Phylogenetic Diversity Over an Abelian Group

Dress, Andreas; Steel, Mike

doi:10.1007/s00026-007-0311-4

Cited by 11 publications

(6 citation statements)

References 16 publications

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“…There is one fly in the ointment, however-for distances, problems arise if G has elements of order 2 (for instance, uniqueness of the tree representation fails; this is apparent from the 15 phylogenetic trees having the shape shown in Figure 2(e) with edges assigned the element 1 of G = ({0, 1}, +) that induce exactly the same distance function). But uniqueness can be restored by moving from distances to diversities, where not just pairs but also triples of leaves are considered [13].…”

Section: Metric Properties Of Treesmentioning

confidence: 99%

Tracing Evolutionary Links between Species

Steel

2014

The American Mathematical Monthly

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The idea that all life on earth traces back to a common beginning dates back at least to Charles Darwin's Origin of Species. Ever since, biologists have tried to piece together parts of this 'tree of life' based on what we can observe today: fossils and the evolutionary signal that is present in the genomes and phenotypes of different organisms. Mathematics has played a key role in helping transform genetic data into phylogenetic (evolutionary) trees and networks. Here, I will explain some of the central concepts and basic results in phylogenetics, which benefit from several branches of mathematics, including combinatorics, probability, and algebra.

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Section: Metric Properties Of Treesmentioning

confidence: 99%

Tracing Evolutionary Links between Species

Steel

2014

The American Mathematical Monthly

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“…The aforementioned measures of diversity are defined for measuring one characteristic at a time: a single locus in genetic studies; a single variable in economics or ecology; a single channel in communications, etc. Areas of application for diversity measures include: phylogenetis (Anselmo and Pinheiro, 2012;Moulton et al, 2007;Dress and Steel, 2007); population genetics (Gillet, 2007;Gilbert et al, 2005, Kussell andLeibler, 2005); time series analysis (Valk and Pinheiro, 2012); and economics (Nayak and Gastwirth, 1989).…”

Section: Introductionmentioning

confidence: 99%

Quasi U-statistics of innite order and applications to the subgroup decomposition of some diversity measures

Pinheiro

Sen

2014

São Paulo J. Math. Sci.

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In several applications, information is drawn from quali- tative variables. In such cases, measures of central tendency and dis- persion may be highly inappropriate. Variability for categorical data can be correctly quantied by the so-called diversity measures. These measures can be modied to quantify heterogeneity between groups (or subpopulations). Pinheiro et al. (2005) shows that Hamming distance can be employed in such way and the resulting estimator of hetero- geneity between populations will be asymptotically normal under mild regularity conditions. Pinheiro et al. (2009) proposes a class of weighted U-statistics based on degenerate kernels of degree 2, called quasi U-statistics, with the property of asymptotic normality under suitable conditions. This is generalized to kernels of degree m by Pinheiro et al. (2011). In this work we generalize this class to an innite order degenerate kernel. We then use this powerful tools and the reverse martingale nature of U-statistics to study the asymptotic behavior of a collection of trans- formed classic diversity measures. We are able to estimate them in a common framework instead of the usual individualized estimation procedures. MSC 2000: primary - 62G10; secondary - 62G20, 92D20.

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“…Note that Dress and Steel (2007) study the related problem of characterizing when a map D from the set of subsets of X of size at most k into some Abelian group G can be represented by a phylogenetic tree on X whose edges are assigned elements from G. However, we consider subsets of X of size precisely k, leading to a quite different characterization.…”

Section: Introductionmentioning

confidence: 99%

Recognizing Treelike k-Dissimilarities

Herrmann¹,

Huber²,

Moulton³

et al. 2012

J Classif

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A k-dissimilarity D on a finite set X, |X| >= k, is a map from the set of size k subsets of X to the real numbers. Such maps naturally arise from edge-weighted trees T with leaf-set X: Given a subset Y of X of size k, D(Y) is defined to be the total length of the smallest subtree of T with leaf-set Y . In case k = 2, it is well-known that 2-dissimilarities arising in this way can be characterized by the so-called "4-point condition". However, in case k > 2 Pachter and Speyer recently posed the following question: Given an arbitrary k-dissimilarity, how do we test whether this map comes from a tree? In this paper, we provide an answer to this question, showing that for k >= 3 a k-dissimilarity on a set X arises from a tree if and only if its restriction to every 2k-element subset of X arises from some tree, and that 2k is the least possible subset size to ensure that this is the case. As a corollary, we show that there exists a polynomial-time algorithm to determine when a k-dissimilarity arises from a tree. We also give a 6-point condition for determining when a 3-dissimilarity arises from a tree, that is similar to the aforementioned 4-point condition.Comment: 18 pages, 4 figure

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Phylogenetic Diversity Over an Abelian Group

Cited by 11 publications

References 16 publications

Tracing Evolutionary Links between Species

Tracing Evolutionary Links between Species

Quasi U-statistics of innite order and applications to the subgroup decomposition of some diversity measures

Recognizing Treelike k-Dissimilarities

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