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1 Introduction 1

1.1 Types of Uncertainty 1

1.2 Uncertainty Modeling and Data Mining 4

1.3 Related Works 6

References 9

2 Induction and Learning 13

2.1 Introduction 13

2.2 Machine Learning 14

2.2.1 Searching in Hypothesis Space 16

2.2.2 Supervised Learning 18

2.2.3 Unsupervised Learning 20

2.2.4 Instance-Based Learning 22

2.3 Data Mining and Algorithms 23

2.3.1 Why Do We Need Data Mining? 24

2.3.2 How Do We do Data Mining? 24

2.3.3 Artificial Neural Networks 25

2.3.4 Support Vector Machines 27

2.4 Measurement of Classifiers 29

2.4.1 ROC Analysis for Classification 30

2.4.2 Area Under the ROC Curve 31

2.5 Summary 34

References 34

3 Label Semantics Theory 39

3.1 Uncertainty Modeling with Labels 39

3.1.1 Fuzzy Logic 39

3.1.2 Computing with Words 41

3.1.3 Mass Assignment Theory 42

3.2 Label Semantics 44

3.2.1 Epistemic View of Label Semantics 45

3.2.2 Random Set Framework 46

3.2.3 Appropriateness Degrees 50

3.2.4 Assumptions for Data Analysis 51

3.2.5 Linguistic Translation 54

3.3 Fuzzy Discretization 57

3.3.1 Percentile-Based Discretization 58

3.3.2 Entropy-Based Discretization 58

3.4 Reasoning with Fuzzy Labels 61

3.4.1 Conditional Distribution Given Mass Assignments 61

3.4.2 Logical Expressions of Fuzzy Labels 62

3.4.3 Linguistic Interpretation of Appropriate Labels 65

3.4.4 Evidence Theory and Mass Assignment 66

3.5 Label Relations 69

3.6 Summary 73

References 74

4 Linguistic Decision Trees for Classification 77

4.1 Introduction 77

4.2 Tree Induction 77

4.2.1 Entropy 79

4.2.2 Soft Decision Trees 82

4.3 Linguistic Decision for Classification 82

4.3.1 Branch Probability 85

4.3.2 Classification by LDT 88

4.3.3 Linguistic ID3 Algorithm 90

4.4 Experimental Studies 92

4.4.1 Influence of the Threshold 93

4.4.2 Overlapping Between Fuzzy Labels 95

4.5 Comparison Studies 98

4.6 Merging of Branches 102

4.6.1 Forward Merging Algorithm 103

4.6.2 Dual-Branch LDTs 105

4.6.3 Experimental Studies for Forward Merging 105

4.6.4 ROC Analysis for Forward Merging 109

4.7 Linguistic Reasoning 111

4.7.1 Linguistic Interpretation of an LDT 111

4.7.2 Linguistic Constraints 113

4.7.3 Classification of Fuzzy Data 115

4.8 Summary 117

References 118

5 Linguistic Decision Trees for Prediction 121

5.1 Prediction Trees 121

5.2 Linguistic Prediction Trees 122

5.2.1 Branch Evaluation 123

5.2.2 Defuzzification 126

5.2.3 Linguistic ID3 Algorithm for Prediction 128

5.2.4 Forward Branch Merging for Prediction 128

5.3 Experimental Studies 130

5.3.1 3D Surface Regression 131

5.3.2 Abalone and Boston Housing Problem 134

5.3.3 Prediction of Sunspots 135

5.3.4 Flood Forecasting 137

5.4 Query Evaluation 143

5.4.1 Single Queries 143

5.4.2 Compound Queries 144

5.5 ROC Analysis for Prediction 145

5.5.1 Predictors and Probabilistic Classifiers 145

5.5.2 AUC Value for Prediction 149

5.6 Summary 152

References 152

6 Bayesian Methods Based on Label Semantics 155

6.1 Introduction 155

6.2 Naive Bayes 156

6.2.1 Bayes Theorem 157

6.2.2 Fuzzy Naive Bayes 158

6.3 Fuzzy Semi-Naive Bayes 159

6.4 Online Fuzzy Bayesian Prediction 161

6.4.1 Bayesian Methods 161

6.4.2 Online Learning 164

6.5 Bayesian Estimation Trees 165

6.5.1 Bayesian Estimation Given an LDT 165

6.5.2 Bayesian Estimation from a Set of Trees 167

6.6 Experimental Studies 168

6.7 Summary 169

References 171

7 Unsupervised Learning with Label Semantics 177

7.1 Introduction 177

7.2 Non-Parametric Density Estimation 178

7.3 Clustering 180

7.3.1 Logical Distance 181

7.3.2 Clustering of Mixed Objects 185

7.4 Experimental Studies 187

7.4.1 Logical Distance Example 187

7.4.2 Images and Labels Clustering 190

7.5 Summary 191

References 192

8 Linguistic FOIL and Multiple Attribute Hierarchy for Decision Making 193

8.1 Introduction 193

8.2 Rule Induction 193

8.3 Multi-Dimensional Label Semantics 196

8.4 Linguistic FOIL 199

8.4.1 Information Heuristics for LFOIL 199

8.4.2 Linguistic Rule Generation 200

8.4.3 Class Probabilities Given a Rule Base 202

8.5 Experimental Studies 203

8.6 Multiple Attribute Decision Making 206

8.6.1 Linguistic Attribute Hierarchies 206

8.6.2 Information Propagation Using LDT 209

8.7 Summary 213

References 213

9 A Prototype Theory Interpretation of Label Semantics 215

9.1 Introduction 215

9.2 Prototype Semantics for Vague Concepts 217

9.2.1 Uncertainty Measures about the Similarity Neighborhoods Determined by Vague Concepts 217

9.2.2 Relating Prototype Theory and Label Semantics 220

9.2.3 Gaussian-Type Density Function 223

9.3 Vague Information Coarsening in Theory of Prototypes 227

9.4 Linguistic Inference Systems 229

9.5 Summary 231

References 232

10 Prototype Theory for Learning 235

10.1 Introduction 235

10.1.1 General Rule Induction Process 235

10.1.2 A Clustering Based Rule Coarsening 236

10.2 Linguistic Modeling of Time Series Predictions 238

10.2.1 Mackey-Glass Time Series Prediction 239

10.2.2 Prediction of Sunspots 244

10.3 Summary 250

References 252

11 Prototype-Based Rule Systems 253

11.1 Introduction 253

11.2 Prototype-Based IF-THEN Rules 254

11.3 Rule Induction Based on Data Clustering and Least-Square Regression 257

11.4 Rule Learning Using a Conjugate Gradient Algorithm 260

11.5 Applications in Prediction Problems 262

11.5.1 Surface Predication 262

11.5.2 Mackev-Glass Time Series Prediction 265

11.5.3 Prediction of Sunspots 269

11.6 Summary 274

References 274

12 Information Cells and Information Cell Mixture Models 277

12.1 Introduction 277

12.2 Information Cell for Cognitive Representation of Vague Concept Semantics 277

12.3 Information Cell Mixture Model(ICMM)for Semantic Representation of Complex Concept 280

12.4 Learning Infcrmation Cell Mixture Model from Data Set 281

12.4.1 Objective Function Based on Positive Density Function 282

12.4.2 Updating Probability Distribution of Information Cells 282

12.4.3 Updating Density Functions of Information Cells 283

12.4.4 Information Cell Updating Algorithm 284

12.4.5 Learning Component Number of ICMM 285

12.5 Experimental Study 286

12.6 Summary 290

References 290