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Abstract: In this study, we present a comprehensive comparative analysis of kernel-based fuzzy clustering and fuzzy clustering. Kernel based clustering has emerged as an interesting and quite visible alternative in fuzzy clustering, however, the effectiveness of this extension vis-à-vis some generic methods of fuzzy clustering has neither been discussed in a complete manner nor the performance of clustering quantified through a convincing comparative analysis. Our focal objective is to understand the performance gains and the importance of parameter selection for kernelized fuzzy clustering. Generic Fuzzy C-Means (FCM) and Gustafson–Kessel (GK) FCM are compared with two typical generalizations of kernel-based fuzzy clustering: one with prototypes located in the feature space (KFCM-F) and the other where the prototypes are distributed in the kernel space (KFCM-K). Both generalizations are studied when dealing with the Gaussian kernel while KFCM-K is also studied with the polynomial kernel. Two criteria are used in evaluating the performance of the clustering method and the resulting clusters, namely classification rate and reconstruction error. Through carefully selected experiments involving synthetic and Machine Learning repository (http://archive.ics.uci.edu/beta/) data sets, we demonstrate that the kernel-based FCM algorithms produce a marginal improvement over standard FCM and GK for most of the analyzed data sets. It has been observed that the kernel-based FCM algorithms are in a number of cases highly sensitive to the selection of specific values of the kernel parameters. [Copyright &y& Elsevier]

Abstract: Ordinal feature values are totally ordered labels that can be considered as fuzzy sets. The formulation of proper fuzzy sets for ordinal labels is important for the systems that deal with the objects of mixed feature types. When a proper ordinal–numerical mapping for an ordinal feature of interest is given, proper fuzzy sets for the labels of the ordinal feature can be easily formulated. In this paper, we propose an adaptive method to learn proper ordinal–numerical mappings for ordinal features of interest from a given objects of mixed features including the ordinal features. The method starts with uniform ordinal–numerical mappings, and performs two steps iteratively. The first step computes a fuzzy partition over the given object set with the ordinal–numerical mappings. The second step learns new ordinal–numerical mappings from the new fuzzy partition in the way that the new mappings make the similarity between two ordinal labels be similar to the average similarity between the objects having the two labels, respectively. Through the alternate repetition of the two steps, both of the ordinal–numerical mappings and the clustering quality become gradually improved. The validity of the proposed method is strongly supported through the experiments with a modified fuzzy C-means clustering algorithm in which the proposed method is implemented. [Copyright &y& Elsevier]

Abstract: The identification of a suitable clustering algorithm to partition data and assessment of the validity of the resultant partitioning are ongoing quests in unsupervised learning. In this study, a fuzzy granulation–degranulation criterion is proposed to evaluate the goodness of a fuzzy partitioning of the data. This, in turn, is used to determine the appropriate clustering algorithm suitable for a particular data set. In general, the quality of a partitioning is measured by computing the variance within it, which is a measure of compactness of the obtained partitioning. Here a new error function, which reflects how well the computed cluster centers represent the whole data set, is used as the goodness measure of the obtained partitioning. Thus a clustering algorithm, providing a good set of cluster centers which approximate well the whole data set, is considered to be the most suited. Thereafter this new fuzzy granulation–degranulation criterion is used to develop six new cluster validity indices. These indices mimic the definitions of the existing and well-known cluster validity indices, such as PBM-index, XB-index, PS-index, FS-index, K-index and SV-index, but use the new fuzzy granulation–degranulation based error function instead of cluster compactness. In order to evaluate the effectiveness of the proposed error function in correctly identifying the appropriate clustering algorithm for a particular data set, eight well-known clustering algorithms, K-means, Fuzzy C-means, GAK-means (genetic algorithm based K-means algorithm), a newly developed genetic point symmetry based clustering technique (GAPS-clustering), Average Linkage clustering algorithm, Expectation Maximization (EM) clustering algorithm, Self-Organizing Map (SOM) and Spectral clustering technique are evaluated on a set of six artificially generated and six real-life data sets. Results show that GAK-means is the most appropriate for most of the data sets used for the experiments. Thereafter the effectiveness of the proposed cluster validity indices in identifying the appropriate number of clusters automatically from different data sets are shown for above mentioned 12 data sets. For the purpose of comparison, results obtained with the original versions of the proposed cluster validity indices and results obtained by a density based clustering technique are also presented. [Copyright &y& Elsevier]