随机圣诞树上叶间成对距离的统计性质
Statistical Properties of Pairwise Distances between Leaves on a Random Yule Tree
摘要：
圣诞树是分支过程的结果，具有恒定的出生率和死亡率。这样一个过程作为许多经验系统的指导性零模型，例如，导致系统发育树的物种进化。然而，在物种发生学中，唯一可用的信息往往是一小部分代表树叶的现存物种之间的成对距离。在这篇文章中，我们研究的统计性质的成对距离在一个圣诞树。使用一种方法的基础上的递归，我们推导出一个精确的，解析的和紧凑的公式的预期数量的对分开一定的时间距离。这个数字是一个递增的指数函数。Yule树的这个性质可以作为一个简单的测试，经验数据可以很好地描述一个Yule过程。我们进一步使用这种递归的方法来计算预期的数量的$n$ -最密切相关的叶对和樱桃的数量分开一定的时间距离。为了使我们的结果更有用的现实情况下，我们明确考虑到，一棵树的叶子可能是不完全采样，并得出一个标准，采样不良的bandgenies。我们表明，我们的结果可以解释的经验数据，使用两个家庭的鸟类物种。
英文摘要：
A Yule tree is the result of a branching process with constant birth and death rates. Such a process serves as an instructive null model of many empirical systems, for instance, the evolution of species leading to a phylogenetic tree. However, often in phylogeny the only available information is the pairwise distances between a small fraction of extant species representing the leaves of the tree. In this article we study statistical properties of the pairwise distances in a Yule tree. Using a method based on a recursion, we derive an exact, analytic and compact formula for the expected number of pairs separated by a certain time distance. This number turns out to follow a increasing exponential function. This property of a Yule tree can serve as a simple test for empirical data to be well described by a Yule process. We further use this recursive method to calculate the expected number of the $n$-most closely related pairs of leaves and the number of cherries separated by a certain time distance. To make our results more useful for realistic scenarios, we explicitly take into account that the leaves of a tree may be incompletely sampled and derive a criterion for poorly sampled phylogenies. We show that our result can account for empirical data, using two families of birds species.
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