## Algorithms for Fuzzy Clustering: Methods in c-Means Clustering with ApplicationsRecently 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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 |

244 | |

### Other editions - View all

### Common terms and phrases

agglomerative arg min artiﬁcial c-means clustering Calculate Cauchy distribution classiﬁcation function cluster centers clustering algorithm clustering criterion Component Analysis consider constraint covariance matrix crisp c-means D(xk D(xk,Vi data points data set deﬁned denote density derived diﬀerent dimensional dissimilarity eigenvalue elements EM algorithm entropy entropy-based method estimate Euclidean distance external criteria fc=i fc=l i=l FCM classiﬁer FCMAS FCRM FCV algorithm FCV clustering Find optimal ﬁrst ﬁxed point fuzzy c-means fuzzy clustering Gaussian golden section search Hence i=l fc=l initial values Iris iteration kernelized least absolute deviation least squares Linear Fuzzy Clustering M-estimation Mahalanobis distance measure membership method of fuzzy minimization missing values mixture models Miyamoto Note number of clusters objective function observed optimal solution parameters partition possibilistic possibilistic clustering principal component Principal Component Analysis problem prototype regression models respectively robust samples technique uki)m updating Wdet

### Popular passages

Page 236 - Fuzzy clustering for the estimation of the parameters of the components of mixtures of normal distributions," Pattern Recognition letter 9, 77-86, N.-Holland, 1989.

Page 236 - Countermeasures (1979) 165171. 6. Fisher, RA, The use of multiple measurements in taxonomic problems, Annals of Eugenics 3 (1936) 179-188. 7. Fukunaga, K. and Olsen, DR , An algorithm for finding intrinsic dimensionality of data, IEEE Trans. Comput. C-20 (1971) 176-183. 8. Hammersley, JM, The distribution of distance in a hypersphere, Ann. Math. Stat. 21 (1950) 447-452. 9. Harman, HH, Modern Factor Analysis (Univ of Chicago Press, Chicago, 1967). 10. Howarth, RJ, Preliminary assessment of a nonlinear...

Page 239 - A possibilistic approach to clustering. IEEE Trans, on Fuzzy Systems 1: 98-110 7.