Summary
The L p -product (1 ≤ p ≤ ∞) of r indexed hierarchies is introduced in connection with the L p -product of the corresponding r ultrametric spaces. The Cartesian procluct of two hierarchies appears to be a quasi-hierarchy. Endowed with an index of L p -type (p < ∞). this quasi-hierarchy is in bijection with the L p -product of two ultrametric spaces. The indexed hierarchy associated with the supremum product of r ultrametric spaces is also characterized.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bandelt, H.J. (1992): Four-point characterization of the dissimilarity functions obtained from indexed closed weak hierarchies, Math. Seminar der Universität, Hamburg.
Bandelt, H.J. and Dress, A.W. (1989): Weak hierarchies associated with similarity measures: an additive clustering technique, Bull. Math. Biology, 51, 113–166.
Batbedat, A. (1989): Les dissimilarités Medas et arbas, Statistique et Analyse des données, 14, 3, 1–18
Benzecri J.P. (1973): L’analyse des données. Tome 1, La Taxinomie., Dunod, Paris.
Bertrand, P. and Diday, E. (1991): Les pyramides classifiantes: une extension de la structure hiérarchique, C.R. Acad. Sci. Paris, Série I, 693–696.
Buneman, P. (1974): A note on metric properties of trees, J. Combin. Theory, Ser. B, 17, 48–50.
Diatta, J. and Fichet, B. (1994): From Apresjan hierarchies and Bandelt-Dress weak hierarchies to quasi-hierarchies, In:New approaches in Classification and Data Analysis, Diday, E. et al. (eds.), 111–118, Springer-Verlag, Berlin.
Durand, C. and Fichet, B. (1988): One-to-one correspondences in pyramidal representation: a unified approach, In: Classification and Related Methods of Data Analysis, Bock, H. (ed.), 80–85, North-Holland, Amsterdam.
Escofier, B. (1969): L’analyse factorielle des correspondances, Cahiers du B. U.R.O., Université de Paris VI, 13, 25–29.
Jardine, C.J., Jardine, N. and Sibson, R. (1967): The structure and construction of taxonomic hierarchies, Mathematical Biosciences, 1, 465–482.
Johnson, S.C. (1967): Hierarchical clustering schemes, Psychometrika, 32, 241–254.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Japan
About this paper
Cite this paper
Fichet, B. (1998). The L p -product of ultrametric spaces and the corresponding product of hierarchies. In: Hayashi, C., Yajima, K., Bock, HH., Ohsumi, N., Tanaka, Y., Baba, Y. (eds) Data Science, Classification, and Related Methods. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Tokyo. https://doi.org/10.1007/978-4-431-65950-1_13
Download citation
DOI: https://doi.org/10.1007/978-4-431-65950-1_13
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-70208-5
Online ISBN: 978-4-431-65950-1
eBook Packages: Springer Book Archive