Bryant David
Department of Mathematics, University of Auckland, Auckland, New Zealand.
Bull Math Biol. 2009 Feb;71(2):339-51. doi: 10.1007/s11538-008-9364-8. Epub 2008 Oct 10.
The Hadamard transform (Hendy and Penny, Syst. Zool. 38(4):297-309, 1989; Hendy, Syst. Zool. 38(4):310-321, 1989) provides a way to work with stochastic models for sequence evolution without having to deal with the complications of tree space and the graphical structure of trees. Here we demonstrate that the transform can be expressed in terms of the familiar P[tau]=e ( Q[tau]) formula for Markov chains. The key idea is to study the evolution of vectors of states, one vector entry for each taxa; we call this the n-taxon process. We derive transition probabilities for the process. Significantly, the findings show that tree-based models are indeed in the family of (multi-variate) exponential distributions.
哈达玛变换（亨迪和彭尼，《系统动物学》38(4):297 - 309，1989年；亨迪，《系统动物学》38(4):310 - 321，1989年）提供了一种处理序列进化随机模型的方法，而无需处理树空间的复杂性和树的图形结构。在此我们证明，该变换可以用马尔可夫链熟悉的P[tau]=e ( Q[tau])公式来表示。关键思想是研究状态向量的进化，每个分类单元对应一个向量条目；我们将此称为n分类单元过程。我们推导了该过程的转移概率。值得注意的是，研究结果表明基于树的模型确实属于（多元）指数分布族。
