Insight into Fuzzy Modeling


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Provides a unique and methodologically consistent treatment of various areas of fuzzy modeling and includes the results of mathematical fuzzy logic and linguistics

This book is the result of almost thirty years of research on fuzzy modeling. It provides a unique view of both the theory and various types of applications. The book is divided into two parts. The first part contains an extensive presentation of the theory of fuzzy modeling. The second part presents selected applications in three important areas: control and decision-making, image processing, and time series analysis and forecasting. The authors address the consistent and appropriate treatment of the notions of fuzzy sets and fuzzy logic and their applications. They provide two complementary views of the methodology, which is based on fuzzy IF-THEN rules. The first, more traditional method involves fuzzy approximation and the theory of fuzzy relations. The second method is based on a combination of formal fuzzy logic and linguistics. A very important topic covered for the first time in book form is the fuzzy transform (F-transform). Applications of this theory are described in separate chapters and include image processing and time series analysis and forecasting. All of the mentioned components make this book of interest to students and researchers of fuzzy modeling as well as to practitioners in industry.


  • Provides a foundation of fuzzy modeling and proposes a thorough description of fuzzy modeling methodology
  • Emphasizes fuzzy modeling based on results in linguistics and formal logic
  • Includes chapters on natural language and approximate reasoning, fuzzy control and fuzzy decision-making, and image processing using the F-transform
  • Discusses fuzzy IF-THEN rules for approximating functions, fuzzy cluster analysis, and time series forecasting

Insight into Fuzzy Modeling is a reference for researchers in the fields of soft computing and fuzzy logic as well as undergraduate, master and Ph.D. students.

Vilém Novák, D.Sc. is Full Professor and Director of the Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Czech Republic.

Irina Perfilieva, Ph.D. is Full Professor, Senior Scientist, and Head of the Department of Theoretical Research at the Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Czech Republic.

Antonín Dvorák, Ph.D. is Associate Professor, and Senior Scientist at the Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, Czech Republic.


Preface xiii

Acknowledgments xv

About the Companion Website xvii


1 What is Fuzzy Modeling 3

1.1 Indeterminacy in Human Life 3

1.2 Fuzzy Modeling: With and Without Words 6

2 Overview of Basic Notions 11

2.1 Relations, Functions, Ordered Sets 11

2.1.1 Relations 11

2.2 Fuzzy Sets and Fuzzy Relations 14

2.2.1 The Concept of a Fuzzy Set 14

2.2.2 Operations with Fuzzy Sets 19

2.2.3 Fuzzy Numbers 28

2.2.4 Fuzzy Partition and Fuzzy Covering 31

2.2.5 Cartesian Product and Fuzzy Relations 32

2.2.6 Fuzzy Equality and Extensional Fuzzy Sets 37

2.3 Elements of Mathematical Fuzzy Logic 41

2.3.1 Structure of Truth Degrees in Mathematical Fuzzy Logic 41

2.3.2 Logical Inference 43

2.3.3 Formal Systems of MFL 45

2.3.4 The Concept of Fuzzy IF-THEN Rule 46

3 Fuzzy IF-THEN Rules in Approximation of Functions 49

3.1 Relational Interpretation of Fuzzy IF-THEN Rules 49

3.1.1 Finite Functions and Their Description 50

3.1.2 Relational Interpretation of Linguistic Descriptions 53

3.1.3 Managing More Variables 59

3.2 Approximation of Functions Using Fuzzy IF-THEN Rules 60

3.2.1 Defuzzification 60

3.2.2 Fuzzy Approximation 63

3.2.3 Choosing between DNF and CNF 69

3.3 Generalized Modus Ponens and Fuzzy Functions 72

3.4 Takagi–Sugeno Rules 74

3.4.1 Basic Concepts 74

3.4.2 Fuzzy Approximation Using TS Rules 75

3.4.3 Identification of TS Rules 78

4 Fuzzy Transform 81

4.1 Fuzzy Partition 81

4.2 The Concept of F-Transform 84

4.2.1 Direct F-Transform 84

4.2.2 Inverse F-Transform 85

4.3 Discrete F-Transform 88

4.4 F-Transform of Functions of Two Variables 89

4.5 F1-Transform 91

4.6 Methodological Remarks to Applications of the F-Transform 94

5 Fuzzy Natural Logic and Approximate Reasoning 97

5.1 Linguistic Semantics and Linguistic Variable 97

5.1.1 Linguistic Variable 98

5.1.2 Intension, Context, Extension 98

5.1.3 Refined Definition of Linguistic Variable 100

5.2 Theory of Evaluative Linguistic Expressions 101

5.2.1 The Concept and Structure of Evaluative Expressions 101

5.2.2 Evaluative Linguistic Predications 105

5.2.3 Mathematical Model of the Semantics of Evaluative Linguistic Expressions 106

5.3 Interpretation of Fuzzy/Linguistic IF-THEN Rules 117

5.3.1 Linguistic Description 117

5.3.2 Intension of Fuzzy/Linguistic IF-THEN Rules 118

5.4 Approximate Reasoning with Linguistic Information 119

5.4.1 Basic Principle of Approximate Reasoning 119

5.4.2 Perception-Based Logical Deduction 120

5.4.3 Formalization of the Perception-Based Logical Deduction 124

5.4.4 Comparison of Two Interpretations of Fuzzy IF-THEN Rules 128

6 Fuzzy Cluster Analysis 137

6.1 Basic Notions 137

6.2 Fuzzy Clustering Algorithms 139

6.3 The Algorithm of Fuzzy c-Means 140

6.4 The Gustafson–Kessel Algorithm 142

6.5 How the Number of Clusters Can Be Determined 144

6.6 Construction of Fuzzy Rules Based on Found Clusters 144


7 Fuzzy/Linguistic Control and Decision-Making 151

7.1 The Principle of Fuzzy Control 151

7.1.1 Control in a Closed Feedback Loop 153

7.1.2 A General Scheme of Fuzzy Controller 154

7.2 Fuzzy Controllers 157

7.2.1 Variables 157

7.2.2 Basic Types of Classical Controllers 158

7.2.3 Basic Types of Fuzzy Controllers 159

7.3 Design of Fuzzy/Linguistic Controller 161

7.3.1 Determination of Variables and Linguistic Context 161

7.3.2 Choosing Fuzzy Action Unit 162

7.3.3 Formation of Knowledge Base 163

7.3.4 Tuning Linguistic Description 166

7.4 Learning 171

7.4.1 Modification and Learning of Linguistic Context 171

7.4.2 Learning Linguistic Description 174

7.4.3 Practical Experiences with Control Using Linguistic Fuzzy Action Unit 177

7.5 Decision-Making Using Linguistic Descriptions 180

7.5.1 Introduction 180

7.5.2 Hierarchy of Linguistic Descriptions in Decision-Making 181

7.5.3 Demonstration of the Decision-Making Methodology Using Linguistic Descriptions 182

8 F-Transform in Image Processing 189

8.1 Image and Its Basic Processing Using F-Transform 189

8.2 F-Transform-Based Image Compression and Reconstruction 190

8.2.1 Basic Principles of Image Compression 190

8.2.2 Simple F-Transform Compression 191

8.2.3 Advanced Image Compression 191

8.3 F1-Transform Edge Detector 193

8.4 F-Transform-Based Image Fusion 195

8.4.1 Basic Idea of Image Fusion 195

8.4.2 Simple F-Transform-Based Fusion Algorithm 197

8.4.3 Complete F-Transform-Based Fusion Algorithm 199

8.4.4 Enhanced Simple Fusion Algorithm 201

8.5 F-Transform-Based Corrupted Image Reconstruction 203

8.5.1 The Reconstruction Problem 203

8.5.2 F-Transform-Based Reconstruction 204

8.5.3 Demonstration Examples 206

9 Analysis and Forecasting of Time Series 209

9.1 Classical Versus Fuzzy Models of Time Series 210

9.1.1 Definition of Time Series 210

9.1.2 Classical Models of Time Series 210

9.1.3 Fuzzy Models of Time Series 211

9.2 Analysis of Time Series Using F-Transform 212

9.2.1 Decomposition of Time Series 212

9.2.2 Extraction of Trend-Cycle and Trend Using F-Transform 214

9.3 Time Series Forecasting 219

9.3.1 Decomposition of Time Domain 219

9.3.2 Forecast of Trend-Cycle 220

9.3.3 Forecast of Seasonal Component 223

9.3.4 Forecast of the Whole Time Series 225

9.4 Characterization of Time Series in Natural Language 225

9.4.1 Sentences Characterizing Trend 226

9.4.2 Automatic Generation of Sentences Characterizing Trend 228

9.4.3 Mining Information from Time Series 230

References 235

Index 243

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