等距树重建的最小二乘方法
Least Squares Methods for Equidistant Tree Reconstruction
摘要：
UPGMA是一种启发式方法，它根据$n$类群之间的经验距离数据，确定最小二乘等距系统发生树。我们研究这个经典的算法，使用的空间的所有等距树与$n$叶，也被称为Bergman复杂的图形拟阵的完整图$K_n$的几何。我们表明，UPGMA进行正交投影的数据到一个最大的细胞的伯格曼复杂。我们还表明，等距树与数据的最小（欧几里德）距离是从这样的正交投影，但不一定由UPGMA。利用这些几何信息，我们给出了一个扩展的UPGMA算法。我们还提出了一个寻找最佳等距树的分支定界法。最后，我们证明了$n$分类群之间存在至少投影到$(n-1)!$等距树上的距离数据。
英文摘要：
UPGMA is a heuristic method identifying the least squares equidistant phylogenetic tree given empirical distance data among $n$ taxa. We study this classic algorithm using the geometry of the space of all equidistant trees with $n$ leaves, also known as the Bergman complex of the graphical matroid for the complete graph $K_n$. We show that UPGMA performs an orthogonal projection of the data onto a maximal cell of the Bergman complex. We also show that the equidistant tree with the least (Euclidean) distance from the data is obtained from such an orthogonal projection, but not necessarily given by UPGMA. Using this geometric information we give an extension of the UPGMA algorithm. We also present a branch and bound method for finding the best equidistant tree. Finally, we prove that there are distance data among $n$ taxa which project to at least $(n-1)!$ equidistant trees.
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