The Geometry of a Segment in a Family of Cluster Partitions of a Finite Set

  • С.В. Дронов Алтайский государственный университет (Барнаул, Россия)
Keywords: cluster metric, partitions of a finite set, partial order, a segment in a metric space

Abstract

The paper considers the metric space of a family of all partitions of a finite set into non-empty disjoint subsets in the cluster distance proposed by the author in one of the previous papers. The relation between this space structure and the partial order generated by the inclusion on a family of partitions is investigated. It is found out that the segment it is coordinated with the partial order when the segment is determined in such space with the boundaries of A and B as the set of those C that the sum of the distances from it to A and B is equal to the distance from A to B. This is expressed by the fact that the distance between partitions corresponds to the smallest path length between them along the chains in the lattice of the corresponding partial order. Nevertheless, the segment defined this way has significant differences from ordinary segments in vector spaces. Therefore, it is not possible to completely carry out the analogy with theorems of usual geometry. The obtained results can be used in the construction of new algorithms for cluster analysis, as well as for finding the exact probability distributions of the distance between, in some sense, a correct partition and a partition constructed from the data of real observations.

DOI 10.14258/izvasu(2018)1-13

Downloads

Download data is not yet available.

Author Biography

С.В. Дронов, Алтайский государственный университет (Барнаул, Россия)
доцент кафедры математического анализа Алтайского государственного университета

References

Айвазян С.А., Бухштабер В.М., Еню-ков И.С., Мешалкин Л.Д. Прикладная статистика: Классификация и снижение размерности. — М., 1989.

Mills P. Efficient statistical classification of satellite measurements. // International Journal of Remote Sensing. — 2011. — № 32 (21). DOI: 10.1080/01431161.2010.507795

Бериков В.С., Лбов Г.С. Современные тенденции в кластерном анализе // Всероссийский конкурсный отбор обзорноаналитических статей по приоритетному направлению «Информационно-телекоммуникационные системы». — Новосибирск, 2008.

Dronov S.V., Dementjeva E.A. A new approach to post-hoc problem in cluster analysis // Model Assisted Statistics and Applications. - 2012. — Vol. 7, № 1. DOI: 10.3233/MAS-2011-02-01.

Биргхоф Г. Теория решеток. — M. ; 1984.

Gratzer G. Lattice Theory: Foundations. — N.Y.: 2011.

Khamsi M.A. An Introduction to Metric Spaces and Fixed Point Theory. — San Francisco, CA: 2001.

Бураго Д.Ю., Бураго Ю.Д., Иванов С.В. Курс метрической геометрии. — М.; Ижевск, 2004.

Гуров С.И. Булевы алгебры, упорядоченные множества, решетки: определения, свойства, примеры. — М., 2013.

Дьёдонне Ж. Линейная алгебра и элементарная геометрия. — М., 1972.

Kaplansky I. Set Theory and Metric Spaces. — Washington, DC: 2001.

Sackett D.L., Rosenberg W.M., Gray J.A., Haynes R.B., Richardson W.S. Evidence Based Medicine: What It Is and What It Isn’t // BMJ — 1996. - № 312 (7023). D0I:10.1136/bmj.312.7023.71

Bryukhanova E.A., Dronov S.V., Chekryzhova O.I. Spatial Approach to the Analysis of the Employment Data in Siberia Based on the 1897 Census (the Experience of the Multivariate Statistical Analysis of the Districts Data) // Journal of Siberian Federal University. Humanities & Social Sciences. — 2016. — № 7. DOI: 10.17516/1997-13702016-9-7-1651-1660.
Published
2018-03-06
How to Cite
Дронов, С. (2018). The Geometry of a Segment in a Family of Cluster Partitions of a Finite Set. Izvestiya of Altai State University, (1(99), 75-80. https://doi.org/https://doi.org/10.14258/izvasu(2018)1-13
Section
Математика и механика