Knowledge-Based Clustering From Data to

A comprehensive coverage of emerging and current technology dealing with heterogeneous sources of information, including data, design hints, reinforcement signals from external datasets, and related topics Covers all necessary prerequisites, and if necessary,additional explanations of more advanced topics, to make abstract concepts more tangible Includes illustrative material andwell-known experimentsto offer hands-on experience

WITOLD PEDRYCZ, PHD, is a Professor and Canada Research Chair at the University of Alberta, Canada. He is also with the Systems Research Institute of The Polish Academy of Sciences, Warsaw, Poland. Dr. Pedrycz is a Fellow of the IEEE, has authored nine research monographs, edited six volumes, and has written numerous papers in computational intelligence, granular computing, pattern recognition, quantitative software engineering, and data mining.

Foreword

xiii

Preface

xv

1 Clustering and Fuzzy Clustering

1

(27)

1.1 Introduction

1

(1)

1.2 Basic Notions and Notation

1

(5)

1.2.1 Types of Data

2

(1)

1.2.2 Distance and Similarity

2

(4)

1.3 Main Categories of Clustering Algorithms

6

(4)

1.3.1 Hierarchical Clustering

6

(2)

1.3.2 Objective Function-Based Clustering

8

(2)

1.4 Clustering and Classification

10

(1)

1.5 Fuzzy Clustering

11

(7)

1.6 Cluster Validity

18

(1)

1.7 Extensions of Objective Function-Based Fuzzy Clustering

19

(4)

1.7.1 Augmented Geometry of Fuzzy Clusters: Fuzzy C Varieties

19

(1)

1.7.2 Possibilistic Clustering

20

(2)

1.7.3 Noise Clustering

22

(1)

1.8 Self-Organizing Maps and Fuzzy Objective Function-Based Clustering

23

(2)

1.9 Conclusions

25

(1)

References

26

(2)

2 Computing with Granular Information: Fuzzy Sets and Fuzzy Relations

28

(22)

2.1 A Paradigm of Granular Computing: Information Granules and Their Processing

28

(3)

2.2 Fuzzy Sets as Human-Centric Information Granules

31

(1)

2.3 Operations on Fuzzy Sets

32

(1)

2.4 Fuzzy Relations

33

(2)

2.5 Comparison of Two Fuzzy Sets

35

(2)

2.6 Generalizations of Fuzzy Sets

37

(1)

2.7 Shadowed Sets

38

(6)

2.8 Rough Sets

44

(2)

2.9 Granular Computing and Distributed Processing

46

(1)

2.10 Conclusions

47

(1)

References

47

(3)

3 Logic-Oriented Neurocomputing

50

(16)

3.1 Introduction

50

(1)

3.2 Main Categories of Fuzzy Neurons

51

(8)

3.2.1 Aggregative Neurons

52

(3)

3.2.2 Referential (Reference) Neurons

55

(4)

3.3 Architectures of Logic Networks

59

(2)

3.4 Interpretation Aspects of the Networks

61

(1)

3.5 Granular Interfaces of Logic Processing

62

(2)

3.6 Conclusions

64

(1)

References

64

(2)

4 Conditional Fuzzy Clustering

66

(21)

4.1 Introduction

66

(2)

4.2 Problem Statement: Context Fuzzy Sets and Objective Function

68

(2)

4.3 The Optimization Problem

70

(10)

4.4 Computational Considerations of Conditional Clustering

80

(1)

4.5 Generalizations of the Algorithm Through the Aggregation Operator

81

(1)

4.6 Fuzzy Clustering with Spatial Constraints

82

(4)

4.7 Conclusions

86

(1)

References

86

(1)

5 Clustering with Partial Supervision

87

(10)

5.1 Introduction

87

(1)

5.2 Problem Formulation

88

(2)

5.3 Design of the Clusters

90

(1)

5.4 Experimental Examples

91

(2)

5.5 Cluster-Based Tracking Problem

93

(3)

5.6 Conclusions

96

(1)

References

96

(1)

6 Principles of Knowledge-Based Guidance in Fuzzy Clustering

97

(32)

6.1 Introduction

97

(2)

6.2 Examples of Knowledge-Oriented Hints and Their General Taxonomy

99

(3)

6.3 The Optimization Environment of Knowledge-Enhanced Clustering

102

(3)

6.4 Quantification of Knowledge-Based Guidance Hints and Their Optimization

105

(2)

6.5 Organization of the Interaction Process

107

(5)

6.6 Proximity-Based Clustering (P-FCM)

112

(5)

6.7 Web Exploration and P-FCM

117

(9)

6.8 Linguistic Augmentation of Knowledge-Based Hints

126

(1)

6.9 Conclusions

127

(1)

References

127

(2)

7 Collaborative Clustering

129

(29)

7.1 Introduction and Rationale

129

(2)

7.2 Horizontal and Vertical Clustering

131

(1)

7.3 Horizontal Collaborative Clustering

132

(8)

7.3.1 Optimization Details

135

(2)

7.3.2 The Flow of Computing of Collaborative Clustering

137

(1)

7.3.3 Quantification of the Collaborative Phenomenon of Clustering

138

(2)

7.4 Experimental Studies

140

(10)

7.5 Further Enhancements of Horizontal Clustering

150

(1)

7.6 The Algorithm of Vertical Clustering

151

(2)

7.7 A Grid Model of Horizontal and Vertical Clustering

153

(2)

7.8 Consensus Clustering

155

(2)

7.9 Conclusions

157

(1)

References

157

(1)

8 Directional Clustering

158

(20)

8.1 Introduction

158

(1)

8.2 Problem Formulation

159

(4)

8.2.1 The Objective Function

160

(1)

8.2.2 The Logic Transformation Between Information Granules

161

(2)

8.3 The Algorithm

163

(3)

8.4 The Development Framework of Directional Clustering

166

(1)

8.5 Numerical Studies

167

(7)

8.6 Conclusions

174

(2)

References

176

(2)

9 Fuzzy Relational Clustering

178

(13)

9.1 Introduction and Problem Statement

178

(1)

9.2 FCM for Relational Data

179

(2)

9.3 Decomposition of Fuzzy Relational Patterns

181

(7)

9.3.1 Gradient-Based Solution to the Decomposition Problem

182

(2)

9.3.2 Neural Network Model of the Decomposition Problem

184

(4)

9.4 Comparative Analysis

188

(1)

9.5 Conclusions

189

(1)

References

189

(2)

10 Fuzzy Clustering of Heterogeneous Patterns

191

(18)

10.1 Introduction

191

(1)

10.2 Heterogeneous Data

192

(2)

10.3 Parametric Models of Granular Data

194

(1)

10.4 Parametric Mode of Heterogeneous Fuzzy Clustering

195

(3)

10.5 Nonparametric Heterogeneous Clustering

198

(9)

10.5.1 A Frame of Reference

198

(2)

10.5.2 Representation of Granular Data Through the Possibility-Necessity Transformation

200

(5)

10.5.3 Dereferencing

205

(2)

10.6 Conclusions

207

(1)

References

208

(1)

11 Hyperbox Models of Granular Data: The Tchebyschev FCM

209

(17)

11.1 Introduction

209

(1)

11.2 Problem Formulation

210

(1)

11.3 The Clustering Algorithm Detailed Considerations

211

(7)

11.4 Development of Granular Prototypes

218

(2)

11.5 Geometry of Information Granules

220

(3)

11.6 Granular Data Description: A General Model

223

(1)

11.7 Conclusions

223

(1)

References

224

(2)

12 Genetic Tolerance Fuzzy Neural Networks

226

(20)

12.1 Introduction

226

(1)

12.2 Operations of Thresholding and Tolerance: Fuzzy Logic-Based Generalizations

227

(4)

12.3 Topology of the Logic Network

231

(4)

12.4 Genetic Optimization

235

(1)

12.5 Illustrative Numeric Studies

236

(8)

12.6 Conclusions

244

(1)

References

245

(1)

13 Granular Prototyping

246

(24)

13.1 Introduction

246

(1)

13.2 Problem Formulation

247

(4)

13.2.1 Expressing Similarity Between Two Fuzzy Sets

247

(1)

13.2.2 Performance Index (Objective Function)

248

(3)

13.3 Prototype Optimization

251

(12)

13.4 Development of Granular Prototypes

263

(5)

13.4.1 Optimization of the Similarity Levels

263

(1)

13.4.2 An Inverse Similarity Problem

264

(4)

13.5 Conclusions

268

(1)

References

268

(2)

14 Granular Mappings

270

(13)

14.1 Introduction and Problem Statement

270

(1)

14.2 Possibility and Necessity Measures as the Computational Vehicles of Granular Representation

271

(1)

14.3 Building the Granular Mapping

272

(3)

14.4 Designing Multivariable Granular Mappings Through Fuzzy Clustering

275

(3)

14.5 Quantification of Granular Mappings

278

(1)

14.6 Experimental Studies

278

(2)

14.7 Conclusions

280

(2)

References

282

(1)

15 Linguistic Modeling

283

(14)

15.1 Introduction

283

(2)

15.2 Cluster-Based Representation of Input-Output Mapping

285

(2)

15.3 Conditional Clustering in the Development of a Blueprint of Granular Models

287

(3)

15.4 The Granular Neuron as a Generic Processing Element in Granular Networks

290

(3)

15.5 The Architecture of Linguistic Models Based on Conditional Fuzzy Clustering

293

(1)

15.6 Refinements of Linguistic Models

294

(1)

15.7 Conclusions

295

(1)

References

296

(1)

Bibliography

297

(18)

Index

315

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Knowledge-Based Clustering From Data to Information Granules

Summary

Author Biography

Table of Contents

Supplemental Materials