Journal of Classification

, Volume 25, Issue 1, pp 43–65 | Cite as

Solving Non-Uniqueness in Agglomerative Hierarchical Clustering Using Multidendrograms

  • Alberto Fernández
  • Sergio GómezEmail author


In agglomerative hierarchical clustering, pair-group methods suffer from a problem of non-uniqueness when two or more distances between different clusters coincide during the amalgamation process. The traditional approach for solving this drawback has been to take any arbitrary criterion in order to break ties between distances, which results in different hierarchical classifications depending on the criterion followed. In this article we propose a variable-group algorithm that consists in grouping more than two clusters at the same time when ties occur. We give a tree representation for the results of the algorithm, which we call a multidendrogram, as well as a generalization of the Lance andWilliams’ formula which enables the implementation of the algorithm in a recursive way.


Agglomerative methods Cluster analysis Hierarchical classification Lance and Williams’ formula Ties in proximity 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ARNAU, V., MARS, S., and MARÍN, I. (2005), “Iterative Cluster Analysis of Protein Interaction Data,” Bioinformatics, 21(3), 364–378.CrossRefGoogle Scholar
  2. BACKELJAU, T., DE BRUYN, L., DE WOLF, H., JORDAENS, K., VAN DONGEN, S., and WINNEPENNINCKX, B. (1996), “Multiple UPGMA and Neighbor-Joining Trees and the Performance of Some Computer Packages,” Molecular Biology and Evolution, 13(2), 309–313.Google Scholar
  3. CORMACK, R.M. (1971), “A Review of Classification” (with discussion), Journal of the Royal Statistical Society, Ser. A, 134, 321–367.CrossRefMathSciNetGoogle Scholar
  4. GORDON, A.D. (1999), Classification (2nd ed.), London/Boca Raton, FL:Chapman & Hall/CRC.zbMATHGoogle Scholar
  5. HART, G. (1983), “The Occurrence of Multiple UPGMA Phenograms,” in Numerical Taxonomy, ed. J. Felsenstein, Berlin Heidelberg: Springer-Verlag, pp. 254–258.Google Scholar
  6. LANCE, G.N., and WILLIAMS, W.T. (1966), “A Generalized Sorting Strategy for Computer Classifications,” Nature, 212, 218.CrossRefGoogle Scholar
  7. MACCUISH, J., NICOLAOU, C., and MACCUISH, N.E. (2001), “Ties in Proximity and Clustering Compounds,” Journal of Chemical Information and Computer Sciences, 41, 134–146.CrossRefGoogle Scholar
  8. MORGAN, B.J.T., and RAY, A.P.G. (1995), “Non-uniqueness and Inversions in Cluster Analysis,” Applied Statistics, 44(1), 117–134.zbMATHCrossRefGoogle Scholar
  9. SNEATH, P.H.A., and SOKAL, R.R. (1973), Numerical Taxonomy: The Principles and Practice of Numerical Classification, San Francisco: W. H. Freeman and Company.zbMATHGoogle Scholar
  10. SZÉKELY, G.J., and RIZZO, M.L. (2005), “Hierarchical Clustering via Joint Between-Within Distances: Extending Ward’s Minimum Variance Method,” Journal of Classification, 22, 151–183.CrossRefMathSciNetGoogle Scholar
  11. VAN DER KLOOT, W.A., SPAANS, A.M.J., and HEISER, W.J. (2005), “Instability of Hierarchical Cluster Analysis Due to Input Order of the Data: The Permu CLUSTER Solution,” Psychological Methods, 10(4), 468–476.CrossRefGoogle Scholar
  12. WARD, J.H., Jr. (1963), “Hierarchical Grouping to Optimize an Objective Function,” Journal of the American Statistical Association, 58, 236–244.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Universitat Rovira i VirgiliTarragonaSpain
  2. 2.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain

Personalised recommendations