The expected sample allele frequencies from populations of changing size via orthogonal polynomials

Lynette Caitlin Mikula*, Claus Vogl

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this article, discrete and stochastic changes in (effective) population size are incorporated into the spectral representation of a biallelic diffusion process for drift and small mutation rates. A forward algorithm inspired by Hidden-Markov-Model (HMM) literature is used to compute exact sample allele frequency spectra for three demographic scenarios: single changes in (effective) population size, boom-bust dynamics, and stochastic fluctuations in (effective) population size. An approach for fully agnostic demographic inference from these sample allele spectra is explored, and sufficient statistics for stepwise changes in population size are found. Further, convergence behaviours of the polymorphic sample spectra for population size changes on different time scales are examined and discussed within the context of inference of the effective population size. Joint visual assessment of the sample spectra and the temporal coefficients of the spectral decomposition of the forward diffusion process is found to be important in determining departure from equilibrium. Stochastic changes in (effective) population size are shown to shape sample spectra particularly strongly.
Original languageEnglish
Pages (from-to)55-85
Number of pages31
JournalTheoretical Population Biology
Early online date8 Apr 2024
Publication statusPublished - 1 Jun 2024


  • Moran model
  • Diffusion operator
  • Orthogonal polynomials
  • Forward–backward algorithm (HMM)
  • Autoregression model
  • Population demography


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