Probabilities on cladograms: introduction to the alpha model

By Daniel J. Ford. 2005

100 citations recorded by the source corpus.

The alpha model, a parametrized family of probabilities on cladograms (rooted binary leaf labeled trees), is introduced. This model is Markovian self-similar, deletion-stable (sampling consistent), and passes through the Yule, Uniform and Comb models. An explicit formula is given to calculate the probability of any cladogram or tree shape under the alpha model. Sackin's and Colless' index are shown to be $O(n^{1+\alpha})$ with asymptotic covariance equal to 1. Thus the expected depth of a random leaf with $n$ leaves is $O(n^\alpha)$. The number of cherries on a random alpha tree is shown to be asymptotically normal with known mean and variance. Finally the shape of published phylogenies is examined, using trees from Treebase.

Subfields: Artificial Intelligence, Mathematical Physics, Nature and Landscape Conservation

View Paper: https://www.semanticscholar.org/paper/a2eef2cea249283891ce2bcd101ca03b6eb5933d

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Citations in the corpus

Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models (2006, unknown): “By Bénédicte Haas, Grégory Miermont, Jim Pitman and Matthias Winkel1”
Gibbs fragmentation trees (2007, background): “Ford [6] gave a characterizaton of consistency for models of unlabelled trees which is easily shown to be equivalent to the following condition for p to be the splitting rule for a consistent binary Markov fragmentation: p(i, j) = p(i+ 1, j) + p(i, j + 1) + p(i+ j, 1)p(i, j) for all i, j ≥ 1.”
Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions (2008, methodology): “For convenience of notation in this proof, denote the nonsymmetric splitting rules, for Ford’s case θ = 1−α by qFn (m,n−m) = ( n− 1 m ) m + (n− 1− 2m)α n− 1 Γ(m− α)Γ(n−m−α) Γ(1− α)Γ(n−α) [see (8)], and for θ = 2− α by qX(m,n−m) = ( n− 1 m ) 2m + (n− 1− 2m)α n− 1 Γ(m− α)Γ(n−m + 1− α) Γ(1−α)Γ(n + 1− α”
A new family of Markov branching trees: the alpha-gamma model (2008, methodology): “…to each of the n − 1 other edges; (ii) F select at random with probabilities proportional to the weights assigned by step (i) F , an edge of T n , say a n → c n directed away from the root ; It was shown in [12] that these trees are Markov branching trees but that the labelling is not exchangeable.”