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Phase-type distributions in population genetics

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  • Hobolth, Asger
  • Siri-Jégousse, Arno
  • Bladt, Mogens

Abstract

Probability modelling for DNA sequence evolution is well established and provides a rich framework for understanding genetic variation between samples of individuals from one or more populations. We show that both classical and more recent models for coalescence (with or without recombination) can be described in terms of the so-called phase-type theory, where complicated and tedious calculations are circumvented by the use of matrix manipulations. The application of phase-type theory in population genetics consists of describing the biological system as a Markov model by appropriately setting up a state space and calculating the corresponding intensity and reward matrices. Formulae of interest are then expressed in terms of these aforementioned matrices. We illustrate this procedure by a number of examples: (a)Â Calculating the mean, (co)variance and even higher order moments of the site frequency spectrum in multiple merger coalescent models, (b)Â Analysing a sample of DNA sequences from the Atlantic Cod using the Beta-coalescent, and (c)Â Determining the correlation of the number of segregating sites for multiple samples in the two-locus ancestral recombination graph. We believe that phase-type theory has great potential as a tool for analysing probability models in population genetics. The compact matrix notation is useful for clarification of current models, and in particular their formal manipulation and calculations, but also for further development or extensions.

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  • Hobolth, Asger & Siri-Jégousse, Arno & Bladt, Mogens, 2019. "Phase-type distributions in population genetics," Theoretical Population Biology, Elsevier, vol. 127(C), pages 16-32.
    • Handle: RePEc:eee:thpobi:v:127:y:2019:i:c:p:16-32
    DOI: 10.1016/j.tpb.2019.02.001
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    References listed on IDEAS

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    1. Hobolth, Asger & Jensen, Jens Ledet, 2014. "Markovian approximation to the finite loci coalescent with recombination along multiple sequences," Theoretical Population Biology, Elsevier, vol. 98(C), pages 48-58.
    2. Miroshnikov, Alexey & Steinrücken, Matthias, 2017. "Computing the joint distribution of the total tree length across loci in populations with variable size," Theoretical Population Biology, Elsevier, vol. 118(C), pages 1-19.
    3. Schweinsberg, Jason, 2003. "Coalescent processes obtained from supercritical Galton-Watson processes," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 107-139, July.
    4. Sargsyan, Ori & Wakeley, John, 2008. "A coalescent process with simultaneous multiple mergers for approximating the gene genealogies of many marine organisms," Theoretical Population Biology, Elsevier, vol. 74(1), pages 104-114.
    5. Klassmann, A. & Ferretti, L., 2018. "The third moments of the site frequency spectrum," Theoretical Population Biology, Elsevier, vol. 120(C), pages 16-28.
    6. Kumagai, Seiji & Uyenoyama, Marcy K., 2015. "Genealogical histories in structured populations," Theoretical Population Biology, Elsevier, vol. 102(C), pages 3-15.
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    4. Freund, Fabian & Siri-Jégousse, Arno, 2021. "The impact of genetic diversity statistics on model selection between coalescents," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    5. Blath, Jochen & Buzzoni, Eugenio & Koskela, Jere & Wilke Berenguer, Maite, 2020. "Statistical tools for seed bank detection," Theoretical Population Biology, Elsevier, vol. 132(C), pages 1-15.

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