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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 | |