论文标题
奇异性扩散的过渡密度的小近似。适用于赖特 - 法派模型
Small-time approximation of the transition density for diffusions with singularities. Application to the Wright-Fisher model
论文作者
论文摘要
Wright-Fisher（W-F）扩散模型是通过随着时间的推移来解释人群进化的基础框架。尽管连续世代之间存在已知的过渡概率，但在任意时间间隔内的过渡密度的精确分析表达仍然难以捉摸。通常使用的分布（例如高斯或β不充分地解决了极端等位基因频率（0或1）的固定问题，尤其是在短时间内。在这项研究中，我们介绍了通过概率方法论得出的两个替代参数函数，即渐近扩张（AE）和高斯近似（Gaussa）（Gaussa），旨在更好地近似于该密度。 AE函数为等位基因频率分布提供了合适的密度，其中包含间隔内的极端值[0,1]。此外，我们概述了高斯近似的有效性范围。尽管我们的主要重点是W-F扩散，但我们演示了我们的发现如何扩展到具有奇异性的其他扩散模型。通过在W-F工艺下对等位基因频率的模拟，并采用了最近开发的自适应密度估计方法，我们进行了比较分析，以评估所提出的密度与Beta和Gaussian分布的拟合。
The Wright-Fisher (W-F) diffusion model serves as a foundational framework for interpreting population evolution through allele frequency dynamics over time. Despite the known transition probability between consecutive generations, an exact analytical expression for the transition density at arbitrary time intervals remains elusive. Commonly utilized distributions such as Gaussian or Beta inadequately address the fixation issue at extreme allele frequencies (0 or 1), particularly for short periods. In this study, we introduce two alternative parametric functions, namely the Asymptotic Expansion (AE) and the Gaussian approximation (GaussA), derived through probabilistic methodologies, aiming to better approximate this density. The AE function provides a suitable density for allele frequency distributions, encompassing extreme values within the interval [0,1]. Additionally, we outline the range of validity for the GaussA approximation. While our primary focus is on W-F diffusion, we demonstrate how our findings extend to other diffusion models featuring singularities. Through simulations of allele frequencies under a W-F process and employing a recently developed adaptive density estimation method, we conduct a comparative analysis to assess the fit of the proposed densities against the Beta and Gaussian distributions.
