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| Preface | p. xv |
| Methodology and Mathematical Framework | |
| Granular Computing as an Emerging Prardigm of Information Processing | p. 1 |
| Introductory comments | p. 1 |
| Information granules are everywhere | p. 1 |
| Spatial granulation: Image processing and GIS | p. 2 |
| Temporal granulation | p. 2 |
| Formal models of information granules | p. 5 |
| Conceptual aspects of information granules | p. 6 |
| Size of information granules and their relevance | p. 6 |
| Usefulness of information granules | p. 7 |
| Defining a granular world | p. 8 |
| Granular computing: An information processing pyramid | p. 9 |
| Communication between granular worlds | p. 11 |
| Fundamental issues of traversing information pyramid: Encoding and decoding | p. 12 |
| Interoperability between different formal platforms of information granules | p. 15 |
| Conclusions | p. 17 |
| References | p. 17 |
| Sets and intervals | p. 19 |
| Historical background | p. 19 |
| The formalism of sets | p. 22 |
| Basic set operations | p. 23 |
| Functional mapping of sets | p. 25 |
| Arithmetical operations on sets | p. 27 |
| Set enclosure | p. 27 |
| Interval analysis | p. 29 |
| Basic interval operations | p. 29 |
| Arithmetical operations on intervals | p. 32 |
| Interval vectors | p. 34 |
| Interval matrices | p. 36 |
| Enclosure of functions | p. 40 |
| Centered enclosures | p. 41 |
| Space subdivision enclosures | p. 42 |
| Conclusions | p. 44 |
| References | p. 45 |
| Fuzzy Sets | p. 47 |
| The concept and formalism | p. 47 |
| The description and geometry of fuzzy sets | p. 51 |
| Main classes of membership functions | p. 54 |
| Operations on fuzzy sets | p. 58 |
| Information granularity and fuzzy sets | p. 62 |
| Relationships between fuzzy sets in the same space | p. 65 |
| Fuzzy sets and linguistic variables | p. 66 |
| Transformations of fuzzy sets in the same space | p. 67 |
| Fuzzy arithmetic | p. 69 |
| Fuzzy relations and relational calculus | p. 71 |
| Fuzzy sets and multivalued logic | p. 74 |
| Calibration of fuzzy sets | p. 75 |
| The embedding principle | p. 76 |
| Conclusions | p. 77 |
| References | p. 78 |
| Rough Sets | p. 81 |
| Introduction | p. 81 |
| The concept | p. 81 |
| Information systems | p. 84 |
| Rough sets as set approximations | p. 87 |
| Characterization of rough sets | p. 88 |
| Set comparisons in the setting of rough sets | p. 90 |
| Reduction of attribute spaces and reducts | p. 92 |
| Rough functions | p. 93 |
| Conclusions | p. 95 |
| References | p. 96 |
| Generalisations of Information Granules | p. 99 |
| Interval-valued fuzzy sets | p. 99 |
| Fuzzy sets of type-2 and higher orders | p. 101 |
| Fuzzy sets of level-2 and higher | p. 103 |
| Fuzzy sets and rough sets | p. 104 |
| Shadowed sets | p. 107 |
| Operations on shadowed sets | p. 112 |
| Transformations of shadowed sets | p. 113 |
| Probabilistic sets | p. 114 |
| Intuitionistic fuzzy sets | p. 115 |
| Probability of granular constructs: Granularity and their experimental relevance | p. 119 |
| Concluding comments | p. 123 |
| References | p. 123 |
| Algorithms of Information Granulation | |
| From Numbers to Information Granules | p. 125 |
| Introductory comments | p. 125 |
| Information granules and information granulation | p. 126 |
| The principle of granular clustering | p. 128 |
| Conceptual design | p. 128 |
| Interpretation and validation of granular clustering | p. 130 |
| The computational aspects of granular computing | p. 131 |
| Defining compatibility between information granules | p. 131 |
| Expressing inclusion of information granules | p. 139 |
| The granular analysis | p. 141 |
| Characterization of hyperboxes | p. 142 |
| Granular feature analysis | p. 142 |
| Experimental studies | p. 144 |
| Synthetic data | p. 144 |
| Boston housing data | p. 151 |
| Conclusions | p. 158 |
| References | p. 159 |
| Recursive Information Granulation | p. 161 |
| Introduction | p. 161 |
| Example application domains | p. 162 |
| Information granules: Design and characterization | p. 164 |
| Building set-based information granules | p. 164 |
| Assessment and interpretation of information granule through fuzzy clustering | p. 174 |
| Granular time series | p. 179 |
| Time-domain granulation | p. 179 |
| Phase-space granulation | p. 183 |
| Numerical studies | p. 184 |
| Conclusions | p. 190 |
| References | p. 190 |
| Granular Prototyping in Fuzzy Clustering | p. 193 |
| Introduction | p. 193 |
| Problem formulation | p. 194 |
| Expressing similarity between two fuzzy sets | p. 194 |
| Performance index (objective function) | p. 196 |
| Prototype optimisation | p. 198 |
| The development of granular prototypes | p. 208 |
| Optimization of the similarity levels | p. 209 |
| An inverse similarity problem | p. 210 |
| Conclusions | p. 213 |
| References | p. 214 |
| Logic-Based Fuzzy Clustering | p. 217 |
| Introduction and problem formulation | p. 217 |
| The algorithm | p. 219 |
| Experimental studies | p. 226 |
| Conclusions | p. 232 |
| References | p. 232 |
| Semantical Stability of Information Granules | p. 235 |
| Introduction | p. 235 |
| Information granulation: Design and validation | p. 237 |
| Set approximation of fuzzy sets | p. 239 |
| Algorithmic issues of information granulation: Design and validation | p. 241 |
| The design of fuzzy sets - information granules | p. 241 |
| The validation phase | p. 244 |
| Experiments | p. 245 |
| Synthetic one-dimensional data | p. 245 |
| Real-world data | p. 248 |
| Conclusions | p. 253 |
| References | p. 253 |
| Granular World Communications | |
| Communications between granular worlds: Fundamentals | p. 255 |
| Introduction | p. 255 |
| Representation of fuzzy sets in the set-theoretic framework | p. 256 |
| Communication with a numeric world | p. 261 |
| Conclusions | p. 265 |
| References | p. 265 |
| Networking of Granular Worlds: Collaborative Clustering | p. 267 |
| Introduction | p. 267 |
| The horizontal collaborative clustering | p. 270 |
| The notation | p. 270 |
| Optimization details of the collaborative clustering | p. 273 |
| The detailed clustering algorithm: A flow of computing | p. 275 |
| Quantification of the collaborative phenomenon of the clustering | p. 276 |
| Numerical examples of horizontal collaboration | p. 277 |
| Vertical collaborative clustering | p. 284 |
| The clustering algorithm | p. 284 |
| Numerical experiments with vertical collaboration | p. 289 |
| Vertical and horizontal clustering: Collaboration space and data confidentiality and security | p. 295 |
| Conclusions | p. 298 |
| References | p. 299 |
| Directional Models of Granular Communication | p. 301 |
| Introduction | p. 301 |
| Problem formulation | p. 302 |
| The objective function and its generalization | p. 303 |
| The logic transformation | p. 304 |
| The algorithm | p. 306 |
| The overall development framework: A flow of optimisation activities | p. 309 |
| Experimental studies | p. 310 |
| Conclusions | p. 321 |
| References | p. 322 |
| Intelligent Agents and Granular Worlds | p. 323 |
| Introduction | p. 323 |
| Communication between the agents in the granular environment | p. 324 |
| A fuzzy state machine as a generic model of an intelligent agent | p. 328 |
| The fuzzy JK flip-flop and its dynamics | p. 330 |
| The development of Moore type fuzzy state machines | p. 334 |
| The architecture | p. 334 |
| A logic processor and its detailed topology | p. 335 |
| A fuzzy Moore state machine | p. 337 |
| The learning scheme | p. 337 |
| Conclusions | p. 346 |
| References | p. 347 |
| Granular Systems Applications | |
| Self-Organising Maps in the Design and Processing of Granular Information | p. 349 |
| Introduction | p. 349 |
| Self-organizing maps | p. 349 |
| Revealing structure in data by cluster growing | p. 354 |
| Associated self-organizing maps | p. 355 |
| Weight maps | p. 355 |
| Region (clustering) map | p. 356 |
| Data distribution map | p. 357 |
| Experiments--Synthetic and Machine Learning data | p. 358 |
| Case study: Analysis of software quality via software measures | p. 364 |
| Software measures | p. 365 |
| Visualising relationships between software measures with SOMs | p. 365 |
| Case study: A granular analysis of ECG data | p. 369 |
| Conclusions | p. 375 |
| References | p. 376 |
| Temporal Granulation and Signal Analysis | p. 377 |
| Introductory notes | p. 377 |
| Granulation of signals in spatial domain | p. 378 |
| The development of data-justifiable information granules: A formulation | p. 378 |
| The detailed granulation algorithm | p. 380 |
| Granular models of signals | p. 387 |
| Predictive description of granular models | p. 388 |
| Condensation of numeric signals | p. 388 |
| Experimental studies | p. 389 |
| Rough sets in signal granulation | p. 395 |
| Conclusions | p. 396 |
| References | p. 397 |
| Granular Data Compression | p. 399 |
| Introduction | p. 399 |
| Fuzzy relational equations: A brief overview | p. 399 |
| Relational calculus in image compression | p. 402 |
| Experiments | p. 407 |
| Conclusions | p. 415 |
| References | p. 416 |
| Interval State Estimation in Systems Modelling | p. 417 |
| Introduction | p. 417 |
| Estimation of the state uncertainty set | p. 419 |
| Monte Carlo method | p. 421 |
| Linear Programming method | p. 422 |
| Ellipsoid method | p. 427 |
| Sensitivity Matrix method | p. 433 |
| Real-life application | p. 436 |
| Conclusions | p. 443 |
| References | p. 444 |
| Epilogue | p. 447 |
| Index | p. 449 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
ISBN: 9781402072734
ISBN-10: 1402072732
Series: KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE
Published: 30th November 2002
Format: Hardcover
Language: English
Number of Pages: 480
Audience: Professional and Scholarly
Publisher: Springer Nature B.V.
Country of Publication: US
Dimensions (cm): 23.5 x 16.64 x 2.54
Weight (kg): 0.84
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