Abstract: Fuzzy clustering methods using objective functions and solving optimization problems for clustering object data have been very developed, and some of them with a great success as the fuzzy c-means families or hybrid clustering models. Even so, we will focus our attention in fuzzy cluster analysis for relational data which presents a more algebraic structure because generally deals with concepts as decomposition of matrices, fuzzy proximity relations or transitive closures.
One of the most applied fuzzy clustering methods for relational data is the single linkage, which coincides with the transitive closure by the t-norm of the minimum. This method establishes very suitable mathematical properties but sometimes presents inappropriate results, keeping all the objects separated or merging all the objects in only a cluster. Some authors have surpassed these difficulties, improving the results, using the transitive closure by another t-norm, but, unfortunately, appearing other inadequate properties.
We have developed another general procedure in order to try to avoid these difficulties, integrating in a homogeneous methodology the three main steps that are compulsory for clustering, namely: to define the similarity between objects, how to relate the similarity between objects and between clusters, and, finally, the own clustering method. Many fuzzy similarity indexes are defined applying crisp properties. Defining the similarity without this requirement we can also establish the theoretical mathematical bases for ensure that the corresponding index of similarity defines a proximity relation, showing that is essential for this purpose the algebraic structure of the t-norm. Defining the clusters as elements of the same referential space where belong the data we are able to implement an algorithm, based only on the fuzzy cardinality of the fuzzy subsets that describe the objects, which shows promising results.

Brief Biography of the Speaker:
Narcis Clara is Associate Professor of the Department of Computing and Applied Mathematics of the Higher Polytechnic School at the University of Girona. He is graduated in Mathematics for the University of Barcelona and he received the Ph. D. degree from the University of Girona. His research experience and interests are diverse and essentially cover the theory of fuzzy connectives, fuzzy additive generators of t-norms, fuzzy similarity measures, fuzzy clustering and complex systems. He is member of the Differential Equations, Modelling and Applications research group although he usually cooperates with other research groups for dealing with uncertainty in Economics and Management, and Chemical Engineering. He has participated in several projects mainly for developing new mathematical techniques for classification and prediction of environmental and economic variables based on fuzzy systems and neural networks. In collaboration with the Laboratory of Chemical and Environmental Engineering he has developed techniques of soft computing for predicting the quality of water at the effluent of a wastewater treatment plant. He has contributed in many subsidized university projects; papers published in edited books, peer-reviewed journals and international conference proceedings, and have served as a reviewer of International Conferences.