Pattern Classification

Boundaries.- 5.3.1 Fuzzy Rules with Pyramidal Membership Functions.- 5.3.2 Ellipsoidal Fuzzy Rules.- 5.3.3 Class Boundaries for the Iris Data.- 5.4 Training Architecture.- 5.4.1 Fuzzy Rule Generation by Preclustering.- 5.4.2 Fuzzy Rule Generation by Postclustering.- 6. Clustering.- 6.1 Fuzzy c-means Clustering Algorithm.- 6.2 The Kohonen Network.- 6.3 Minimum Volume Clustering Algorithm.- 6.4 Fuzzy Min-max Clustering Algorithm.- 6.5 Overlap Resolving Clustering Algorithm.- 6.5.1 Approximation of Overlapping Regions.- 6.5.2 Extraction of Data from the Overlapping Regions.- 6.5.3 Clustering Algorithm.- 7. Tuning of Membership Functions.- 7.1 Problem Formulation.- 7.2 Direct Methods.- 7.2.1 Tuning of Slopes.- 7.2.2 Tuning of Locations.- 7.2.3 Order of Tuning.- 7.3 Indirect Methods.- 7.3.1 Tuning of Slopes Using the Least-squares Method.- 7.3.2 Tuning by the Steepest Descent Method.- 7.4 Performance Evaluation.- 7.4.1 Performance Evaluation of the Fuzzy Classifier with Pyramidal Membership Functions.- 7.4.2 Performance Evaluation of the Fuzzy Classifier with Polyhedral Regions.- 7.4.3 Performance Evaluation of the Fuzzy Classifier with Ellipsoidal Regions.- 8. Robust Pattern Classification.- 8.1 Why Robust Classification Is Necessary?.- 8.2 Robust Classification.- 8.2.1 The First Stage.- 8.2.2 The Second Stage.- 8.2.3 Tuning Slopes near Class Boundaries.- 8.2.4 Upper and Lower Bounds Determined by Correctly Classified Data.- 8.2.5 Range of the Interclass Tuning Parameter that Resolves Misclassification.- 8.3 Performance Evaluation.- 8.3.1 Classification Performance without Outliers.- 8.3.2 Classification Performance with Outliers.- 9. Dynamic Fuzzy Rule Generation.- 9.1 Fuzzy Min-max Classifiers.- 9.1.1 Concept.- 9.1.2 Approximation of Input Regions.- 9.1.3 Fuzzy Rule Extraction.- 9.1.4 Performance Evaluation.- 9.2 Fuzzy Min-max Classifiers with Inhibition.- 9.2.1 Concept.- 9.2.2 Fuzzy Rule Extraction.- 9.2.3 Fuzzy Rule Inference.- 9.2.4 Performance Evaluation.- 10. Comparison of Classifier Performance.- 10.1 Evaluation Conditions.- 10.2 Iris Data.- 10.3 Numeral Data.- 10.4 Thyroid Data.- 10.5 Blood Cell Data.- 10.6 Hiragana Data.- 10.7 Discussions.- 11. Optimizing Features.- 11.1 Scaling Features.- 11.1.1 Scale and Translation Invariance of the Fuzzy Classifier with Pyramidal Membership Functions.- 11.1.2 Invariance of the Fuzzy Classifiers with Hyperbox Regions.- 11.1.3 Invariance of the Fuzzy Classifier with Ellipsoidal Regions.- 11.2 Feature Extraction.- 11.2.1 Principal Component Analysis.- 11.2.2 Discriminant Analysis.- 11.2.3 Neural-network-based Feature Extraction.- 11.3 Feature Selection.- 11.3.1 Criteria for Feature Selection.- 11.3.2 Monotonicity of the Selection Criteria.- 11.3.3 Selection Search.- 11.4 Performance Evaluation of Feature Selection.- 11.4.1 Evaluation Conditions.- 11.4.2 Selected Features.- 11.4.3 Iris Data.- 11.4.4 Numeral Data.- 11.4.5 Thyroid Data.- 11.4.6 Blood Cell Data.- 11.4.7 Discussions.- 12. Generation of Training and Test Data Sets.- 12.1 Resampling Techniques.- 12.1.1 Estimation of Classification Performance.- 12.1.2 Effect of Resampling on Classifier Performance.- 12.2 Division of Data by Pairing.- 12.3 Similarity Measures.- 12.3.1 Relative Difference of Center Norms.- 12.3.2 Relative Difference of Covariance Matrix Norms.- 12.4 Performance Evaluation.- 12.4.1 Evaluation Conditions.- 12.4.2 Blood Cell Data.- 12.4.3 Hiragana-50 Data.- 12.4.4 Discussions.- II. Function Approximation.- 13. Introduction.- 13.1 Function Approximators.- 13.1.1 Neural Networks.- 13.1.2 Conventional Fuzzy Function Approximators.- 13.1.3 Fuzzy Function Approximators with Learning Capability.- 14. Fuzzy Rule Representation and Inference.- 14.1 Fuzzy Rule Representation.- 14.2 Defuzzification Methods.- 15. Fuzzy Rule Generation.- 15.1 Fuzzy Rule Generation by Preclustering.- 15.1.1 Clustering of Input Space.- 15.1.2 Clustering of Input and Output Spaces.- 15.2 Fuzzy Rule Generation by Postclustering.- 15.2.1 Concept.- 15.2.2 Rule Generation for FACG.- 15.2.3 Rule Generation for FALC.- 15.3 Fuzzy Rule Tuning.- 15.3.1 Concept.- 15.3.2 Evaluation Functions.- 15.3.3 Tuning of Parameters.- 15.4 Performance Evaluation.- 15.4.1 Mackey-Glass Differential Equation.- 15.4.2 Water Purification Plant.- 15.4.3 Discussions.- 16. Robust Function Approximation.- 16.1 Introduction.- 16.2 Fuzzy Function Approximator with Ellipsoidal Regions.- 16.2.1 Fuzzy Rule Representation.- 16.2.2 Fuzzy Rule Generation.- 16.3 Robust Parameter Estimation.- 16.3.1 Robust Estimation by the Least-median-of-squares Method.- 16.3.2 Robust Estimation by the Least-squares Method with Partial Data.- 16.4 Performance Evaluation.- 16.4.1 Determination of the Multiplier.- 16.4.2 Performance Comparison.- 16.5 Discussions.- III. Appendices.- A. Conventional Classifiers.- A.1 Bayesian Classifiers.- A.2 Nearest Neighbor Classifiers.- A.2.1 Classifier Architecture.- A.2.2 Performance Evaluation.- B. Matrices.- B.1 Matrix Properties.- B.2 Least-squares Method and Singular Value Decomposition.- B.3 Covariance Matrix.- References.