A Geometrical Framework for f-Statistics

A detailed derivation of the f-statistics formalism is made from a geometrical framework. It is shown that the f-statistics appear when a genetic distance matrix is constrained to describe a four population phylogenetic tree. The choice of genetic metric is crucial and plays an outstanding role as regards the tree-like-ness criterion. The case of lack of treeness is interpreted in the formalism as the presence of population admixture. In this respect, four formulas are given to estimate the admixture proportions. One of them is the so-called f4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_4$$\end{document}-ratio estimate and we show that a second one is related to a known result developed in terms of the fixation index FST\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_{\mathrm{ST}}$$\end{document}. An illustrative numerical simulation of admixture proportion estimates is included. Relationships of the formalism with coalescence times and pairwise sequence differences are also provided.