Algorithms for clustering data
Algorithms for clustering data
An order theoretic framework for overlapping clustering
Discrete Mathematics - Special issue: trends in discrete mathematics
Quasi-ultrametrics and their 2-ball hypergraphs
Proceedings of the conference on Discrete metric spaces
Set systems and dissimilarities
European Journal of Combinatorics
The k-weak hierarchical representations: an extension of the indexed closed weak hierarchies
Discrete Applied Mathematics - Special issue: The 1998 conference on ordinal and symbolic data analysis (OSDA '98)
The k-weak hierarchical representations: an extension of the indexed closed weak hierarchies
Discrete Applied Mathematics - Special issue: The 1998 conference on ordinal and symbolic data analysis (OSDA '98)
Clustering pair-wise dissimilarity data into partially ordered sets
Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining
Discrete Applied Mathematics
Cluster structures and collections of Galois closed entity subsets
Discrete Applied Mathematics
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There are several well known bijections between classes of dissimilarity coefficients and structures such as indexed or weakly indexed pyramids, as well as indexed closed weak hierarchies. Our goal will be to approach these results from the viewpoint developed by Jardine and Sibson (Mathematical Taxonomy, Wiley, New York, 1971). Properties of dissimilarity coefficients will be related to properties of the maximal linked subsets defined by the family of relations associated with the underlying dissimilarity coefficient. Our approach also involves a close study of the inclusion and diameter conditions introduced by Diatta and Fichet (in: E. Diday et al. (Eds.), New Approaches in Classification and Data Analysis, Springer, Berlin, 1994, p. 111). Typical results include showing that the diameter condition is equivalent to a weakening of the Bandelt four-point characterization that appears in Bandelt (Mathematisches Seminar, Universität Hamburg, Germany, 1992) as well as Bandelt and Dress (Discrete Math. 136 (1994) 21), and this in turn is equivalent to the maximal linked subsets being closed under nonempty intersections; the inclusion condition is equivalent to the 2-balls coinciding with the weak clusters; the Bandelt four-point characterization is equivalent to the maximal linked subsets coinciding with the weak clusters; and a Robinsonian dissimilarity coefficient is strongly Robinsonian (in the sense of Fichet (in: Y.A. Prohorov, V.V. Sazonov (Eds.), Proceedings of the First World Congress of the BERNOULLI SOCIETY, Tachkent, 1986, V.N.U. Science Press, Vol. 2, 1987, p. 123)) if and only if it satisfies the inclusion condition.