120,534 research outputs found
A Beta-splitting model for evolutionary trees
In this article, we construct a generalization of the Blum-Fran\c{c}ois
Beta-splitting model for evolutionary trees, which was itself inspired by
Aldous' Beta-splitting model on cladograms. The novelty of our approach allows
for asymmetric shares of diversification rates (or diversification `potential')
between two sister species in an evolutionarily interpretable manner, as well
as the addition of extinction to the model in a natural way. We describe the
incremental evolutionary construction of a tree with n leaves by splitting or
freezing extant lineages through the Generating, Organizing and Deleting
processes. We then give the probability of any (binary rooted) tree under this
model with no extinction, at several resolutions: ranked planar trees giving
asymmetric roles to the first and second offspring species of a given species
and keeping track of the order of the speciation events occurring during the
creation of the tree, unranked planar trees, ranked non-planar trees and
finally (unranked non-planar) trees. We also describe a continuous-time
equivalent of the Generating, Organizing and Deleting processes where tree
topology and branch-lengths are jointly modeled and provide code in
SageMath/python for these algorithms.Comment: 23 pages, 3 figures, 1 tabl
Variable Selection Bias in Classification Trees Based on Imprecise Probabilities
Classification trees based on imprecise probabilities provide an advancement of classical classification trees. The Gini Index is the default splitting criterion in classical classification trees, while in classification trees based on imprecise probabilities, an extension of the Shannon entropy has been introduced as the splitting criterion. However, the use of these empirical entropy measures as split selection criteria can lead to a bias in variable selection, such that variables are preferred for features other than their information content. This bias is not eliminated by the imprecise probability approach. The source of variable selection bias for the estimated Shannon entropy, as well as possible corrections, are outlined. The variable selection performance of the biased and corrected estimators are evaluated in a simulation study. Additional results from research on variable selection bias in classical classification trees are incorporated, implying further investigation of alternative split selection criteria in classification trees based on imprecise probabilities
Creature forcing and large continuum: The joy of halving
For let be the minimal number of
uniform -splitting trees needed to cover the uniform -splitting tree,
i.e., for every branch of the -tree, one of the -trees contains
. Let be the dual notion: For every branch , one of
the -trees guesses infinitely often. We show that it is consistent
that
for continuum many pairwise different cardinals and suitable
pairs . For the proof we introduce a new mixed-limit
creature forcing construction
A new family of Markov branching trees: the alpha-gamma model
We introduce a simple tree growth process that gives rise to a new
two-parameter family of discrete fragmentation trees that extends Ford's alpha
model to multifurcating trees and includes the trees obtained by uniform
sampling from Duquesne and Le Gall's stable continuum random tree. We call
these new trees the alpha-gamma trees. In this paper, we obtain their splitting
rules, dislocation measures both in ranked order and in sized-biased order, and
we study their limiting behaviour.Comment: 23 pages, 1 figur
Decisive creatures and large continuum
For f,g\in\omega\ho let \mycfa_{f,g} be the minimal number of uniform
-splitting trees needed to cover the uniform -splitting tree, i.e. for
every branch of the -tree, one of the -trees contains .
\myc_{f,g} is the dual notion: For every branch , one of the -trees
guesses infinitely often.
It is consistent that
\myc_{f_\epsilon,g_\epsilon}=\mycfa_{f_\epsilon,g_\epsilon}=\kappa_\epsilon
for \al1 many pairwise different cardinals and suitable
pairs .
For the proof we use creatures with sufficient bigness and halving. We show
that the lim-inf creature forcing satisfies fusion and pure decision. We
introduce decisiveness and use it to construct a variant of the countable
support iteration of such forcings, which still satisfies fusion and pure
decision.Comment: major revisio
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