Abstract
This paper describes a solution to the following problem: Given a set of weighted data points, find the cluster center points, which minimize the least squared errors. The k-means-type methods produce good results, but usually the quality of the representation depends on an initial cluster configuration. Also this does not allow a variable number of clusters for a given error tolerance.
The proposed method removes these disadvantages by an adaptive sequential insertion of new clusters in those areas, where the largest errors occur. This can be done more efficiently by using multidimensional Voronoi diagrams and local procedures. The data points can be weighted and arbitrarily distributed in the Euclidean space. The weight of each point may be chosen by the user depending on the importance or correctness of that point. At the same time the method produces a hierarchical multidimensional triangulation of the data at different levels of accuracy.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
A. Bowyer: Computing Dirichlet tesselations, Comp. Journal, Vol. 24, No. 2, 1981, 162–166.
T. Dierks: The Modelling and Visualization of Scattered Volumetric Data, Masters Thesis, Arizona State University, USA, Dec. 1990.
J. A. Hartigan: Clustering Algorithms, Wiley, New York, 1975.
P. Heckbert: Color image quantization for frame buffer display, ACM Trans. Computer Graphics 16, 3 (July 1982), 297–304.
I. Hyafil, R. L. Rivest: Construction optimal binary decision trees is NP-complete, Inf. Process. Lett. 5, May 1976, 15–17.
J. R. McMahon: Knot Selection for Least Squares Approximation using Thin Plate Splines, Masters Thesis, Naval Postgraduate School, Monterey, USA, June 1986.
J. R. McMahon, R. Franke: An Enhanced Knot Selection Algorithm for Least Squares Approximation using Thin Plate Splines, ARO Report 90-1, Trans. of the Seventh Army, Conf. on Applied Math. and Computing.
O. Palacios-Velez, B. C. Renaud: A Dynamic Hierarchical Subdivision Algorithm for Computing Delaunay Triangulations and Other Closest-Point Problems, ACM Trans. on Math. Soft., Vol. 16, No. 3, Sept. 1990, 275–292.
F. P. Preparata, M. I. Shamos: Computational Geometry, Springer, 1985.
S. Z. Selim, M. A. Ismail: K-means-type algorithms: A generalized convergence theorem and characterization of local optimality, IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 1 (1986), 81–87.
S. J. Wan, S. K. M. Wong, P. Prusinkiewicz: An Algorithm for Multidimensional Data Clustering, ACM Trans. on Math. Soft., Vol. 14, No. 2, June 1988, 153–162.
D. F. Watson: Computing the n-dimensional Delaunay tesselation with application to Voronoi polytops, Comp. Journal, Vol. 24, No. 2, 1981, 162–166.
X. Wu, I. H. Witten: A fast k-means type clustering algorithm, Dept. Computer Science, Univ. of Calgary, Canada, May 1985.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schreiber, T. (1991). A Voronoi diagram based adaptive k-means-type clustering algorithm for multidimensional weighted data. In: Bieri, H., Noltemeier, H. (eds) Computational Geometry-Methods, Algorithms and Applications. CG 1991. Lecture Notes in Computer Science, vol 553. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54891-2_20
Download citation
DOI: https://doi.org/10.1007/3-540-54891-2_20
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-54891-1
Online ISBN: 978-3-540-46459-4
eBook Packages: Springer Book Archive