Abstract
Directed binary hierarchies have been introduced in order to give a graphical reduced representation of a family of association rules. This type of structure extends the classical binary hierarchical classification in a very specific way. In this paper an accurate formalization of this new structure is studied. A directed hierarchy is defined as a set of ordered pairs of subsets of the initial individual set satisfying specific conditions. A new notion of directed ultrametricity is studied. The main result consists in establishing a bijective correspondence between a directed ultrametric space and a directed binary hierarchy. Finally, an algorithm is proposed in order to transform a directed ultrametric structure into a graphical representation associated with a directed binary hierarchy.
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We are indebted to Professor Benjamin Enriquez of the University Louis Pasteur of Strasbourg for his relevant remarks and suggestions which have improved the presentation of a first version of this article.
We are also very indebted to the anonymous reviewers for their comments and suggestions which lead us to a more explicit and to a richer version.
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Lerman, IC., Kuntz, P. Directed Binary Hierarchies and Directed Ultrametrics. J Classif 28, 272–296 (2011). https://doi.org/10.1007/s00357-011-9091-y
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DOI: https://doi.org/10.1007/s00357-011-9091-y