Abstract
Integral pseudometrics arise in a certain type of numerical taxonomy as a dissimilarity coefficient. To each partition on a finite set corresponds a pseudometric on the set in which the distance between points in the same subset is zero, and between points in different subsets is one. Each dissimilarity coefficient is a sum of partition pseudometrics.
We may naturally then ask whether given a dissimilarity coefficient we can recover the partitions, and whether every integral pseudometric on a finite set can be a dissimilarity coefficient. We answer both these questions in the negative.
We give a necessary condition for a decomposition into partition pseudometrics where every partition has two subsets, and an example illustrating the method of attempting a decomposition.
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References
Jardine, N. and Sibson, R. Mathematical Taxonomy. Wiley, New York 1971.
Kelley, J. L. General Topology. Van Nostrand, Princeton, N.J. 1955.
Sokal, R. R. and Sneath P.H.A. Principles of Numerical Taxonomy. W. H. Freeman, San Francisco, 1963.
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© 1979 Springer-Verlag
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Robinson, D.F. (1979). Decomposition of integral pseudometrics. In: Horadam, A.F., Wallis, W.D. (eds) Combinatorial Mathematics VI. Lecture Notes in Mathematics, vol 748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0102689
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DOI: https://doi.org/10.1007/BFb0102689
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-09555-2
Online ISBN: 978-3-540-34857-3
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