ROUGH-FUZZY PATTERN RECOGNITION......Page 3
CONTENTS......Page 9
Foreword......Page 15
Preface......Page 17
About the Authors......Page 21
1.1 Introduction......Page 24
1.2 Pattern Recognition......Page 26
1.2.2 Feature Selection......Page 27
1.2.3 Classification and Clustering......Page 28
1.3 Data Mining......Page 29
1.3.1 Tasks, Tools, and Applications......Page 30
1.3.2 Pattern Recognition Perspective......Page 31
1.4 Relevance of Soft Computing......Page 32
1.5 Scope and Organization of the Book......Page 33
References......Page 37
2.1 Introduction......Page 44
2.2 Fuzzy Sets......Page 45
2.3 Rough Sets......Page 46
2.4.2 Computational Theory of Perception and f -Granulation......Page 49
2.4.3 Rough-Fuzzy Computing......Page 51
2.5 Generalized Rough Sets......Page 52
2.6 Entropy Measures......Page 53
2.7 Conclusion and Discussion......Page 59
References......Page 60
3.1 Introduction......Page 70
3.2.1 Hard c-Means......Page 72
3.2.2 Fuzzy c-Means......Page 73
3.2.3 Possibilistic c-Means......Page 74
3.2.4 Rough c-Means......Page 75
3.3 Rough-Fuzzy-Possibilistic c-Means......Page 76
3.3.1 Objective Function......Page 77
3.3.2 Cluster Prototypes......Page 78
3.3.3 Fundamental Properties......Page 79
3.3.4 Convergence Condition......Page 80
3.3.5 Details of the Algorithm......Page 82
3.3.6 Selection of Parameters......Page 83
3.4.1 RFCM: Rough-Fuzzy c-Means......Page 84
3.4.2 RPCM: Rough-Possibilistic c-Means......Page 85
3.4.3 RCM: Rough c-Means......Page 86
3.4.6 PCM: Possibilistic c-Means......Page 87
3.5.1 Average Accuracy, a Index......Page 88
3.5.3 Accuracy of Approximation, a* Index......Page 90
3.6.1 Quantitative Indices......Page 91
3.6.2 Synthetic Data Set: X32......Page 92
3.6.3 Benchmark Data Sets......Page 93
3.7 Conclusion and Discussion......Page 103
References......Page 104
4.1 Introduction......Page 108
4.2 Pattern Classification Model......Page 110
4.2.1 Class-Dependent Fuzzy Granulation......Page 111
4.2.2 Rough-Set-Based Feature Selection......Page 113
4.3.1 Dispersion Measure......Page 118
4.3.3 k Coefficient......Page 119
4.4 Description of Data Sets......Page 120
4.4.1 Completely Labeled Data Sets......Page 121
4.4.2 Partially Labeled Data Sets......Page 122
4.5 Experimental Results......Page 123
4.5.1 Statistical Significance Test......Page 125
4.5.3 Performance on Completely Labeled Data......Page 126
4.5.4 Performance on Partially Labeled Data......Page 133
4.6 Conclusion and Discussion......Page 135
References......Page 137
5.1 Introduction......Page 140
5.2 Fuzzy-Rough Sets......Page 143
5.3.1 Fuzzy Equivalence Partition Matrix and Entropy......Page 144
5.3.2 Mutual Information......Page 146
5.4.1 V -Information......Page 148
5.4.2 Ia-Information......Page 149
5.4.4 ca-Information......Page 150
5.4.6 Renyi Distance......Page 151
5.5.1 Feature Selection Using f -Information......Page 152
5.5.2 Computational Complexity......Page 153
5.5.3 Fuzzy Equivalence Classes......Page 154
5.6.2 Existing Feature Evaluation Indices......Page 156
5.7 Experimental Results......Page 158
5.7.1 Description of Data Sets......Page 159
5.7.2 Illustrative Example......Page 160
5.7.3 Effectiveness of the FEPM-Based Method......Page 161
5.7.5 Optimum Value of Multiplicative Parameter h......Page 164
5.7.6 Performance of Different f -Information Measures......Page 168
5.7.7 Comparative Performance of Different Algorithms......Page 175
References......Page 179
6.1 Introduction......Page 184
6.2.1 Bio-Basis Function......Page 187
6.2.2 Selection of Bio-Basis Strings Using Mutual Information......Page 189
6.2.3 Selection of Bio-Basis Strings Using Fisher Ratio......Page 190
6.3.1 Hard c-Medoids......Page 191
6.3.2 Fuzzy c-Medoids......Page 192
6.3.3 Possibilistic c-Medoids......Page 193
6.3.4 Fuzzy-Possibilistic c-Medoids......Page 194
6.4.1 Rough c-Medoids......Page 195
6.4.2 Rough-Fuzzy c-Medoids......Page 197
6.5 Relational Clustering for Bio-Basis String Selection......Page 199
6.6.1 Using Homology Alignment Score......Page 201
6.6.2 Using Mutual Information......Page 202
6.7.1 Description of Data Sets......Page 204
6.7.2 Illustrative Example......Page 206
6.7.3 Performance Analysis......Page 207
References......Page 219
7.1 Introduction......Page 224
7.2.2 Self-Organizing Map......Page 226
7.2.4 Graph-Theoretical Approach......Page 227
7.2.5 Model-Based Clustering......Page 228
7.2.8 Rough-Fuzzy Clustering......Page 229
7.3.2 Eisen and Cluster Profile Plots......Page 230
7.3.4 Gene-Ontology-Based Analysis......Page 231
7.4.1 Fifteen Yeast Data......Page 232
7.4.5 Reduced Cell Cycle Data......Page 234
7.5.2 Comparative Analysis of Different c-Means......Page 235
7.5.4 Comparative Analysis of Different Algorithms......Page 238
7.6 Conclusion and Discussion......Page 240
References......Page 243
8.1 Introduction......Page 248
8.2 Evaluation Criteria for Gene Selection......Page 250
8.2.2 Euclidean Distance......Page 251
8.2.4 Mutual Information......Page 252
8.3 Approximation of Density Function......Page 253
8.3.2 Parzen Window Density Estimator......Page 254
8.3.3 Fuzzy Equivalence Partition Matrix......Page 256
8.4 Gene Selection using Information Measures......Page 257
8.5.1 Support Vector Machine......Page 258
8.5.3 Performance Analysis of the FEPM......Page 259
8.6 Conclusion and Discussion......Page 273
References......Page 275
9.1 Introduction......Page 280
9.2 Pixel Classification of Brain MR Images......Page 282
9.2.1 Performance on Real Brain MR Images......Page 283
9.2.2 Performance on Simulated Brain MR Images......Page 286
9.3 Segmentation of Brain MR Images......Page 287
9.3.1 Feature Extraction......Page 288
9.3.2 Selection of Initial Prototypes......Page 297
9.4.1 Illustrative Example......Page 300
9.4.2 Importance of Homogeneity and Edge Value......Page 301
9.4.3 Importance of Discriminant Analysis-Based Initialization......Page 302
9.4.4 Comparative Performance Analysis......Page 303
References......Page 306
Index......Page 310