# Algorithms for Fuzzy Clustering: Methods in c-Means Clustering with Applications

Springer Science & Business Media, Apr 15, 2008 - Computers - 247 pages
Recently many researchers are working on cluster analysis as a main tool for exploratory data analysis and data mining. A notable feature is that specialists in di?erent ?elds of sciences are considering the tool of data clustering to be useful. A major reason is that clustering algorithms and software are ?exible in thesensethatdi?erentmathematicalframeworksareemployedinthealgorithms and a user can select a suitable method according to his application. Moreover clusteringalgorithmshavedi?erentoutputsrangingfromtheolddendrogramsof agglomerativeclustering to more recent self-organizingmaps. Thus, a researcher or user can choose an appropriate output suited to his purpose,which is another ?exibility of the methods of clustering. An old and still most popular method is the K-means which use K cluster centers. A group of data is gathered around a cluster center and thus forms a cluster. The main subject of this book is the fuzzy c-means proposed by Dunn and Bezdek and their variations including recent studies. A main reasonwhy we concentrate on fuzzy c-means is that most methodology and application studies infuzzy clusteringusefuzzy c-means,andfuzzy c-meansshouldbe consideredto beamajortechniqueofclusteringingeneral,regardlesswhetheroneisinterested in fuzzy methods or not. Moreover recent advances in clustering techniques are rapid and we requirea new textbook that includes recent algorithms.We should also note that several books have recently been published but the contents do not include some methods studied herein.

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### Contents

 Introduction 1 11 Fuzziness and Neural Networks in Clustering 3 12 An Illustrative Example 4 Basic Methods for cMeans Clustering 8 22 A Basic Algorithm of cMeans 11 23 Optimization Formulation of Crisp cMeans Clustering 12 24 Fuzzy cMeans 16 25 EntropyBased Fuzzy cMeans 20
 552 Robustness of Algorithms 117 Application to Classifier Design 119 611 A Generalized Objective Function 120 612 Connections with kHarmonic Means 123 613 Graphical Comparisons 125 62 Clustering with Iteratively Reweighted Least Square Technique 130 63 FCM Classifier 133 631 Parameter Optimization with CV Protocol and Deterministic Initialization 134

 26 Addition of a Quadratic Term 23 261 Derivation of Algorithm in the Method of the Quadratic Term 24 27 Fuzzy Classification Rules 25 28 Clustering by Competitive Learning 29 29 Fixed Point Iterations General Consideration 30 210 Heuristic Algorithms of Fixed Point Iterations 31 211 Direct Derivation of Classification Functions 33 212 Mixture Density Model and the EM Algorithm 36 2121 The EM Algorithm 37 2122 Parameter Estimation in the Mixture Densities 39 Variations and Generalizations I 43 311 EntropyBased Possibilistic Clustering 44 312 Possibilistic Clustering Using a Quadratic Term 46 32 Variables for Controlling Cluster Sizes 47 321 Solutions for JefcaU V A 50 33 Covariance Matrices within Clusters 51 331 Solutions for FCMAS by the GKGustafsonKessel Method 53 34 The KL KullbackLeibler Information Based Method 55 35 Defuzzified Methods of cMeans Clustering 56 351 Defuzzified cMeans with Cluster Size Variable 57 352 Defuzzification of the KLInformation Based Method 58 354 Efficient Calculation of Variables 59 36 Fuzzy cVarieties 60 361 Multidimensional Linear Varieties 62 38 Noise Clustering 65 Variations and Generalizations II 67 411 Transformation into HighDimensional Feature Space 68 412 Kernelized Crisp cMeans Algorithm 71 413 Kernelized Learning Vector Quantization Algorithm 73 414 An Illustrative Example 74 42 Similarity Measure in Fuzzy cMeans 77 421 Variable for Controlling Cluster Sizes 80 422 Kernelization Using Cosine Correlation 81 423 Clustering by Kernelized Competitive Learning Using Cosine Correlation 84 43 Fuzzy cMeans Based on L₁ Metric 86 431 Finite Termination Property of the L₁ Algorithm 88 432 Classification Functions in the L₁ Case 89 433 Boundary between Two Clusters in the L₁ Case 90 44 Fuzzy cRegression Models Based on Absolute Deviation 91 441 Termination of Algorithm Based on Least Absolute Deviation 93 442 An Illustrative Example 96 Miscellanea 99 52 Other Methods of Fuzzy Clustering 100 522 Relational Clustering 101 53 Agglomerative Hierarchical Clustering 102 531 The Transitive Closure of a Fuzzy Relation and the Single Link 106 54 A Recent Study on Cluster Validity Functions 108 542 Kernelized Measures of Cluster Validity 110 544 Kernelized XieBeni Index 111 55 Numerical Examples 112
 632 Imputation of Missing Values 136 633 Numerical Experiments 139 64 Receiver Operating Characteristics 144 65 Fuzzy Classifier with Crisp cMeans Clustering 150 652 Numerical Experiments 153 Fuzzy Clustering and Probabilistic PCA Model 156 712 Another Interpretation of Mixture Models 159 713 FCMtype Counterpart of Gaussian Mixture Models 160 72 Probabilistic PCA Mixture Models and Regularized Fuzzy Clustering 162 722 Linear Fuzzy Clustering with Regularized Objective Function 164 723 An Illustrative Example 167 Local Multivariate Analysis Based on Fuzzy Clustering 171 812 Switching Linear Regression by Standard Fuzzy cRegression Models 174 813 Local Regression Analysis with Centered Data Model 175 814 Connection of the Two Formulations 177 82 Local Principal Component Analysis and Fuzzy cVarieties 179 822 Local PCA Based on Fitting LowDimensional Subspace 182 823 Linear Clustering with Variance Measure of Latent Variables 183 824 Local PCA Based on Lower Rank Approximation of Data Matrix 184 825 Local PCA Based on Regression Model 186 83 Fuzzy ClusteringBased Local Quantification of Categorical Variables 188 832 Local Quantification Method and FCV Clustering of Categorical Data 190 833 Application to Classification of Variables 192 834 An Illustrative Example 193 Extended Algorithms for Local Multivariate Analysis 195 912 Linear Fuzzy Clustering with Partial Distance Strategy 197 913 Linear Fuzzy Clustering with Optimal Completion Strategy 199 914 Linear Fuzzy Clustering with Nearest Prototype Strategy 201 915 A Comparative Experiment 202 92 Componentwise Robust Clustering 203 922 Robust Local Principal Component Analysis 204 923 Handling Missing Values and Application to Missing Value Estimation 207 Collaborative Filtering 208 93 Local Minor Component Analysis Based on Least Absolute Deviations 211 932 Calculation of Optimal Cluster Centers 214 933 An Illustrative Example 215 94 Local PCA with External Criteria 216 942 Local PCA with External Criteria 219 95 Fuzzy Local Independent Component Analysis 220 951 ICA Formulation and Fast ICA Algorithm 221 952 Fuzzy Local ICA with FCV Clustering 222 953 An Illustrative Example 224 96 Fuzzy Local ICA with External Criteria 226 962 Extraction of Local Independent Components Uncorrelated to External Criteria 227 97 Fuzzy ClusteringBased Variable Selection in Local PCA 228 972 Graded Possibilistic Variable Selection 231 973 An Illustrative Example 232 References 234 Index 244 Copyright

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