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

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Springer Science & Business Media, Apr 15, 2008 - Computers - 247 pages
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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
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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...
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