A Central Limit Theorem for Punctuated Equilibrium

Bartoszek, Krzysztof

doi:10.1101/039867

Cited by 4 publications

(3 citation statements)

References 48 publications

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“…In Fig. 1 if we took the pair of tip species i = 6 and j = 8, then X i,j would be the value at the internal node labelled (6,8) and τ i,j the sum of branch lengths on the path between tip 6 (equivalently tip 8) and node (6,8). Because a jump happens just after speciation, any jumps associated with this ancestral node are not shared by the pair of tip species and hence cannot contribute to the covariance between them.…”

Section: Simulation Setup and Resultsmentioning

confidence: 99%

“…Interestingly, there does seem to be any visible dependency of this speed of convergence on α. At the first moment level there is a phase transition in the Central Limit Theorems for the average of the contemporary sample [1,6,8] at α = λ/2. In the mESS factors' case rapid convergence seems present when α ≥ 1, but for lower values of α the situation is not obvious.…”

Section: Simulation Setup and Resultsmentioning

confidence: 99%

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The phylogenetic effective sample size and jumps

Bartoszek¹

2018

MathAppl

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The phylogenetic effective sample size is a parameter that has as its goal the quantification of the amount of independent signal in a phylogenetically correlated sample. It was studied for Brownian motion and Ornstein-Uhlenbeck models of trait evolution. Here, we study this composite parameter when the trait is allowed to jump at speciation points of the phylogeny. Our numerical study indicates that there is a non-trivial limit as the effect of jumps grows. The limit depends on the value of the drift parameter of the Ornstein-Uhlenbeck process.

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Section: Simulation Setup and Resultsmentioning

confidence: 99%

Section: Simulation Setup and Resultsmentioning

confidence: 99%

The phylogenetic effective sample size and jumps

Bartoszek¹

2018

MathAppl

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“…We notice (as Bartoszek and Sagitov, 2015b;Bartoszek, 2016, in Lemmata 11 and 2 respectively) that we may write…”

Section: Proofmentioning

confidence: 99%

Exact and approximate limit behaviour of the Yule tree’s cophenetic index

Bartoszek

2018

Mathematical Biosciences

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In this work we study the limit distribution of an appropriately normalized cophenetic index of the pure-birth tree conditioned on n contemporary tips. We show that this normalized phylogenetic balance index is a submartingale that converges almost surely and in L. We link our work with studies on trees without branch lengths and show that in this case the limit distribution is a contraction-type distribution, similar to the Quicksort limit distribution. In the continuous branch case we suggest approximations to the limit distribution. We propose heuristic methods of simulating from these distributions and it may be observed that these algorithms result in reasonable tails. Therefore, we propose a way based on the quantiles of the derived distributions for hypothesis testing, whether an observed phylogenetic tree is consistent with the pure-birth process. Simulating a sample by the proposed heuristics is rapid, while exact simulation (simulating the tree and then calculating the index) is a time-consuming procedure. We conduct a power study to investigate how well the cophenetic indices detect deviations from the Yule tree and apply the methodology to empirical phylogenies.

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Exact and approximate limit behaviour of the Yule tree’s cophenetic index

Bartoszek

2017

Preprint

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In this work we study the limit distribution of an appropriately normalized cophenetic index of the pure birth tree on n contemporary tips. We show that this normalized phylogenetic balance index is a submartingale that converges almost surely and in L 2 . We link our work with studies on trees without branch lengths and show that in this case the limit distribution is a contraction type distribution, similar to the Quicksort limit distribution. In the continuous branch case we suggest approximations to the limit distribution. We propose heuristic methods of simulating from these distributions and it may be observed that these algorithms result in reasonable tails. Therefore, we postulate using quantiles of the derived distributions for hypothesis testing, whether an observed phylogenetic tree is consistent with the pure birth process. Simulating a sample by the proposed heuristics is rapid while exact simulation (simulating the tree and then calculating the index) is a time-consuming procedure.

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A Central Limit Theorem for Punctuated Equilibrium

Cited by 4 publications

References 48 publications

The phylogenetic effective sample size and jumps

The phylogenetic effective sample size and jumps

Exact and approximate limit behaviour of the Yule tree’s cophenetic index

Exact and approximate limit behaviour of the Yule tree’s cophenetic index

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