Preface |
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xiii | |
The Author |
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xvii | |
Foreword |
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xix | |
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1 Information Granularity, Information Granules, and Granular Computing |
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1 | (18) |
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1.1 Information Granularity and the Discipline of Granular Computing |
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1 | (4) |
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1.2 Formal Platforms of Information Granularity |
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5 | (3) |
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1.3 Information Granularity and Its Quantification |
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8 | (1) |
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1.4 Information Granules and a Principle of the Least Commitment |
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9 | (1) |
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1.5 Information Granules of Higher Type and Higher Order |
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10 | (2) |
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1.6 Hybrid Models of Information Granules |
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12 | (1) |
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1.7 A Design of Information Granules |
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12 | (1) |
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1.8 The Granulation-Degranulation Principle |
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13 | (1) |
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1.9 Information Granularity in Data Representation and Processing |
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14 | (2) |
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1.10 Optimal Allocation of Information Granularity |
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16 | (1) |
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16 | (3) |
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17 | (2) |
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2 Key Formalisms for Representation of Information Granules and Processing Mechanisms |
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19 | (28) |
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2.1 Sets and Interval Analysis |
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19 | (2) |
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21 | (3) |
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2.3 Fuzzy Sets: A Departure from the Principle of Dichotomy |
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24 | (12) |
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2.3.1 Membership Functions and Classes of Fuzzy Sets |
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26 | (2) |
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2.3.2 Selected Descriptors of Fuzzy Sets |
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28 | (3) |
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2.3.3 Fuzzy Sets as a Family of α - Cuts |
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31 | (3) |
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2.3.4 Triangular Norms and Triangular Conorms as Models of Operations on Fuzzy Sets |
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34 | (2) |
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36 | (3) |
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2.5 Shadowed Sets as a Three-Valued Logic Characterization of Fuzzy Sets |
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39 | (5) |
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2.5.1 Defining Shadowed Sets |
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40 | (2) |
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2.5.2 The Development of Shadowed Sets |
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42 | (2) |
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44 | (3) |
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45 | (2) |
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3 Information Granules of Higher Type and Higher Order, and Hybrid Information Granules |
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47 | (14) |
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3.1 Fuzzy Sets of Higher Order |
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47 | (3) |
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3.2 Rough Fuzzy Sets and Fuzzy Rough Sets |
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50 | (1) |
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51 | (2) |
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3.4 Interval-Valued Fuzzy Sets |
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53 | (1) |
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54 | (1) |
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3.6 Hybrid Models of Information Granules: Probabilistic and Fuzzy Set Information Granules |
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55 | (2) |
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3.7 Realization of Fuzzy Models with Information Granules of Higher Type and Higher Order |
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57 | (2) |
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59 | (2) |
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60 | (1) |
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4 Representation of Information Granules |
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61 | (14) |
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4.1 Description of Information Granules by a Certain Vocabulary of Information Granules |
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61 | (5) |
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4.2 Information Granulation-Degranulation Mechanism in the Presence of Numeric Data |
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66 | (4) |
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4.3 Granulation-Degranulation in the Presence of Triangular Fuzzy Sets |
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70 | (2) |
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72 | (3) |
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72 | (3) |
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5 The Design of Information Granules |
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75 | (32) |
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5.1 The Principle of Justifiable Granularity |
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75 | (14) |
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5.1.1 Some Illustrative Examples |
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79 | (2) |
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5.1.2 A Determination of Feasible Values of α |
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81 | (3) |
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5.1.3 Formation of Shadowed Sets or Rough Sets of Information Granules |
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84 | (1) |
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5.1.4 Weighted Data in the Construction of Information Granules |
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85 | (1) |
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5.1.5 From a Family of Interval Information Granules to a Fuzzy Set |
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86 | (1) |
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5.1.6 Development of Fuzzy Sets of Type 2 |
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86 | (1) |
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5.1.7 A Design of Multidimensional Information Granules |
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87 | (1) |
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5.1.8 A General View of the Principle of Information Granularity: A Diversity of Formal Setups of Information Granularity |
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87 | (2) |
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5.2 Construction of Information Granules through Clustering of Numeric Experimental Evidence |
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89 | (6) |
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5.3 Knowledge-Based Clustering: Bringing Together Data and Knowledge |
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95 | (4) |
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5.4 Refinement of Information Granules through Successive Clustering |
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99 | (2) |
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5.5 Collaborative Clustering and Higher-Level Information Granules |
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101 | (4) |
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105 | (2) |
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105 | (2) |
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6 Optimal Allocation of Information Granularity: Building Granular Mappings |
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107 | (18) |
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6.1 From Mappings and Models to Granular Mappings and Granular Models |
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107 | (5) |
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112 | (2) |
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6.3 Protocols of Allocation of Information Granularity |
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114 | (1) |
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6.4 Design Criteria Guiding the Realization of the Protocols for Allocation of Information Granularity |
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115 | (2) |
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6.5 Granular Neural Networks as Examples of Granular Nonlinear Mappings |
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117 | (4) |
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6.6 Further Problems of Optimal Allocation of Information Granularity |
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121 | (2) |
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6.6.1 Specificity Maximization through Allocation of Information Granularity |
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122 | (1) |
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6.6.2 Optimal Allocation of Granularity in the Input Space: A Construction of the Granular Input Space |
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122 | (1) |
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123 | (2) |
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124 | (1) |
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7 Granular Description of Data and Pattern Classification |
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125 | (28) |
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7.1 Granular Description of Data---A Shadowed Sets Approach |
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125 | (1) |
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7.2 Building Granular Representatives of Data |
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126 | (11) |
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7.2.1 A Two-Phase Formation of Granular Representatives |
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128 | (3) |
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7.2.2 Optimization of Information Granularity with the Use of the Particle Swarm Optimization (PSO) Algorithm |
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131 | (2) |
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7.2.3 Some Illustrative Examples |
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133 | (4) |
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7.3 A Construction of Granular Prototypes with the Use of the Granulation-Degranulation Mechanism |
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137 | (3) |
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7.3.1 Granulation and Degranulation Mechanisms in G(x, V1, V2, ..., Vc, U) |
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139 | (1) |
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7.4 Information Granularity as a Design Asset and Its Optimal Allocation |
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140 | (2) |
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7.5 Design Considerations |
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142 | (4) |
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7.5.1 Granular Core and Granular Data Description |
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142 | (1) |
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7.5.2 Selection of a Suitable Value of Information Granularity |
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142 | (4) |
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7.6 Pattern Classification with Information Granules |
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146 | (1) |
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7.7 Granular Classification Schemes |
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147 | (4) |
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7.7.1 Classification Content of Information Granules |
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149 | (1) |
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7.7.2 Determination of Interval-Valued Class Membership Grades |
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149 | (1) |
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7.7.3 Computing Granular Classification Results |
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150 | (1) |
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151 | (2) |
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152 | (1) |
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8 Granular Models: Architectures and Development |
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153 | (32) |
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8.1 The Mechanisms of Collaboration and Associated Architectures |
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153 | (3) |
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8.2 Realization of Granular Models in a Hierarchical Modeling Topology |
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156 | (1) |
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8.3 The Detailed Considerations: From Fuzzy Rule-Based Models to Granular Fuzzy Models |
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157 | (7) |
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8.4 A Single-Level Knowledge Reconciliation: Mechanisms of Collaboration |
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164 | (6) |
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8.5 Collaboration Scheme: Information Granules as Sources of Knowledge and a Development of Information Granules of a Higher Type |
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170 | (3) |
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8.6 Structure-Free Granular Models |
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173 | (1) |
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8.7 The Essence of Mappings between Input and Output Information Granules and the Underlying Processing |
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174 | (1) |
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8.8 The Design of Information Granules in the Output Space and the Realization of the Aggregation Process |
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175 | (2) |
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8.9 The Development of the Output Information Granules with the Use of the Principle of Justifiable Granularity |
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177 | (1) |
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8.10 Interpretation of Granular Mappings |
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178 | (1) |
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8.11 Illustrative Examples |
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179 | (4) |
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183 | (2) |
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184 | (1) |
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185 | (18) |
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185 | (1) |
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9.2 Information Granules and Time Series |
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186 | (1) |
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9.3 A Granular Framework of Interpretation of Time Series: A Layered Approach to the Interpretation of Time Series |
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186 | (6) |
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9.3.1 Formation of Interval Information Granules |
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190 | (1) |
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9.3.2 Optimization of Temporal Intervals |
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190 | (1) |
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9.3.3 Clustering Information Granules---A Formation of Linguistic Landmarks |
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191 | (1) |
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9.3.4 Matching Information Granules and a Realization of Linguistic Description of Time Series |
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191 | (1) |
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9.4 A Classification Framework of Granular Time, Series |
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192 | (6) |
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9.4.1 Building a Feature Space for Time Series Representation and Classification |
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196 | (1) |
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9.4.2 Formation of a Granular Feature Space |
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197 | (1) |
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198 | (3) |
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9.5.1 Underlying Architecture of the Classifier |
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198 | (2) |
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9.5.2 A Construction of the Fuzzy Relation of the Classifier |
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200 | (1) |
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201 | (2) |
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202 | (1) |
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10 From Models to Granular Models |
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203 | (36) |
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10.1 Knowledge Transfer in System Modeling |
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203 | (3) |
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10.2 Fuzzy Logic Networks---Architectural Considerations |
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206 | (7) |
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10.2.1 Realization of a Fuzzy Logic Mapping |
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206 | (1) |
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10.2.2 Main Categories of Aggregative Fuzzy Neurons: AND and OR Neurons |
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207 | (3) |
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10.2.3 An Architecture of the Fuzzy Logic Networks |
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210 | (2) |
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10.2.4 Allocation of Information Granularity |
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212 | (1) |
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10.3 Granular Logic Descriptors |
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213 | (8) |
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10.3.1 Logic Descriptors: Quantified and and or Logic Structures |
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215 | (2) |
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10.3.2 The Development of Granular Logic: A Holistic and Unified View of a Collection of Logic Descriptors |
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217 | (2) |
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10.3.3 The Development of the Granular Logic Descriptor in a Feedback Mode |
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219 | (2) |
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10.4 Granular Neural Networks |
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221 | (7) |
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10.4.1 Design Issues of the Granular Neural Networks |
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224 | (4) |
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10.5 The Design of Granular Fuzzy Takagi---Sugeno Rule-Based Models: An Optimal Allocation of Information Granularity |
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228 | (8) |
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10.5.1 General Observations |
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228 | (2) |
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10.5.2 Design of Takagi-Sugeno Fuzzy Models: Some General Views and Common Development Practices |
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230 | (1) |
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10.5.3 Granular Fuzzy Clusters |
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231 | (5) |
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10.5.4 Optimization of Granular Fuzzy Models |
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236 | (1) |
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236 | (3) |
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237 | (2) |
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11 Collaborative and Linguistic Models of Decision Making |
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239 | (32) |
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11.1 Analytic Hierarchy Process (AHP) Method and Its Granular Generalization |
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239 | (2) |
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11.2 Analytic Hierarchy Process Model---The Concept |
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241 | (1) |
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11.3 Granular Reciprocal Matrices |
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242 | (8) |
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11.3.1 The Objective Function |
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244 | (6) |
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11.4 A Quantification (Granulation) of Linguistic Terms as Their Operational Realization |
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250 | (6) |
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11.4.1 Evaluation of the Mapping from Linguistic Terms to Information Granules |
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252 | (4) |
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11.5 Granular Logic Operators |
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256 | (10) |
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11.5.1 Construction of Information Granules of Membership Function Representation: An Optimization Problem |
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259 | (3) |
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11.5.2 The Optimization Criterion |
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262 | (2) |
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11.5.3 Logic-Consistent Granular Representations of Fuzzy Sets: Experiments |
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264 | (2) |
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11.6 Modes of Processing with Granular Characterization of Fuzzy Sets |
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266 | (2) |
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268 | (3) |
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269 | (2) |
Index |
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