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| Preface | |
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| Methodology and Mathematical Framework | |
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| Granular Computing as an Emerging Prardigm of Information Processing | |
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| Introductory comments | |
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| Information granules are everywhere | |
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| Spatial granulation: Image processing and GIS | |
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| Temporal granulation | |
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| Formal models of information granules | |
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| Conceptual aspects of information granules | |
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| Size of information granules and their relevance | |
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| Usefulness of information granules | |
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| Defining a granular world | |
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| Granular computing: An information processing pyramid | |
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| Communication between granular worlds | |
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| Fundamental issues of traversing information pyramid: Encoding and decoding | |
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| Interoperability between different formal platforms of information granules | |
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| Conclusions | |
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| References | |
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| Sets and intervals | |
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| Historical background | |
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| The formalism of sets | |
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| Basic set operations | |
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| Functional mapping of sets | |
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| Arithmetical operations on sets | |
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| Set enclosure | |
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| Interval analysis | |
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| Basic interval operations | |
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| Arithmetical operations on intervals | |
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| Interval vectors | |
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| Interval matrices | |
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| Enclosure of functions | |
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| Centered enclosures | |
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| Space subdivision enclosures | |
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| Conclusions | |
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| References | |
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| Fuzzy Sets | |
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| The concept and formalism | |
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| The description and geometry of fuzzy sets | |
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| Main classes of membership functions | |
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| Operations on fuzzy sets | |
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| Information granularity and fuzzy sets | |
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| Relationships between fuzzy sets in the same space | |
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| Fuzzy sets and linguistic variables | |
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| Transformations of fuzzy sets in the same space | |
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| Fuzzy arithmetic | |
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| Fuzzy relations and relational calculus | |
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| Fuzzy sets and multivalued logic | |
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| Calibration of fuzzy sets | |
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| The embedding principle | |
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| Conclusions | |
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| References | |
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| Rough Sets | |
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| Introduction | |
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| The concept | |
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| Information systems | |
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| Rough sets as set approximations | |
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| Characterization of rough sets | |
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| Set comparisons in the setting of rough sets | |
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| Reduction of attribute spaces and reducts | |
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| Rough functions | |
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| Conclusions | |
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| References | |
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| Generalisations of Information Granules | |
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| Interval-valued fuzzy sets | |
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| Fuzzy sets of type-2 and higher orders | |
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| Fuzzy sets of level-2 and higher | |
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| Fuzzy sets and rough sets | |
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| Shadowed sets | |
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| Operations on shadowed sets | |
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| Transformations of shadowed sets | |
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| Probabilistic sets | |
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| Intuitionistic fuzzy sets | |
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| Probability of granular constructs: Granularity and their experimental relevance | |
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| Concluding comments | |
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| References | |
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| Algorithms of Information Granulation | |
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| From Numbers to Information Granules | |
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| Introductory comments | |
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| Information granules and information granulation | |
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| The principle of granular clustering | |
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| Conceptual design | |
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| Interpretation and validation of granular clustering | |
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| The computational aspects of granular computing | |
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| Defining compatibility between information granules | |
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| Expressing inclusion of information granules | |
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| The granular analysis | |
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| Characterization of hyperboxes | |
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| Granular feature analysis | |
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| Experimental studies | |
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| Synthetic data | |
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| Boston housing data | |
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| Conclusions | |
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| References | |
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| Recursive Information Granulation | |
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| Introduction | |
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| Example application domains | |
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| Information granules: Design and characterization | |
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| Building set-based information granules | |
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| Assessment and interpretation of information granule through fuzzy clustering | |
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| Granular time series | |
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| Time-domain granulation | |
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| Phase-space granulation | |
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| Numerical studies | |
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| Conclusions | |
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| References | |
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| Granular Prototyping in Fuzzy Clustering | |
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| Introduction | |
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| Problem formulation | |
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| Expressing similarity between two fuzzy sets | |
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| Performance index (objective function) | |
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| Prototype optimisation | |
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| The development of granular prototypes | |
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| Optimization of the similarity levels | |
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| An inverse similarity problem | |
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| Conclusions | |
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| References | |
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| Logic-Based Fuzzy Clustering | |
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| Introduction and problem formulation | |
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| The algorithm | |
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| Experimental studies | |
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| Conclusions | |
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| References | |
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| Semantical Stability of Information Granules | |
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| Introduction | |
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| Information granulation: Design and validation | |
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| Set approximation of fuzzy sets | |
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| Algorithmic issues of information granulation: Design and validation | |
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| The design of fuzzy sets - information granules | |
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| The validation phase | |
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| Experiments | |
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| Synthetic one-dimensional data | |
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| Real-world data | |
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| Conclusions | |
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| References | |
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| Granular World Communications | |
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| Communications between granular worlds: Fundamentals | |
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| Introduction | |
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| Representation of fuzzy sets in the set-theoretic framework | |
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| Communication with a numeric world | |
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| Conclusions | |
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| References | |
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| Networking of Granular Worlds: Collaborative Clustering | |
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| Introduction | |
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| The horizontal collaborative clustering | |
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| The notation | |
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| Optimization details of the collaborative clustering | |
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| The detailed clustering algorithm: A flow of computing | |
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| Quantification of the collaborative phenomenon of the clustering | |
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| Numerical examples of horizontal collaboration | |
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| Vertical collaborative clustering | |
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| The clustering algorithm | |
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| Numerical experiments with vertical collaboration | |
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| Vertical and horizontal clustering: Collaboration space and data confidentiality and security | |
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| Conclusions | |
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| References | |
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| Directional Models of Granular Communication | |
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| Introduction | |
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| Problem formulation | |
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| The objective function and its generalization | |
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| The logic transformation | |
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| The algorithm | |
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| The overall development framework: A flow of optimisation activities | |
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| Experimental studies | |
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| Conclusions | |
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| References | |
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| Intelligent Agents and Granular Worlds | |
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| Introduction | |
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| Communication between the agents in the granular environment | |
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| A fuzzy state machine as a generic model of an intelligent agent | |
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| The fuzzy JK flip-flop and its dynamics | |
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| The development of Moore type fuzzy state machines | |
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| The architecture | |
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| A logic processor and its detailed topology | |
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| A fuzzy Moore state machine | |
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| The learning scheme | |
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| Conclusions | |
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| References | |
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| Granular Systems Applications | |
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| Self-Organising Maps in the Design and Processing of Granular Information | |
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| Introduction | |
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| Self-organizing maps | |
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| Revealing structure in data by cluster growing | |
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| Associated self-organizing maps | |
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| Weight maps | |
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| Region (clustering) map | |
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| Data distribution map | |
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| Experiments--Synthetic and Machine Learning data | |
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| Case study: Analysis of software quality via software measures | |
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| Software measures | |
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| Visualising relationships between software measures with SOMs | |
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| Case study: A granular analysis of ECG data | |
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| Conclusions | |
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| References | |
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| Temporal Granulation and Signal Analysis | |
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| Introductory notes | |
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| Granulation of signals in spatial domain | |
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| The development of data-justifiable information granules: A formulation | |
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| The detailed granulation algorithm | |
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| Granular models of signals | |
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| Predictive description of granular models | |
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| Condensation of numeric signals | |
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| Experimental studies | |
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| Rough sets in signal granulation | |
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| Conclusions | |
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| References | |
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| Granular Data Compression | |
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| Introduction | |
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| Fuzzy relational equations: A brief overview | |
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| Relational calculus in image compression | |
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| Experiments | |
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| Conclusions | |
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| References | |
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| Interval State Estimation in Systems Modelling | |
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| Introduction | |
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| Estimation of the state uncertainty set | |
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| Monte Carlo method | |
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| Linear Programming method | |
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| Ellipsoid method | |
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| Sensitivity Matrix method | |
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| Real-life application | |
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| Conclusions | |
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| References | |
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| Epilogue | |
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| Index | |