WŁADYSŁAW HOMENDA, MSc., PhD, DSc., is an Associate Professor with the Faculty of Mathematics and Information Science at the Warsaw University of Technology, Poland, and an Associate Professor with the Faculty of Economics and Informatics in Vilnius at the University of Białystok, Lithuania. WITOLD PEDRYCZ is a Professor with the Systems Research Institute, Polish Academy of Sciences Warsaw, Poland and Department of Electrical & Computer Engineering, University of Alberta, Edmonton, AB, Canada.

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• Preface ix Part I Fundamentals 1 Chapter 1 Pattern Recognition: Feature Space Construction 3 1.1 Concepts 3 1.2 From Patterns to Features 8 1.3 Features Scaling 17 1.4 Evaluation and Selection of Features 23 1.5 Conclusions 47 Appendix 1.A 48 Appendix 1.B 50 References 50 Chapter 2 Pattern Recognition: Classifiers 53 2.1 Concepts 53 2.2 Nearest Neighbors Classification Method 55 2.3 Support Vector Machines Classification Algorithm 57 2.4 Decision Trees in Classification Problems 65 2.5 Ensemble Classifiers 78 2.6 Bayes Classifiers 82 2.7 Conclusions 97 References 97 Chapter 3 Classification with Rejection Problem Formulation And An Overview 101 3.1 Concepts 102 3.2 The Concept of Rejecting Architectures 107 3.3 Native Patterns-Based Rejection 112 3.4 Rejection Option in the Dataset of Native Patterns: A Case Study 118 3.5 Conclusions 129 References 130 Chapter 4 Evaluating Pattern Recognition Problem 133 4.1 Evaluating Recognition with Rejection: Basic Concepts 133 4.2 Classification with Rejection with No Foreign Patterns 145 4.3 Classification with Rejection: Local Characterization 149 4.4 Conclusions 156 References 156 Chapter 5 Recognition with Rejection: Empirical Analysis 159 5.1 Experimental Results 160 5.2 Geometrical Approach 175 5.3 Conclusions 191 References 192 Part II Advanced Topics: a Framework of Granular Computing 195 Chapter 6 Concepts and Notions of Information Granules 197 6.1 Information Granularity and Granular Computing 197 6.2 Formal Platforms of Information Granularity 201 6.3 Intervals and Calculus of Intervals 205 6.4 Calculus of Fuzzy Sets 208 6.5 Characterization of Information Granules: Coverage and Specificity 216 6.6 Matching Information Granules 219 6.7 Conclusions 220 References 221 Chapter 7 Information Granules: Fundamental Constructs 223 7.1 The Principle of Justifiable Granularity 223 7.2 Information Granularity as a Design Asset 230 7.3 Single-Step and Multistep Prediction of Temporal Data in Time Series Models 235 7.4 Development of Granular Models of Higher Type 236 7.5 Classification with Granular Patterns 241 7.6 Conclusions 245 References 246 Chapter 8 Clustering 247 8.1 Fuzzy C-Means Clustering Method 247 8.2 k-Means Clustering Algorithm 252 8.3 Augmented Fuzzy Clustering with Clusters and Variables Weighting 253 8.4 Knowledge-Based Clustering 254 8.5 Quality of Clustering Results 254 8.6 Information Granules and Interpretation of Clustering Results 256 8.7 Hierarchical Clustering 258 8.8 Information Granules in Privacy Problem: A Concept of Microaggregation 261 8.9 Development of Information Granules of Higher Type 262 8.10 Experimental Studies 264 8.11 Conclusions 272 References 273 Chapter 9 Quality of Data: Imputation and Data Balancing 275 9.1 Data Imputation: Underlying Concepts and Key Problems 275 9.2 Selected Categories of Imputation Methods 276 9.3 Imputation with the Use of Information Granules 278 9.4 Granular Imputation with the Principle of Justifiable Granularity 279 9.5 Granular Imputation with Fuzzy Clustering 283 9.6 Data Imputation in System Modeling 285 9.7 Imbalanced Data and their Granular Characterization 286 9.8 Conclusions 291 References 291 Index 293

WŁADYSŁAW HOMENDA, MSc., PhD, DSc., is an Associate Professor with the Faculty of Mathematics and Information Science at the Warsaw University of Technology, Poland, and an Associate Professor with the Faculty of Economics and Informatics in Vilnius at the University of Białystok, Lithuania. WITOLD PEDRYCZ is a Professor with the Systems Research Institute, Polish Academy of Sciences Warsaw, Poland and Department of Electrical & Computer Engineering, University of Alberta, Edmonton, AB, Canada.

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• Preface ix Part I Fundamentals 1 Chapter 1 Pattern Recognition: Feature Space Construction 3 1.1 Concepts 3 1.2 From Patterns to Features 8 1.3 Features Scaling 17 1.4 Evaluation and Selection of Features 23 1.5 Conclusions 47 Appendix 1.A 48 Appendix 1.B 50 References 50 Chapter 2 Pattern Recognition: Classifiers 53 2.1 Concepts 53 2.2 Nearest Neighbors Classification Method 55 2.3 Support Vector Machines Classification Algorithm 57 2.4 Decision Trees in Classification Problems 65 2.5 Ensemble Classifiers 78 2.6 Bayes Classifiers 82 2.7 Conclusions 97 References 97 Chapter 3 Classification with Rejection Problem Formulation And An Overview 101 3.1 Concepts 102 3.2 The Concept of Rejecting Architectures 107 3.3 Native Patterns-Based Rejection 112 3.4 Rejection Option in the Dataset of Native Patterns: A Case Study 118 3.5 Conclusions 129 References 130 Chapter 4 Evaluating Pattern Recognition Problem 133 4.1 Evaluating Recognition with Rejection: Basic Concepts 133 4.2 Classification with Rejection with No Foreign Patterns 145 4.3 Classification with Rejection: Local Characterization 149 4.4 Conclusions 156 References 156 Chapter 5 Recognition with Rejection: Empirical Analysis 159 5.1 Experimental Results 160 5.2 Geometrical Approach 175 5.3 Conclusions 191 References 192 Part II Advanced Topics: a Framework of Granular Computing 195 Chapter 6 Concepts and Notions of Information Granules 197 6.1 Information Granularity and Granular Computing 197 6.2 Formal Platforms of Information Granularity 201 6.3 Intervals and Calculus of Intervals 205 6.4 Calculus of Fuzzy Sets 208 6.5 Characterization of Information Granules: Coverage and Specificity 216 6.6 Matching Information Granules 219 6.7 Conclusions 220 References 221 Chapter 7 Information Granules: Fundamental Constructs 223 7.1 The Principle of Justifiable Granularity 223 7.2 Information Granularity as a Design Asset 230 7.3 Single-Step and Multistep Prediction of Temporal Data in Time Series Models 235 7.4 Development of Granular Models of Higher Type 236 7.5 Classification with Granular Patterns 241 7.6 Conclusions 245 References 246 Chapter 8 Clustering 247 8.1 Fuzzy C-Means Clustering Method 247 8.2 k-Means Clustering Algorithm 252 8.3 Augmented Fuzzy Clustering with Clusters and Variables Weighting 253 8.4 Knowledge-Based Clustering 254 8.5 Quality of Clustering Results 254 8.6 Information Granules and Interpretation of Clustering Results 256 8.7 Hierarchical Clustering 258 8.8 Information Granules in Privacy Problem: A Concept of Microaggregation 261 8.9 Development of Information Granules of Higher Type 262 8.10 Experimental Studies 264 8.11 Conclusions 272 References 273 Chapter 9 Quality of Data: Imputation and Data Balancing 275 9.1 Data Imputation: Underlying Concepts and Key Problems 275 9.2 Selected Categories of Imputation Methods 276 9.3 Imputation with the Use of Information Granules 278 9.4 Granular Imputation with the Principle of Justifiable Granularity 279 9.5 Granular Imputation with Fuzzy Clustering 283 9.6 Data Imputation in System Modeling 285 9.7 Imbalanced Data and their Granular Characterization 286 9.8 Conclusions 291 References 291 Index 293

WŁADYSŁAW HOMENDA, MSc., PhD, DSc., is an Associate Professor with the Faculty of Mathematics and Information Science at the Warsaw University of Technology, Poland, and an Associate Professor with the Faculty of Economics and Informatics in Vilnius at the University of Białystok, Lithuania. WITOLD PEDRYCZ is a Professor with the Systems Research Institute, Polish Academy of Sciences Warsaw, Poland and Department of Electrical & Computer Engineering, University of Alberta, Edmonton, AB, Canada.

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• Preface ix Part I Fundamentals 1 Chapter 1 Pattern Recognition: Feature Space Construction 3 1.1 Concepts 3 1.2 From Patterns to Features 8 1.3 Features Scaling 17 1.4 Evaluation and Selection of Features 23 1.5 Conclusions 47 Appendix 1.A 48 Appendix 1.B 50 References 50 Chapter 2 Pattern Recognition: Classifiers 53 2.1 Concepts 53 2.2 Nearest Neighbors Classification Method 55 2.3 Support Vector Machines Classification Algorithm 57 2.4 Decision Trees in Classification Problems 65 2.5 Ensemble Classifiers 78 2.6 Bayes Classifiers 82 2.7 Conclusions 97 References 97 Chapter 3 Classification with Rejection Problem Formulation And An Overview 101 3.1 Concepts 102 3.2 The Concept of Rejecting Architectures 107 3.3 Native Patterns-Based Rejection 112 3.4 Rejection Option in the Dataset of Native Patterns: A Case Study 118 3.5 Conclusions 129 References 130 Chapter 4 Evaluating Pattern Recognition Problem 133 4.1 Evaluating Recognition with Rejection: Basic Concepts 133 4.2 Classification with Rejection with No Foreign Patterns 145 4.3 Classification with Rejection: Local Characterization 149 4.4 Conclusions 156 References 156 Chapter 5 Recognition with Rejection: Empirical Analysis 159 5.1 Experimental Results 160 5.2 Geometrical Approach 175 5.3 Conclusions 191 References 192 Part II Advanced Topics: a Framework of Granular Computing 195 Chapter 6 Concepts and Notions of Information Granules 197 6.1 Information Granularity and Granular Computing 197 6.2 Formal Platforms of Information Granularity 201 6.3 Intervals and Calculus of Intervals 205 6.4 Calculus of Fuzzy Sets 208 6.5 Characterization of Information Granules: Coverage and Specificity 216 6.6 Matching Information Granules 219 6.7 Conclusions 220 References 221 Chapter 7 Information Granules: Fundamental Constructs 223 7.1 The Principle of Justifiable Granularity 223 7.2 Information Granularity as a Design Asset 230 7.3 Single-Step and Multistep Prediction of Temporal Data in Time Series Models 235 7.4 Development of Granular Models of Higher Type 236 7.5 Classification with Granular Patterns 241 7.6 Conclusions 245 References 246 Chapter 8 Clustering 247 8.1 Fuzzy C-Means Clustering Method 247 8.2 k-Means Clustering Algorithm 252 8.3 Augmented Fuzzy Clustering with Clusters and Variables Weighting 253 8.4 Knowledge-Based Clustering 254 8.5 Quality of Clustering Results 254 8.6 Information Granules and Interpretation of Clustering Results 256 8.7 Hierarchical Clustering 258 8.8 Information Granules in Privacy Problem: A Concept of Microaggregation 261 8.9 Development of Information Granules of Higher Type 262 8.10 Experimental Studies 264 8.11 Conclusions 272 References 273 Chapter 9 Quality of Data: Imputation and Data Balancing 275 9.1 Data Imputation: Underlying Concepts and Key Problems 275 9.2 Selected Categories of Imputation Methods 276 9.3 Imputation with the Use of Information Granules 278 9.4 Granular Imputation with the Principle of Justifiable Granularity 279 9.5 Granular Imputation with Fuzzy Clustering 283 9.6 Data Imputation in System Modeling 285 9.7 Imbalanced Data and their Granular Characterization 286 9.8 Conclusions 291 References 291 Index 293