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Decomposition of integral pseudometrics

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Part of the Lecture Notes in Mathematics book series (LNM,volume 748)

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

  1. Jardine, N. and Sibson, R. Mathematical Taxonomy. Wiley, New York 1971.

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  2. Kelley, J. L. General Topology. Van Nostrand, Princeton, N.J. 1955.

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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

  • eBook Packages: Springer Book Archive