Abstract
Let I denote a totally ordered set of n ≥ 1 elements. It is notationally convenient to identify I with the set of the first n integers or, on occasion, with the row vector (1,…,n). In the former case the order on I is embodied in the natural order on the reals, and in the latter case in the ordering amongst the elements of a vector
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References
Batbedat, A. (1989), Les dissimilarites medas ou arbas, Statist. Anal. Données, 14, pp. 1–18.
Batbedat, A. (1991), Phylogenie et dendrogrammes, Journées de Statistique, Strasbourg, France.
Batbedat, A. (1992), Les distances quadrangulaires qui ont une orientation, RAIRO- Rech. Opér, 26, pp. 15–29.
Bertrand, P., Diday, E. (1985), A visual representation of the compatibility between an order and a dissimilarity index: the pyramids, Comput. Statist. Quart, 2, pp. 31–42.
Blumenthai, L.M. (1953), Theory and Applications of Distance Geometry, Clarendon Press, Oxford.
Bock, H.H., ed. (1988), Classification and Related Methods of Data Analysis, North- Holland, Amsterdam.
Buneman, P. (1971), The recovery of trees from measures of dissimilarity, In Hodson, F.R., Kendall, D.G., Taŭtu, P., eds., Mathematics in the Archaeological and Historical Sciences, The University Press, Edinburgh, pp. 387–395.
Critchley, F. (1988a), On certain linear mappings between inner-product and squared- distance matrices, Linear Algebra Appl., 150, pp. 91–107.
Critchley, F. (1988b), Oil exchangeability-based equivalence relations induced by strongly Robinson and, in particular, by quadripolar Robinson dissimilarity matrices, Warwick Statistics Research Report n° 152, U.K.
Critchley, F . (1988c), On quadripolar Robinson dissimilarity matrices, Warwick Statistics Research Report nº 153, U.K., To appear in Diday, E., et al., eds (1994), Proc. IFCS93 meeting, Springer-Verlag, New York.
Critchley, F., Fichet, B. (1994), The partial order by inclusion of the principal classes of dissimilarity on a finite set, and some of their basic properties, In Van Cutsem, B., ed., Classification and Dissimilarity Analysis, Lecture Notes in Statistics, Ch. 2, Springer-Verlag, New York.
de Leeuw, J., Heiser, W. (1982), Theory of multidimensional scaling, In Krishnaiah, P.R., ed., Handbook of Statistics, Vol. 2, North-Holland, Amsterdam, Ch. 13.
de Leeuw, J., Heiser, W., Meulman, J., Critchley, F., eds. (1986), Multidimensional Data Analysis, DSWO Press, Leiden.
Diday, E. (1984), Une représentation visuelle des classes empietantes: les pyramides, Rapport de Recherche nº 291, INRIA, Rocquencourt, France.
Diday, E. (1986), Orders and overlapping clusters in pyramids, In de Leeuw, J., Heiser, W., Meulman, J., Critchley, F., eds., Multidimensional Data Analysis, DSWO Press, Leiden, pp. 201–234.
Diday, E., Escoufier, Y., Lebart, L., Pagés, J.P., Schektman, Y., Tomassone, R., eds. (1986), Data Analysis and Informatics 4, North-Holland, Amsterdam.
Durand, C. (1988), Une approximation de Robinson inférieure maximale, Rapport de Recherche Laboratoire de Mathématiques Appliquées et Informatique, nº 88-02, Université de Provence, Marseille, France.
Durand, C., Fichet, B. (1988), One-to-one correspondences in pyramidal representation: a unified approach, In Bock, H.H., ed., Classification and Related Methods of Data Analysis, North-Holland, Amsterdam, pp. 85–90.
Fichet, B. (1984), Sur une extension de la notion de hierarchie et son équivalence avec certaines matrices de Robinson, Journees de Statistique, Montpellier, France.
Gordon, A.D. (1987), A review of hierarchical classification, J. Roy. Statist. Soc. A, 150, pp. 119–137.
Gower, J.C. (1985), Properties of Euclidean and non-Euclidean distance matrices, Linear Algebra Appl, 67, pp. 81–97.
Hayden, T.L., Wells, J. (1988), Approximation by matrices positive semidefinite on a subspace, Linear Algebra Appl, 109, pp. 115–130.
Johnson, S.C. (1967), Hierarchical clustering schemes, Psychometrika, 32, pp. 241–253.
Leclerc, B. (1993), Minimum spanning trees for tree metrics: abridgements and adjustments, Research Report C.M.S. P.084, Centre d’Analyse et de Mathematique Sociales, Paris, Prance.
Mathar, R. (1985), The best Euclidean fit to a given distance matrix in prescribed dimensions, Linear Algebra Appl, 67, pp. 1–6.
Robinson, W.S. (1951), A method for chronological ordering of archaeological deposits, American Antiquity, 16, pp. 293–301.
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Critchley, F. (1994). On exchangeability-based equivalence relations induced by strongly Robinson and, in particular, by quadripolar Robinson dissimilarity matrices. In: Van Cutsem, B. (eds) Classification and Dissimilarity Analysis. Lecture Notes in Statistics, vol 93. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2686-4_7
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