Cluster Analysis Based on Data Depth

  • Richard Hoberg
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)


A data depth depth(y, χ) measures how deep a point y lies in a set χ. The corresponding α-trimmed regions Dα(χ) = y : depth(y,χ) ≤ α are monotonely decreasing with α, that is a α > β implies Dα ⊂ Dβ. We introduce clustering procedures based on weighted averages of volumes of α-trimmed regions.The hypervolume method turns out to be a special case of these procedures.We investigate the performance in a simulation study.


Convex Hull Cluster Procedure Data Depth Cluster Criterion Normal Mixture modeL 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BOCK, H.-H. (1974): Automatische Klassifikation. Vandenhoeck & Ruprecht, Göttingen.Google Scholar
  2. DYCKERHOFF, R., KOSHEVOY, G. and MOSLER, K. (1996): Zonoid Data Depth: Theory and Computation. In A. Pratt (Ed.): Proceedings in Computational Statistics, Physica, Heidelberg, 235–240.Google Scholar
  3. DYCKERHOFF, R. (2000): Computing Zonoid Trimmed Regions of Bivariate Data Sets, COMPSTAT 2000 - Proceedings in Computational Statistics (to appear).Google Scholar
  4. FISHER, L. and VAN NESS, J.W. (1971): Admissible Clustering Procedures. Biometrika, 58, 91–104.CrossRefGoogle Scholar
  5. HARDY, A. and RASSON, J.-P. (1982): Une Nouvelle Approche des Problèmes de Classification Automatique. Statistique et Analyse des Données, 7, 41–56.Google Scholar
  6. KOSHEVOY, G. and MOSLER, K. (1997a): Multivariate Gini Indices. Journal of Multivariate Analysis, 60, 252–276.CrossRefGoogle Scholar
  7. KOSHEVOY, G. and MOSLER, K. (1997b): Lift Zonoid Trimming for Multivariate Distributions. Annals of Statistics, 25, 1998–2017.CrossRefGoogle Scholar
  8. KOSHEVOY, G. and MOSLER, K. (1998): Lift Zonoids, Random Convex Hulls and the Variability of Random Vectors. Bernoulli, 4 377–399.CrossRefGoogle Scholar
  9. LIU, R.Y., PARELIUS, J.M., and SINGH, K. (1990): On a Notion of Data Depth Based on Random Simplices. Annals of Statistics, 18, 405–414.CrossRefGoogle Scholar
  10. MAHALANOBIS, P.C. (1936): On the Generalized Distance in Statistics, Proceedings of National Academy India, 12, 49–55.Google Scholar
  11. RASSON, J.-P. and GRANVILLE, V. (1996): Geometrical Tools in Classification, Computational Statistics and Data Analysis, 23, 105–123.CrossRefGoogle Scholar
  12. RUTS, I. and ROUSSEEUW, P.J. (1996): Computing Depth Contours of Bivariate Point Clouds, Computational Statistics and Data Analysis, 23, 153–168.CrossRefGoogle Scholar
  13. TUKEY, J.W. (1975): Mathematics and Picturing of Data, In: R.D. James (Ed.): The Proceedings of the International Congress of Mathematicians Vancouver, 523–531.Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 2000

Authors and Affiliations

  • Richard Hoberg
    • 1
  1. 1.Seminar für Wirtschafts- und SozialstatistikUniversität zu KölnKölnGermany

Personalised recommendations