Granular Computing: An Introduction

Front Cover
Springer Science & Business Media, Nov 30, 2002 - Computers - 452 pages
This book is about Granular Computing (GC) - an emerging conceptual and of information processing. As the name suggests, GC concerns computing paradigm processing of complex information entities - information granules. In essence, information granules arise in the process of abstraction of data and derivation of knowledge from information. Information granules are everywhere. We commonly use granules of time (seconds, months, years). We granulate images; millions of pixels manipulated individually by computers appear to us as granules representing physical objects. In natural language, we operate on the basis of word-granules that become crucial entities used to realize interaction and communication between humans. Intuitively, we sense that information granules are at the heart of all our perceptual activities. In the past, several formal frameworks and tools, geared for processing specific information granules, have been proposed. Interval analysis, rough sets, fuzzy sets have all played important role in knowledge representation and processing. Subsequently, information granulation and information granules arose in numerous application domains. Well-known ideas of rule-based systems dwell inherently on information granules. Qualitative modeling, being one of the leading threads of AI, operates on a level of information granules. Multi-tier architectures and hierarchical systems (such as those encountered in control engineering), planning and scheduling systems all exploit information granularity. We also utilize information granules when it comes to functionality granulation, reusability of information and efficient ways of developing underlying information infrastructures.
 

Contents

GRANULAR COMPUTING AS AN EMERGING PARADIGM OF INFORMATION PROCESSING
1
Image Processing and GIS
2
13 FORMAL MODELS OF INFORMATION GRANULES
5
14 CONCEPTUAL ASPECTS OF INFORMATION GRANULES
6
Usefulness of Information Granules
7
15 DEFINING A GRANULAR WORLD
8
AN INFORMATION PROCESSING PYRAMID
9
17 COMMUNICATION BETWEEN GRANULAR WORLDS
11
An Inverse Similarity Problem
210
85 CONCLUSIONS
213
REFERENCES
214
LOGICBASED FUZZY CLUSTERING
217
92 The Algorithm
219
93 EXPERIMENTAL STUDIES
226
94 CONCLUSIONS
232
SEMANTICAL STABILITY OF INFORMATION GRANULES
235

Encoding and Decoding
12
Interoperability Between Different Formal Platforms of Information Granules
15
18 CONCLUSIONS
17
SETS AND INTERVALS
19
22 THE FORMALISM OF SETS
22
Basic Set Operations
23
Functional Mapping of Sets
25
Arithmetical Operations on Sets
27
24 INTERVAL ANALYSIS
29
Arithmetical Operations on Intervals
32
25 INTERVAL VECTORS
34
26 INTERVAL MATRICES
36
27 ENCLOSURE OF FUNCTIONS
40
Centered Enclosures
41
Space Subdivision Enclosures
42
28 CONCLUSIONS
44
REFERENCES
45
FUZZY SETS
47
32 THE DESCRIPTION AND GEOMETRY OF FUZZY SETS
51
33 MAIN CLASSES OF MEMBERSHIP FUNCTIONS
54
34 OPERATIONS ON FUZZY SETS
58
35 INFORMATION GRANULARITY AND FUZZY SETS
62
36 RELATIONSHIPS BETWEEN FUZZY SETS IN THE SAME SPACE
65
37 FUZZY SETS AND LINGUISTIC VARIABLES
66
38 TRANSFORMATIONS OF FUZZY SETS BETWEEN SPACES
67
39 FUZZY ARITHMETIC
69
310 FUZZY RELATIONS AND RELATIONAL CALCULUS
71
311 FUZZY SETS AND MULTIVALUED LOGIC
74
ESTIMATION OF MEMBERSHIP FUNCTIONS AND AN ADJUSTMENT OF UNIVERSE OF DISCOURSE
75
313 THE EMBEDDING PRINCIPLE
76
314 CONCLUSIONS
77
References
78
ROUGH SETS
81
43 INFORMATION SYSTEMS
84
44 ROUGH SETS AS SET APPROXIMATIONS
87
45 CHARACTERIZATION OF ROUGH SETS
88
46 SET COMPARISONS IN THE SETTING OF ROUGH SETS
90
47 REDUCTION OF ATTRIBUTE SPACES AND REDUCTS
92
48 ROUGH FUNCTIONS
93
49 CONCLUSIONS
95
References
96
GENERALIZATIONS OF INFORMATION GRANULES
99
52 FUZZY SETS OF TYPE2 AND HIGHER ORDERS
101
53 FUZZY SETS OF LEVEL 2 AND HIGHER
103
54 FUZZY SETS AND ROUGH SETS
104
55 SHADOWED SETS
107
Operations on Shadowed Sets
112
Transformations of Shadowed Sets
113
56 PROBABILISTIC SETS
114
57 INTUITIONISTIC FUZZY SETS
115
GRANULARITY AND THEIR EXPERIMENTAL RELEVANCE
119
59 CONCLUDING COMMENTS
123
ALGORITHMS OF INFORMATION GRANULATION
124
FROM NUMBERS TO INFORMATION GRANULES
125
62 INFORMATION GRANULES AND INFORMATION GRANULATION
126
63 THE PRINCIPLE OF GRANULAR CLUSTERING
128
Interpretation and Validation of Granular Clustering
130
64 THE COMPUTATIONAL ASPECTS OF GRANULAR COMPUTING
131
Expressing Inclusion of Information Granules
139
65 THE GRANULAR ANALYSIS
141
Characterization of Hyperboxes
142
66 EXPERIMENTAL STUDIES
144
Boston Housing Data
151
67 CONCLUSIONS
158
References
159
RECURSIVE INFORMATION GRANULATION
161
72 EXAMPLE APPLICATION DOMAINS
162
DESIGN AND CHARACTERIZATION
164
74 ASSESSMENT AND INTERPRETATION OF INFORMATION GRANULES THROUGH FUZZY CLUSTERING
174
75 GRANULAR TIME SERIES
179
PhaseSpace Granulation
183
66 NUMERICAL STUDIES
184
67 CONCLUSIONS
190
GRANULAR PROTOTYPING IN FUZZY CLUSTERING
193
82 PROBLEM FORMULATION
194
Performance Index objective function
196
83 PROTOTYPE OPTIMIZATION
198
84 THE DEVELOPMENT OF GRANULAR PROTOTYPES
208
Optimization of the Similarity Levels
209
DESIGN AND VALIDATION
237
103 SET APPROXIMATION OF FUZZY SETS
239
DESIGN AND VALIDATION
241
The validation phase
244
105 EXPERIMENTS
245
Realworld data
248
106 CONCLUSIONS
253
GRANULAR WORLD COMMUNICATIONS
254
COMMUNICATIONS BETWEEN GRANULAR WORLDS FUNDAMENTALS
255
112 REPRESENTATION OF FUZZY SETS IN THE SETTHEORETIC FRAMEWORK
256
113 COMMUNICATION WITH A NUMERIC WORLD
261
114 CONCLUSIONS
265
NETWORKING OF GRANULAR WORLDS COLLABORATIVE CLUSTERING
267
122 THE HORIZONTAL COLLABORATIVE CLUSTERING
270
Optimization Details of the Collaborative Clustering
273
A Flow of Computing
275
Quantification of the Collaborative Phenomenon of the Clustering
276
Numerical Examples of Horizontal Collaboration
277
123 VERTICAL COLLABORATIVE CLUSTERING
284
Numeric Experiments with Vertical Collaboration
289
COLLABORATION SPACE AND DATA CONFIDENTIALITY AND SECURITY
295
125 CONCLUSIONS
298
REFERENCES
299
DIRECTIONAL MODELS OF GRANULAR COMMUNICATION
301
132 PROBLEM FORMULATION
302
The Objective Function and its Generalization
303
The Logic Transformation
304
133 THE ALGORITHM
306
A FLOW OF OPTIMIZATION ACTIVITIES
309
135 EXPERIMENTAL STUDIES
310
136 CONCLUSIONS
321
REFERENCES
322
INTELLIGENT AGENTS AND GRANULAR WORLDS
323
142 COMMUNICATION BETWEEN THE AGENTS IN THE GRANULAR ENVIRONMENT
324
143 A FUZZY STATE MACHINE AS A GENERIC MODEL OF AN INTELLIGENT AGENT
328
144 THE FUZZY JK FLIPFLOP AND ITS DYNAMICS
330
145 THE DEVELOPMENT OF MOORE TYPE FUZZY STATE MACHINES
334
A Logic Processor and its Detailed Topology
335
A fuzzy Moore State Machine
337
147 CONCLUSIONS
346
REFERENCES
347
GRANULAR SYSTEMS APPLICATIONS
348
SELFORGANIZING MAPS IN THE DESIGN AND PROCESSING OF GRANULAR INFORMATION
349
Revealing Structure in Data by Cluster Growing
354
153 ASSOCIATED SELFORGANIZING MAPS
355
Region clustering Map
356
Data Distribution Map
357
154 EXPERIMENTSSYNTHETIC AND MACHINE LEARNING DATA
358
ANALYSIS OF SOFTWARE QUALITY VIA SOFTWARE MEASURES
364
Software Measures
365
A GRANULAR ANALYSIS OF ECG DATA
369
157 CONCLUSIONS
375
REFERENCES
376
TEMPORAL GRANULATION AND SIGNAL ANALYSIS
377
162 GRANULATION OF SIGNALS IN SPATIAL DOMAIN
378
163 The detailed granulation algorithm
380
164 GRANULAR MODELS OF SIGNALS
387
Predictive Description of Granular Models
388
165 EXPERIMENTAL STUDIES
389
166 ROUGH SETS IN SIGNAL GRANULATION
395
167 CONCLUSIONS
396
References
397
GRANULAR DATA COMPRESSION
399
173 RELATIONAL CALCULUS IN IMAGE COMPRESSION
402
174 EXPERIMENTS
407
175 CONCLUSIONS
415
References
416
INTERVAL STATE ESTIMATION IN SYSTEMS MODELLING
417
182 ESTIMATION OF THE STATE UNCERTAINTY SET
419
Monte Carlo Method
421
Linear Programming Method
422
Ellipsoid Method
427
Sensitivity Matrix Method
433
183 REALLIFE APPLICATION
436
184 CONCLUSIONS
443
REFERENCES
444
EPILOGUE
447
INDEX
449
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