Directed Binary Hierarchies and Directed Ultrametrics

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...

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Bibliographic Details
Published in:Journal of classification Vol. 28; no. 3; pp. 272 - 296
Main Authors: Lerman, I-C., Kuntz, P.
Format: Journal Article
Language:English
Published: New York Springer-Verlag 01.10.2011
Springer
Springer Nature B.V
Springer Verlag
Subjects:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Automatic classification
Bioinformatics
Classification
Computer Science
Computer science; control theory; systems
Exact sciences and technology
Family structure
Hierarchic clustering
Hierarchies
Logic and foundations
Machine Learning
Marketing
Mathematical analysis
Mathematical logic, foundations, set theory
Mathematics
Mathematics and Statistics
Methods of scientific computing (including symbolic computation, algebraic computation)
Multivariate analysis
Numerical analysis. Scientific computation
Pattern Recognition
Probability and statistics
Psychometrics
Recursion theory
Sciences and techniques of general use
Signal,Image and Speech Processing
Statistical Theory and Methods
Statistics
Subsets
Theoretical computing
Directed hierarchical classification
Association rule representation
Data analysis
Factor analysis
Initial condition
Hierarchical classification
Statistical association
Multivariate analysis
Algorithm
Correspondence analysis
Graphic method
Directed graph
Principal component analysis
ISSN:0176-4268, 1432-1343
Online Access:Get full text
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Summary: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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ISSN:0176-4268
1432-1343
DOI:10.1007/s00357-011-9091-y