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Metric and Euclidean properties of dissimilarity coefficients. (English) Zbl 0592.62048
Summary: We assemble here properties of certain dissimilarity coefficients and are specially concerned with their metric and Euclidean status. No attempt is made to be exhaustive as far as coefficients are concerned, but certain mathematical results that we have found useful are presented and should help establish similar properties for other coefficients. The response to different types of data is investigated, leading to guidance on the choice of an appropriate coefficient.

MSC:
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62-07 Data analysis (statistics) (MSC2010)
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[1] BAKER, F.B. (1974), ”Stability of Two Hierarchical Grouping Techniques. Case 1: Sensitivity to Data Errors,”Journal of the American Statistical Association, 69, 440–445.
[2] BLASHFIELD, R.K. (1976), ”Mixture Model Tests of Cluster Analysis: Accuracy of Four Agglomerative Hierarchical Methods,”Psychological Bulletin, 83, 377–388.
[3] BLOOM, S.A. (1981), ”Similarity Indices in Community Studies: Potential Pitfalls,”Marine Ecology Progress Series, 5, 125–128.
[4] CAILLIEZ, F. (1983), ”The Analytical Solution to the Additive Constant Problem,”Psychometrika, 48, 305–308. · Zbl 0534.62079
[5] CAILLIEZ, F., and PAGES, J.-P. (1976),Introduction à l’analyse des données, Paris: Société de Mathématiques appliquées et de Sciences humaines.
[6] CHARLTON, J.R.H., and WYNN, H.P. (1985), ”Metric Scaling and Infinitely Divisible Distributions: Schoenberg’s Theorem,” Personal Communication.
[7] CUNNINGHAM, K.M., and OGILVIE, J.C. (1972), ”Evaluation of Hierarchical Grouping Techniques: A Preliminary Study,”Computer Journal, 15, 209–213.
[8] EESTABROOK, G.F., and ROGERS, D.J. (1966), ”A General Method of Taxonomic Description for a Computed Similarity Measure,”BioScience, 16, 789–793.
[9] EVERITT, B. (1974),Cluster Analysis, London: Heinemann Educational Books. · Zbl 0507.62060
[10] FAITH, D.P. (1985), ”Distance Methods and the Approximation of Most-Parsimonious Trees,”Systematic Zoology, 34, 312–325.
[11] FISHER, L., and VAN NESS, J.W. (1971), ”Admissible Clustering Procedures,”Biometrika, 58, 91–104. · Zbl 0224.62030
[12] GOWER, J.C. (1971), ”A General Coefficient of Similarity and Some of its Properties,”Biometrics, 27, 857–871.
[13] GOWER, J.C. (1982), ”Euclidean Distance Geometry,”Mathematical Scientist, 7, 1–14. · Zbl 0492.51017
[14] GOWER, J.C. (1984a), ”Multivariate Analysis: Ordination, Multidimensional Scaling and Allied Topics,” inHandbook of Applicable Mathematics, Vol. VI: Statistics, Part B, Ed. E. Lloyd, Chichester: John Wiley and Sons, 727–781.
[15] GOWER, J.C. (1984b), ”Distance Matrices and Their Euclidean Approximation,” inData Analysis and Informatics, 3, Eds. E. Diday, M. Jambu, L. Lebart, J. Pagès and R. Tomassone, Amsterdam: North-Holland, 3–21.
[16] GOWER, J.C. (1985), ”Measures of Similarity, Dissimilarity, and Distance,” inEncyclopedia of Statistical Sciences, Vol. 5, Eds. S. Kotz, N.L. Johnson and C.B. Read, New York: John Wiley and Sons, 397–405.
[17] HAJDU, L.J. (1981), ”Graphical Comparison of Resemblance Measures in Phytosociology,”Vegetatio, 48, 47–59.
[18] HUBERT, L. (1974), ”Approximate Evaluation Techniques for the Single-Link and Complete-Link Hierarchical Clustering Procedures,”Journal of the American Statistical Association, 69, 698–704. · Zbl 0291.62071
[19] JACCARD, P. (1901), ”Etude Comparative de la Distribution Florale dans une Portion des Alpes et des Jura,”Bulletin de la Société vaudoise des Sciences Naturelles, 37, 547–579.
[20] JARDINE, N., and SIBSON, R. (1968), ”The Construction of Hierarchic and Non-Hierarchic Classifications,”Computer Journal, 11, 177–184. · Zbl 0164.46207
[21] KULCZYNSKI, S. (1928), ”Die Pflanzenassoziationen der Pieninen,”Bulletin international de l’Académie polonaise des Sciences et des Lettres, Classe des Sciences mathématiques et naturelles, Série B, Supplément II (1927), 57–203.
[22] LEGENDRE, P., and CHODOROWSKI, A. (1977), ”A Generalization of Jaccard’s Association Coefficient for Q Analysis of Multi-State Ecological Data Matrices,”Ekologia Polska, 25, 297–308.
[23] LEGENDRE, P., DALLOT, S., and LEGENDRE, L. (1985), ”Succession of Species Within a Community: Chronological Clustering, with Applications to Marine and Freshwater Zooplankton,”American Naturalist, 125, 257–288.
[24] LEGENDRE, L., and LEGENDRE, P. (1983a),Numerical Ecology, Developments in Environmental Modelling, Vol. 3, Amsterdam: Elsevier Scientific Publishing Company. · Zbl 0588.92021
[25] LEGENDRE, L. and LEGENDRE, P. (1983b), ”Partitioning Ordered Variables into Discrete States for Discriminant Analysis of Ecological Classifications,”Canadian Journal of Zoology, 61, 1002–1010.
[26] LINGOES, J.C. (1971), ”Some Boundary Conditions for a Monotone Analysis of Symmetric Matrices,”Psychometrika, 36, 195–203. · Zbl 0222.15011
[27] MIRSKY, L. (1955).Introduction to Linear Algebra, Oxford: Oxford University Press. · Zbl 0066.26305
[28] ORLOCI, L. (1978),Multivariate Analysis in Vegetation Research, Second Edition, The Hague: Dr. W. Junk B.V.
[29] RAND, W.M. (1971), ”Objective Criteria for the Evaluation of Clustering Methods,”Journal of the American Statistical Association, 66, 846–850.
[30] RENKONEN, O. (1938), ”Statistisch-ökologische Untersuchungen über die terrestische Käferwelt der finnischen Bruchmoore,”Annales Zoologici Societatis Zoologicae-Botanicae Fennicae ’Vanamo’, 6, 1–231.
[31] SCHOENBERG, I.J. (1935), ”Remarks to Maurice Fréchet’s article ’Sur la définition axiomatique d’une classe d’espaces vectoriels distanciés applicables vectoriellement sur l’espace de Hilbert’,”Annals of Mathematics, 36, 724–732. · Zbl 0012.30703
[32] SIBSON, R. (1971), ”Some Observations on a Paper by Lance and Williams,”Computer Journal, 14, 156–157. · Zbl 0227.68049
[33] SIBSON, R. (1979), ”Studies in the Robustness of Multidimensional Scaling: Perturbational Analysis of Classical Scaling,”Journal of the Royal Statistical Society, Series B, 41, 217–229. · Zbl 0413.62046
[34] SPATH, H. (1980),Cluster Analysis Algorithms for Data Reduction and Classification of Objects, translated by Ursula Bull, Chichester: Ellis Horwood Ltd., and New York: John Wiley and Sons.
[35] WILLIAMS, W.T., CLIFFORD, H.T., and LANCE, G.N. (1971a), ”Group-size Dependence: A Rationale for Choice Between Numerical Classifications,”Computer Journal, 14, 157–162. · Zbl 0257.68095
[36] WILLIAMS, W.T., LANCE, G.N., DALE, M.B., and CLIFFORD, H.T. (1971b), ”Controversy Concerning the Criteria for Taxonometric Strategies,”Computer Journal, 14, 162–165. · Zbl 0234.68041
[37] WOLDA, H. (1981), ”Similarity Indices, Sample Size and Diversity,”Oecologia (Berl.), 50, 296–302.
[38] ZEGERS, F.E. (1986), ”Two Classes of Element-Wise Transformations Preserving the Positive Semi-Definite Nature of Coefficient Matrices,”Journal of Classification, 3, 49–53. · Zbl 0592.62049
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