Granular Computing: An Introduction

463 Pages·2003·28.12 MB·English

by  Andrzej Bargiela

GRANULAR COMPUTING
An Introduction
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
GRANULAR COMPUTING
An Introduction
by
Andrzej Bargiela
The Nottingham Trent University
Nottingham, United Kingdom
Witold Pedrycz
University ofA /berta
Edmonton, AB, Canada
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging-in-Publication Data
A C.I.P. Catalogue record for this book is avaiIable
from the Library of Congress.
Bargiela, Andrzej and Pedrycz, Witold
GRANULAR COMPUTING: An Introduction
ISBN 978-1-4613-5361-4 ISBN 978-1-4615-1033-8 (eBook)
DOI 10.1007/978-1-4615-1033-8
Copyright © 2003 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 2003
Softcover reprint of the hardcover 1s t edition 2003
All rights reserved. No part ofthis work may be reproduced, stored in a retrieval
system, or transmitted in any form or by any means, electronic, mechanical,
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Printed on acid-free paper.
TO OUR FAMILIES
The Authors
CONTENTS
Preface xv
PART I -METHODOLOGY AND MATHEMATICAL FRAMEWORK
Chapter 1 GRANULAR COMPUTING AS AN EMERGING PRARDIGM OF
INFORMATION PROCESSING 1
1.1 Introductory comments 1
1.2 Information granules are everywhere 1
Spatial granulation: Image processing and GIS 2
Temporal granulation 2
1.3 Formal models of information granules 5
1.4 Conceptual aspects of information granules 6
Size ofi nformation granules and their relevance 6
Usefulness of information granules 7
1.5 Defining a granular world 8
1.6 Granular computing: An information processing pyramid 9
1.7 Communication between granular worlds 11
Fundamental issues oft raversing information pyramid:
Encoding and decoding 12
Interoperability between different formal platforms of
information granules 15
1.8 Conclusions 17
References 17
Chapter 2 SETS AND INTERVALS 19
2.1 Historical background 19
2.2 The formalism of sets 22
Basic set operations 23
Functional mapping ofs ets 25
Arithmetical operations on sets 27
2.3 Set enclosure 27
2.4 Interval analysis 29
Basic interval operations 29
Arithmetical operations on intervals 32
2.5 Interval vectors 34
viii Contents
2.6 Interval matrices 36
2.7 Enclosure of functions 40
Centered enclosures 41
Space subdivision enclosures 42
2.8 Conclusions 44
References 45
Chapter 3 Fuzzy SETS 47
3.1 The concept and formalism 47
3.2 The description and geometry of fuzzy sets 51
3.3 Main classes of membership functions 54
3.4 Operations on fuzzy sets 58
3.5 Information granularity and fuzzy sets 62
3.6 Relationships between fuzzy sets in the same space 65
3.7 Fuzzy sets and linguistic variables 66
3.8 Transformations of fuzzy sets in the same space 67
3.9 Fuzzy arithmetic 69
3.10 Fuzzy relations and relational calculus 71
3.11 Fuzzy sets and multivalued logic 74
3.12 Calibration of fuzzy sets 75
3.13 The embedding principle 76
3.14 Conclusions 77
References 78
Chapter 4 ROUGH SETS 81
4.1 Introduction 81
4.2 The concept 81
4.3 Information systems 84
4.4 Rough sets as set approximations 87
4.5 Characterization of rough sets 88
4.6 Set comparisons in the setting of rough sets 90
4.7 Reduction of attribute spaces and reducts 92
4.8 Rough functions 93
4.9 Conclusions 95
References 96
Chapter 5 GENERALISATIONS OF INFORMATION GRANULES 99
5.1 Interval-valued fuzzy sets 99
5.2 Fuzzy sets oftype-2 and higher orders 101
5.3 Fuzzy sets oflevel-2 and higher 103
5.4 Fuzzy sets and rough sets 104
5.5 Shadowed sets 107
Operations on shadowed sets 112
Transformations ofs hadowed sets 113
Contents ix
5.6 Probabilistic sets 114
5.7 Intuitionistic fuzzy sets 115
5.8 Probability of granular constructs: Granularity and their
experimental relevance 119
5.9 Concluding comments 123
References 123
PART II -ALGORITHMS OF INFORMATION GRANULATION
Chapter 6 FROM NUMBERS TO INFORMATION GRANULES 125
6.1 Introductory comments 125
6.2 Information granules and information granulation 126
6.3 The principle of granular clustering 128
Conceptual design 128
Interpretation and validation ofg ranular clustering 130
6.4 The computational aspects of granular computing 131
Defining compatibility between information granules 131
Expressing inclusion of information granules 139
6.5 The granular analysis 141
Characterization ofh yperboxes 142
Granular feature analysis 142
6.6 Experimental studies 144
Synthetic data 144
Boston housing data 151
6.7 Conclusions 158
References 159
Chapter 7 RECURSIVE INFORMATION GRANULATION 161
7.1 Introduction 161
7.2 Example application domains 162
7.3 Information granules: Design and characterization 164
Building set-based information granules 164
7.4 Assessment and interpretation of information granule
through fuzzy clustering 174
7.5 Granular time series 179
Time-domain granulation 179
Phase-space granulation 183
7.6 Numerical studies 184
7.7 Conclusions 190
References 190
x Contents
Chapter 8 GRANULAR PROTOTYPING IN FuzZY CLUSTERING 193
8.1 Introduction 193
8.2 Problem formulation 194
Expressing similarity between two fuzzy sets 194
Performance index (objective function) 196
8.3 Prototype optimisation 198
8.4 The development of granular prototypes 208
Optimization oft he similarity levels 209
An inverse similarity problem 210
8.5 Conclusions 213
References 214
Chapter 9 LOGIC-BASED FUZZY CLUSTERING 217
9.1 Introduction and problem formulation 217
9.2 The algorithm 219
9.3 Experimental studies 226
9.4 Conclusions 232
References 232
Chapter 10 SEMANTICAL STABILITY OF INFORMATION GRANULES 235
10.1 Introduction 235
10.2 Information granulation: Design and validation 237
10.3 Set approximation of fuzzy sets 239
10.4 Algorithmic issues of information granulation: Design
and validation 241
The design off uzzy sets - information granules 241
The validation phase 244
10.5 Experiments 245
Synthetic one-dimensional data 245
Real-world data 248
10.6 Conclusions 253
References 253
PART III - GRANULAR WORLD COMMUNICATIONS
Chapter 11 COMMUNICATIONS BETWEEN GRANULAR WORLDS:
FUNDAMENTALS 255
11.1 Introduction 255
11.2 Representation of fuzzy sets in the set-theoretic framework 256
11.3 Communication with a numeric world 261
11.4 Conclusions 265
References 265
Contents xi
Chapter 12 NETWORKING OF GRANULAR WORLDS:
COLLABORATIVE CLUSTERING 267
12.1 Introduction 267
12.2 The horizontal collaborative clustering 270
The notation 270
Optimization details oft he collaborative clustering 273
The detailed clustering algorithm: Ajlow ofc omputing 275
Quantification oft he collaborative phenomenon oft he clustering 276
Numerical examples ofh orizontal collaboration 277
12.3 Vertical collaborative clustering 284
The clustering algorithm 284
Numerical experiments with vertical collaboration 289
12.4 Vertical and horizontal clustering: Collaboration space and
data confidentiality and security 295
12.5 Conclusions 298
References 299
Chapter 13 DIRECTIONAL MODELS OF GRANULAR COMMUNICATION 301
13.1 Introduction 301
13.2 Problem formulation 302
The objective function and its generalization 303
The logic transformation 304
13.3 The algorithm 306
13.4 The overall development framework: A flow of optimisation
activities 309
13.5 Experimental studies 310
13.6 Conclusions 321
References 322
Chapter 14 INTELLIGENT AGENTS AND GRANULAR WORLDS 323
14.1 Introduction 323
14.2 Communication between the agents in the granular environment 324
14.3 A fuzzy state machine as a generic model of an intelligent agent 328
14.4 The fuzzy JK flip-flop and its dynamics 330
14.5 The development of Moore type fuzzy state machines 334
The architecture 334
A logic processor and its detailed topology 335
A fuzzy Moore state machine 337
14.6 The learning scheme 337
14.7 Conclusions 346
References 347