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Probabilistic Aspects in Cluster Analysis

  • H. H. Bock

Abstract

Cluster analysis provides methods and algorithms for partitioning a set of objects O = 1,…, n (or data vectors x1,…, xnR p ) into a suitable number of classes C1,…,Cm ⊆ O such that these classes are homogeneous and each of them comprizes only objects which are’similar’ in some sense. The historical evolution shows a surprising trend from an algorithmic, heuristic and applications oriented point of view (Sokal/Sneath 1963) to a more basic, theory oriented investigation of the structural, mathematical and statistical properties of clustering methods. Nowadays, the questions to be answered are of the type’How many clusters are there ?’,’Is there a classification structure ?’,’Is the calculated classification adequate ?’,’Which are the strongest clusters ?’ etc.

Keywords

Cluster Algorithm Mixture Model Cluster Model Minimum Span Tree Cluster Criterion 
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.

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References

  1. Aitkin, M., Anderson, D., Hinde, J. Statistical modelling of data on teaching style (with discussion). J. Roy. Statist. Soc. A 144 (1981) 419–461.CrossRefGoogle Scholar
  2. Ambrosi, K. Klassifikation mit Dichteschätzungen. Oper. Res. Verf. 22 (1976) 1–11.MathSciNetGoogle Scholar
  3. Anderson, J. J. Normal mixtures and the number of clusters problems. Computational Statistics Quarterly 2 (1985) 3–14.MATHGoogle Scholar
  4. Arnold, S. J. A test for clusters. J. Marketing Research 16 (1979) 545–551.CrossRefGoogle Scholar
  5. Aubuchon, J. C., Hettmansperger, T. P. A note on the estimation of the integral of f 2 (x) . J. Statist. Planing and Inference 9 (1984) 321–332.MathSciNetMATHCrossRefGoogle Scholar
  6. Baubkus, W. Minimizing the variance criterion in cluster analysis: Optimal configurations in the multidimensional normal case. Diplomarbeit, Institute of Statistics, Technical University Aachen, 1984.Google Scholar
  7. Beale, E. M. L. Euclidean cluster analysis. Bull. Intern. Statist. Inst. 43 (1969), Vol. 2, 82–94.Google Scholar
  8. Bezdek, J. C. Pattern recognition with fuzzy objective function algorithms. Plenum Press, New York, 1981.MATHGoogle Scholar
  9. Bickel, P. J., Breiman, L. Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann. Probab. 11 (1983) 185–214.MathSciNetMATHCrossRefGoogle Scholar
  10. Binder, D. A. Bayesian cluster analysis. Biometrika 65 (1978) 31–38.MathSciNetMATHCrossRefGoogle Scholar
  11. Binder, D. A. Approximations to Bayesian clustering rules. Biometrika 68 (1981) 275–286.MathSciNetCrossRefGoogle Scholar
  12. Bock, H. H. Statistische Modelle für die einfache und doppelte Klassifikation von normalverteilten Beobachtungen. Dissertation, Univ. Freiburg i. Brsg., 1968.Google Scholar
  13. Bock, H. H. Statistische Modelle und Bayes’sche Verfahren zur Bestimmung einer unbekannten Klassifikation normalverteilter zufälliger Vektoren. Metrika 18 (1972) 120–132.MathSciNetMATHCrossRefGoogle Scholar
  14. Bock, H.H. Automatische Klassifikation. Theoretische und praktische Methoden zur Gruppierung und Strukturierung von Daten (Clusteranalyse). Vandenhoeck & Ruprecht, Göttingen, 1974.Google Scholar
  15. Bock, H. H. On tests concerning the existence of a classification. In: IRIA, vol. 2, 1977, 449–464.Google Scholar
  16. Bock, H. H. Clusteranalyse mit unscharfen Partitionen. In: Bock, H. H. (ed.): Klassifikation und Erkenntnis III: Numerische Klassifikation. Studien zur Klassifikation SK-6. Indeks-Verlag, Frankfurt, 1979a, 137–163.Google Scholar
  17. Bock, H. H. (1979b) Clustering by density estimation. In: Tomassone, R. (ed.), 1979, 173–186.Google Scholar
  18. Bock, H. H. (1979c) Fuzzy clustering procedures. In: Tomassone, R. (ed.), 1979, 205–218.Google Scholar
  19. Bock, H, H. Dichteschätzung und Clusteranalyse (abstract). Koll.’Dichteschätzung und verwandte Themen’, Univ.-GHS Siegen, FB Math., 20./21. Nov. 1980.Google Scholar
  20. Bock, H. H. Statistical testing and evaluation methods in cluster analysis. (1981) In: Ghosh, J. K., Roy, J. (eds.), 1984, 116–146.Google Scholar
  21. Bock, H. H. Statistische Testverfahren im Rahmen der Clusteranalyse. In: Dahlberg, L, Schader, M. (Hrsg.): Studien zur Klassifikation SK-13. Indeks- Verlag, Frankfurt, 1983, 161–176.Google Scholar
  22. Bock, H. H. On some significance tests in cluster analysis. J. of Classification 2 (1985) 77–108.MathSciNetMATHCrossRefGoogle Scholar
  23. Bock, H. H. Loglinear models and entropy clustering methods for qualitative data. (1986a) In: Gaul, W., Schader, M. (eds.), 1986, 19–26.Google Scholar
  24. Bock, H. H. Multidimensional scaling in the framework of cluster analysis. In: Degens, P. O., Hermes H.-J., Opitz, O. (Hrsg.): Classification and its environment. Indeks-Verlag, Frankfurt, 1986b, 247–258.Google Scholar
  25. Bock, H. H. On the interface between cluster analysis, principal component analysis, and multidimensional scaling. In: Bozdogan, H., Gupta, A. K. (eds.): Multivariate statistical modeling and data analysis. Reidel, Dordrecht, 1987, 17–34.CrossRefGoogle Scholar
  26. Bock, H. H. (ed.) Classification and related methods of data analysis. Proc. First Conference of the International Federation of Classification Societies (IFCS-87), June 29 — July 1, 1987, Aa-chen/FRG. North Holland, Amsterdam, 1988.MATHGoogle Scholar
  27. Boswell, St. B. Nonparametric estimation of the modes of high-dimensional densities. In: Billard, L. (ed.), Computer Science and Statistics. Proc. 16th Symposium on the Interface. North Holland, Amsterdam, 1985, 217–226.Google Scholar
  28. Bryant, P.G., Williamson, J. A. Asymptotic behaviour of classification maximum likelihood estimates. Biometrika 65 (1978) 273–281.MATHCrossRefGoogle Scholar
  29. Bryant, P. G., Williamson, J. A. Maximum likelihood and classification. A comparison of three approaches. In: Gaul, W. et al., 1986, 35–46.Google Scholar
  30. Butler, R. W. Optimal stratification and clustering on the line using the Li-norm. J. Multiv. Analysis 18 (1986) 142–155.CrossRefGoogle Scholar
  31. Butler, R. W. Optimal clustering in the real line. J. Multiv. Analysis 24 (1988) 88–108.MATHCrossRefGoogle Scholar
  32. Céleux, G., Diebolt, J. The SEM algorithm: A probabilistic teacher algorithm derived from the EM algorithm for the mixture problem. Computational Statistics Quarterly 2 (1985) 73–82.Google Scholar
  33. Céleux, G., Diebolt, J. L’algorithme SEM: un algorithme d’apprentissage probabiliste pour la reconnaissance de mélange de densités. Revue Statist. Appliquée 34 (1986).Google Scholar
  34. Cressie, N. An optimal statistic based on higher order gaps. Biometrika 66 (1979) 619–628.MathSciNetMATHCrossRefGoogle Scholar
  35. Davies, P. L. Consistent estimates for finite mixtures of well separated elliptical distributions. In: Bock, H. H. (ed.): IFCS-87, 1988, 195–202.Google Scholar
  36. Degens, P.O. Clusteranalyse auf topologisch-maßtheoretischer Grundlage. Dissertation, Universität München, Fachbereich Mathematik, 1978.Google Scholar
  37. Deheuvels, P., Einmahl, J. H. J., Mason, D. M., Ruymgaart, F. H. The almost sure behavior of maximal and minimal multivariate k n-spacings. J. Multiv. Analysis 24 (1988) 155–176.MathSciNetMATHCrossRefGoogle Scholar
  38. Dette, H., Henze, N. The limit distribution of the largest nearest neighbour link in the unit d-cube. J. Appl. Probab. 26 (1988).Google Scholar
  39. Diday, E. et al. (eds.) Optimisation en classification automatique. Vol. 1, 2. Institut National de Recherche en Informatique et en Automatique (INRIA), Le Chesnay, 1979.Google Scholar
  40. Dubes, R. Jain, A.K. Validity studies in clustering methodologies. Pattern Recognition 11 (1979) 235–254.MATHCrossRefGoogle Scholar
  41. Duda, R. O., Hart, P. E. Pattern classification and scene analysis. Wiley, New York, 1973.MATHGoogle Scholar
  42. Engelman, L., Hartigan, J. A. Percentage points of a test for clusters. J. Amer. Statist. Assoc. 64 (1969) 1647–1649.CrossRefGoogle Scholar
  43. Everitt, B. S. A Monte Carlo investigation of the likelihood ratio test for the number of components in a mixture of normal distributions. Multivariate Behavioural Research 16 (1981a) 171–180.CrossRefGoogle Scholar
  44. Everitt, B. S. Contribution to the discussion of the paper by M. Aitkin, D. Anderson and J. Hinde. J. Roy. Statist. Soc. A 144 (1981b) 457–458.Google Scholar
  45. Everitt, B. S., Hand, D. Finite mixture distributions. Chapman and Hall, London, 1981.MATHGoogle Scholar
  46. Felsenstein, J. Numerical taxonomy. Springer-Verlag, Heidelberg, New York, 1983.MATHCrossRefGoogle Scholar
  47. Fukunaga, K., Hostetler, L. D. The estimation of the gradient of a density function with applications in pattern recognition. IEEE Trans. Information Theory IT-21 (1975) 32–40.MathSciNetCrossRefGoogle Scholar
  48. Fukunaga, K., Koontz, W. G. A nonparametric valley-seeking technique for cluster analysis. IEEE Trans. Computers C-21 (1972) 171–178.MathSciNetCrossRefGoogle Scholar
  49. Gänßler, P. On a modification of the k-means clustering procedure. Preprint No. 39, Math. Inst., Univ. München, 1986.Google Scholar
  50. Gates, D. J., Westcott, M. On the distribution of scan statistics. J. Amer. Statist. Assoc. 79 (1984) 423–429.MathSciNetMATHCrossRefGoogle Scholar
  51. Gaul, W, Schader, M. (eds.) Classification as a tool of research. Proc. 9th Annual Meeting of the Gesellschaft für Klassifikation. Karlsruhe, 26 – 28 June 1986. North Holland, Amsterdam, 1987.Google Scholar
  52. Geisser, S., Cornfield, J. Porterior distributions for multivariate normal parameters. J. Roy. Statist. Soc. B 25 (1963) 368–376.MathSciNetMATHGoogle Scholar
  53. Ghosh, J. K., Roy, J. (eds.) Golden Jubilee Conference in Statistics: Applications and new directions. Calcutta, December 1981. Indian Statistical Institute, Calcutta, 1984.Google Scholar
  54. Ghosh, J. K., Sen, P. K. On the asymptotic performance of the log likelihood ratio statistic for the mixture model and related results. In: LeCam, L. M., Ohlsen, R. A. (eds.), 1985, 789–806.Google Scholar
  55. Giacomelli, F. et. al. (eds.) Subpopulations of blood lymphocytes demonstrated by quantitative chemistry. J. Histochemistry and Cytochemistry 19 (1971) 426–433.Google Scholar
  56. Glaz, J., Naus, J. Multiple clusters on the line. Comm. Statist., Theory and Methods 12 (1983) 1961–1986.MathSciNetMATHCrossRefGoogle Scholar
  57. Godehardt, E. Graphs as structural models. The application of graphs and multigraphs in cluster analysis. Vieweg, Braunschweig, 1988.MATHGoogle Scholar
  58. Gray, R.M., Kamin, E.D. Multiple local optima in vector quantizers. IEEE Trans. Information Theory IT-28 (1982) 256–261.CrossRefGoogle Scholar
  59. Hall, P. On powerful distributional tests based on sample spacings. J. Multiv. Analysis 19 (1986) 201–224.MATHCrossRefGoogle Scholar
  60. Hartigan, J. A. Clustering algorithms. Wiley, New York, 1975.MATHGoogle Scholar
  61. Hartigan, J. A. Clusters as modes. (1977a) In: IRIA, vol. II, 1977, 433–448.Google Scholar
  62. Hartigan, J. A. Distribution problems in clustering. (1977b) In: Ryzin, J. van (ed.), 1977, 45–72Google Scholar
  63. Hartigan, J. A. Asymptotic distributions for clustering criteria. Ann. Statist. 6 (1978) 117–131.MathSciNetMATHCrossRefGoogle Scholar
  64. Hartigan, J. A. Consistency of single linkage for high-density clusters. J. Amer. Statist. Assoc. 76 (1981) 388–394.MathSciNetMATHCrossRefGoogle Scholar
  65. Hartigan, J. A. Statistical theory in clustering. J. of Classification 2 (1985a) 63–76.MathSciNetMATHCrossRefGoogle Scholar
  66. Hartigan, J. A. A failure of likelihood asymptotics for normal mixtures. (1985b) In: LeCam, L. M., Ohlsen, R. A. (eds.), 1985, 807–810.Google Scholar
  67. Hartigan, J. A. The span test for unimodality. In: Bock, H.H. (ed.), IFCS-87, 1988, 229–236.Google Scholar
  68. Hartigan, J. A., Hartigan, P. M. The dip test of unimodality. Ann. Statist. 13 (1985) 70–84.MathSciNetMATHCrossRefGoogle Scholar
  69. Henze, N. The limit distribution for maxima of”weighted” rth-nearest neighbour distances. J. Appl. Probab. 19 (1982) 344–354.MathSciNetMATHCrossRefGoogle Scholar
  70. Henze, N. Ein asymptotischer Satz über den maximalen Minimalabstand von unabhängigen Zufallsvektoren mit Anwendung auf einen Anpassungstest im R p und auf der Kugel. Metrika 30 (1983) 245–260.MathSciNetMATHCrossRefGoogle Scholar
  71. Hill. L. R., Silvestri, L. G., Ihm, P., Farchi, G., Lanciani, P. Automatic classification of staphylococci by principal component analysis and a gradient method. J. Bacteriology 89 (1965) 1393–1401.Google Scholar
  72. Hüsler, J. Minimal spacings of non-uniform densities. Stoch. Proc. Appl. 25 (1987) 73–82.MATHCrossRefGoogle Scholar
  73. IRIA Proc. 1st Symp. Data Analysis and Informatics. Versailles, 1977. Institut de Recherche, d’Informatique et d’Automatique (IRIA), Le Chesnay, 1977.Google Scholar
  74. Jahnke, H., Clusteranalyseverfahren als Verfahren der schließenden Statistik. Vandenhoeck & Ruprecht, Göttingen, 1988.MATHGoogle Scholar
  75. Jain A.K., Dubes, R.C. Algorithms for clustering data. Prentice Hall, Englewood Cliffs, 1988.MATHGoogle Scholar
  76. Kopp, B. Über ein Verfahren zur Gruppenbildung durch Dichtefunktionen. Biometrische Zeitschrift 18 (1976) 291–296.MathSciNetMATHCrossRefGoogle Scholar
  77. Krauth, J. An improved upper bound for the tail probabilities of the scan statistic for testing non-random clustering. In: H.H. Bock, (ed.): IFCS-87, 1988, 237–244.Google Scholar
  78. Kuo, M., Rao, J. S. Limit theory and efficiences for tests based on higher order spacings. (1981) In: J.K. Gosh, J. Roy (eds.), 1984, 333–352.Google Scholar
  79. LeCam, L. M., Ohlsen, R. A. (eds.) Proc. Berkely Conference in honor of Jerzy Neyman and Jack Kiefer, Vol.11, Wadsworth, Mon-tery, 1985.Google Scholar
  80. Lee, K. L. Multivariate tests for clusters. J. Amer. Statist. Assoc. 74 (1979) 708–714.MathSciNetMATHCrossRefGoogle Scholar
  81. Lewis, T., Thompson, J. W. Dispersive distributions, and the connection between dispersivity and strong unimodality. J. Appl. Prob. 18 (1981) 76–90.MathSciNetMATHCrossRefGoogle Scholar
  82. Li, L. A., Sedransk, N. Mixtures of distributions: A topological approach. Ann. Statist. 16 (1988) 1623–1634.MathSciNetMATHCrossRefGoogle Scholar
  83. Lynch, J. Mixtures, generalized convexity and balayages. Scand. J. Statist. 15 (1988) 203–210.MathSciNetMATHGoogle Scholar
  84. Marriott, F. H. C. Separating mixtures of normal distributions. Biometrics 31 (1975) 767–769.MATHCrossRefGoogle Scholar
  85. McLachlan, G. J. The classification and mixture maximum likelihood approaches to cluster analysis. In: Krishnaiah, P.R., Kanal, L.N. (eds.): Handbook of Statistics, vol. 2. North Holland, Amsterdam, 1982, 199–208.Google Scholar
  86. McLachlan, G. J. On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Appl. Statist. 36 (1987) 318–324.CrossRefGoogle Scholar
  87. McLachlan, G. J., Basford, K. E. Mixture models. Inference and applications to clustering. Dekker, New York, 1988.MATHGoogle Scholar
  88. Menzefricke, U. Bayesian clustering of data sets. Comm. Statist., Theory and Methods A 10 (1981) 65–77.MathSciNetCrossRefGoogle Scholar
  89. Milligan, G. W. A Monte Carlo study of thirty internal criterion measures for cluster analysis. Psychometrika 46 (1981a) 187–199.MathSciNetMATHCrossRefGoogle Scholar
  90. Milligan, G. W. A review of Monte Carlo tests of cluster analysis. Multivariate Behavioral Research 16 (1981b) 379–401.CrossRefGoogle Scholar
  91. Milligan, G. W., Cooper, M. C. An examination of procedures for determining the number of clusters in a data set. Psychometrika 50 (1985) 159–179.CrossRefGoogle Scholar
  92. Milligan, G. W., Soon, S. C., Sokal, L. M. The effect of cluster size, dimensionality, and the number of clusters on recovery of true cluster structure. IEEE Trans. PAMI-5 (1983) 40–47.Google Scholar
  93. Molenaar, W., Zwet, W. R. van On mixtures of distributions. Ann. Math. Statist. 37 (1966) 281–283.MathSciNetMATHCrossRefGoogle Scholar
  94. Müller, D.W., Sawitzki, G. Using excess mass estimates to investigate the modality of a distribution. Preprint Nr. 398, Univ. Heidelberg, Sonderforschungsbereich 123, Heidelberg, 1987.Google Scholar
  95. Narendra, P. M., Goldberg, M. A non-parametric clustering schema for Landsat. Pattern Recognition 9 (1977) 207–215.CrossRefGoogle Scholar
  96. Naus, J. I. The distribution of the size of the maximum cluster of points on a line. J. Amer. Statist. Assoc. 60 (1965a) 532–538.MathSciNetCrossRefGoogle Scholar
  97. Naus, J. I. Clustering of random points in two dimensions. Biometrika 52 (1965b) 263–267.MathSciNetMATHCrossRefGoogle Scholar
  98. Naus, J. I. An indexed bibliography of clusters, clumps and coincidences. Intern. Statist. Review 47 (1979) 47–78.MathSciNetMATHGoogle Scholar
  99. Naus, J. I. Approximations for distributions of scan statistics. J. Amer. Statist. Assoc. 7 (1982) 177–183.MathSciNetCrossRefGoogle Scholar
  100. Newell, G. F. Distribution for the smallest distance between any pair of kth nearest neighbor random points on a line. In: Rosenblatt, M. (ed.): Proc. Symp. Time Series Analysis. Wiley, New York, 1963, 89–103.Google Scholar
  101. Pärna, K. Strong consistency of k-means clustering criterion in separable metric spaces. Tartu Riikliku Ülikooli, TOIMEISED 733 (1986) 86–96.Google Scholar
  102. Perruchet, Ch. Significance tests for clusters: Overview and comments. In: J. Felsenstein (ed.), Berlin, 1983, 199–208.Google Scholar
  103. Pino, G. E. del On the asymptotic distribution of k-spacings with applications to goodness-of-fit tests. Ann. Statist. 7 (1979) 1058–1065.MathSciNetMATHCrossRefGoogle Scholar
  104. Pollard, D. Strong consistency of k-means clustering. Ann. Statist. 9 (1981) 135–140.MathSciNetMATHCrossRefGoogle Scholar
  105. Pollard, D. A central limit theorem for k-means clustering. Ann. Pro-bab. 10 (1982a) 919–926.MathSciNetMATHCrossRefGoogle Scholar
  106. Pollard, D. Quantization and the method of k-means. IEEE Trans. Information Theory T-28 (1982b) 199–205.MathSciNetCrossRefGoogle Scholar
  107. Rao, J.S., Sethuraman, J. Pitman efficiencies of tests based on spacings. In: M.L. Puri (ed.): Nonparametric techniques in statistical inference. Cambridge Univ. Press, Cambridge/Mass., 1970, 267–273.Google Scholar
  108. Rasson, J. P., Hardy, A., Weverbergh, D. Point process, classification and data analysis. In: Bock, H. H. (ed.): IFCS-87, 1988, 245–256.Google Scholar
  109. Ripley, B. D. Spatial Statistics. Wiley, New York, 1981.MATHCrossRefGoogle Scholar
  110. Rohlf, F.J. Generalization of the gap test for multivariate outliers. Biometrics 31 (1975) 93–101.MATHCrossRefGoogle Scholar
  111. Ryzin, J. van (ed.) Classification and clustering. Academic Press, New York, 1977.Google Scholar
  112. Sarle, W. S. Cubic clustering criterion. SAS Technical Report A-108. SAS Institute Inc., Cary, NC, 15 November, 1983.Google Scholar
  113. Saunders, R., Funk, G. M. Poisson limits for a clustering model of Strauss. J. Appl. Prob. 14 (1977) 776–784.MathSciNetMATHCrossRefGoogle Scholar
  114. Schilling, M. F. Goodness of fit testing in Rm based on the weighted empirical distribution of certain nearest neighbor statistics. Ann. Statist. 11 (1983) 1–12.MathSciNetMATHCrossRefGoogle Scholar
  115. Schroeder, A. Analyse d’un mélange de distributions de probabilité de même type. Revue de Statistique Appliquée 24 (1976), no.1, 39–62.MathSciNetGoogle Scholar
  116. Schweder, T. On the dispersion of mixtures. Scand. J. Statist., Theory and Applications 9 (1982) 165–170.MathSciNetMATHGoogle Scholar
  117. Sclove, S. L. Population mixture models and clustering algorithms. Communications in Statistics, Theory and Methods A 6 (1977) 417–434.MathSciNetCrossRefGoogle Scholar
  118. Scott, A. J., Knott, M. A cluster analysis method for grouping means in the analysis of variance. Biometrics 30 (1974) 507–512.MATHCrossRefGoogle Scholar
  119. Scott, A. J., Symon, M. J. Clustering methods based on likelihood ratio criteria. Biometrics 27 (1971) 387–397.CrossRefGoogle Scholar
  120. Shaked, M. On mixtures from exponential families. J. Roy. Statist. Soc. B 42 (1980) 192–198.MathSciNetMATHGoogle Scholar
  121. Silverman, B.W. Limit theorems for dissociated random variables. Adv. Appl. Prob. 8 (1976) 806–819.MATHCrossRefGoogle Scholar
  122. Silverman, B. W. Using kernel density estimates to investigate multimodality. J. Roy. Statist. Soc. B 43 (1981) 97–99.Google Scholar
  123. Silverman, B., Brown, T.Short distances, flat triangles and Poisson limits. J. Appl. Prob. 15 (1978) 816–826.MathSciNetGoogle Scholar
  124. Sokal, R. R., Sneath, P. H. A. Principles of numerical taxonomy. Freeman, San Francisco-London, 1963.Google Scholar
  125. Späth, H. Cluster dissection and analysis. Wiley, Chichester, 1985.MATHGoogle Scholar
  126. Symons, M. J. Clustering criteria and multivariate normal mixtures. Biometrics 37 (1981) 35–43.MathSciNetMATHCrossRefGoogle Scholar
  127. Titterington, D. M. Contribution to the discussion of the paper by M. Aitkin, D. Anderson and J. Hinde. J. Roy. Statist. Soc. A 144 (1981) 459.MathSciNetCrossRefGoogle Scholar
  128. Titterington, D. M., Smith, A. F. M., Makov, U. E. Statistical analysis of finite mixture distributions. Wiley, Chichester, 1985.MATHGoogle Scholar
  129. Tomassone, R. (ed.) Analyse de données et informatique. Institut National de Recherche en Informatique et en Automatique (IN-RIA), Le Chesnay, 1979.Google Scholar
  130. Trémolières, R. The percolation method for an efficient grouping of data. Pattern Recognition 11 (1979) 255–262.MATHCrossRefGoogle Scholar
  131. Windham, M. P. Parameter modification for clustering criteria. J. of Classification 4 (1987) 191–214.MathSciNetMATHCrossRefGoogle Scholar
  132. Wolfe, J. H. A Monte-Carlo study of the sampling distribution of the likelihood ratio for mixtures of multinormal distributions. Tech. Bull. STB 72–2. Naval Personnel and Training Research Laboratory, San Diego, 1971.Google Scholar
  133. Wong, M. A. A hybrid clustering method for identifying high-desity clusters. J. Amer. Statist. Assoc. 77 (1982) 841–847.MathSciNetMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin · Heidelberg 1989

Authors and Affiliations

  • H. H. Bock
    • 1
  1. 1.Institut für Statistik und WirtschaftsmathematikTechnical University AachenGermany

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