zzy ifrc ym ha hav b widy mpoyd i variou appicaio. Th dirc bw h hr zzy ifrc ym i i h coqu of hir zzy ru, ad hu hir agggaio d dfuzzicaio procdur dir accordigy. Th w wi iroduc d compar hr dir way of pariioig h ipu pac; h pariioig mhod ca b adopd by y fuzzy ifrc ym, rgard of h rucur of h coqu of i ru. Fiay, w wi addr briy h faur d h probm of fuzzy modig, which i cocrd wih h corucio of zzy ifrc ym for modig a giv arg ym. 4.2
MAM DAN I FUZZY MOD ELS
Th Mamda fzzy fc systm [6 w propod h r amp o coro a am gi d boir combiaio by a of iguiic coro ru obaid fom xpricd huma opraor. Figur 4.2 i a iuraio of how a woru Mdi fuzzy ifrc ym driv h ovra oupu z wh ubjcd o wo crip ipu x d y . w adop max d agbraic produc our choic for h Torm ad Tcoorm opraor, rpcivy, d u maxproduc compoiio iad of h origia maxmi compoiio, h h ruig zzy roig i how i Fig ur 4., whr h ifrd oupu of ach ru i a zzy cad dow by rig rgh via agbraic produc. Ahough hi yp of zzy roig w o mpoyd i Mamdai' origia papr, i h o b ud i h iraur. Ohr variaio ar poib if w u dir Torm ad Tcoorm opraor. I Mamdai' appicaio [6], wo zzy ifrc ym wr ud wo coror o gra h ha ipu o h boir d hro opig of h gi cyidr, rpcivy, o rgua h am prur i h boir ad h
75
S amdai Fzzy ods mn
8,
C
x
x
C
Z
y
C' Z COA
Fig 2 he mdi zzy iferee system sig mi d m for -o d -oorm opertors respetivey
pd of th gi Sic th pat ta oy crip vau a iput w hav to u a dfuzzir to covrt a fuzzy t to a crip u. Dfzzicatio
Dfuzzicatio rfr to th way a crip vau i xtractd om a fuzzy t a a rprtativ vau. gra thr ar v mthod for dzziig a zzy t of a uivr of dicour Z a how i Figur 4.4 r th zzy t i uuay rprtd by a aggrgatd output MF uch a C' i Figur 4 2 ad 43 ) A brif xpaatio of h dzzicatio tratgy foow.
• Ctroid of ara COA (4.1) whr i th aggrgatd output MF Thi i th mot widy adoptd dzzicatio tratgy which i rmiict of th cacuatio of xpctd vau of probabiity ditributio
76
Fy Ifrc Systms
Ch
produc
C
8,
X
C C
z
X x
y
z COA
Fie he md zzy feree system sg pouc d max for orm d -oorm opertors respetvey
• Bictor of ara zBOA :
zBOA satisfes
BOA =
4.2
BOA
j
whr = mi{ } ad = max{ }. That i th vrtica i = BOA partitio th rgio btw = = y = 0 ad Y = ito two rgio with th am ara .
j
• Ma of mimum MOM MOM i th avrag of th maximizig at which th MF rach a maximum . I ymbo
Jz' z Jz' whr { }. I particuar h a ig mimum at th . Morovr if rach it mimum whvr MOM =
=
(4 . 3)
=
= MOM = right] thi i th c i Figur 4 4) th MOM =
+
right /2. Th ma of mimum i th dzzicatio tratgy mpoyd i Mamdai zzy ogic cotror 6].
77
Sc amda Fzzy ods
Smales of Max. Largest of Max
Centroid of Area Bisecter of Area Mean of Max.
Figure rios dezzitio shemes for obtiig risp otpt
• Smat of mimum SOM SOM i th miimum i trm of magitude of th mimizig .
• argt of mimum OM OM i th mimum i trm of magitud
of th mimizig . Bcau of thir obviou bi SOM ad OM ar ot ud oft th othr thr dfuzzicatio mthod.
Th cacuatio dd to carry out ay of th v dzzicatio opratio i timcoumig u pcia hardwar upport i avaiab. rthrmor th dfuzzicatio opratio ar ot iy ubjct to rigorou mathmatica aayi o mot of th tudi ar bd o xprimta rut. Thi ad to th propoi tio of othr typ of zzy ifrc ytm that do ot d dzzicatio at a; two of thm ar itroducd i th xt ctio. Othr mor xib dfuzzicatio mthod ca b foud i [7, 8 2] Th foowig two xamp ar igiput ad twoiput Mamdai zzy mod . Exle 1 Sige-ipt ige-otpt mdi zzy mode
A xamp of a igiput igoutput Mamdai fuzzy mod with thr ru c b xprd
{
X i ma th i ma. X i mdium th i mdium. X i arg th i arg.
Figur 4.5a pot th mmbrhip fuctio of iput X ad output whr th iput ad output uivr ar [0, 0] ad [0, 0, rpctivy. With max mi compoitio ad ctroid dfuzzicatio w ca d th ovra iputoutput curv how i Figur 4.5b . ot that th output variab vr rach th mimum (0) ad miimum (0) of th output uivr. tad th rachab miimum ad maximum of th output variab ar dtrmid by th ctroid of th mot ad rightmot coqut MF, rpctivy.
7
zzy Ifrc Sems
Ch
I16
I 4 7 X 2 4 6 I 6 4 3 OI6 2 4 >
I
1
5
-
X
b
Fie 5 Sige-ipt sige-o tpt mdi zzy mode i Exmpe . teedet d oseqet Fs b over iptotpt e.
(MATLA l
. ) o
Exle 2 wo-ipt sige-otpt mdi zzy mode
A xamp of a twoiput igoutput Mamdai zzy mod with four rul ca b xprd a X i ma ad Y i ma th Z i gativ larg. X i ma ad Y i arg th Z i gativ mal. X i arg ad Y i mal th Z i poitiv mal. X i arg ad Y i arg th Z i poitiv larg. Figur 4.6a pot th mmbrhip ctio of iput X ad Y ad output Z al with th am uivr [5, 5 With maxmi compoitio ad ctroid dzzi catio, w ca d th ovra iputoutput urfac a how i Figur 4.6b. For a mutipiput zzy mod omtim it i hlpl to hav a too for viwig th proc of zzy ifrc; Figur 4.7 i th fuzzy ifrc viwr avaiab i th zzy ogic Toolbox whr you ca chag th iput vau by cic ad drag th iput vrtica i ad th th itractiv chag of quaid coqut MF ad ovral output MF. o
S amda Fzzy os
y
x
b
Figure 6 wo-ipt sige-otpt mdi zzy mode i Exmpe teedet d oseqet Fs b over ipt-otpt srfe.
(MATLA l:
mm . m)
4 2 1 Other Variants Figur 4.2 ad 4.3 coform to th zzy roig dd prviouly. owvr i coidratio of computatio cicy or mathmatical tractability a zzy i frc ytem i practic may hav a crtai roig mchaim that do ot follow th trict ditio of th compoitioal rul of ifrc. For itac o might u product for computig rig trgth for rul with AD d a tcdt) mi for computig qualid coqut MF ad max for aggrgatig thm ito a ovrall output MF. Thrfor to compltly pci th opratio of a Mamdai zzy ifrc ytm w d to ig a fuctio for ach of th followig oprator
•
AND oerator uually Torm) for calculatig th rig trgth of a rul
•
OR oerator uually Tcoorm) for calculatig th rig trgth of a rul
with ADd atcdt. with Od atcdt.
•
Imlication oerator uually Torm) for calculatig qualid coqut
•
Aggregate oerator uually Tcoorm) for aggrgatig quid cos
•
Defuzzication oerator for traformig a output MF to a crip igl
MF bd o giv rig trgth.
qut MF to grate a ovrall output MF. output valu.
0
Fzzy Ifrc Systms
Ch
Figure 7 Fzzy iferee viewer i the zzy ogi oobox his is obtied by typig fuy 2 withi MATLA
O uch xamp i to u product for th impicatio oprator ad poit wi ummatio um for th agggat oprator ot that um i ot v a Tcoorm oprator A advatag of thi sumroduct comosition [3 i that th a crip output via ctroid dfuzzicatio i qua to th wightd avrag of th ctroid of coqut MF whr th wightig factor for ach ru i qua to it rig trgh mutipid by th ara of th coqut MF Thi i xprd a th foowig thorm eorem 1 Compttio shortt for mdi zzy iferee systems
Udr umproduct compoitio th output of a Mamdai zzy ifrc ytm with ctroid dzzicatio i qua to th wightd avrag of th ctroid of coqut MF whr ach of th wightig factor i qua to th product of a rig trgh ad th coqut MF ara Proof W ha prov thi thorm for a zzy ifrc ym with two ru
Figur 43). By uig product ad um for impicatio ad aggrgat oprator rpcivy w hav
81
Sc Sgo Fzzy ods in r Prduct
A
j Av_
Figure 8 he Sgeo zzy mode.
ot tha th prcdig MF coud hav vau gratr tha at crtai poit.) Th crip output udr ctroid dfuzzicatio i z
Jz c' (z)zdz c' (z)dz
W J C (z)zdz + W2 J C2 (z)zdz W J Cl (z)dz + W2 J C2 (z)dz + W a + W 2 a2
z
. zC(z)zdz ) ar th ara ad ctrd of th (z )dz
whr ai = J Ci (z )dz) ad Zi = coqut MF C (z), rpctivly.
o
By uig thi thorm computatio i mor cit if w ca obtai th ara ad ctroid of ach coqut MF i advac 4.3
SUG ENO FUZZY MOD ELS
Th Sugeno fuzzy model ao ow th SK fuzzy model) w propod by Taagi Sugo ad Kag [0, 9 i a ort to dvop a ytmatic approach to gratig fuzzy ru from a giv iputoutput data t. typica fuzzy ru i a Sugo zzy mod h th form if i ad y i B th z = y), whr ad B ar fuzzy t i th atcdt whi Z = y) i a crip fuctio i th coqut Uuly y) i a poyomia i th iput variabl ad y,
Fzzy Ifrc Systms
Ch
but t ca b ay cto og t ca appropraty dcrb th output of th mod wth th zzy rgo pcd by th atcdt of th ru Wh f(x, y) a rordr poyoma th rutg zzy frc ytm cad a rst-order Sugeno fuzzy model, whch w orgay propod [0, 9]. Wh a cotat w th hav a zerorder Sugeno fuzzy model, whch ca b vwd thr a pca c of th Mamda fuzzy frc ytm whch ach ru coqut pcd by a zzy gto or a prdfuzzd coqut) or a pca c of th Tuoto zzy mod to b troducd xt) whch ach ru coqut pcd by a MF of a tp cto ctr at th cotat. Morovr a zroordr Sugo fuzzy mod ctoay quvat to a rada b cto twor udr cra mor cotrat [, w b dtad Chaptr 12. Th output of a zroordr Sugo mod a mooth cto of t put varab og th ghborg MF th atcdt hav ough ovrap. I othr word th ovrap of MF th coqut of a Mamda mod do ot hav a dcv ct o th mooth; t th ovrap of th atcdt MF that dtrm th mooth of th rutg putoutput bhavor. Fgur 4.8 how th zzy rog procdur for a rtordr Sugo fuzzy mod. Sc ach ru h a crp output th ovra output obtad va weigted average, thu avodg th tmcoumg proc of dfuzzcato rqurd a Mamda mod. I practc th wghtd avrag oprator om tm rpacd wth th weigted sum oprator that Z = WI ZI W2 Z2 Fgur 4.8) to rduc computato rthr pcay th trag of a fuzzy frc ytm. owvr th mpcato coud ad to th o of MF gutc mag u th um of rg trgh that i Wi co to uty. Sc th oy zzy part of a Sugo mod t atcdt t y to dmotrat th dtcto btw a t of zzy ru ad ozzy o.
+
L
Examle zzy d ozzy e set ompso
{
A xamp of a gput Sugo zzy mod ca b xprd
+
X ma th = .X 6.4. f X mdum th = 0.5X + 4. X arg th = X 2.
"ma "mdum ad "arg ar ozzy t wth mmbrhp cto how Fgur 4.9a) th th ovra putoutput curv pcw ar how Fgur 4.9b). O th othr had f w hav mooth mmbrhp fuco Fgur 4.9c)] tad th ovra putoutput curv Fgur 4.9d)] bcom a moothr o. o
Somtm a mp Sugo zzy mod ca grat compx bhavor. Th foowg a xamp of a twoput ytm.
3
S Suo uzzy o
(b)
a) Atecedent MFs o Csp Rues small
"
medum
E 0 4
large
5
\
X
'
:
5 X d) Ovell O Cue o Fuzy Rue 5
6
-5
,,
lage
medum
E O 4
,,
2
small
,,
4
5 X c) Antedet MFs o Fuzy Rules
"
Overall O Cue or C Rul es
\
4
2
5
5
X
5
Fige 4. Compriso betwee zzy d ozzy res i Exmpe teedet Fs d b ipt-otpt e for ozzy res; teedet Fs d d ipt-otpt rve for zzy res (MATLA )
Exle 44 wo-ipt sigeotpt Sgeo zzy mode
A xamp of a twoiput igoutput Sugo fuzzy mod with four ru ca b xprd X i ma ad i ma th z = x + y + 1. X i ma ad i arg th z = y + 3. X i arg ad i ma th z = x + 3. X i arg ad i arg th z = x + y + 2
Fgur 4.10 a pot th mmbrhip fuctio of iput X ad ad Figur 4.10 b i th rutig iputoutput urfac. Th urfac i compx but it i ti obviou that th urfac i compod of four pa ach of which i pcid by th output quatio of a fuzzy ru. o
Ui th Mamdai fuzzy mod th Sugo fuzzy mod caot foow th compoitioa ru of ifrc Sctio 3.4.1 tricty i it zzy ronig mch im Thi po om dicuti wh th iput to a Sugo zzy mod ar
zzy Ifrce Sms
1f0o.·64 0.20
Ch
5
I 08 04
-1
4 5 N Lg
".5 2 0 1
0.2
Y
4 5
5 Y
b)
Fie 10 wo-ipt sige-otpt Sgeo zzy mode i Exmpe . teedet d oseqet Fs b over iptotpt srfe.
(MATLA l
sug2.)
fuzzy Spcicay w ca til mpoy th matchig of zzy t how i th atcdt part of Figur 3.4, to d th rig trgth of ach ru. owvr th rutig ovra output via ithr wightd avrag or wightd um i away crip; thi i coutrituitiv ic a fuzzy mod houd b ab to propagat th fuzzi om iput to output i a appropriat mar Without th timcoumig ad mathmaticaly itractab dfuzzicatio op ratio th Sugo zzy mod i by far th mot popuar cadidat for ampl databd zy modig which i itroducd i Chaptr 2 4.4
TSU KAMOTO FU Y MO DLS
I th sukamoto fuzzy models [], th coqut of ach zzy ifth ru
i rprtd by a zzy t with a mootoica MF how i Figur 4.11. A a rult th ifrrd output of ach ru i dd a crip vau iducd by th rul rig trgth Th ovra output i ta th wightd avrag of ach ru output Figur 4. ilutrat th roig procdur for a twoiput tworu ytm. Sic ach ru ifr a crip output th Tuoto zzy mod aggrgat ach rul output by th mthod of wightd avrag ad thu avoid th timcoumig proc of dzzicatio. owvr th Tuoto zzy mod i ot ud o ic it i ot trapart ithr th Mamdai or Sugo fuzzy mod. Th foowig i a igiput xamp. Examle 5 Sige-ipt skmoto zzy mode
85
S Othr Cnsdratns
Mnr Prduct
8
A
C
---- ---------- Wj---- --
X
Z
----
X
lht Z2
Z
Av
+ W2Z2 Z = WZ + W W2
Fge 11 he skmoto zzy mode
{II I
A xamp of a ig-ipu Tumoo fuzzy mod ca b xprd X i ma h Y i C1 X i mdium h Y i C2 X i arg h Y i C3,
whr h acd MF for "m, "mdium, d "arg ar how i Fig ur 412a) ad h coqu MF for C, C2 , ad C3 ar how i Fig ur 4.12b) . Th ovr ipu-oupu curv, how i Figur 412d) , i qua o (E�= wi i)/(E� Wi ) , whr Ii i h oupu of ach ru iducd by h rig rgth Wi ad MF for Ci . w po ach ru oupu Ii a cio of x w obai Figur 4.12c), whic i o qui obviou from h origi ru b ad MF po o
Sic h roig mchaim of h Tuoo zzy mod do o foow ricy h compoiio ru of ifrc, h oupu i way crip v wh h ipu ar zzy 4.5
OHER CONS ID ERAI ONS
Thr ar crai commo iu cocrig a h hr fuzzy ifrc ym iroducd prviouy, uch how o pariio ipu pac ad how o coruc a fuzzy ifrc ym for a paricuar appicaio. W ha xami h iu i hi cio
86
zzy Infrence Sysems
(b)
a) Atedet MFs
"C Q� CC
sma
0.8
-
-
-
'
"C Q � CC
\
E 0.4 � 0.
'
0 5 X c) Eac Res Ot
-5
0 8
> 4
0 0
0 5 Y d) Ove ptOt Ce
0 8
. . . . . . . . . .
0 0
\
06
E 04 � 0.
0
Coseqet MFs
08
: , , ,
06
0 -0
ae
medm
C
-5
>
' 0 X
6 4
5
0
0 -0
-5
0 X
5
10
Figre 12 Sigeipt sige otpt skmoto zzy mode i Exmpe .
teedet Fs; b oseqet Fs; eh e s otpt rve; d over iptotpt rve. M tsu )
4. 5. 1 In put Space Partitioning ow i houd b car ha h piri of fuzzy ifrc ym rmb ha of "divid ad coqurh acd of a fuzzy ru d a oc fuzzy r go, whi h coqu dcrib h bhavior wihi h rgio via variou coiu Th coqu coiu ca b a coqu MF Mamdai ad Tumoo fuzzy mod), a coa u zrordr Sugo mod) or a ar quaio r-ordr Sugo mod) Dir coqu coiu ru i dir zzy ifrc ym , bu hir acd ar away h am Thr for, h foowig dicuio of mhod of pariioig ipu pac o form h acd of fuzzy ru i appicab o a hr yp of fuzzy ifrc ym
•
Grid artition Figur 413a) iura a ypica grid pariio i a w
dimio ipu pac Thi pariio mhod i o cho i digig a fuzzy coror, which uuay ivov oy vra a variab a h ipu o h coror Thi pariio ragy d oy a m umbr of MF for ach ipu owvr, i cour probm wh w hav a modray arg umbr of ipu For iac, a zzy mod wih 10 ipu ad 2 MF o ach ipu woud ru i 2 = 1024 zzy if-h
87
Sc Othr Cnsdratns
Fire 1 rios methods for prtitioig the ipt spe grid prtitio; b tree prtitio; s tter prtitio.
ru, which i prohibiivy arg. Thi probm, uuay rfrrd o h curse of dimensionality, ca b aviad by h ohr pariio ragi.
•
•
ee artition Figur 4.13b) how a ypic r pariio, i which ach
rgio ca b uiquy pcid og a corrpodig dciio r. Th r pariio riv h probm of a xpoi icr i h umbr of ru. owvr, mor MF for ach ipu ar dd o d h zzy rgio, ad h MF do o uuy bar car iguiic maig uch "ma, "big, ad o o. ohr word, orhogoaiy hod roughy X x Y, bu o i ihr X or Y o. pariio i ud by h CAT cicaio ad rgrio r) gorihm, dicud i Chapr 14. Scatter atition A how i Figur 4.13c), by covrig a ub of
h who ipu pac ha characriz a rgio of poib occurrc of h ipu vcor, h car pariio ca o imi h umbr of ru o a roab amou. owvr, h car pariio i uuay dicad by dird ipu-oupu daa pair ad hu i gr, orhogoaiy do o hod i X, Y or X x Y. Thi ma i hard o ima h ovra mappig dircy from h coqu of ach ru oupu.
I
o ha Figur 4.13 i bd o h umpio ha MF ar dd o h ipu variab dircy. MF ar dd o crai raformaio of h ipu variab, w coud d up i a mor xib partiio y. Figur 4. 14 i a xamp of h ipu pariio wh MF ar dd o iar raformaio of h ipu variab. 4.5.2 Fuzzy Modeling By ow h radr houd hav arady dvopd a car picur of boh h ruc ur ad opraio of vra yp of zzy ifrc ym. gr, w dig a zzy ifrc ym bd o h p ow bhavior of a arg y m. Th zzy ym i h xpcd o b ab o rproduc h bhavior of h
88
uzzy Infrnc Systms
Ch
Figure 1 pt spe prtto whe Fs deed o er trsfotos of pt vrbes: grd prtto b tree prtto b stter prtto.
arg ym. For xamp, if h arg ym i a huma opraor i charg of a chmic racio proc, h h zzy ifrc ym bcom a fuzzy ogic coror ha ca rgua ad coro h proc Simiary, if h arg ym i a mdic docor, h h fuzzy ifrc bcom a zzy xpr ym for mdic diagoi. L u ow coidr how w migh coruc a zzy ifrc ym for a pcic appicaio. Gray paig, h adard mhod for corucig a zzy ifrc ym, a proc uuay cad fzzy modeling, h h foowig faur • Th ru rucur of a fuzzy ifrc ym ma i y o icorpora huma xpri abou h arg ym dircy io h modig proc. amy, zzy modig a advaag of domain knowledge ha migh o b iy or dircy mpoyd i ohr modig approach. • Wh h ipu-oupu daa of a arg ym i avaiab, covio ym idicaio chiqu c b ud for zzy modig. I ohr word h u of numerical data o pay a impora ro i fuzzy modig, ju i ohr mahmaic modig mhod I wha foow, w ha ummariz om gr guidi cocrig zzy modig. Spcic xamp of zzy modig for variou appicaio ca b foud i ubqu chapr. Cocpuay, zzy modig ca b purud i wo ag which ar o oy dijoi. Th r ag i h idicaio of h surface structre, which icud h foowig 1 Sc r ipu ad oupu variab. 2. Choo a pcic yp of zzy ifrc ym 3. Drmi h umbr of iguiic rm ociad wih ach ipu ad oupu variab. For a Sugo mod drmi h ordr of coqu quaio. )
S Summa
8
4. Dig a coctio of fuzzy if-th ru.
ot that to accompih th prcdig t, w ry o our ow owdg commo , imp phyica aw, ad o o) of th targt ytm iformatio providd by huma xprt who ar famiiar with th targt ytm which coud b th huma xprt thmv), or impy tri ad rror. Ar th rt tag of fuzzy modig, w obtai a ru b that ca mor or dcrib th bhavior of th targt ytm by ma of iguitic trm. Th maig of th iguitic trm i dtrmid i th cod tag, th idtica tio of dee structure, which dtrmi th MF of ach iguitic trm d th cocit of ach ru output poyomi if a Sugo fuzzy mod i ud) Spcicay, th idticatio of dp tructur icud th foowig t 1 Choo a appropriat famiy of paramtrizd MF Sctio 2.4) 2 trviw hum xprt famiiar with th targt ytm to dtrmi th
paramtr of th MF ud i th ru b.
3 th paramtr of th MF uig rgrio ad optimizatio tch
iqu.
T 1 ad 2 um th avaiabiity of huma xprt, whi t 3 um th avaiabiity of a dird iput-output data t. Variou ytm idticatio d optimizatio tchiqu for paramtr idticatio i t 3 ar dtaid i Chaptr 5 , 6, ad 7 A pcic twor tructur that faciitat t 3 i covrd i Chaptr 12 Wh a fuzzy ifrc ytm i ud a cotror for a giv pat, th th objctiv i t 3 houd b chagd to that of archig for paramtr that wi grat th bt prformac of th pat; thi pct of fuzzy ogc cotror dig i xpord i Chaptr 17 ad 18. 4.6
SUMMARY
Thi chaptr prt thr of th frquty ud fuzzy ifrc ytm th Mamdai, Sugo, ad Tumoto zzy mod W dicu thir trgth ad wa ad othr ratd iu uch iput pac partitioig ad fuzzy modig Fzzy ifrc ytm ar th mot importat modig too bd o zzy t thory Covtioa fuzzy ifrc ytm ar typicay buit by domai xprt ad hav b ud i automatic cotro dciio aayi, ad xprt ytm. Optimizatio ad adaptiv tchiqu xpad th appicatio of fuzzy ifrc ytm to d uch adaptiv cotro, adaptiv ig procig, oiar rgrio, ad pattr rcogitio Chaptr 12 dicu adaptiv fuzzy ifrc ytm thir appicatio ar covrd i Chaptr 19
90
uzzy Infrnc Systms
Ch
EXERCSES 1 Driv th zzy roig mchaim how i Figur 4.3 by chooig th m
ad th gbraic product for th T-orm ad T-coorm oprator, rpctivly.
2 Giv formu for th thr dfuzzicatio tratgi Equatio (4) to (43)]
wh w hav a it uivr of dicour X = { X • • X n } whr X < < Xn ·
U th thr dfuzzicatio tratgi i Equatio (4.1), (4.2), ad (4.3) to
d th rprtativ valu of a zzy t dd by A (X ) = trapzoid, 10, 30, 50, 90).
4.
pat th prviou xrci, but um that th uivr of dicour X cotai itgr om 0 to 100.
5 Modify th program mm .m i Examp 41 uch that you ca cic ad drag a
corr of a trapzoid MF to chag it hap ad th itractiv chag of th ovr iput-output curv.
6 Chag th MF i Examp 4.2 to trapzoid o ad pot th ovra iput
output urfac.
7 Modi Examp 4.3 uch that oy cott trm ar rtaid i th co
qut. pat th pot
8
i Figur 49.
Modi Examp 4.3 by addig a cod-ordr trm to th coqut quatio of ach ru pat th pot i Figur 4.9.
9 I Examp 4.4, u th foowig dirt ditio of MF
ig, [5,0), ma arg - ig, [5,0), ma - ig, [2,0), arg - ig, [2,0]), (4.4)
ad rpat th pot
i Figur 410.
10 pat th prviou xrci but u wightd um itad wightd avrag
to driv th a output Do you gt xacty th am iput-output urfac that i th prviou xrci? Why?
91
EFEENCE
REFERENCES
1] JS Roger Jang and CT Sun Fnctional equivalence between radial bis unc tion networs and uzzy inerence systems IEEE ansactons on Nul Ntworks (1)156159 January 1993 2] A Kandel editor Fuzzy xprt systms CRC ress Inc Boca Raton FL 1992 3] B Koso Nul ntworks and zzy systms: a dynamcal systms approach rentice Hall Upper Saddle River NJ 1991 ] CC Lee Fzzy logic in control systems uzzy logic controllerpart 1 IEEE ns actons on Systms Man and Cybtcs 20(2) 018 1990 5] CC Lee Fzzy logic in contro systems uzzy logic controllerpt 2 IEEE ans actons on Systms Man and Cybtcs 20(2) 1935 1990 6] E H Mamdani and S Assilian An experiment in linguistic synthesis with a uzzy logic controller Intatonal Joual of Man-Machn Studs 7(1)113 1975 7] N uger J Yen and R Langari A deuzzifcation strate or a uzzy logic con troller employing prohibitive inormation in command ormulation In Procdngs of IEEE Intatonal Confrnc on zzy Systms pages 717723 San Diego March 1992 8] T A Runler and M Glesner Deuzzifcation and ring in the context o member ship value semantics rule modality and meurement theory In Europan Congrss on Fuzzy and Intllgnt Tchnologs Aachen September 199 9] M Sugeno and G T Kang Structure identifcation o uzzy model Fuzzy Sts and Systms 28 1 533 1988 10] T Taagi and M Sugeno Fzzy identifcation o systems and its applications to modeling and control IEEE ansactons on Systms Man and Cybtcs 15 116 132 1985 11] Y Tsuamoto An approah to uzzy reoning method In Madan M Gupta Ram mohan K Ragade and Ronald R Yager editors Advancs n zzy st thory and applcatons pages 13719 NorthHolland Amsterdam 1979 12] R R Yager and D Filev SLIDE A simpe adaptive deuzzifcation method IEEE ansactons on zzy Systms 1(1)6978 Februy 1993
