Probabilistic models in cluster analysis

Computational Statistics and Data Analysis

Volumn 23, Issue 1, 1996, Pages 5-28

  Bock, Hans H ^a  

^a RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

Hierarchical clustering models; Partition type clustering; Phylogenetic inference; Probabilistic cluster analysis; Testing for a clustering structure

Indexed keywords

MARKOV PROCESSES; MATHEMATICAL MODELS; PROBABILITY; TREES (MATHEMATICS); VECTORS;

ASYMPTOTIC DISTRIBUTION; CLUSTER ANALYSIS; CLUSTERING CRITERIA; GROUPING METHODS; HYPOTHESIS TESTING; INFERENTIAL FRAMEWORK; MOLECULAR SEQUENCE DATA; PARTITION TYPE CLUSTERING; PHYLOGENETIC INFERENCE;

STATISTICAL TESTS;

CLUSTER ANALYSIS; PROBABILISTIC MODELLING;

EID: 0030289011     PISSN: 01679473     EISSN: None     Source Type: Journal    
DOI: 10.1016/0167-9473(96)88919-5     Document Type: Article

References (173)

1. Anderson, J.J., Normal mixtures and the number of clusters problem, Comput. Statist. Quart., 2 (1985) 3-14.
2. Archie, J.W., A randomization test for phylogenetic information in systematic data, Systematic Zoology, 38 (1989) 239-252.
3. Banks, D. and K. Carley, Metric inference for social networks, J. Classification, 11 (1994) 121-149.
4. Benkaraache, T. and B. Van Cutsem, Comparison of hierarchical classifications, Technical Report RT-100 (Laboratoire de Modélisation et Calcul, Institut IMAG, CNRS, Grenoble, 1993) 10 pp.
5. Bernardo, J.M., Optimizing prediction with hierarchical models: Bayesian clustering, in: P.R. Freeman and A.F.M. Smith (Eds.), Aspects of uncertainty (Wiley, New York, 1994) 67-76.
6. Bhattacharya, R.N. and J.K. Ghosh, A class of U-statistics and asymptotic normality of the number of k-clusters. J. Multivariate Analysis, 43 (1992) 300-330.
7. Bishop, M.J. and A.E. Friday, Evolutionary trees from nucleic acid and protein sequences, Proc. Royal Soc. London, B 226 (1985) 271-302.
8. Bock, H.H., The equivalence of two extremal problems and its application to the iterative classification of multivariate data, Report of the Conf. on "Medizinische Statistik" (Forschungsinstitut Oberwolfach, February 1969).
9. Bock, H.H., Automatische Klassifikation (Clusteranalyse) (Vandenhoeck and Ruprecht, Göttingen, 480 pp., 1974).
10. Bock, H.H., On tests concerning the existence of a classification, in: Proc. First Symposium on Data Analysis and Informatics, Versailles, 1977, Vol. II (Institut de Recherche d'Informatique et d'Automatique (IRIA), Le Chesnay, 1977) 449-464.
11. Bock, H.H., On some significance tests in cluster analysis, J. Classification, 2 (1985) 77-108.
12. Bock, H.H., Loglinear models and entropy clustering methods for qualitative data, in: W. Gaul, M. Schader (Eds.), Classification as a tool of research (North-Holland, Amsterdam, 1986) 19-26.
13. Bock, H.H. (1987): On the interface between cluster analysis, principal component analysis, and multidimensional scaling, in: H. Bozdogan and A.K. Gupta (Eds.), Multivariate statistical modeling and data analysis (Reidel, Dordrecht, 1987) 17-34.
14. Bock, H.H. (Ed.), Classification and related methods of data analysis (North-Holland, Amsterdam, 1988) 749 pp.
15. Bock, H.H., Probabilistic aspects in cluster analysis, in: O. Opitz (Ed.), Conceptual and numerical analysis of data (Springer, Heidelberg, 1989a) 12-44.
16. Bock, H.H., A probabilistic clustering model for graphs and similarity relations. Paper presented at the Fall Meeting 1989 of the Working Group 'Numerical Classification and Data Analysis' of the Gesellschaft für Klassifikation (Essen, November 1989b).
17. Bock, H.H., Information and entropy in cluster analysis, in: H. Bozdogan et al. (Eds.), Multivariate statistical modeling. Vol. II. Proc. 1st US/Japan Conference on the Frontiers of Statistical Modeling: An Informational Approach. Univ. of Tennessee, Knoxville, 1992 (Kluwer, Dordrecht, 1994) Vol. II, 115-117.
18. Bock, H.H., Probabilistic approaches and hypotheses testing in partitioning cluster analysis, in: Ph. Arabie, G. De Soete and L. Hubert (Eds.), Clustering and classification (World Science Publishers, Singapore and River Edge, NJ/USA, 1996a) 377-453.
19. Bock, H.H., Probabilistic models in partitional cluster analysis, in: A. Ferligoj and A. Kramberger (Eds.), Developments in data analysis (Metodološki zvezki, 12, FDV, Ljubejana, 1996b).
20. Bock, H.H., Probability models for convex clusters, in: O. Opitz and R. Klar (Eds.), Classification and data analysis (Springer, Heidelberg, to appear), 11 pp. (1996c).
21. Bock, H.H. and P. Ihm (Eds.), Classification, data analysis and knowledge organization (Springer, Heidelberg, 1991).
22. Bock, H.H., W. Lenski, M.M. Richter (Eds.), Information systems and data analysis: Prospects, foundations, applications (Springer, Heidelberg, 1994) 462 pp.
23. Bock, H.H. and W. Polasek (Eds.), Data analysis and information systems (Springer, Heidelberg, 1996) 548 pp.
24. Bollobás, B., Random graphs (Academic Press, London, 1985).
25. Boorman, S.A. and D.C. Olivier, Metrics on spaces of finite trees, J. Math. Psychology, 10 (1973) 26-59.
26. Bozdogan, H., Choosing the number of clusters, subset selection of variables, and outlier detection in the standard mixture-model cluster analysis, in: E. Diday et al. (Eds.), 1994, 169-177.
27. Brown, E.K. and W.H.E. Day, A computationally efficient approximation to the nearest neighbor interchange metric, J. Classification, 1 (1984) 93-124.
28. Bryant, P.G. and J.A. Williamson, Asymptotic behaviour of classification maximum likelihood estimates, Biometrika, 65 (1978) 273-281.
29. Camin, J.H. and R.R. Sokal, A method for deducing branching sequences in phylogeny, Evolution, 19 (1965) 311-326.
30. Carter, M., M. Hendy, D. Penny, L.A. Székely and N.C. Wormald, On the distribution of lengths of evolutionary trees, SIAM J. Discrete Math., 3 (1990) 38-47.
31. Cavender, J.A., Necessary conditions for the method of inferring phylogeny by linear invariants, Molecular Biosciences, 103 (1991) 69-75.
32. Cavender, J.A. and J. Felsenstein, Invariants of phylogenies in a simple case with discrete states, J. Classification, 4 (1987) 57-71.
33. Cayley, A., A theorem on trees, Quarterly J. Pure Appl. Math., 23 (1889) 376-378.
34. Céleux, G. and G. Govaert, Clustering criteria for discrete data and latent class models, J. Classification, 8 (1991) 157-176.
35. Cressie, N., Statistics for spatial data (Wiley, New York, 1991).
36. Dale, M.R.T. and J.W. Moon, Statistical tests on two characteristics of the shapes of cluster diagrams, J. Classification, 5 (1988) 21-38.
37. Day, W.H.E., Distributions of distances between pairs of classifications, in: J. Felsenstein (Ed.), 1983c, 127-131.
38. Day, W.H.E., Analysis of quartet dissimilarity measures between undirected phylogentic trees, Systematic Zoology, 35 (1986) 325-333.
39. Day, W.H.E., Estimating phylogenies with invariant functions of data, in: H.H. Bock and P. Ihm (eds.), 1991, 248-253.
40. Day, W.H.E., D.S. Johnston and D. Sankoff, The computational complexity of inferring rooted phylogenies by parsimony, Math. Biosciences, 81 (1986) 33-42.
41. Degens, P.O., Hierarchical cluster methods as maximum likelihood estimators, in: J. Felsenstein (ed.), 1983c, 249-253.
42. Degens, P.O., Ultrametric approximation to distances, Computat. Statist. Quart., 2(1) (1985) 93-101.
43. Degens, P.O., Reconstruction of phylogenies by weighted genetic distances, in: H.H. Bock (Ed.), 1988, 727-739.
44. Degens, P.O., B. Lausen and W. Vach, Reconstruction of phylogenies by distance data: Mathematical framework and statistical analysis. Research report 88/8 (Department of Statistics, University of Dortmund, 1988) 34 pp.
45. De Soete, G., Ultrametric tree representations of incomplete similarity data, J. Classification, 1 (1984) 235-242.
46. De Soete, Tree representations of proximity data by least squares methods, in: H.H. Bock (Ed.). 1988, 147-157.
47. De Soete, J.D. Carroll and W.S. Desarbo, Least squares algorithms for constructing constrained ultrametric and additive tree representations of symmetric proximity data, J. Classification, 4 (1987) 155-173.
48. Dette, H. and N. Henze, The limit distribution of the largest nearest neighbour link in the unit d-cube, J. Appl. Probab., 26 (1988).
49. Diday, E., Introduction à l'analyse factorielle typologique, (Rapport de Recherche No. 27, IRIA, LeChesnay, France, 1973).
50. Diday, E., Y. Lechevallier, M. Schader, P. Bertrand and B. Burtschy (Eds.), New approaches in classification and data analysis, Proc. 4th Conf. Internat. Federation of Classification Societies (IFCS-93), Paris, 1993 (Springer, Heidelberg, 1994) 693 pp.
51. Farris, J.S., A probability model for inferring phylogenetic trees, Systematic Zoology, 22 (1973) 250-256.
52. Felsenstein, J., Cases in which parsimony or compatibility methods will be positively misleading, Systematic Zoology, 27 (1978) 401-410.
53. Felsenstein, J., Evolutionary trees from DNA sequences: A maximum likelihood approach, J. Molecular Evolution, 17 (1981) 368-376.
54. Felsenstein, J., Statistical inference of phytogenies, J. Royal Statist. Soc., A 146 (1983a) 246-272.
55. Felsenstein, J., Parsimony in systematics: Biological and statistical issues, Ann. Rev. Ecological Systems, 14 (1983b) 313-333.
56. Felsenstein, J., Ed., Numerical taxonomy, (NATO Advanced Studies Institute, Ser. G. (Ecological Sciences) 1, Springer, Berlin, 1983c).
57. Felsenstein, J., The statistical approach to inferring evolutionary trees and what it tells us about parsimony and compatibility, in: T. Duncan and T.F. Steussy (Eds.), Cladistics: perspectives on the reconstruction of evolutionary history (Columbia Univ. Press, New York, 1984) 169-191.
58. Felsenstein, J., Confidence limits on phytogenies: An approach using the bootstrap, Evolution, 39 (1985) 783-791.
59. Felsenstein, J., Phytogenies from molecular sequences: inference and reliability, Annual Review of Genetics, 22 (1988) 521-565.
60. Felsenstein, J., PHYLIP (Phylogenetic inference package), Version 3.3 (University of Washington, Seattle, 1990).
61. Felsenstein, J., and E. Sober, Parsimony and likelihood: an exchange, Systematic Zoology, 35 (1986) 617-626.
62. Fienberg, S.E., M.M. Meyer and S.S. Wasserman, Statistical analysis of multiple sociometric relations, J. Amer. Statist. Assoc., 80 (1985) 51-67.
63. Fitch, W.M., Towards defining the course of evolution: Minimum change for a specific tree topology, Systematic Zoology, 20 (1971) 406-416.
64. Fowlkes, E.B. and C.L. Mallows, A method for comparing two hierarchical clusterings, J. Amer. Statist. Assoc., 78 (1983) 553-569.
65. Frank, O., Inferences concerning cluster structure, in: COMPSTAT 1978 (Physica-Verlag, Würzburg, 1978) 259-265.
66. Frank, O. and D. Strauss, Markov graphs, J. Amer. Statist. Assoc., 81 (1986) 832-842.
67. Frank, O. and K. Svensson, On probability distributions of single-linkage dendrograms, J. Statist. Comput. Simul., 12 (1981) 121-131.
68. Furman, W.D. and B.G. Lindsay, Testing for the number of components in a mixture of normal distributions using moment estimators, Comput. Statist. Data Anal., 17 (1994) 473-492.
69. Furnas, G.W., The generation of random, binary unordered trees, J. Classification, 1 (1984) 187-234.
70. Gaul, W. and D. Pfeifer (Eds.), From data to knowledge. Proc. 18th Ann. Conf. Gesellschaft für Klassifikation, Oldenburg, 1994 (Springer-Verlag, Heidelberg, 1996) 472 pp.
71. 2) 240 pp.
72. Godehardt, E., Multigraphs for the uncovering and testing of structures, in: H.H. Bock and P. Ihm (Eds.), 1991, 43-52.
73. Godehardt, E. and A. Horsch, The testing of data structures with graph-theoretical models, in: H.H. Bock, W. Lenski and M.M. Richter (Eds.), 1994, 226-241.
74. Goldman, N., Maximum likelihood inference of phylogenetic trees, with special reference to a Poisson process model of DNA substitution and to parsimony analyses, Systematic Zoology, 39 (1990) 345-361.
75. Gordon, A.D., Clustering algorithms and cluster validation, in: P. Dirschedl and R. Ostermann (Eds.), Computational Statistics (Physica-Verlag, Heidelberg, 1994), 497-512.
76. Gordon, A.D., Null models in cluster validation, in: W. Gaul and D. Pfeifer (Eds.), 1996a, 32-44.
77. Gordon, A.D., Hierarchical classification, in: P. Arabie, L.J. Hubert and G. De Soete (Eds.), Clustering and classification (World Science Publishers, Singapore and River Edge, NJ/USA, 1996b) 65-121.
78. Guénoche, A., Hiérarchies conceptuelles de données binaires, Math. Inform. Sci. Hum., 121 (1993a) 23-34.
79. Guénoche, A., Alignment and hierarchical clustering method for strings, in: O. Opitz, B. Lausen and R. Klar (Eds.), Information and classification: concepts, methods and applications (Springer, Heidelberg, 1993b) 403-412.
80. Hansen, P., B. Jaumard and E. Sanlaville, Partitioning problems in cluster analysis: a review of mathematical programming approaches, in: E. Diday et al. (Eds.), 1994, 228-240.
81. Hansen, P. and B. Jaumard, Computational methods in clustering from a mathematical programming viewpoint, in: H.H. Bock and W. Polasek (Eds.), 1996, 24-40.
82. Harding, E.F., The probabilities of rooted tree-shapes generated by random bifurcation, Advances in Applied Probability, 3 (1971) 44-77.
83. Hardy, A., An examination of procedures for determining the number of clusters in a data set, in: E. Diday et al. (Eds.), 1994, 178-185.
84. Hartigan, J.A., Mimimum mutation fits to a given tree, Biometrics, 29 (1973) 53-65.
85. Hartigan, J.A., Asymptotic distributions for clustering criteria, Ann. Statist., 6 (1978) 117-131.
86. Hartigan, J.A., The SPAN test for multimodality, in: H.H. Bock (Ed.), 1988, 229-236.
87. Hartigan, J.A. and P.M. Hartigan, The DIP test for multimodality, Ann. Statist., 13 (1985) 70-84.
88. Hartigan, J.A. and S. Mohanty, The RUNT test for multimodality, J. Classification, 9 (1992) 63-70.
89. Hasegawa, M., H. Kishino and T. Yano, Dating of the human-ape splitting by a molecular clock of mitochondrial DNA, J. Molecular Evolution, 22 (1985) 160-174.
90. Hasegawa, M., H. Kishino and T. Yano, Phylogenetic inference from DNA sequence data, in: K. Matusita (Ed.), Statistical theory and data analysis II (North Holland, Amsterdam, 1988) 1-13.
91. Hendy, M.D. and D. Penny, Branch and bound algorithms to determine evolutionary trees, Math. Biosciences, 59 (1982) 277-290.
92. Hendy, M.D. and D. Penny, A framework for the quantitative study of evolutionary trees, Systematic Zoology, 38 (1989) 297-309.
93. Henze, N., The limit distribution for maxima of "weighted" rth-nearest neighbour distances, J. Appl. Probab., 19 (1982) 344-354.
94. Hoffman, R. and A.K. Jain, A test of randomness based on the minimal spanning tree, Pattern Recognition Letters, 1 (1983) 175-180.
95. Holland, P.W. and S. Leinhardt, An exponential family of probability distributions for directed graphs, J. Amer. Statist. Assoc., 76 (1981) 33-65.
96. Hubert, L., Assignment methods in combinatorial data analysis (Marcel Dekker, New York, 1987).
97. Jain, A.K. and R.C. Dubes, Algorithms for clustering data (Prentice-Hall, Englewood Cliffs, NJ, 1988).
98. 2 und die Verteilung zugehöriger Gütekriterien (Diploma thesis, Institute of Statistics, Technical University of Aachen, Germany) 204 pp.
99. Janson, S., Maximal spacings in several dimensions, Ann. Probab., 15 (1987) 274-280.
100. Kimura, M., A simple method for estimating evolutionary rates of base substitutions trough comparative studies of nucleotide sequences, J. Molecular Evolution, 2 (1980) 87-90.
101. Kishino, H. and M. Hasegawa, Evaluation of the maximum likelihood estimate of the evolutionary tree topologies from DNA sequence data, and the branching order of Hominoidea, J. Molecular Evolution, 29 (1989) 170-179.
102. Krajowski, C. and A.W. Dickerman, Bootstrap analysis of phylogenetic trees derived from DNA hybridization distances, Systematic Zoology, 39 (1990) 383-390.
103. Lake, J.A., A rate-independent technique for analysis of nucleic acid sequences: Evolutionary parsimony, Molecular Biology and Evolution, 4 (1987) 167-191.
104. Lapointe, F.-J. and P. Legendre, A statistical framework to test the consensus of two nested classifications, Systematic Zoology, 39 (1990) 1-13.
105. Lapointe, F.-J. and P. Legendre, The generation of random ultrametric matrices representing dendrograms, J. Classification, 8 (1991) 177-200.
106. Lapointe, F.-J. and P. Legendre, A statistical framework to test the consensus among additive trees (cladograms), Systematic Biology, 41 (1992a) 158-171.
107. Lapointe, F.-J. and P. Legendre, Statistical significance of the matrix correlation coefficient for comparing independent phylogenetic trees, Systematic Biology, 41 (1992b) 378-384.
108. Lausen, B., Exploring homologuous tRNA sequence data: Positional mutation rates and genetic distance, in: O. Opitz (Ed.), Conceptual and numerical analysis of data (Springer, Heidelberg, 1989) 481-488.
109. Lausen, B., Statistical analysis of genetic distance data, in: H.H. Bock and P. Ihm (Eds.), 1991, 254-261.
110. Lausen, B. and P.O. Degens, Variance estimation and the reconstruction of phylogenies, in: P.O. Degens, H.-J. Hermes and O. Opitz (Eds.), Classification and its environment. Studien zur Klassifikation SK-17 (Indeks-Verlag, Frankfurt a. M., 1986) 306-314.
111. Lausen, B. and P.O. Degens, Evaluation of the reconstruction of phylogenies with DNA-DNA hybridization, in: H.H. Bock (Ed.), 1988, 367-374.
112. Le Quesne, W.J., The uniquely involved character concept and its cladistic application, Systematic Zoology, 23 (1974) 513-517.
113. Lee, K. L., Multivariate tests for clusters, J. Amer. Statist. Assoc., 74 (1979) 708-714.
114. Li, W.-H., A statistical test of phylogenies estimated from sequence data, Molecular Biology and Evolution, 6 (1989) 424-435.
115. Ling, R.F., A probability theory of clustering, J. Amer. Statist. Assoc., 68 (1973) 159-164.
116. Ling, R.F., An exact probability distribution on the connectivity of random graphs, J. Math. Psychology, 12 (1975) 90-98.
117. Ling, R. F. and G.G. Killough, Probability tables for cluster analysis based on a theory of random graphs, J. Amer. Statist. Assoc., 71 (1976) 293-300.
118. Matula, D.W., The employee party problem, Notices Amer. Math. Soc., 19 (1972) A-382.
119. Matula, D.W., The largest clique in a random graph, Technical report CS-7608 (Dept. Computer Science, Southern Methodist Univ., Dallas TX, April 1976) 22 pp.
120. McLachlan, G.J. and K.E. Basford, Mixture models: Inference and applications to clustering (Marcel Dekker, New York and Basel, 1988).
121. Mountford, M.D., A test of the difference between clusters, in: G.P. Patil et al. (Eds.), Statistical ecology. Vol. 3 (Pennsylvania State Univ. Press, University Park, Pa., 1970) 237-257.
122. Müller, D.W. and G. Sawitzki, Excess mass estimates and tests for unimodality, J. Amer. Statist. Assoc., 86 (1991) 738-746.
123. Murtagh, F., A probability theory of hierarchic clustering using random dendrograms, J. Statist. Comput. Simul., 18 (1983) 145-157.
124. Murtagh, F., Counting dendrograms: A survey, Discrete Appl. Math., 7 (1984) 191-199.
125. Navidi, W.C., G.A. Churchill and A. von Haeseler, Phylogenetic inference: Linear invariants and maximum likelihood. Biometrics, 49 (1993) 543-555.
126. Nemec, A.F.L. and R.O. Brinkhurst, The Fowlkes-Mallows statistic and the comparison of two independently determined dendrograms, Canad. J. Fish. Aquat. Sci., 45 (1988) 971-975.
127. Nowicki, K., Asymptotic Poisson distributions with applications to statistical analysis of graphs, Advances Appl. Probab., 20 (1988) 315-330.
128. Page, R.D.M., Random dendrograms and null hypotheses in cladistic biogeography, Systematic Zoology, 40 (1991) 54-62.
129. Palmer, E.M., Graphical evolution: An introduction to the theory of random graphs (Wiley, New York, 1985).
130. Perruchet, C., Une analyse bibliographique des épreuves de classifiabilité en analyse des données, Statistique et Analyse des Données, 8 (1983) 18-41.
131. Pociecha, J. and A. Sokolowski, Empirical tests of multidimensional uniformity, Control and Cybernetics, 18 (1) (1989) 81-86.
132. Pollard, D., A central limit theorem for k-means clustering, Ann. Probab., 10 (1982) 919-926.
133. Postaire, J.G., R.D. Zhang and C. Botte-Lecocq, Cluster analysis by binary morphology, IEEE Trans. Pattern Anal. Machine Intell., 15 (2) (1993) 170-180.
134. Quiroz, A.J., Fast random generation of binary, t-ary and other types of trees, J. Classification, 6 (1989) 223-231.
135. Rasson, J.P., A. Hardy and D. Weverbergh, Point process, classification and data analysis, in: H.H. Bock (Ed.), 1988, 245-256.
136. Redner, R.A. and H.F. Walker, Mixture densities, maximum likelihood and the EM algorithm, SIAM Review, 26 (1984) 195-239.
137. Ripley, B.D., Spatial statistics (Wiley, New York, 1981).
138. Roeder, K., A graphical technique for determining the number of components in a mixture of normals, J. Amer. Statist. Assoc., 89 (1994) 487-495.
139. Rohlf, F.J., A randomization test of the nonspecificity hypothesis in numerical taxonomy, Taxon, 14 (1965) 262-267.
140. Rohlf, F.J., Numbering binary trees with labeled terminal vertices, Bull. Math. Biology, 45 (1983) 33-40.
141. Rozál, G.P.M. and J.A. Hartigan, The MAP test for multimodality, J. Classification, 11 (1994) 3-36.
142. Saitou, N., Property and efficiency of the maximum likelihood method for molecular phylogeny, J. Molecular Evolution, 27 (1988) 261-273.
143. Sanderson, M.J., Confidence limits on phytogenies: The Bootstrap revisited, Cadistics, 5 (1989) 113-129.
144. Sankoff, D., Minimal mutation trees of sequences, SIAM J. Appl. Math., 28 (1975) 35-42.
145. Sawitzki, G., The excess-mass approach and the analysis of multi-modality, in: W. Gaul and D. Pfeifer (Eds.), 1996, 203-211.
146. Sbihi, A. and J.-G. Postaire, Mode extraction by multivalue morphology for cluster analysis, in: W. Gaul and D. Pfeifer (Eds.), 1996, 212-221.
147. Schöniger, M., A. Janke and A. von Haeseler, How to deal with third codon positions in phylogenetic analysis, in: H.H. Bock, W. Lenski, M.M. Richter (Eds.), 1994, 376-383.
148. Schröder, E., Vier combinatorische Probleme, Zeitschrift für Mathematik und Physik, 15 (1870) 361-376.
149. Schroeder, A., Analyse d'un mélange de distributions de probabilité de même type, Revue de Statistique Appliquée, 24 (1) (1976) 39-62.
150. Shao, K. and F.J. Rohlf, Sampling distribution of consensus indices when all bifurcating trees are equally likely, in: J. Felsenstein (Ed.), 1983e, 132-137.
151. Shao, K. and R.R. Sokal, Significance tests of consensus indices, Systematic Zoology, 35 (1986) 582-590.
152. Silverman, B.W., Using kernel density estimates to investigate multimodality, J. Royal Statist. Soc., B 43 (1981) 97-99.
153. Silverman, B.W. and T. Brown, Short distances, flat triangles and Poisson limits, J. Appl. Probab., 15 (1978) 816-826.
154. Sneath, P.H.A., Predictivity in taxonomy and the probability of a tree, Pl. Syst. Evol, 167 (1989) 43-57.
155. Simberloff, D., Calculating the probabilities that cladograms match: A method of biogeographic inference, Systematic Zoology, 36 (1987) 175-195.
156. Sober, E., A likelihood justification of parsimony, Cladistics, 1 (1985) 209-233.
157. Späth, H., Cluster dissection and analysis. Theory, FORTRAN programs, examples (Ellis Horwood, Chichester, 1985).
158. Sriram, N. and S. Lewis, Constructing optimal ultrametrics, J. Classification, 10 (1993) 241-268.
159. Steel, M.A., Distributions on bicoloured binary trees from the principle of parsimony, SIAM J. Discrete App. Math., 1992.
160. Steel, M. and D. Penny, Probability distributions of tree comparison metrics-some new results, Systematic Biology, 42 (1993).
161. Steel, M.A., M.D. Hendy and D. Penny, Significance of the length of the shortest tree, J. Classification, 9 (1992) 71-90.
162. Steele, J.M., Growth rates of euclidean minimal spanning trees with power weighted edges, Annals of Probability, 16 (1988) 1767-1787.
163. Steele, J.M. and L. Tierney, Boundary domination and the distribution of the largest nearest neigbor link in higher dimensions, J. Appl. Probab., 23 (1988) 524-528.
164. Tabakis, E., On the longest edge of the minimal spanning tree, in: W. Gaul and D. Pfeifer (Eds.), 1996, 222-230.
165. Titterington, D.M., A.F.M. Smith and U.E. Makov, Statistical analysis of finite mixture distributions (Wiley, New York, 1985).
166. Van Cutsem, B., Combinatorial structures and structures for classification, Comput. Statistics and Data Analysis, 23 (1996) This volume.
167. Van Cutsem, B. and B. Yeart, Probability distributions on indexed dendrograms and related problems of classifiability, in: H.H. Bock and W. Polasek (Eds.), 1996, 73-87.
168. Wasserman, S. and C. Anderson, Stochastic a posteriori blockmodels: construction and assessment, Social Networks, 9 (1987) 1-36.
169. Waterman, M.S. and T.F. Smith, On the similarity of dendrograms, J. Theoret. Biology, 73 (1978) 789-800.
170. Windham, M.P., Parameter modification for clustering criteria, J. Classification, 4 (1987) 191-214.
171. Windham, M.P. and A. Cutler, Information ratios for validating mixture analyses, J. Amer. Statist. Assoc., 87 (1992) 1188-1192.
172. Windham, M.P. and A. Cutler, Mixture analysis with noisy data, in: E. Diday et al. (Eds.), 1994, 155-160.
173. Wolf, K. and P.O. Degens, Variance estimation in the additive tree model, in: H.H. Bock and P. Ihm (Eds.), 1991, 262-269.