Fuzzy Expert Systems and Fuzzy Reasoning

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ISBN: 9780471388593 | 0471388599
Cover: Hardcover
Copyright: 12/13/2004

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Fuzzy sets were for a long time not accepted by the AI community. Now they have become highly evolved and their techniques are well established. This book will teach the reader how to construct a fuzzy expert system to solve real-world problems. After a general discussion of expert systems, the basic fuzzy math required is presented first, requiring little more math background than high-school algebra. This book will fill a void in the market because although there are many books on expert systems, none devote more than a few pages to the notion of fuzzy sets and their applications in this domain. Therefore their use in this book is timely and should be well received.The book is designed as a text and has ample problems with solutions, a solutions manual and an accompanying program on our ftp site. Coverage is accessible to practitioners and academic readers alike.

Preface

xiii

1 Introduction

(14)

1.1 Characteristics of Expert Systems

(4)

1.1.1 Production Systems

(1)

1.1.2 Data-Driven Systems

(1)

1.1.3 Special Features of Fuzzy Systems

(1)

1.1.4 Expert Systems for Fuzzy Control and for Fuzzy Reasoning

(1)

1.2 Neural Nets

(1)

1.3 Symbolic Reasoning

(1)

1.4 Developing a Rule-Based Expert System

(1)

1.5 Fuzzy Rule-Based Systems

(1)

1.6 Problems in Learning How to Construct Fuzzy Expert Systems

(1)

1.7 Tools for Learning How to Construct Fuzzy Expert Systems

(1)

1.8 Auxiliary Reading

(1)

1.9 Summary

(1)

1.10 Questions

(2)

2 Rule-Based Systems: Overview

(14)

2.1 Expert Knowledge: Rules and Data

(1)

2.2 Rule Antecedent and Consequent

(2)

2.2.1 Antecedents

(1)

2.2.2 Admissible Data Types

(1)

2.2.3 Consequents

(1)

2.3 Data-Driven Systems

(1)

2.4 Run and Command Modes

(5)

2.4.1 Run Mode: Serial and Parallel Rule Firing

(1)

2.4.2 Checking which Rules are Fireable: The RETE Algorithm

(1)

2.4.3 Serial Rule Firing

(1)

2.4.4 Parallel Rule-Firing

(1)

2.5 Forward and Backward Chaining

(1)

2.6 Program Modularization and Blackboard Systems

(1)

2.7 Handling Uncertainties in an Expert System

(1)

2.8 Summary

(2)

2.9 Questions

(1)

3 Fuzzy Logic, Fuzzy Sets, and Fuzzy Numbers: I

(28)

3.1 Classical Logic

(7)

3.1.1 Evaluating A AND B and A OR B

(4)

3.1.2 The Implication Operator

(2)

3.2 Elementary Fuzzy Logic and Fuzzy Propositions

(2)

3.3 Fuzzy Sets

(5)

3.3.1 Discrete Fuzzy Sets

(1)

3.3.2 Fuzzy Numbers

(2)

3.3.3 Linguistic Variables and Membership Functions

(1)

3.4 Fuzzy Relations

(4)

3.4.1 Matrices of Truth Values

(1)

3.4.2 Relations Between Sets

(3)

3.5 Truth Value of Fuzzy Propositions

(2)

3.5.1 Comparing Single-Valued Data

(1)

3.5.2 Comparing Fuzzy Numbers

(1)

3.6 Fuzzification and Defuzzification

(5)

3.6.1 Fuzzification

(2)

3.6.2 Defuzzification

(3)

3.7 Questions

(3)

4 Fuzzy Logic, Fuzzy Sets, and Fuzzy Numbers: II

(28)

4.1 Introduction

(1)

4.2 Algebra of Fuzzy Sets

(8)

4.2.1 T-Norms and t-Conorms: Fuzzy AND and OR Operators

(3)

4.2.2 Correlation Fuzzy Logic

(3)

4.2.3 Combining Fuzzy Numbers

(2)

4.3 Approximate Reasoning

(4)

4.4 Hedges

(3)

4.5 Fuzzy Arithmetic

(2)

4.5.1 Extension Principle

(1)

4.5.2 Alpha-Cut and Interval Arithmetic

(1)

4.5.3 Comparison of Alpha-Cut and Interval Arithmetic Methods

(1)

4.6 Comparisons between Fuzzy Numbers

(4)

4.6.1 Using the Extension Principle

(2)

4.6.2 Alternate Method

(2)

4.7 Fuzzy Propositions

(4)

4.8 Questions

(3)

5 Combining Uncertainties

(14)

5.1 Generalizing AND and OR Operators

(3)

5.1.1 Correlation Logic: A Family of Fuzzy Logical Operators that Obeys Excluded Middle and Non-Contradiction Laws

(2)

5.2 Combining Single Truth Values

(1)

5.3 Combining Fuzzy Numbers and Membership Functions

(3)

5.4 Bayesian Methods

(2)

5.5 The Dempster-Shafer Method

(2)

5.6 Summary

(1)

5.7 Questions

(2)

6 Inference in an Expert System I

(16)

6.1 Overview

(1)

6.2 Types of Fuzzy Inference

100

(1)

6.3 Nature of Inference in a Fuzzy Expert System

100

(2)

6.4 Modification and Assignment of Truth Values

102

(3)

6.4.1 Monotonic Inference

102

(1)

6.4.2 Non-monotonic Inference

103

(1)

6.4.3 Downward Monotonic Inference

103

(2)

6.5 Approximate Reasoning

105

(1)

6.6 Tests of Procedures to Obtain the Truth Value of a Consequent from the Truth Value of Its Antecedent

105

(5)

6.6.1 Desirable Properties

105

(1)

6.6.2 Summary of Candidate Methods

106

(1)

6.6.3 Tests of Methods Against Desirable Properties

107

(2)

6.6.4 Implementation of Choices Among Types of Reasoning

109

(1)

6.7 Summary

110

(3)

6.7.1 Data Types and Truth Values

111

(1)

6.7.2 Types of Fuzzy Reasoning

112

(1)

6.8 Questions

113

(2)

7 Inference in a Fuzzy Expert System II: Modification of Data and Truth Values

115

(26)

7.1 Modification of Existing Data by Rule Consequent Instructions

116

(1)

7.2 Modification of Numeric Discrete Fuzzy Sets: Linguistic Variables and Linguistic Terms

117

(1)

7.3 Selection of Reasoning Type and Grade-of-Membership Initialization

118

(1)

7.4 Fuzzification and Defuzzification

119

(3)

7.4.1 Fuzzification and Evaluation of Antecedent Confidence

119

(1)

7.4.2 Modification of Consequent Membership Functions

119

(1)

7.4.3 Aggregation of Consequent Membership Functions for Each Consequent Linguistic Variable

120

(1)

7.4.4 Determination of Defuzzified Value for Consequent Attribute

121

(1)

7.5 Non-numeric Discrete Fuzzy Sets

122

(2)

7.6 Discrete Fuzzy Sets: Fuzziness, Ambiguity, and Contradiction

124

(3)

7.6.1 Fuzziness and Ambiguity

124

(1)

7.6.2 Ambiguities and Contradictions

125

(1)

7.6.3 Handling Ambiguities and Contradictions

126

(1)

7.7 Invalidation of Data: Non-monotonic Reasoning

127

(2)

7.8 Modification of Values of Data

129

(1)

7.9 Modeling the Entire Rule Space

129

(126)

7.9.1 Conventional Method: The Intersection Rule Configuration (IRC)

131

(1)

7.9.2 The Combs Union Rule Configuration

132

(1)

7.9.3 Performance of the Combs Method

133

(1)

7.9.4 Sample IRC and URC Programs

133

(1)

7.9.5 Exercises Iris.par and IrisCombs.par

134

(1)

7.9.6 Data Mining and the Combs Method

135

(1)

7.9.7 Combs Method Summary

135

(1)

7.10 Reducing the Number of Classification Rules Required in the Conventional Intersection Rule Configuration

135

(1)

7.11 Summary

136

(2)

7.11.1 Data Types and Their Truth Values

137

(1)

7.11.2 Types of Fuzzy Reasoning

137

(1)

7.12 Questions

138

(3)

8 Resolving Contradictions: Possibility and Necessity

141

(10)

8.1 Definition of Possibility and Necessity

142

(1)

8.2 Possibility and Necessity Suitable for MultiStep Rule-Based Fuzzy Reasoning

142

(2)

8.2.1 Data Types and Their Truth Values

142

(2)

8.2.2 Initialization of Data and Truth Values

144

(1)

8.3 Modification of Truth Values During a Fuzzy Reasoning Process

144

(2)

8.4 Formulation of Rules for Possibility and Necessity

146

(1)

8.5 Resolving Contradictions Using Possibility in a Necessity-Based System

146

(3)

8.5.1 Input Data: Useful Ambiguities

147

(1)

8.5.2 Output Data: Contradictions

147

(1)

8.5.3 Reaching Preliminary Classifications Using Supporting Evidence and Necessities

147

(1)

8.5.4 Resolving Contradictions Using Refuting Evidence and Possibilities

148

(1)

8.6 Summary

149

(1)

8.7 Questions

149

(2)

9 Expert System Shells and the Integrated Development Environment (IDE)

151

(8)

9.1 Overview

151

(2)

9.2 Help Files

153

(1)

9.3 Program Editing

153

(1)

9.4 Running the Program

154

(1)

9.5 Features of General-Purpose Fuzzy Expert Systems

154

(1)

9.6 Program Debugging

155

(1)

9.7 Summary

156

(1)

9.8 Questions

157

(2)

10 Simple Example Programs

159

(22)

10.1 Simple FLOPS Programs

159

(1)

10.2 Numbers.fps

159

(5)

10.2.1 Program Listing

159

(2)

10.2.2 Program Structure

161

(1)

10.2.3 Running the Program

161

(1)

10.2.4 Using Basic Debugging Commands

162

(1)

10.2.5 Confidence Terminology: Antecedent Confidence, Rule Confidence, and Posterior Confidence (pconf)

163

(1)

10.3 Sum.fps

164

(4)

10.3.1 Program Listing

164

(2)

10.3.2 Running sum.fps

166

(1)

10.3.3 Running sum.fps with Debugging Commands prstack; and Run 1;

167

(1)

10.4 Sum.par

168

(5)

10.4.1 Program Listing

169

(1)

10.4.2 Running sum.par

170

(1)

10.4.3 Running sum.par with prstack; and Run 1; Commands

171

(2)

10.5 Comparison of Serial and Parallel FLOPS

173

(1)

10.6 Membership Functions, Fuzzification and Defuzzification

173

(6)

10.6.1 Membership Functions in FLOPS/

173

(2)

10.6.2 Fuzzifying Numbers in FLOPS

175

(2)

10.6.3 Defuzzification in FLOPS

177

(2)

10.7 Summary

179

(1)

10.8 Questions

180

(1)

11 Running and Debugging Fuzzy Expert Systems I: Parallel Programs

181

(18)

11.1 Overview

181

(1)

11.2 Debugging Tools

181

(1)

11.3 Debugging Short Simple Programs

182

(7)

11.4 Isolating the Bug: System Modularization

189

(1)

11.5 The Debug Run

189

(1)

11.6 Interrupting the Program for Debug Checks

190

(1)

11.7 Locating Program Defects with Debug Commands

191

(5)

11.7.1 Program Structure

191

(1)

11.7.2 Sample Debugging Session

192

(4)

11.8 Summary

196

(1)

11.9 Questions

197

(2)

12 Running and Debugging Expert Systems II: Sequential Rule-Firing

199

(20)

12.1 Data Acquisition: From a User Versus Automatically Acquired

199

(2)

12.2 Ways of Solving a Tree-Search Problem

201

(1)

12.3 Expert Knowledge in Rules; auto1.fps

202

(5)

12.3.1 auto1.fps Program: Partial Listing

202

(2)

12.3.2 Running auto1.fps

204

(3)

12.4 Expert Knowledge in a Database: auto2.fps

207

(7)

12.4.1 Expert Knowledge Database Structure

207

(1)

12.4.2 auto2.fps Program Listing

208

(2)

12.4.3 Running and Debugging auto2.fps

210

(4)

12.4.4 Advantages of Database-Defined Tree Search Method

214

(1)

12.5 Other Applications of Sequential Rule Firing

214

(3)

12.5.1 Missionaries and Cannibals

214

(3)

12.6 Rules that Make Themselves Refireable: Runaway Programs and Recursion

217

(1)

12.7 Summary

218

(1)

12.8 Questions

218

(1)

13 Solving "What?" Problems when the Answer is Expressed in Words

219

(24)

13.1 General Methods

219

(1)

13.2 Iris.par: What Species Is It?

220

(7)

13.2.1 Planning Our Data Elements and Membership Functions

221

(2)

13.2.2 Writing Our Rules: Getting Started

223

(1)

13.2.3 Writing Our Rules: Classifying the Specimens

224

(1)

13.2.4 Writing Our Rules: Reporting Results

225

(1)

13.2.5 Setting Our Initial Data

225

(1)

13.2.6 Running iris.par

226

(1)

13.2.7 Improving iris.par

227

(1)

13.3 Echocardiogram Pattern Recognition

227

(8)

13.3.1 Echocardiogram Pattern Recognition: Data Declarations

228

(2)

13.3.2 Echocardiogram Pattern Recognition: Creating Classification Rules from a Database

230

(1)

13.3.3 The Organization of echo.par

231

(1)

13.3.4 Metarules to Control Activation of Rule Blocks

231

(2)

13.3.5 New Features of Rules in echo.par

233

(2)

13.3.6 Running echo.par

235

(1)

13.4 Schizo.par

235

(5)

13.4.1 Data Declarations in schizo.par

235

(1)

13.4.2 Structure of schizo.par

236

(1)

13.4.3 Rules for schizo.par; Block 0, Inputting Data

237

(1)

13.4.4 Rules for schizo.par; Block 1, Getting Truth Values of Generalized Symptoms

237

(1)

13.4.5 Rules for schizo.par; Block 2, Getting Basic Diagnoses

238

(1)

13.4.6 Rules for schizo.par; Block 3, Getting Final Diagnoses

238

(1)

13.4.7 Rules for schizo.par; Block 4, Output of Diagnoses

239

(1)

13.4.8 Rules for schizo.par; Block 5, Block Firing Control

239

(1)

13.4.9 Running schizo.par with Debugging Commands

240

(1)

13.5 Discussion

240

(1)

13.6 Questions

240

(3)

14 Programs that Can Learn from Experience

243

(14)

14.1 General Methods

243

(1)

14.2 Pavlov1.par: Learning by Adding Rules

244

(6)

14.2.1 Data for Pavlov1.par

244

(2)

14.2.2 Rules for Pavlov1.par

246

(3)

14.2.3 Exercise: Pavlov1.par

249

(1)

14.3 Pavlov2.par: Learning by Adding Facts to Long-Term Memory

250

(3)

14.3.1 Data for Pavlov1.par

250

(1)

14.3.2 Rules for Pavlov2.par

251

(1)

14.3.3 Running: Pavlov2.par

252

(1)

14.4 Defining New Data Elements and New: RULEGEN.FPS

253

(1)

14.4.1 Running: RULEGEN.FPS

253

(1)

14.5 Most General Way of Creating New Rules and Data Descriptors

254

(1)

14.6 Discussion

254

(1)

14.7 Questions

255

(2)

15 Running On-Line in Real-Time

257

(18)

15.1 Overview of On-Line Real-Time Work

257

(1)

15.2 Input/Output On-Line in Real-Time

258

(1)

15.3 On-Line Real-Time Processing

259

(6)

15.3.1 Detection of Obvious Artifacts in the Input Data Stream

260

(1)

15.3.2 Data Smoothing and Time Delays and Baselines: Exponential Smoothing

260

(3)

15.3.3 Rates of Change: Differentiating Noisy Data

263

(1)

15.3.4 Rates of Change: Differentiating Wandering Data

264

(1)

15.4 Types of Rules Useful in Real-Time On-Line Work

265

(5)

15.4.1 Fuzzy Control Rules

265

(2)

15.4.2 Focused Control-Type Rules and Focused Membership Functions

267

(2)

15.4.3 Fuzzy Reasoning with Approximate Numerical Comparisons

269

(1)

15.5 Memory Management

270

(1)

15.6 Development of On-Line Real-Time Programs

270

(2)

15.7 Speeding Up a Program

272

(1)

15.8 Debugging Real-Time Online Programs

272

(1)

15.9 Discussion

273

(1)

15.10 Questions

273

(2)

Appendix

275

(102)

Answers

377

(22)

References

399

(4)

Index

403

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Fuzzy Expert Systems and Fuzzy Reasoning

Summary

Author Biography

Table of Contents

Supplemental Materials