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 step-wise 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.

翻译：本文通过正交多项式谱表示，将（有效）群体大小的离散与随机变化纳入漂移和小突变率双等位基因扩散过程的谱表示中。借鉴隐马尔可夫模型文献，提出一种前向算法，用于计算三种人口统计学场景下的精确样本等位基因频率谱：有效群体大小的单次变化、繁荣-衰退动态以及有效群体大小的随机波动。本文探索了从这些样本等位基因谱进行完全不可知的人口统计学推断方法，并找到了群体大小逐步变化的充分统计量。进一步地，研究了不同时间尺度下群体大小变化的多态样本谱的收敛行为，并在有效群体大小推断的背景下进行了讨论。联合可视化评估样本谱与前向扩散过程谱分解的时间系数，对于判断是否偏离平衡态具有重要意义。研究表明，有效群体大小的随机变化对样本谱的塑造作用尤为显著。