zbMATH — the first resource for mathematics

The median procedure for n-trees. (English) Zbl 0617.62066
One approach to produce a consensus of several classifications constructed for a set of objects is to produce a reasonable-looking method and then seek to discover those properties that characterize it. In the present paper this is made by giving an axiomatic characterization of the median procedure for n-trees. Let (X,d) be a metric space. The function \(M: X^ k\to 2^ X\) defined by \[ M(x_ 1,...,x_ k)=\{x\in X:\sum^{k}_{j>1}d(x,x_ j)\quad is\quad \min imum\} \] is called the median procedure. Axioms are presented that characterize M when X is a certain class of trees (hierarchical classification), and d is the symmetric difference metric. The median complete multiconsensus function (CMF) is shown to be the unique CMF that is efficient, stable on clusters, consistent, symmetric, and quasi-Condorcet.
Reviewer: V.Yu.Urbakh

62H30 Classification and discrimination; cluster analysis (statistical aspects)
Full Text: DOI
[1] BANDELT, H J, and BARTHELEMY, J P (1984), ”Medians in Median Graphs,”Discrete Applied Mathematics, 8, 131–142 · Zbl 0536.05057
[2] BARTHELEMY, J P, and MONJARDET, B (1981), ”The Median Procedure in CLuster Analysis and Social Choice Theory,”Mathematical Social Sciences, 1, 235–267 · Zbl 0486.62057
[3] BOBISUD, H M, and BOBISUD, L E (1972), ”A Metric for Classifications,”Taxon, 21, 607–613
[4] MARGUSH, T, and MCMORRIS, F R (1981), ”Consensus n-Trees,”Bulletin of Mathematical Biology, 43, 239–244 · Zbl 0455.92019
[5] PENNY, D, FOULDS, L R, and HENDY, M D (1982), ”Testing the Theory of Evolution by Comparing Phylogenetic Trees Constructed from Five Different Protein Sequences,”Nature, 297, 197–200
[6] YOUNG, H P, and LEVENGLICK, A (1978), ”A Consistent Extension of Condorcet’s Election Principle,”SIAM Journal of Applied Mathematics, 35, 285–300 · Zbl 0385.90010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.