PRADIPTA MAJI, PhD, is Assistant Professor in the Machine Intelligence Unit of the Indian Statistical Institute. His research explores pattern recognition, bioinformatics, medical image processing, cellular automata, and soft computing.SANKAR K. PAL, PhD, is Director and Distinguished Scientist of the Indian Statistical Institute. He is also a J. C. Bose Fellow of the Government of India. Dr. Pal founded both the Machine Intelligence Unit and the Center for Soft Computing Research at the Indian Statistical Institute. He is a Fellow of the IEEE, IAPR, IFSA, TWAS, and Indian National Science Academy.

• smallForeword xiiiPreface xvAbout the Authors xix1 Introduction to Pattern Recognition and Data Mining 11.1 Introduction 11.2 Pattern Recognition 31.2.1 Data Acquisition 41.2.2 Feature Selection 41.2.3 Classification and Clustering 51.3 Data Mining 61.3.1 Tasks, Tools, and Applications 71.3.2 Pattern Recognition Perspective 81.4 Relevance of Soft Computing 91.5 Scope and Organization of the Book 10References 142 Rough-Fuzzy Hybridization and Granular Computing 212.1 Introduction 212.2 Fuzzy Sets 222.3 Rough Sets 232.4 Emergence of Rough-Fuzzy Computing 262.4.1 Granular Computing 262.4.2 Computational Theory of Perception and f-Granulation 262.4.3 Rough-Fuzzy Computing 282.5 Generalized Rough Sets 292.6 Entropy Measures 302.7 Conclusion and Discussion 36References 373 Rough-Fuzzy Clustering: Generalized c-Means Algorithm 473.1 Introduction 473.2 Existing c-Means Algorithms 493.2.1 Hard c-Means 493.2.2 Fuzzy c-Means 503.2.3 Possibilistic c-Means 513.2.4 Rough c-Means 523.3 Rough-Fuzzy-Possibilistic c-Means 533.3.1 Objective Function 543.3.2 Cluster Prototypes 553.3.3 Fundamental Properties 563.3.4 Convergence Condition 573.3.5 Details of the Algorithm 593.3.6 Selection of Parameters 603.4 Generalization of Existing c-Means Algorithms 613.4.1 RFCM: Rough-Fuzzy c-Means 613.4.2 RPCM: Rough-Possibilistic c-Means 623.4.3 RCM: Rough c-Means 633.4.4 FPCM: Fuzzy-Possibilistic c-Means 643.4.5 FCM: Fuzzy c-Means 643.4.6 PCM: Possibilistic c-Means 643.4.7 HCM: Hard c-Means 653.5 Quantitative Indices for Rough-Fuzzy Clustering 653.5.1 Average Accuracy, a Index 653.5.2 Average Roughness, Index 673.5.3 Accuracy of Approximation, a Index 673.5.4 Quality of Approximation, y Index 683.6 Performance Analysis 683.6.1 Quantitative Indices 683.6.2 Synthetic Data Set: X32 693.6.3 Benchmark Data Sets 703.7 Conclusion and Discussion 80References 814 Rough-Fuzzy Granulation and Pattern Classification 854.1 Introduction 854.2 Pattern Classification Model 874.2.1 Class-Dependent Fuzzy Granulation 884.2.2 Rough-Set-Based Feature Selection 904.3 Quantitative Measures 954.3.1 Dispersion Measure 954.3.2 Classification Accuracy, Precision, and Recall 964.3.3 κ Coefficient 964.3.4 β Index 974.4 Description of Data Sets 974.4.1 Completely Labeled Data Sets 984.4.2 Partially Labeled Data Sets 994.5 Experimental Results 1004.5.1 Statistical Significance Test 1024.5.2 Class Prediction Methods 1034.5.3 Performance on Completely Labeled Data 1034.5.4 Performance on Partially Labeled Data 1104.6 Conclusion and Discussion 112References 1145 Fuzzy-Rough Feature Selection using f-Information Measures 1175.1 Introduction 1175.2 Fuzzy-Rough Sets 1205.3 Information Measure on Fuzzy Approximation Spaces 1215.3.1 Fuzzy Equivalence Partition Matrix and Entropy 1215.3.2 Mutual Information 1235.4 F-Information and Fuzzy Approximation Spaces 1255.4.1 V -Information 1255.4.2 I-Information 1265.4.3 M-Information 1275.4.4 X-Information 1275.4.5 Hellinger Integral 1285.4.6 Renyi Distance 1285.5 F-Information for Feature Selection 1295.5.1 Feature Selection Using f-Information 1295.5.2 Computational Complexity 1305.5.3 Fuzzy Equivalence Classes 1315.6 Quantitative Measures 1335.6.1 Fuzzy-Rough-Set-Based Quantitative Indices 1335.6.2 Existing Feature Evaluation Indices 1335.7 Experimental Results 1355.7.1 Description of Data Sets 1365.7.2 Illustrative Example 1375.7.3 Effectiveness of the FEPM-Based Method 1385.7.4 Optimum Value of Weight Parameter B 1415.7.5 Optimum Value of Multiplicative Parameter n 1415.7.6 Performance of Different f-Information Measures 1455.7.7 Comparative Performance of Different Algorithms 1525.8 Conclusion and Discussion 156References 1566 Rough Fuzzy c-Medoids and Amino Acid Sequence Analysis 1616.1 Introduction 1616.2 Bio-Basis Function and String Selection Methods 1646.2.1 Bio-Basis Function 1646.2.2 Selection of Bio-Basis Strings Using Mutual Information 1666.2.3 Selection of Bio-Basis Strings Using Fisher Ratio 1676.3 Fuzzy-Possibilistic c-Medoids Algorithm 1686.3.1 Hard c-Medoids 1686.3.2 Fuzzy c-Medoids 1696.3.3 Possibilistic c-Medoids 1706.3.4 Fuzzy-Possibilistic c-Medoids 1716.4 Rough-Fuzzy c-Medoids Algorithm 1726.4.1 Rough c-Medoids 1726.4.2 Rough-Fuzzy c-Medoids 1746.5 Relational Clustering for Bio-Basis String Selection 1766.6 Quantitative Measures 1786.6.1 Using Homology Alignment Score 1786.6.2 Using Mutual Information 1796.7 Experimental Results 1816.7.1 Description of Data Sets 1816.7.2 Illustrative Example 1836.7.3 Performance Analysis 1846.8 Conclusion and Discussion 196References 1967 Clustering Functionally Similar Genes from Microarray Data 2017.1 Introduction 2017.2 Clustering Gene Expression Data 2037.2.1 k-Means Algorithm 2037.2.2 Self-Organizing Map 2037.2.3 Hierarchical Clustering 2047.2.4 Graph-Theoretical Approach 2047.2.5 Model-Based Clustering 2057.2.6 Density-Based Hierarchical Approach 2067.2.7 Fuzzy Clustering 2067.2.8 Rough-Fuzzy Clustering 2067.3 Quantitative and Qualitative Analysis 2077.3.1 Silhouette Index 2077.3.2 Eisen and Cluster Profile Plots 2077.3.3 Z-Score 2087.3.4 Gene-Ontology-Based Analysis 2087.4 Description of Data Sets 2097.4.1 Fifteen Yeast Data 2097.4.2 Yeast Sporulation 2117.4.3 Auble Data 2117.4.4 Cho et al. Data 2117.4.5 Reduced Cell Cycle Data 2117.5 Experimental Results 2127.5.1 Performance Analysis of Rough-Fuzzy c-Means 2127.5.2 Comparative Analysis of Different c-Means 2127.5.3 Biological Significance Analysis 2157.5.4 Comparative Analysis of Different Algorithms 2157.5.5 Performance Analysis of Rough-Fuzzy-Possibilistic c-Means 2177.6 Conclusion and Discussion 217References 2208 Selection of Discriminative Genes from Microarray Data 2258.1 Introduction 2258.2 Evaluation Criteria for Gene Selection 2278.2.1 Statistical Tests 2288.2.2 Euclidean Distance 2288.2.3 Pearson’s Correlation 2298.2.4 Mutual Information 2298.2.5 F-Information Measures 2308.3 Approximation of Density Function 2308.3.1 Discretization 2318.3.2 Parzen Window Density Estimator 2318.3.3 Fuzzy Equivalence Partition Matrix 2338.4 Gene Selection using Information Measures 2348.5 Experimental Results 2358.5.1 Support Vector Machine 2358.5.2 Gene Expression Data Sets 2368.5.3 Performance Analysis of the FEPM 2368.5.4 Comparative Performance Analysis 2508.6 Conclusion and Discussion 250References 2529 Segmentation of Brain Magnetic Resonance Images 2579.1 Introduction 2579.2 Pixel Classification of Brain MR Images 2599.2.1 Performance on Real Brain MR Images 2609.2.2 Performance on Simulated Brain MR Images 2639.3 Segmentation of Brain MR Images 2649.3.1 Feature Extraction 2659.3.2 Selection of Initial Prototypes 2749.4 Experimental Results 2779.4.1 Illustrative Example 2779.4.2 Importance of Homogeneity and Edge Value 2789.4.3 Importance of Discriminant Analysis-Based Initialization 2799.4.4 Comparative Performance Analysis 2809.5 Conclusion and Discussion 283References 283Index 287

PRADIPTA MAJI, PhD, is Assistant Professor in the Machine Intelligence Unit of the Indian Statistical Institute. His research explores pattern recognition, bioinformatics, medical image processing, cellular automata, and soft computing.SANKAR K. PAL, PhD, is Director and Distinguished Scientist of the Indian Statistical Institute. He is also a J. C. Bose Fellow of the Government of India. Dr. Pal founded both the Machine Intelligence Unit and the Center for Soft Computing Research at the Indian Statistical Institute. He is a Fellow of the IEEE, IAPR, IFSA, TWAS, and Indian National Science Academy.

• smallForeword xiiiPreface xvAbout the Authors xix1 Introduction to Pattern Recognition and Data Mining 11.1 Introduction 11.2 Pattern Recognition 31.2.1 Data Acquisition 41.2.2 Feature Selection 41.2.3 Classification and Clustering 51.3 Data Mining 61.3.1 Tasks, Tools, and Applications 71.3.2 Pattern Recognition Perspective 81.4 Relevance of Soft Computing 91.5 Scope and Organization of the Book 10References 142 Rough-Fuzzy Hybridization and Granular Computing 212.1 Introduction 212.2 Fuzzy Sets 222.3 Rough Sets 232.4 Emergence of Rough-Fuzzy Computing 262.4.1 Granular Computing 262.4.2 Computational Theory of Perception and f-Granulation 262.4.3 Rough-Fuzzy Computing 282.5 Generalized Rough Sets 292.6 Entropy Measures 302.7 Conclusion and Discussion 36References 373 Rough-Fuzzy Clustering: Generalized c-Means Algorithm 473.1 Introduction 473.2 Existing c-Means Algorithms 493.2.1 Hard c-Means 493.2.2 Fuzzy c-Means 503.2.3 Possibilistic c-Means 513.2.4 Rough c-Means 523.3 Rough-Fuzzy-Possibilistic c-Means 533.3.1 Objective Function 543.3.2 Cluster Prototypes 553.3.3 Fundamental Properties 563.3.4 Convergence Condition 573.3.5 Details of the Algorithm 593.3.6 Selection of Parameters 603.4 Generalization of Existing c-Means Algorithms 613.4.1 RFCM: Rough-Fuzzy c-Means 613.4.2 RPCM: Rough-Possibilistic c-Means 623.4.3 RCM: Rough c-Means 633.4.4 FPCM: Fuzzy-Possibilistic c-Means 643.4.5 FCM: Fuzzy c-Means 643.4.6 PCM: Possibilistic c-Means 643.4.7 HCM: Hard c-Means 653.5 Quantitative Indices for Rough-Fuzzy Clustering 653.5.1 Average Accuracy, a Index 653.5.2 Average Roughness, Index 673.5.3 Accuracy of Approximation, a Index 673.5.4 Quality of Approximation, y Index 683.6 Performance Analysis 683.6.1 Quantitative Indices 683.6.2 Synthetic Data Set: X32 693.6.3 Benchmark Data Sets 703.7 Conclusion and Discussion 80References 814 Rough-Fuzzy Granulation and Pattern Classification 854.1 Introduction 854.2 Pattern Classification Model 874.2.1 Class-Dependent Fuzzy Granulation 884.2.2 Rough-Set-Based Feature Selection 904.3 Quantitative Measures 954.3.1 Dispersion Measure 954.3.2 Classification Accuracy, Precision, and Recall 964.3.3 κ Coefficient 964.3.4 β Index 974.4 Description of Data Sets 974.4.1 Completely Labeled Data Sets 984.4.2 Partially Labeled Data Sets 994.5 Experimental Results 1004.5.1 Statistical Significance Test 1024.5.2 Class Prediction Methods 1034.5.3 Performance on Completely Labeled Data 1034.5.4 Performance on Partially Labeled Data 1104.6 Conclusion and Discussion 112References 1145 Fuzzy-Rough Feature Selection using f-Information Measures 1175.1 Introduction 1175.2 Fuzzy-Rough Sets 1205.3 Information Measure on Fuzzy Approximation Spaces 1215.3.1 Fuzzy Equivalence Partition Matrix and Entropy 1215.3.2 Mutual Information 1235.4 F-Information and Fuzzy Approximation Spaces 1255.4.1 V -Information 1255.4.2 I-Information 1265.4.3 M-Information 1275.4.4 X-Information 1275.4.5 Hellinger Integral 1285.4.6 Renyi Distance 1285.5 F-Information for Feature Selection 1295.5.1 Feature Selection Using f-Information 1295.5.2 Computational Complexity 1305.5.3 Fuzzy Equivalence Classes 1315.6 Quantitative Measures 1335.6.1 Fuzzy-Rough-Set-Based Quantitative Indices 1335.6.2 Existing Feature Evaluation Indices 1335.7 Experimental Results 1355.7.1 Description of Data Sets 1365.7.2 Illustrative Example 1375.7.3 Effectiveness of the FEPM-Based Method 1385.7.4 Optimum Value of Weight Parameter B 1415.7.5 Optimum Value of Multiplicative Parameter n 1415.7.6 Performance of Different f-Information Measures 1455.7.7 Comparative Performance of Different Algorithms 1525.8 Conclusion and Discussion 156References 1566 Rough Fuzzy c-Medoids and Amino Acid Sequence Analysis 1616.1 Introduction 1616.2 Bio-Basis Function and String Selection Methods 1646.2.1 Bio-Basis Function 1646.2.2 Selection of Bio-Basis Strings Using Mutual Information 1666.2.3 Selection of Bio-Basis Strings Using Fisher Ratio 1676.3 Fuzzy-Possibilistic c-Medoids Algorithm 1686.3.1 Hard c-Medoids 1686.3.2 Fuzzy c-Medoids 1696.3.3 Possibilistic c-Medoids 1706.3.4 Fuzzy-Possibilistic c-Medoids 1716.4 Rough-Fuzzy c-Medoids Algorithm 1726.4.1 Rough c-Medoids 1726.4.2 Rough-Fuzzy c-Medoids 1746.5 Relational Clustering for Bio-Basis String Selection 1766.6 Quantitative Measures 1786.6.1 Using Homology Alignment Score 1786.6.2 Using Mutual Information 1796.7 Experimental Results 1816.7.1 Description of Data Sets 1816.7.2 Illustrative Example 1836.7.3 Performance Analysis 1846.8 Conclusion and Discussion 196References 1967 Clustering Functionally Similar Genes from Microarray Data 2017.1 Introduction 2017.2 Clustering Gene Expression Data 2037.2.1 k-Means Algorithm 2037.2.2 Self-Organizing Map 2037.2.3 Hierarchical Clustering 2047.2.4 Graph-Theoretical Approach 2047.2.5 Model-Based Clustering 2057.2.6 Density-Based Hierarchical Approach 2067.2.7 Fuzzy Clustering 2067.2.8 Rough-Fuzzy Clustering 2067.3 Quantitative and Qualitative Analysis 2077.3.1 Silhouette Index 2077.3.2 Eisen and Cluster Profile Plots 2077.3.3 Z-Score 2087.3.4 Gene-Ontology-Based Analysis 2087.4 Description of Data Sets 2097.4.1 Fifteen Yeast Data 2097.4.2 Yeast Sporulation 2117.4.3 Auble Data 2117.4.4 Cho et al. Data 2117.4.5 Reduced Cell Cycle Data 2117.5 Experimental Results 2127.5.1 Performance Analysis of Rough-Fuzzy c-Means 2127.5.2 Comparative Analysis of Different c-Means 2127.5.3 Biological Significance Analysis 2157.5.4 Comparative Analysis of Different Algorithms 2157.5.5 Performance Analysis of Rough-Fuzzy-Possibilistic c-Means 2177.6 Conclusion and Discussion 217References 2208 Selection of Discriminative Genes from Microarray Data 2258.1 Introduction 2258.2 Evaluation Criteria for Gene Selection 2278.2.1 Statistical Tests 2288.2.2 Euclidean Distance 2288.2.3 Pearson’s Correlation 2298.2.4 Mutual Information 2298.2.5 F-Information Measures 2308.3 Approximation of Density Function 2308.3.1 Discretization 2318.3.2 Parzen Window Density Estimator 2318.3.3 Fuzzy Equivalence Partition Matrix 2338.4 Gene Selection using Information Measures 2348.5 Experimental Results 2358.5.1 Support Vector Machine 2358.5.2 Gene Expression Data Sets 2368.5.3 Performance Analysis of the FEPM 2368.5.4 Comparative Performance Analysis 2508.6 Conclusion and Discussion 250References 2529 Segmentation of Brain Magnetic Resonance Images 2579.1 Introduction 2579.2 Pixel Classification of Brain MR Images 2599.2.1 Performance on Real Brain MR Images 2609.2.2 Performance on Simulated Brain MR Images 2639.3 Segmentation of Brain MR Images 2649.3.1 Feature Extraction 2659.3.2 Selection of Initial Prototypes 2749.4 Experimental Results 2779.4.1 Illustrative Example 2779.4.2 Importance of Homogeneity and Edge Value 2789.4.3 Importance of Discriminant Analysis-Based Initialization 2799.4.4 Comparative Performance Analysis 2809.5 Conclusion and Discussion 283References 283Index 287

PRADIPTA MAJI, PhD, is Assistant Professor in the Machine Intelligence Unit of the Indian Statistical Institute. His research explores pattern recognition, bioinformatics, medical image processing, cellular automata, and soft computing.SANKAR K. PAL, PhD, is Director and Distinguished Scientist of the Indian Statistical Institute. He is also a J. C. Bose Fellow of the Government of India. Dr. Pal founded both the Machine Intelligence Unit and the Center for Soft Computing Research at the Indian Statistical Institute. He is a Fellow of the IEEE, IAPR, IFSA, TWAS, and Indian National Science Academy.

• smallForeword xiiiPreface xvAbout the Authors xix1 Introduction to Pattern Recognition and Data Mining 11.1 Introduction 11.2 Pattern Recognition 31.2.1 Data Acquisition 41.2.2 Feature Selection 41.2.3 Classification and Clustering 51.3 Data Mining 61.3.1 Tasks, Tools, and Applications 71.3.2 Pattern Recognition Perspective 81.4 Relevance of Soft Computing 91.5 Scope and Organization of the Book 10References 142 Rough-Fuzzy Hybridization and Granular Computing 212.1 Introduction 212.2 Fuzzy Sets 222.3 Rough Sets 232.4 Emergence of Rough-Fuzzy Computing 262.4.1 Granular Computing 262.4.2 Computational Theory of Perception and f-Granulation 262.4.3 Rough-Fuzzy Computing 282.5 Generalized Rough Sets 292.6 Entropy Measures 302.7 Conclusion and Discussion 36References 373 Rough-Fuzzy Clustering: Generalized c-Means Algorithm 473.1 Introduction 473.2 Existing c-Means Algorithms 493.2.1 Hard c-Means 493.2.2 Fuzzy c-Means 503.2.3 Possibilistic c-Means 513.2.4 Rough c-Means 523.3 Rough-Fuzzy-Possibilistic c-Means 533.3.1 Objective Function 543.3.2 Cluster Prototypes 553.3.3 Fundamental Properties 563.3.4 Convergence Condition 573.3.5 Details of the Algorithm 593.3.6 Selection of Parameters 603.4 Generalization of Existing c-Means Algorithms 613.4.1 RFCM: Rough-Fuzzy c-Means 613.4.2 RPCM: Rough-Possibilistic c-Means 623.4.3 RCM: Rough c-Means 633.4.4 FPCM: Fuzzy-Possibilistic c-Means 643.4.5 FCM: Fuzzy c-Means 643.4.6 PCM: Possibilistic c-Means 643.4.7 HCM: Hard c-Means 653.5 Quantitative Indices for Rough-Fuzzy Clustering 653.5.1 Average Accuracy, a Index 653.5.2 Average Roughness, Index 673.5.3 Accuracy of Approximation, a Index 673.5.4 Quality of Approximation, y Index 683.6 Performance Analysis 683.6.1 Quantitative Indices 683.6.2 Synthetic Data Set: X32 693.6.3 Benchmark Data Sets 703.7 Conclusion and Discussion 80References 814 Rough-Fuzzy Granulation and Pattern Classification 854.1 Introduction 854.2 Pattern Classification Model 874.2.1 Class-Dependent Fuzzy Granulation 884.2.2 Rough-Set-Based Feature Selection 904.3 Quantitative Measures 954.3.1 Dispersion Measure 954.3.2 Classification Accuracy, Precision, and Recall 964.3.3 κ Coefficient 964.3.4 β Index 974.4 Description of Data Sets 974.4.1 Completely Labeled Data Sets 984.4.2 Partially Labeled Data Sets 994.5 Experimental Results 1004.5.1 Statistical Significance Test 1024.5.2 Class Prediction Methods 1034.5.3 Performance on Completely Labeled Data 1034.5.4 Performance on Partially Labeled Data 1104.6 Conclusion and Discussion 112References 1145 Fuzzy-Rough Feature Selection using f-Information Measures 1175.1 Introduction 1175.2 Fuzzy-Rough Sets 1205.3 Information Measure on Fuzzy Approximation Spaces 1215.3.1 Fuzzy Equivalence Partition Matrix and Entropy 1215.3.2 Mutual Information 1235.4 F-Information and Fuzzy Approximation Spaces 1255.4.1 V -Information 1255.4.2 I-Information 1265.4.3 M-Information 1275.4.4 X-Information 1275.4.5 Hellinger Integral 1285.4.6 Renyi Distance 1285.5 F-Information for Feature Selection 1295.5.1 Feature Selection Using f-Information 1295.5.2 Computational Complexity 1305.5.3 Fuzzy Equivalence Classes 1315.6 Quantitative Measures 1335.6.1 Fuzzy-Rough-Set-Based Quantitative Indices 1335.6.2 Existing Feature Evaluation Indices 1335.7 Experimental Results 1355.7.1 Description of Data Sets 1365.7.2 Illustrative Example 1375.7.3 Effectiveness of the FEPM-Based Method 1385.7.4 Optimum Value of Weight Parameter B 1415.7.5 Optimum Value of Multiplicative Parameter n 1415.7.6 Performance of Different f-Information Measures 1455.7.7 Comparative Performance of Different Algorithms 1525.8 Conclusion and Discussion 156References 1566 Rough Fuzzy c-Medoids and Amino Acid Sequence Analysis 1616.1 Introduction 1616.2 Bio-Basis Function and String Selection Methods 1646.2.1 Bio-Basis Function 1646.2.2 Selection of Bio-Basis Strings Using Mutual Information 1666.2.3 Selection of Bio-Basis Strings Using Fisher Ratio 1676.3 Fuzzy-Possibilistic c-Medoids Algorithm 1686.3.1 Hard c-Medoids 1686.3.2 Fuzzy c-Medoids 1696.3.3 Possibilistic c-Medoids 1706.3.4 Fuzzy-Possibilistic c-Medoids 1716.4 Rough-Fuzzy c-Medoids Algorithm 1726.4.1 Rough c-Medoids 1726.4.2 Rough-Fuzzy c-Medoids 1746.5 Relational Clustering for Bio-Basis String Selection 1766.6 Quantitative Measures 1786.6.1 Using Homology Alignment Score 1786.6.2 Using Mutual Information 1796.7 Experimental Results 1816.7.1 Description of Data Sets 1816.7.2 Illustrative Example 1836.7.3 Performance Analysis 1846.8 Conclusion and Discussion 196References 1967 Clustering Functionally Similar Genes from Microarray Data 2017.1 Introduction 2017.2 Clustering Gene Expression Data 2037.2.1 k-Means Algorithm 2037.2.2 Self-Organizing Map 2037.2.3 Hierarchical Clustering 2047.2.4 Graph-Theoretical Approach 2047.2.5 Model-Based Clustering 2057.2.6 Density-Based Hierarchical Approach 2067.2.7 Fuzzy Clustering 2067.2.8 Rough-Fuzzy Clustering 2067.3 Quantitative and Qualitative Analysis 2077.3.1 Silhouette Index 2077.3.2 Eisen and Cluster Profile Plots 2077.3.3 Z-Score 2087.3.4 Gene-Ontology-Based Analysis 2087.4 Description of Data Sets 2097.4.1 Fifteen Yeast Data 2097.4.2 Yeast Sporulation 2117.4.3 Auble Data 2117.4.4 Cho et al. Data 2117.4.5 Reduced Cell Cycle Data 2117.5 Experimental Results 2127.5.1 Performance Analysis of Rough-Fuzzy c-Means 2127.5.2 Comparative Analysis of Different c-Means 2127.5.3 Biological Significance Analysis 2157.5.4 Comparative Analysis of Different Algorithms 2157.5.5 Performance Analysis of Rough-Fuzzy-Possibilistic c-Means 2177.6 Conclusion and Discussion 217References 2208 Selection of Discriminative Genes from Microarray Data 2258.1 Introduction 2258.2 Evaluation Criteria for Gene Selection 2278.2.1 Statistical Tests 2288.2.2 Euclidean Distance 2288.2.3 Pearson’s Correlation 2298.2.4 Mutual Information 2298.2.5 F-Information Measures 2308.3 Approximation of Density Function 2308.3.1 Discretization 2318.3.2 Parzen Window Density Estimator 2318.3.3 Fuzzy Equivalence Partition Matrix 2338.4 Gene Selection using Information Measures 2348.5 Experimental Results 2358.5.1 Support Vector Machine 2358.5.2 Gene Expression Data Sets 2368.5.3 Performance Analysis of the FEPM 2368.5.4 Comparative Performance Analysis 2508.6 Conclusion and Discussion 250References 2529 Segmentation of Brain Magnetic Resonance Images 2579.1 Introduction 2579.2 Pixel Classification of Brain MR Images 2599.2.1 Performance on Real Brain MR Images 2609.2.2 Performance on Simulated Brain MR Images 2639.3 Segmentation of Brain MR Images 2649.3.1 Feature Extraction 2659.3.2 Selection of Initial Prototypes 2749.4 Experimental Results 2779.4.1 Illustrative Example 2779.4.2 Importance of Homogeneity and Edge Value 2789.4.3 Importance of Discriminant Analysis-Based Initialization 2799.4.4 Comparative Performance Analysis 2809.5 Conclusion and Discussion 283References 283Index 287