Abstract
Assume that n k-dimensional data points have been obtained and subjected to a cluster analysis algorithm. A potential concern is whether the resulting clusters have a “causal” interpretation or whether they are merely consequences of a “random” fluctuation. In this report, the asymptotic properties of a number of potentially useful combinatorial tests based on the theory of random interval graphs are described. Some preliminary numerical results illustrating their possible application as a method of resolving the above question are provided.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
BOCK, H.H. (1974): Automatische Klassifikation. Vandenhoeck & Ruprecht, Göttingen.
DAVID, H.A. (1981): Order statistics. John Wiley & Sons, New York.
EBERL, W., HAFNER, R. (1971): Die asymptotische Verteilung von Koinzidenzen. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 18, 322–332.
GODEHARDT, E. (1990): Graphs as structural models. Vieweg, Braunschweig.
GODEHARDT, E., HARRIS, B. (1995): Asymptotic properties of random interval graphs and their use in cluster analysis. University of Wisconsin Statistics Department Technical Report (submitted for publication).
HAFNER, R. (1972): Die asymptotische Verteilung von mehrfachen Koinzidenzen. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 21, 96–108.
JAMMALAMADAKA, S.R., JANSON, S. (1986) Limit theorems for a triangular scheme of U-statistics with applications to interpoint distances. Annals of Probability 14, 1317–1358.
JAMMALAMADAKA, S.R., ZHOU, X. (1990): Some goodness of fit tests in higher dimensions based on interpoint distances. In: Proceedings of the R.C. Bose Symposium on Probability, Statistics and Design of Experiments, Delhi 1988. Wiley Eastern, New Delhi, 391–404.
MAEHARA, H. (1990): On the intersection graph of random arcs on the cycle. In: M. Karoński, J. Jaworski, A. Ruciński (eds.): Random Graphs ‘87. John Wiley & Sons, New York — Chichester — Brisbane, 159–173.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin · Heidelberg
About this paper
Cite this paper
Harris, B., Godehardt, E. (1998). Probability Models and Limit Theorems for Random Interval Graphs with Applications to Cluster Analysis. In: Balderjahn, I., Mathar, R., Schader, M. (eds) Classification, Data Analysis, and Data Highways. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-72087-1_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-72087-1_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63909-1
Online ISBN: 978-3-642-72087-1
eBook Packages: Springer Book Archive