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First two moments and cross-moments of some coalescent times of the pure birth tree

Krzysztof Bartoszek, Bayu Brahmantio, Woodrow Hao Chi Kiang

TL;DR

This work provides a thorough analytical treatment of the first two moments and cross-moments for the height $U^{(n)}$ of a pure-birth (Yule) tree and the coalescent time $\tau^{(n)}$ of a random tip pair, under a rate-$1$ specification. Using harmonic numbers and a Laplace-transform approach, it derives explicit closed-form expressions for moments up to $m=2$ and key cross-moments, including $\mathrm{E}[U^{(n)}],\ \mathrm{E}[U^{(n)^{2}}],\ \mathrm{E}[\tau^{(n)}],\ \mathrm{E}[U^{(n)}\tau^{(n)}]$, and $\mathrm{E}[\tau^{(n)^{2}}]$, along with $\mathrm{E}[\tau^{(n)}|\mathcal{Y}_{n}]$ and its square, all in terms of $H_{n,r}$ and related quantities. The authors also establish asymptotic limits, e.g., $\mathrm{Var}[\tau^{(n)}]\to\pi^{2}/6$ and $\mathrm{E}[\mathrm{E}[\tau^{(n)}|\mathcal{Y}_{n}]^{2}]\to 2\pi^{2}/9+3$, and quantify the behavior of shared-path lengths $U^{(n)}-\tau^{(n)}$, which converge in mean to $2$ with variance $4(\pi^{2}/6-1)$. The results are validated by simulations, and the work outlines practical applications to phylogenetic indices such as the cophenetic index, with available code for reproducibility. Overall, the paper furnishes a ready-to-use suite of moment formulas for tree heights and pairwise coalescent times in Yule trees, enabling precise statistical analyses of phylogenetic timing structures.

Abstract

We present here a thorough study of the first two moments and cross-moments for the pure birth tree's height and for the coalescent time of a randomly sampled pair of tips. We consider also the first two moments of the conditional, on the tree, expectation of this coalescent time.

First two moments and cross-moments of some coalescent times of the pure birth tree

TL;DR

This work provides a thorough analytical treatment of the first two moments and cross-moments for the height of a pure-birth (Yule) tree and the coalescent time of a random tip pair, under a rate- specification. Using harmonic numbers and a Laplace-transform approach, it derives explicit closed-form expressions for moments up to and key cross-moments, including , and , along with and its square, all in terms of and related quantities. The authors also establish asymptotic limits, e.g., and , and quantify the behavior of shared-path lengths , which converge in mean to with variance . The results are validated by simulations, and the work outlines practical applications to phylogenetic indices such as the cophenetic index, with available code for reproducibility. Overall, the paper furnishes a ready-to-use suite of moment formulas for tree heights and pairwise coalescent times in Yule trees, enabling precise statistical analyses of phylogenetic timing structures.

Abstract

We present here a thorough study of the first two moments and cross-moments for the pure birth tree's height and for the coalescent time of a randomly sampled pair of tips. We consider also the first two moments of the conditional, on the tree, expectation of this coalescent time.

Paper Structure

This paper contains 5 sections, 18 theorems, 109 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Random variables associated with heights of the pure--birth tree
  3. Summary of results concerning moments of heights of the pure--birth tree
  4. Moments of heights of the pure--birth tree
  5. Comparison with simulations

Key Result

Theorem Y4.1

Define the set of integer valued vectors i.e., $\mathcal{A}_{m}$ is the set of all possible ways to represent $m$ as a sum of positive integers. The $m$--th moment of the tree height of a Yule tree with speciation rate $1$ is where for a non--negative integer valued vector $\mathbf{k}=(k_{1},k_{2},\ldots)$, such that $m=\sum_{i=1}^{m}k_{i}i$ we define $c_{\mathbf{k}}$ recursively as and The bo

Figures (4)

  • Figure 1: A pure–birth tree, stopped just before the fifth speciation event, illustrating the various coalescent components we consider. We have four tips, $n=4$, and we "randomly sample" tips $x_{3}$ and $x_{4}$, and $\tau^{(4)}$ is their coalescent time. The interspeciation times, $T_{i}$s are distributed as $T_{i}\sim \exp(i)$ in our case. The pairs $\{(x_{1},x_{2}),(x_{1},x_{3}),(x_{1},x_{4})\}$ coalesced at $\kappa_{n}=1$; at $\kappa_{n}=2$ the pairs $\{(x_{2},x_{4}),(x_{3},x_{4})\}$; and at $\kappa_{n}=3$ only the pairs $\{(x_{2},x_{3})\}$.
  • Figure 2: Comparison of simulated and theoretical values of Lemmata \ref{['BartoszekBrahmantioKiang_YuleMoments:lemUnmTaun2']} to \ref{['BartoszekBrahmantioKiang_YuleMoments:lemETaunYn2']}. For each lemma, the left-hand side plot shows values between 2 to 10, while the right-hand side plot shows values between 25 to 2500.
  • Figure 3: Comparison of simulated and theoretical values of Lemmata \ref{['BartoszekBrahmantioKiang_YuleMoments:lemtaunmETaunYn2']} to \ref{['BartoszekBrahmantioKiang_YuleMoments:lemvarUnmETaunYn']}. For each lemma, the left-hand side plot shows values between 2 to 10, while the right-hand side plot shows values between 25 to 2500.
  • Figure 4: Comparison of simulated and theoretical values of Lemmata \ref{['BartoszekBrahmantioKiang_YuleMoments:lemcovTaunETaunYn']} to \ref{['BartoszekBrahmantioKiang_YuleMoments:lemcovUnmTaunTaunmETaunYn']}, where each lemma has a covariance and correlation part. For each lemma, the left-hand side plot shows values between 2 to 10, while the right-hand side plot shows values between 25 to 2500.

Theorems & Definitions (20)

  • Theorem Y4.1: Appendix A in KBarSSag2015aart
  • Example Y4.1
  • Theorem Y4.2: Appendix A KBarSSag2015aart
  • Example Y4.2
  • Lemma Y4.1: See also the proof of Thm. $5.1$ in KBar2018art
  • Lemma Y4.2
  • Lemma Y4.3
  • Lemma Y4.4
  • Lemma Y4.5
  • Lemma Y4.6
  • ...and 10 more