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**Hardcover**|

**$75.00 Short**|

**£51.95**| ISBN: 9780262161718 | 490 pp. | 7.1 x 10.1 in | April 1998

## Essential Info

- Table of Contents

## Table of Contents

FOREWORD | ||||||

PREFACE | ||||||

Philosophy | ||||||

Significant Features | ||||||

Chapter Descriptions | ||||||

Intended Readership and Use of the Book | ||||||

INTRODUCTION | ||||||

Note | ||||||

References | ||||||

FUNDAMENTALS OF FUZZY SETS | ||||||

Basic Notions and Concepts of Fuzzy Sets | ||||||

Set Membership and Fuzzy Sets | ||||||

Basic Definition of a Fuzzy Set | ||||||

Types of Membership Functions | ||||||

Characteristics of a Fuzzy Set | ||||||

Basic Relationships between Fuzzy Sets: Equality and Inclusion | ||||||

Fuzzy Sets and Sets: The Representation Theorem | ||||||

The Extension Principle | ||||||

Membership Function Determination | ||||||

Horizontal Method of Membership Estimation | ||||||

Vertical Method of Membership Estimation | ||||||

Pairwise-Comparison Method of Membership Function Estimation | ||||||

Problem Specification-Based Membership Determination | ||||||

Membership Estimation as a Problem of Parametric Optimization | ||||||

Membership Estimation via Fuzzy Clustering | ||||||

Generalizations of Fuzzy Sets | ||||||

Interval-Valued Fuzzy Sets and Second-Order Fuzzy Sets | ||||||

Type-Two Fuzzy Sets | ||||||

Chapter Summary | ||||||

Problems | ||||||

References | ||||||

Fuzzy Set Operations | ||||||

Set Theory Operations and Their Properties | ||||||

Triangular Norms | ||||||

Several Classes of Triangular Norms | ||||||

Triangular Norms as Models of Operations on Fuzzy Sets | ||||||

Aggregation Operations on Fuzzy Sets | ||||||

Compensatory Operators | ||||||

Symmetric Sums | ||||||

Averaging Operation | ||||||

Ordered Weighted Averaging Operations | ||||||

Sensitivity of Fuzzy Set Operators | ||||||

Negations | ||||||

Comparison Operations on Fuzzy Sets | ||||||

Distance Measures | ||||||

Equality Indexes | ||||||

Possibility and Necessity Measures | ||||||

Compatibility Measures | ||||||

Chapter Summary | ||||||

Problems | ||||||

References | ||||||

Information-Based Characterization of Fuzzy Sets | ||||||

Entropy Measures of Fuzziness | ||||||

Energy Measures of Fuzziness | ||||||

Specificity of a Fuzzy Set | ||||||

Frames of Cognition | ||||||

Basic Definition | ||||||

Main Properties | ||||||

Specificity | ||||||

Focus of Attention | ||||||

Information Hiding | ||||||

Information Encoding and Decoding Using Linguistic Landmarks | ||||||

Encoding Schemes in the Fuzzy Communication Channel | ||||||

Decoding Mechanisms | ||||||

Decoding Mechanisms for Pointwise Data | ||||||

Decoding Based on Modal Values of the Codebook | ||||||

Center-of-Gravity Decoding | ||||||

Polynomial Expansion | ||||||

Linguistic Expansion | ||||||

Decoding Using Membership Functions of the Linguistic Terms of the Codebook | ||||||

General Possibility-Necessity Decoding | ||||||

Distance between Fuzzy Sets Based on Their Internal, Linguistic Representation | ||||||

Chapter Summary | ||||||

Problems | ||||||

References | ||||||

Fuzzy Relations and Their Calculus | ||||||

Relations and Fuzzy Relations | ||||||

Operations on Fuzzy Relations | ||||||

Compositions of Fuzzy Relations | ||||||

Projections and Cylindric Extensions of Fuzzy Relations | ||||||

Binary Fuzzy Relations | ||||||

Some Classes of Fuzzy Relations | ||||||

Equivalence and Similarity Relations | ||||||

Compatibility and Proximity Relations | ||||||

Fuzzy-Relational Equations | ||||||

Introductory Comments | ||||||

Interpretations of Composition Operators | ||||||

sup-t (max-t) Composition | ||||||

inf-s Composition | ||||||

Composition Operators Involving Implication (j-operator) | ||||||

Estimation and Inverse Problem in Fuzzy-Relational Equations | ||||||

Solving Fuzzy-Relational Equations with the sup-t Composition | ||||||

Properties of the Implication Operator | ||||||

Extended Topologies of Fuzzy-Relational Equations | ||||||

Solvability Conditions | ||||||

Relation-Relation Fuzzy Equations | ||||||

Solutions to Dual Fuzzy-Relational Equations | ||||||

Adjoint Fuzzy-Relational Equations | ||||||

Generalizations of Fuzzy-Relational Equations | ||||||

Fuzzy-Relational Equations with an Equality Composition Operator | ||||||

Multilevel Fuzzy-Relational Equations | ||||||

Fuzzy-Relational Equations with s-t and t-s Composition Operations | ||||||

Approximate Solutions to Fuzzy-Relational Equations | ||||||

Modifications of Relational Constraints via Thresholding | ||||||

Preprocessing Fuzzy Data via Clustering | ||||||

Use of Auxiliary Variables | ||||||

Solving Fuzzy-Relational Equations via Logic Filtering | ||||||

Solving Fuzzy-Relational Equations as a Problem of Learning of Fuzzy Neurons | ||||||

Chapter Summary | ||||||

Problems | ||||||

Note | ||||||

References | ||||||

Fuzzy Numbers | ||||||

Defining Fuzzy Numbers | ||||||

Interval Analysis and Fuzzy Numbers | ||||||

Computing with Fuzzy Numbers | ||||||

Triangular Fuzzy Numbers and Basic Operations | ||||||

Addition | ||||||

Multiplication | ||||||

Division | ||||||

Inverse | ||||||

Fuzzy Minimum | ||||||

General Formulas for LR Fuzzy Numbers | ||||||

Accumulation of Fuzziness in Computing with Fuzzy Numbers | ||||||

Inverse Problem in Computation with Fuzzy Numbers | ||||||

Fuzzy Numbers and Approximate Operations | ||||||

Chapter Summary | ||||||

Problems | ||||||

References | ||||||

Fuzzy Sets and Probability | ||||||

Introduction | ||||||

Probability and Fuzzy Sets | ||||||

Hybrid Fuzzy-Probabilistic Models of Uncertainty | ||||||

Probability of Fuzzy Events | ||||||

Linguistic Probabilities | ||||||

Probability-Possibility Transformations | ||||||

Probabilistic Sets and Fuzzy Random Variables | ||||||

Probabilistic Sets | ||||||

Fuzzy Random Variables | ||||||

Chapter Summary | ||||||

Problems | ||||||

Notes | ||||||

References | ||||||

Linguistic Variables | ||||||

Introduction | ||||||

Linguistic Variables: Formalization | ||||||

Computing with Linguistic Variables: Hedges, Connectives, and Negation | ||||||

Linguistic Approximation | ||||||

Linguistic Quantifiers | ||||||

Applications of Linguistic Variables | ||||||

Chapter Summary | ||||||

Problems | ||||||

References | ||||||

Fuzzy Logic | ||||||

Introduction | ||||||

Propositional Calculus | ||||||

Predicate Logic | ||||||

Many-Valued Logic | ||||||

Fuzzy Logic | ||||||

Computing with Fuzzy Logic | ||||||

Truth Space Methods | ||||||

Fuzzy Truth Values and Fuzzy Truth Qualification | ||||||

Inverse Truth Qualification | ||||||

Operations in Fuzzy Logic | ||||||

Reasoning in the Framework of Truth Space | ||||||

Compositional Rule of Inference | ||||||

Some Remarks about Inference Methods | ||||||

Chapter Summary | ||||||

Problems | ||||||

References | ||||||

Fuzzy Measures and Fuzzy Integrals | ||||||

Fuzzy Measures | ||||||

Fuzzy Integrals | ||||||

Basic Properties of Fuzzy Integration | ||||||

Optimization Aspects of the Fuzzy Integral | ||||||

Chapter Summary | ||||||

Problems | ||||||

Note | ||||||

References | ||||||

COMPUTATIONAL MODELS | ||||||

Rule-Based Computations | ||||||

Rules in Knowledge Representation | ||||||

Qualified Propositions | ||||||

Quantified Propositions | ||||||

Unless Rules | ||||||

Gradual Rules | ||||||

Potential Inconsistency and Conflicting Rules | ||||||

Categorical and Dispositional Propositions | ||||||

Syntax of Fuzzy Rules | ||||||

Semantics of Fuzzy Rules and Inference | ||||||

Semantics of Unless Rules | ||||||

Semantics of Gradual Rules | ||||||

Computing with Fuzzy Rules | ||||||

Some Properties of Fuzzy Rule-Based Systems | ||||||

Rule Consistency and Completeness | ||||||

Chapter Summary | ||||||

Problems | ||||||

References | ||||||

Fuzzy Neurocomputation | ||||||

Neural Networks: Basic Notions, Architectures, and Learning | ||||||

Logic-Based Neurons | ||||||

Aggregative OR and AND Logic Neurons | ||||||

OR/AND Neurons | ||||||

Conceptual and Computational Augmentations of Fuzzy Neurons | ||||||

Representing Inhibitory Information | ||||||

Computational Enhancements of Fuzzy Neurons | ||||||

Logic Neurons and Fuzzy Neural Networks with Feedback | ||||||

Referential Logic-Based Neurons | ||||||

Fuzzy Threshold Neurons | ||||||

Classes of Fuzzy Neural Networks | ||||||

Approximation of Logical Relationships-Development of the Logic Processor | ||||||

Referential Processor | ||||||

Fuzzy Cellular Automata | ||||||

Learning | ||||||

Learning in a Single Neuron | ||||||

Self-Organization Mechanisms in Fuzzy Neural Networks | ||||||

Selected Aspects of Knowledge Representation in Fuzzy Neural Networks | ||||||

Representing and Processing Uncertainty | ||||||

Induced Boolean and Core Neural Networks | ||||||

Chapter Summary | ||||||

Problems | ||||||

References | ||||||

Fuzzy Evolutionary Computation | ||||||

Evolution and Computing | ||||||

Genetic Algorithms | ||||||

Reproduction | ||||||

Crossover | ||||||

Mutation | ||||||

Design of Fuzzy Rule-Based Systems with Genetic Algorithms | ||||||

Learning in Fuzzy Neural Networks with Genetic Algorithms | ||||||

Evolution Strategies | ||||||

Hybrid and Cooperating Approaches | ||||||

Chapter Summary | ||||||

Problems | ||||||

References | ||||||

Fuzzy Modeling | ||||||

Fuzzy Models: Beyond Numerical Computations | ||||||

Main Phases of System Modeling | ||||||

Fundamental Design Objectives in System Modeling | ||||||

General Topology of Fuzzy Models | ||||||

Compatibility of Encoding and Decoding Modules | ||||||

Classes of Fuzzy Models | ||||||

Tabular Format of the Fuzzy Model | ||||||

Fuzzy-Relation Equations | ||||||

Fuzzy Grammars | ||||||

Local Fuzzy Models | ||||||

Verification and Validation of Fuzzy Models | ||||||

Verification Algorithms for Fuzzy Models | ||||||

Validation of Fuzzy Models | ||||||

Chapter Summary | ||||||

Problems | ||||||

References | ||||||

PROBLEM SOLVING WITH FUZZY SETS | ||||||

Methodology | ||||||

Analysis and Design | ||||||

Fuzzy Controllers and Fuzzy Control | ||||||

Knowledge Acquisition in a Simple Control Problem | ||||||

Construction of the Fuzzy Controller: Algorithmic Aspects | ||||||

Rules and Rule Base | ||||||

Inference | ||||||

Encoding | ||||||

Decoding | ||||||

Fuzzy Controllers and PI and PD Controllers | ||||||

Possibility-Necessity Computations in the Fuzzy Controller | ||||||

Fuzzy Hebbian Learning | ||||||

Design Considerations and Controller Adjustments | ||||||

Relational Partition of the Input Space | ||||||

Controller Adjustments | ||||||

Scaling Factors | ||||||

Context-Dependent Adjustment of the Universe of Discourse | ||||||

Windowing Effect | ||||||

Fuzzy Logic Controller | ||||||

Mathematical Programming and Fuzzy Optimization | ||||||

General Optimization Problems | ||||||

Fuzzy Optimization Problems | ||||||

Fuzzy Linear Programming | ||||||

Fuzzy Objective Functions and Fuzzy Constraints | ||||||

Fuzzy Constraints with Fuzzy Coefficients | ||||||

Fuzzy Coefficients in the Objective Function | ||||||

Chapter Summary | ||||||

Problems | ||||||

Note | ||||||

References | ||||||

Case Studies | ||||||

Traffic Intersection Control | ||||||

Fuzzy Traffic Controller | ||||||

Fuzzy Controller | ||||||

State Machine | ||||||

Adaptation Methods | ||||||

Distributed Traffic Control | ||||||

Distributed Control System Architecture | ||||||

Distributed Traffic Control System | ||||||

Local Problem Solver | ||||||

Evolutive Case-Based Mechanism | ||||||

Elevator Group Control | ||||||

Elevator Group Control System | ||||||

Fuzzy Group Controller | ||||||

Simulation Experiments | ||||||

Induction Motor Control | ||||||

Speed Control of AC Machines | ||||||

Fuzzy Control Strategy | ||||||

Controller Design | ||||||

Communication Network Planning | ||||||

Communication Network Model | ||||||

Clustering Procedure | ||||||

Neurocomputation in Fault Diagnosis of Dynamic Systems | ||||||

Neurofuzzy Network Structure and Learning | ||||||

Learning Algorithm | ||||||

Fault Detection and Diagnosis | ||||||

Simulation Results | ||||||

Multicommodity Transportation Planning in Railways | ||||||

References | ||||||

INDEX |

## Instructor Resources

# An Introduction to Fuzzy Sets

## Overview

The concept of fuzzy sets is one of the most fundamental and influential tools in computational intelligence. Fuzzy sets can provide solutions to a broad range of problems of control, pattern classification, reasoning, planning, and computer vision. This book bridges the gap that has developed between theory and practice. The authors explain what fuzzy sets are, why they work, when they should be used (and when they shouldn't), and how to design systems using them.

The authors take an unusual top-down approach to the design of detailed algorithms. They begin with illustrative examples, explain the fundamental theory and design methodologies, and then present more advanced case studies dealing with practical tasks. While they use mathematics to introduce concepts, they ground them in examples of real-world problems that can be solved through fuzzy set technology. The only mathematics prerequisites are a basic knowledge of introductory calculus and linear algebra.

## Instructor Resources for This Title:

## About the Author

Witold Pedrycz is Professor and Chair of Electrical and Computer Engineering at the University of Alberta.

## Endorsements

“The Pedrycz and Gomide text is superb in all respects. Its exposition of fuzzy-neural networks and fuzzy-genetic systems adds much to its value as a textbook.”

—**Lotfi A. Zadeh**, University of California, Berkeley