Clustering Methods: A History of k-Means Algorithms

  • Hans-Hermann Bock
Part of the Studies in Classification, Data Analysis, and Knowledge Organization book series (STUDIES CLASS)


This paper surveys some historical issues related to the well-known k-means algorithm in cluster analysis. It shows to which authors the different versions of this algorithm can be traced back, and which were the underlying applications. We sketch various generalizations (with references also to Diday’s work) and thereby underline the usefulness of the k-means approach in data analysis.


Cluster Criterion Supply Point Class Centroid Class Prototype Comptes Rendus Acad 
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. ANDERBERG, M.R. (1973): Cluster analysis for applications. Academic Press, New York.zbMATHGoogle Scholar
  2. BIJNEN, E.J. (1973): Cluster analysis. Tilburg University Press, Tilburg, Netherlands.zbMATHGoogle Scholar
  3. BOCK, H.-H. (1969): The equivalence of two extremal problems and its application to the iterative classification of multivariate data. Paper presented at the Workshop ‘Medizinische Statistik’, February 1969, Forschungsinstitut Oberwolfach.Google Scholar
  4. BOCK, H.-H. (1974): Automatische Klassifikation. Theoretische und praktische Methoden zur Strukturierung von Daten (Clusteranalyse). Vandenhoeck & Ruprecht, Göttingen.Google Scholar
  5. BOCK, H.-H. (1985): On some significance tests in cluster analysis. Journal of Classification 2, 77–108.zbMATHCrossRefMathSciNetGoogle Scholar
  6. BOCK, H.-H. (1983): A clustering algorithm for choosing optimal classes for the chi-square test. Bull. 44th Session of the International Statistical institute, Madrid, Contributed Papers, Vol 2, 758–762.MathSciNetGoogle Scholar
  7. BOCK, H.-H. (1986): Loglinear models and entropy clustering methods for qualitative data. In: W. Gaul, M. Schader (Eds.): Classification as a tool of research. North Holland, Amsterdam, 19–26.Google Scholar
  8. BOCK, H.-H. (1987): On the interface between cluster analysis, principal component analysis, and multidimensional scaling. In: H. Bozdogan, A.K. Gupta (Eds.): Multivariate statistical modeling and data analysis. Reidel, Dordrecht, 17–34.Google Scholar
  9. BOCK, H.-H. (1992): A clustering technique for maximizing Ø-divergence, noncentrality and discriminating power. In: M. Schader (Ed.): Analyzing and modeling data and knowledge. Springer, Heidelberg, 19–36.Google Scholar
  10. BOCK, H.-H. (1996a): Probability models and hypotheses testing in partitioning cluster analysis. In: P. Arabie, L.J. Hubert, G. De Soete (Eds.): Clustering and classification. World Scientific, Singapore, 377–453.Google Scholar
  11. BOCK, H.-H. (1996b): Probabilistic models in partitional cluster analysis. Computational Statistics and Data Analysis 23, 5–28.zbMATHCrossRefGoogle Scholar
  12. BOCK, H.-H. (1996c): Probabilistic models in cluster analysis. In: A. Ferligoj, A. Kramberger (Eds.): Developments in data analysis. Proc. Intern.Conf. on’ statistical data collection and analysis’, Bled, 1994. FDV, Metodoloski zvezki, 12, Ljubljana, Slovenia, 3–25.Google Scholar
  13. BOCK, H.-H. (2003): Convexity-based clustering criteria: theory, algorithms, and applications in statistics. Statistical Methods & Applications 12, 293–317.zbMATHMathSciNetGoogle Scholar
  14. BRYANT, P. (1988): On characterizing optimization-based clustering methods. Journal of Classification 5, 81–84.CrossRefMathSciNetGoogle Scholar
  15. CHARLES, C. (1977): Regression typologique. Rapport de Recherche no. 257. IRIALABORIA, Le Chesnay.Google Scholar
  16. COX, D.R. (1957) Note on grouping. J. Amer. Statist. Assoc. 52, 543–547.zbMATHCrossRefGoogle Scholar
  17. DALENIUS, T. (1950): The problem of optimum stratification I. Skandinavisk Aktuarietidskrift 1950, 203–213.MathSciNetGoogle Scholar
  18. DALENIUS, T., GURNEY, M. (1951): The problem of optimum stratification. II. Skandinavisk Aktuarietidskrift 1951, 133–148.zbMATHMathSciNetGoogle Scholar
  19. DIDAY, E. (1971): Une nouvelle méthode de classification automatique et reconnaissance des formes: la méthode des nuées dynamiques. Revue de Statistique Appliquée XIX(2), 1970, 19–33.Google Scholar
  20. DIDAY, E. (1972): Optimisation en classification automatique et reconnaissance des formes. Revue Française d’Automatique, Informatique et Recherche Opérationelle (R.A.I.R.O.) VI, 61–96.MathSciNetGoogle Scholar
  21. DIDAY, E. (1973): The dynamic clusters method in nonhierarchical clustering. Intern. Journal of Computer and Information Sciences 2(1), 61–88.CrossRefzbMATHMathSciNetGoogle Scholar
  22. DIDAY, E. et al. (1979): Optimisation en classification automatique. Vol. I, II. Institut National der Recherche en Informatique et en Automatique (INRIA), Le Chesnay, France.zbMATHGoogle Scholar
  23. DIDAY, E., GOVAERT, G. (1974): Classification avec distance adaptative. Comptes Rendus Acad. Sci. Paris 278 A, 993–995.MathSciNetGoogle Scholar
  24. DIDAY, E., GOVAERT, G. (1977): Classification automatique avec distances adaptatives. R.A.I.R.O. Information/Computer Science 11(4), 329–349.MathSciNetGoogle Scholar
  25. DIDAY, E., SCHROEDER, A. (1974a): The dynamic clusters method in pattern recognition. In: J.L. Rosenfeld (Ed.): Information Processing 74. Proc. IFIP Congress, Stockholm, August 1974. North Holland, Amsterdam, 691–697.Google Scholar
  26. DIDAY, E., SCHROEDER, A. (1974b): A new approach in mixed distribution detection. Rapport de Recherche no. 52, Janvier 1974. INRIA, Le Chesnay.Google Scholar
  27. DIDAY, E., SCHROEDER, A. (1976): A new approach in mixed distribution detection. R.A.I.R.O. Recherche Opérationelle 10(6), 75–1060.MathSciNetGoogle Scholar
  28. FISHER, W.D. (1958): On grouping for maximum heterogeneity. J. Amer. Statist. Assoc. 53, 789–798.zbMATHCrossRefMathSciNetGoogle Scholar
  29. FORGY, E.W. (1965): Cluster analysis of multivariate data: efficiency versus interpretability of classifications. Biometric Society Meeting, Riverside, California, 1965. Abstract in Biometrics 21 (1965) 768.Google Scholar
  30. GALLEGOS, M.T. (2002): Maximum likelihood clustering with outliers. In: K. Jajuga, A. Sokolowski, H.-H. Bock (Eds.): Classification, clustering, and data analysis. Springer, Heidelberg, 248–255.Google Scholar
  31. GALLEGOS, M.T., RITTER, G. (2005): A robust method for cluster analysis. Annals of Statistics 33, 347–380.zbMATHCrossRefMathSciNetGoogle Scholar
  32. GRÖTSCHEL, M., WAKABAYASHI, Y. (1989): A cutting plane algorithm for a clustering problem. Mathematical Programming 45, 59–96.zbMATHCrossRefMathSciNetGoogle Scholar
  33. HANSEN, P., JAUMARD, B. (1997): Cluster analysis and mathematical programming. Mathematical Programming 79, 191–215.MathSciNetGoogle Scholar
  34. HARTIGAN, J.A. (1975): Clustering algorithms. Wiley, New York.zbMATHGoogle Scholar
  35. HARTIGAN, J.A., WONG, M.A. (1979): A k-means clustering algorithm. Applied Statistics 28, 100–108.zbMATHCrossRefGoogle Scholar
  36. JANCEY, R.C. (1966a): Multidimensional group analysis. Australian J. Botany 14, 127–130.CrossRefGoogle Scholar
  37. JANCEY, R. C. (1966b): The application of numerical methods of data analysis to the genus Phyllota Benth. in New South Wales. Australian J. Botany 14, 131–149.CrossRefGoogle Scholar
  38. JARDINE, N., SIBSON, R. (1971): Mathematical taxonomy. Wiley, New York.zbMATHGoogle Scholar
  39. JENSEN, R.E. (1969): A dynamic programming algorithm for cluster analysis. Operations Research 17, 1034–1057.zbMATHGoogle Scholar
  40. KAUFMAN, L., ROUSSEEUW, P.J. (1987): Clustering by means of medoids. In: Y. Dodge (Ed.): Statistical data analysis based on the L 1-norm and related methods. North Holland, Amsterdam, 405–416.Google Scholar
  41. KAUFMAN, L., ROUSSEEUW, P.J. (1990): Finding groups in data. Wiley, New York.Google Scholar
  42. LERMAN, I.C. (1970): Les bases de la classification automatique. Gauthier-Villars, Paris.zbMATHGoogle Scholar
  43. LLOYD, S.P. (1957): Least squares quantization in PCM. Bell Telephone Labs Memorandum, Murray Hill, NJ. Reprinted in: IEEE Trans. Information Theory IT-28 (1982), vol. 2, 129–137.Google Scholar
  44. MACQUEEN, J. (1967): Some methods for classification and analysis of multivariate observations. In: L.M. LeCam, J. Neyman (eds.): Proc. 5th Berkeley Symp. Math. Statist. Probab. 1965/66. Univ. of California Press, Berkeley, vol. I, 281–297.Google Scholar
  45. MARANZANA, F.E. (1963): On the location of supply points to minimize transportation costs. IBM Systems Journal 2, 129–135.CrossRefGoogle Scholar
  46. MULVEY, J.M., CROWDER, H.P. (1979): Cluster analysis: an application of Lagrangian relaxation. Management Science 25, 329–340.zbMATHCrossRefGoogle Scholar
  47. PÖTZELBERGER, K., STRASSER, H. (2001): Clustering and quantization by MSP partitions. Statistics and Decision 19, 331–371.zbMATHGoogle Scholar
  48. POLLARD, D. (1982): A central limit theorem for k-means clustering. nnals of Probability 10, 919–926.zbMATHMathSciNetGoogle Scholar
  49. RAO, M.R. (1971): Cluster analysis and mathematical programming. J. Amer. Statist. Assoc. 66, 622–626.zbMATHCrossRefGoogle Scholar
  50. SCHNEEBERGER, H. (1967): Optimale Schichtung bei proportionaler Aufteilung mit Hilfe eines iterativen Analogrechners. Unternehmensforschung 11, 21–32.CrossRefGoogle Scholar
  51. SCLOVE, S.L. (1977): Population mixture models and clustering algorithms. Commun. in Statistics, Theory and Methods, A6, 417–434.CrossRefMathSciNetGoogle Scholar
  52. SODEUR, W. (1974): Empirische Verfahren zur Klassifikation. Teubner, Stuttgart.Google Scholar
  53. SOKAL, R.R., SNEATH, P. H. (1963): Principles of numerical taxonomy. Freeman, San Francisco-London.Google Scholar
  54. SPÄ TH, H. (1975): Cluster-Analyse-Algorithmen zur Objektklassifizierung und Datenreduktion. Oldenbourg Verlag, München-Wien.Google Scholar
  55. SPÄ TH, H. (1979): Algorithm 39: Clusterwise linear regression. Computing 22, 367–373. Correction in Computing 26 (1981), 275.CrossRefMathSciNetGoogle Scholar
  56. SPÄ TH, H. (1985): Cluster dissection and analysis. Wiley, Chichester.Google Scholar
  57. STANGE, K. (1960): Die zeichnerische Ermittlung der besten Schätzung bei proportionaler Aufteilung der Stichprobe. Zeitschrift für Unternehmensforschung 4, 156–163.zbMATHCrossRefGoogle Scholar
  58. STEINHAUS, H. (1956): Sur la division des corps matériels en parties. Bulletin de l’Académie Polonaise des Sciences, Classe III, vol. IV, no. 12, 801–804.MathSciNetGoogle Scholar
  59. STRECKER, H. (1957): Moderne Methoden in der Agrarstatistik. Physica, Würzburg, p. 80 etc.Google Scholar
  60. VICHI, M. (2005): Clustering including dimensionality reduction. In: D. Baier, R. Decker, L. Schmidt-Thieme (Eds.): Data analysis and decision support. Springer, Heidelberg, 149–156.CrossRefGoogle Scholar
  61. VINOD, H.D. (1969): Integer programming and the theory of grouping. J. Amer. Statist. Assoc. 64, 506–519.zbMATHCrossRefGoogle Scholar
  62. VOGEL, F. (1975): Probleme und Verfahren der Numerischen Klassifikation. Vandenhoeck & Ruprecht, Göttingen.zbMATHGoogle Scholar
  63. WINDHAM, M.P. (1986): A unification of optimization-based clustering algorithms. In: W. Gaul, M. Schader (Eds.): Classification as a tool of research. North Holland, Amsterdam, 447–451.Google Scholar
  64. WINDHAM, M.P. (1987): Parameter modification for clustering criteria. Journal of Classification 4, 191–214.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Hans-Hermann Bock
    • 1
  1. 1.Institute of StatisticsRWTH Aachen UniversityAachenGermany

Personalised recommendations