Modelling the substitution of nucleotides along a phylogenetic tree is usually done by a hidden Markov process. This allows to define a distribution of characters at the leaves of the trees and one might be able to obtain polynomial relationships among the probabilities of different characters. The study of these polynomials and the geometry of the algebraic varieties defined by them can be used to reconstruct phylogenetic trees. However, not all points in these algebraic varieties have biological sense. In this paper, we explore the extent to which adding semi-algebraic conditions arising from the restriction to parameters with statistical meaning can improve existing methods of phylogenetic reconstruction. To this end, our aim is to compute the distance of data points to algebraic varieties and to the stochastic part of these varieties. Computing these distances involves optimization by nonlinear programming algorithms. We use analytical methods to find some of these distances for quartet trees evolving under the Kimura 3-parameter or the Jukes-Cantor models. Numerical algebraic geometry and computational algebra play also a fundamental role in this paper.

沿着系统发育树对核苷酸的取代进行建模通常是通过隐马尔可夫过程完成的。这允许在树的叶子处定义字符的分布，并且可能能够获得不同字符的概率之间的多项式关系。对这些多项式及其所定义的代数变体的几何形状的研究可用于重建系统发育树。但是，并非这些代数变体中的所有点都具有生物学意义。在本文中，我们探讨了在一定程度上具有限制意义的参数添加具有统计学意义的半代数条件可以改善现有的系统发育重建方法。为此，我们的目的是计算数据点到代数变体以及这些变体的随机部分的距离。计算这些距离需要通过非线性编程算法进行优化。我们使用分析方法找到在木村3参数或Jukes-Cantor模型下进化的四方树的某些距离。数值代数几何和计算代数在本文中也起着基本作用。