<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.gif" overflow="scroll"><mml:mi>Δ</mml:mi></mml:math> additive and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si8.gif" overflow="scroll"><mml:mi>Δ</mml:mi></mml:math> ultra-additive maps, Gromov's trees, and the Farris transform

In this note, it is shown that by applying ranking procedures to data that allow, for any three objects a 1 , a 2 , b in a collection X of objects of interest, to make consistent decisions about which of the two objects a 1 or a 2 is more similar to b, a family of cluster systems A (k) (k = 0, 1, . . .) can be constructed that start with the associated Apresjan Hierarchy and keep growing until, for k = #X − 1, the full set P (X) of all subsets of X is reached. Various ideas regarding canonical modifications of the similarity values so that these cluster systems contain as many clusters as possible for small values of k (and in particular for k := 0) and/or are rooted at a specific element in X, possible applications, e.g. concerning (i) the comparison of distinct dissimilarity data defined on the same set X or (ii) diversity optimization, and new tasks arising in ranking statistics are also discussed.

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Constructing Hierarchical Set Systems

Devauchelle¹,

Dress

Grossmann³

et al. 2005

Ann. Comb.

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Some Uses of the Farris Transform in Mathematics and Phylogenetics—A Review

2007

Self Cite

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In 1970, Farris introduced a procedure that can be used to transform a tree metric into an ultra metric. Since its discovery, Farris' procedure has been used extensively within phylogenetics where it has become commonly known as the Farris transform. Remarkably, the Farris transform has not only been rediscovered several times within phylogenetics, but also in other fields. In this paper, we will review some of its various properties and uses.The paper is divided into four parts and, altogether, 12 sections. In the first part, we introduce a standardized scheme for classifying those dissimilarity mappings to which the Farris transform can be applied -a scheme that has evolved over the years, but has apparently not been spelled out before in sufficient detail. In the second part, we will discuss how a straightforward generalization of the Farris transform naturally arises in T-Theory. The third part describes how this generalized Farris transform can be used to approximate dissimilarities by tree metrics. And in the final part, we describe some further, "non-standard" applications of the Farris transform.

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Phylogenetic Diversity Over an Abelian Group

Dress

Steel

2007

Ann. Comb.

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There is a natural way to associate to any tree T with leaf set X, and with edges weighted by elements from an abelian group G, a map from the power set of X into G -simply add the elements on the edges that connect the leaves in that subset. This map has been wellstudied in the case where G has no elements of order 2 (particularly when G is the additive group of real numbers) and, for this setting, subsets of leaves of size two play a crucial role. However, the existence and uniqueness results in that setting do not extend to arbitrary abelian groups. We study this more general problem here, and by working instead with both, pairs and triples of leaves, we obtain analogous existence and uniqueness results. Some particular results for elementary abelian 2-groups are also described.

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Δ additive and Δ ultra-additive maps, Gromov's trees, and the Farris transform

Cited by 16 publications

References 15 publications

Constructing Hierarchical Set Systems

Constructing Hierarchical Set Systems

Some Uses of the Farris Transform in Mathematics and Phylogenetics—A Review

Phylogenetic Diversity Over an Abelian Group

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