Summary
Lerman [1970] has demonstrated, that the dissimilarity indices normally used in data analysis are identical up to strictly monotone transformationsf:R +→R + if the data are nominal and each set of attribute scores is finite.
In that case he proposes to use a preorder between pairs of objects to express similarity or dissimilarity, in order to avoid inconsistent classification results that might occur, if clustering schemes which are not monotone invariant are applied to a quantitative index. Here it is shown, how a hierarchy on the objects can be calculated if such a preorder relation is given.
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References
Barbut, M., andB. Monjardet: Ordre et Classification 1, 2. Paris 1970.
Birkhoff, G.: Lattice Theory (3.ed.). American Mathematical Society, Providence 1973.
Comyn, G., andJ.C. van Dorpe: Valuation et semi-modularité dans les demi-treillis. Math. Sci. hum.56, 1976, 63–75.
Lerman, I.C.: Les Bases de la Classification Automatique. Paris 1970.
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Schader, M. Hierarchical analysis: Classification with ordinal object dissimilarities. Metrika 27, 127–132 (1980). https://doi.org/10.1007/BF01893584
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DOI: https://doi.org/10.1007/BF01893584