Degenerating Families of Dendrograms
- 102 Downloads
Dendrograms used in data analysis are ultrametric spaces, hence objects of nonarchimedean geometry. It is known that there exist p-adic representations of dendrograms. Completed by a point at infinity, they can be viewed as subtrees of the Bruhat-Tits tree associated to the p-adic projective line. The implications are that certain moduli spaces known in algebraic geometry are in fact p-adic parameter spaces of dendrograms, and stochastic classification can also be handled within this framework. At the end, we calculate the topology of the hidden part of a dendrogram.
KeywordsDendrograms p-adic numbers Bruhat-Tits tree Moduli spaces
Unable to display preview. Download preview PDF.
- BAKER, M. (2004), “Analysis and Dynamics on the Berkovich Projective Line,” Lecture notes, http://www.math.gatech.edu/~mbaker/pdf/BerkNotes.pdf
- BAKER, M., and RUMELY, R. (2007), “Potential Theory on the Berkovich Projective Line,” Book in preparation, http://www.math.gatech.edu/~mbaker/pdf/BerkBook.pdf
- BERKOVICH, V.G. (1990), Spectral Theory and Analytic Geometry over Non-archimedean Fields, Mathematical Surveys and Monographs Number 33, American Mathematical Society.Google Scholar
- BOCK, H.H. (1974), Automatische Klassifikation, StudiaMathematica/Mathematische Lehrbücher, Band XXIV, Göttingen: Vandenhoek & Ruprecht.Google Scholar
- BRADLEY, P.E. (2007a), “Families of Dendrograms,” to appear in Proceedings of the 31st Annual Conference of the German Classification Society on Data Analysis, Machine Learning, and Applications, Freiburg im Breisgau, Springer series Studies in Classification, Data Analysis, and Knowledge Organization, 2007.Google Scholar
- BRADLEY, P.E. (2007b), “Mumford Dendrograms,” to appear in Computer Journal.Google Scholar
- DRAGOVICH, B., and DRAGOVICH, A. (2006), “A p-adic Model of DNA-sequence and Genetic Code,” Preprint arXiv:q-bio.GN/0607018.Google Scholar
- GRIFFITHS, P.A. (1989), Introduction to Algebraic Curves, American Mathematical Society, Translations of Mathematical Monographs, 76.Google Scholar
- HARRIS, J., and MORRISON, I. (1998), Moduli of Curves, Graduate Texts in Mathematics, 187, Berlin: Springer.Google Scholar
- MUMFORD, D. (1999), The Red Book of Varieties and Schemes, (2nd ed., expanded) Lecture Notes in Mathematics, 1358, Berlin: Springer.Google Scholar
- MURTAGH, F. (2004b), “Thinking Ultrametrically,” in Classification, Clustering and Data Mining Applications, Berlin: Springer, pp. 3–14.Google Scholar