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A characterisation of the genealogy of a birth-death process through time rescaling
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Ignatieva, Anastasia, Hein, Jotun and Jenkins, Paul (2020) A characterisation of the genealogy of a birth-death process through time rescaling. Theoretical Population Biology, 134 . pp. 61-76. doi:10.1016/j.tpb.2020.05.001
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Official URL: https://doi.org/10.1016/j.tpb.2020.05.001
Abstract
The dynamics of a population exhibiting exponential growth can be modelled as a birth-death process, which naturally captures the stochastic variation in population size over time. In this article, we consider a supercritical birth-death process, started at a random time in the past, and conditioned to have n sampled individuals at the present. The genealogy of individuals sampled at the present time is then described by the reversed reconstructed process (RRP), which traces the ancestry of the sample backwards from the present. We show that a simple, analytic, time rescaling of the RRP provides a straightforward way to derive its inter-event times. The same rescaling characterises other distributions underlying this process, obtained elsewhere in the literature via more cumbersome calculations. We also consider the case of incomplete sampling of the population, in which each leaf of the genealogy is retained with an independent Bernoulli trial with probability \psi, and we show that corresponding results for Bernoulli-sampled RRPs can be derived using time rescaling, for any values of the underlying parameters. A central result is the derivation of a scaling limit as \psi approaches 0, corresponding to the underlying population growing to infinity, using the time rescaling formalism. We show that in this setting, after a linear time rescaling, the event times are the order statistics of n logistic random variables with mode \log(1/\psi); moreover, we show that the inter-event times are approximately exponentially distributed.
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science > Computer Science Faculty of Science > Statistics |
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| Library of Congress Subject Headings (LCSH): | Birth and death processes (Stochastic processes), Sampling (Statistics), Error analysis (Mathematics) | |||||||||
| Journal or Publication Title: | Theoretical Population Biology | |||||||||
| Publisher: | Academic Press | |||||||||
| ISSN: | 0040-5809 | |||||||||
| Official Date: | August 2020 | |||||||||
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| Volume: | 134 | |||||||||
| Page Range: | pp. 61-76 | |||||||||
| DOI: | 10.1016/j.tpb.2020.05.001 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| RIOXX Funder/Project Grant: |
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