Fundamentals of Fuzzy Clustering Methods

2012 ◽  
pp. 37-61
Author(s):  
Edy Portmann
Keyword(s):  
Fuzzy Clustering ◽  
Clustering Methods ◽  
Fuzzy Clustering Methods

scholarly journals Fuzzy Clustering Methods with Rényi Relative Entropy and Cluster Size

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1423
Author(s):  
Javier Bonilla ◽  
Daniel Vélez ◽  
Javier Montero ◽  
J. Tinguaro Rodríguez
Keyword(s):  
Relative Entropy ◽  
Fuzzy Clustering ◽  
Cluster Size ◽  
Computational Study ◽  
Gaussian Kernel ◽  
Clustering Methods ◽  
Rényi Divergence ◽  
Divergence Measures ◽  
Fuzzy Clustering Methods ◽  
Rényi Relative Entropy

In the last two decades, information entropy measures have been relevantly applied in fuzzy clustering problems in order to regularize solutions by avoiding the formation of partitions with excessively overlapping clusters. Following this idea, relative entropy or divergence measures have been similarly applied, particularly to enable that kind of entropy-based regularization to also take into account, as well as interact with, cluster size variables. Particularly, since Rényi divergence generalizes several other divergence measures, its application in fuzzy clustering seems promising for devising more general and potentially more effective methods. However, previous works making use of either Rényi entropy or divergence in fuzzy clustering, respectively, have not considered cluster sizes (thus applying regularization in terms of entropy, not divergence) or employed divergence without a regularization purpose. Then, the main contribution of this work is the introduction of a new regularization term based on Rényi relative entropy between membership degrees and observation ratios per cluster to penalize overlapping solutions in fuzzy clustering analysis. Specifically, such Rényi divergence-based term is added to the variance-based Fuzzy C-means objective function when allowing cluster sizes. This then leads to the development of two new fuzzy clustering methods exhibiting Rényi divergence-based regularization, the second one extending the first by considering a Gaussian kernel metric instead of the Euclidean distance. Iterative expressions for these methods are derived through the explicit application of Lagrange multipliers. An interesting feature of these expressions is that the proposed methods seem to take advantage of a greater amount of information in the updating steps for membership degrees and observations ratios per cluster. Finally, an extensive computational study is presented showing the feasibility and comparatively good performance of the proposed methods.

Fuzzy Clustering Methods for Categorical Multivariate Data Based on q-Divergence

2018 ◽  
Vol 22 (4) ◽  
pp. 524-536 ◽  
Author(s):  
Tadafumi Kondo ◽  
◽  
Yuchi Kanzawa
Keyword(s):  
Conventional Method ◽  
Fuzzy Clustering ◽  
Optimization Problems ◽  
Clustering Algorithms ◽  
Multivariate Data ◽  
Clustering Methods ◽  
Kl Divergence ◽  
Fuzzy Clustering Methods ◽  
Conventional Methods ◽  
Vectorial Data

This paper presents two fuzzy clustering algorithms for categorical multivariate data based on q-divergence. First, this study shows that a conventional method for vectorial data can be explained as regularizing another conventional method using q-divergence. Second, based on the known results that Kullback-Leibler (KL)-divergence is generalized into the q-divergence, and two conventional fuzzy clustering methods for categorical multivariate data adopt KL-divergence, two fuzzy clustering algorithms for categorical multivariate data that are based on q-divergence are derived from two optimization problems built by extending the KL-divergence in these conventional methods to the q-divergence. Through numerical experiments using real datasets, the proposed methods outperform the conventional methods in term of clustering accuracy.

A review of fuzzy clustering methods

1984 ◽  
Vol 6 (4) ◽  
pp. 189-191 ◽  
Author(s):  
R.H. Davis ◽  
C.E. Economou
Keyword(s):  
Fuzzy Clustering ◽  
Clustering Methods ◽  
Fuzzy Clustering Methods
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