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Minkowski Generalizations of Ward’s Method in Hierarchical Clustering

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Abstract

In this paper, we consider several generalizations of the popular Ward’s method for agglomerative hierarchical clustering. Our work was motivated by clustering software, such as the R function hclust, which accepts a distance matrix as input and applies Ward’s definition of inter-cluster distance to produce a clustering. The standard version of Ward’s method uses squared Euclidean distance to form the distance matrix. We explore the effect on the clustering of using other definitions of distance, such as the Minkowski distance.

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References

  • ALBIAC, F., and KALTON, N.J. (2006), Topics in Banach Space Theory, New York: Springer.

    Google Scholar 

  • CHEN, Z., and VAN NESS, J.W. (1996), “Space Conserving and Agglomerative Algorithms,” Journal of Classification, 13, 157-168.

    Article  MATH  MathSciNet  Google Scholar 

  • CHRITCHLOW, D.E., PEARL, D.K., and QIAN, C. (1996), “The Triples Distance for Rooted Bifurcating Trees,” Systematic Biology, 3, 323-334.

    Google Scholar 

  • ELSNER, L., HAN, L., KOLTRACHT, I., NEUMANN, M., and ZIPPEN, M. (1997), “On a Polygon Inequality by Bernius and Blanchard”, unpublished manuscript, available at www.ma.huji.ac.il/~landau/preprint97/polygon.ps.

  • EVERITT, B.S., LANDAU, S., and LEESE, M. (2001), Cluster Analysis (4th ed.), New York: Oxford University Press.

    MATH  Google Scholar 

  • HUBERT, L., and ARABIE, P. (1985), “Comparing Partitions”, Journal of Classification, 2, 193-218.

    Article  Google Scholar 

  • JACCARD, P. (1912), “The Distribution of Flora in the Alpine Zone”, New Phytologist, 11, 37-50.

    Article  Google Scholar 

  • KUHNER, M. K., and FELSENSTEIN, J. (1994), “A Simulation Comparison of Phylogeny Algorithms Under Equal and Unequal Evolutionary Rates”, Molecular Biology and Evolution, 11, 459-468.

    Google Scholar 

  • LANCE, G.N., and WILLIAMS, W.T. (1967), “A General Theory of Classificatory Sorting Strategies. 1. Hierarchical Systems,” Computer Journal, 9, 373-380.

    Article  Google Scholar 

  • LEISCH, F., and DIMITRIADOU, E. (2010), “mlbench: Machine Learning Benchmark Problems”, R package version 2.1-0, available at http://cran.r-project.org/web/packages/mlbench/mlbench.pdf.

  • MEILA, M. (2007), “Comparing Clusterings - An Information Based Distance,” Journal of Multivariate Analysis, 98, 873-895.

    Article  MATH  MathSciNet  Google Scholar 

  • PENNY, D., and HENDY, M.D. (1985), “The Use of Tree Comparison Metrics,” Systematic Zoology, 34, 75-82.

    Article  Google Scholar 

  • RAND, W.M. (1971), “Objective Crieteria for the Evaluation of Clustering Methods”, Jounral of the American Statiscal Association, 66, 846-850.

    Article  Google Scholar 

  • ROBINSON, D.F., and FOULDS, L.R. (1981), “Comparison of Phylogenetic Trees,” Mathematical Biosciences, 53, 131-147.

    Article  MATH  MathSciNet  Google Scholar 

  • SEBER, G.A.F. (1984), Multivariate Observations, New York: Wiley.

    Book  MATH  Google Scholar 

  • SOKAL, R.R., and ROHLF, F.J. (1962), “The Comparison of Dendrograms by Objective Methods,” Taxon, 11, 33-40.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • WELLS, J.H., and WILLIAMS, L.R. (1970), Embeddings and Extensions in Analysis, New York: Springer.

    Google Scholar 

  • WILLIAMS, W.T., and CLIFFORD, H.T. (1971), “On the Comparison of Two Classifications of the Same Set of Elements”, Taxon, 20, 519-522.

    Article  Google Scholar 

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Correspondence to Alan Lee.

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Lee, A., Willcox, B. Minkowski Generalizations of Ward’s Method in Hierarchical Clustering. J Classif 31, 194–218 (2014). https://doi.org/10.1007/s00357-014-9157-8

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  • DOI: https://doi.org/10.1007/s00357-014-9157-8

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