Abstract—Most of fuzzy clustering algorithms have some discrepancies, e.g. they are not able to detect clusters with convex shapes, the number of the clusters should be a priori known, they suffer from numerical problems, like sensitiveness to the initialization, etc. This paper studies the synergistic combination of the hierarchical and graph theoretic minimal spanning tree based clustering algorithm with the partitional Gath-Geva fuzzy clustering algorithm. The aim of this hybridization is to increase the robustness and consistency of the clustering results and to decrease the number of the heuristically defined parameters of these algorithms to decrease the influence of the user on the clustering results. For the analysis of the resulted fuzzy clusters a new fuzzy similarity measure based tool has been presented. The calculated similarities of the clusters can be used for the hierarchical clustering of the resulted fuzzy clusters, which information is useful for cluster merging and for the visualization of the clustering results. As the examples used for the illustration of the operation of the new algorithm will show, the proposed algorithm can detect clusters from data with arbitrary shape and does not suffer from the numerical problems of the classical Gath-Geva fuzzy clustering algorithm. Keywords—Clustering, fuzzy clustering, minimal spanning tree, cluster validity, fuzzy similarity. I. INTRODUCTION AST and robust clustering algorithms play an important role in extracting useful information from large databases. The aim of cluster analysis is to partition a set of N objects in c clusters such that objects within clusters should be similar to each other and objects in different clusters should be dissimilar from each other. Clustering can be used to quantize the available data, to extract a set of cluster prototypes for the compact representation of the dataset, to select the relevant features, to segment the dataset into homogenous subsets, and to initialize regression and classification models. There are two main approaches in the clustering: Hard clustering algorithms allocate each object to a single cluster during their operation and in its output. Fuzzy clustering methods assign degrees of membership in several clusters to each input pattern. So, the fuzzy clustering methods result more dynamic separation of the patterns. Manuscript received October 18, 2005. Ágnes Vathy-Fogarassy, University of Veszprém, Department of Mathematics an Computing Science, P.O. Box 158, Veszprém, H-8201 Hungary (e-mail: vathya@almos.vein.hu). Balázs Feil, University of Veszprém, Department of Process Engineering, P.O. Box 158, Veszprém, H-8201 Hungary (e-mail: feilb@fmt.vein.hu). János Abonyi, University of Veszprém, Department of Process Engineering, P.O. Box 158, Veszprém, H-8201 Hungary (e-mail: abonyij@fmt.vein.hu). In the literature a wide variety of algorithms (partitional, hierarchical, density-based, graph-based, model-based, etc.) have been proposed, but it is a difficult challenge to find a general and powerful method that is quite robust and that does not require the fine-tuning of the user. Most of these algorithms have some discrepancies. For example the basic partitional methods are not able to detect convex clusters; when using hierarchical methods the number of the clusters should be a priori known, and they are not efficient enough for large datasets; while linkage-based methods often suffer from the chaining effect. A problem accompanying the use of a partitional algorithm is that the number of the desired clusters should be given in advance. The partitional techniques usually produce clusters by optimizing a criterion function defined either locally (on a subset of the patterns) or globally (defined over all of the patterns). Generally, different cluster shapes (orientations, volumes) are required for the different clusters (partitions), but there is no guideline as to how to choose them a priori. The norm-inducing matrix of the cluster prototypes can be adapted by using estimates of the data covariance, and can be used to estimate the statistical dependence of the data in each cluster. The Gaussian mixture based fuzzy maximum likelihood estimation algorithm (Gath-Geva algorithm (GG)) is based on such an adaptive distance measure, it can adapt the distance norm to the underlying distribution of the data which is reflected in the different sizes of the clusters, hence it is able to detect clusters with different orientation and volume. Unfortunately the GG algorithm is very sensitive to initialization, hence often it cannot be directly applied to the data. The hierarchical clustering approaches are related to graph- theoretic clustering. These algorithms are able to detect clusters of various shapes and sizes, and they do not require initialization. One of the best-known graph-based divisive clustering algorithm is based on the construction of the minimal spanning tree (MST) of the objects [3,7,9,13,16]. By the elimination of any edge from the MST we get subtrees which correspond to clusters. Clustering methods using a minimal spanning tree take advantages of the MST. The MST ignores many possible connections between the data patterns, so the cost of clustering can be decreased. Single-link clusters are subgraphs of the minimum spanning tree of the data [10,11] which are also the connected components. Complete- link clusters are maximal complete subgraphs, and are related to the node colorability of graphs [2]. The maximal complete subgraph was considered the strictest definition of a cluster in [1,15]. Clustering, as an unsupervised learning, is mainly Minimal Spanning Tree based Fuzzy Clustering Ágnes Vathy-Fogarassy, Balázs Feil, and János Abonyi F PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND TECHNOLOGY VOLUME 8 OCTOBER 2005 ISSN 1307-6884 PWASET VOLUME 8 OCTOBER 2005 ISSN 1307-6884 7 © 2005 WASET.ORG