School of Physical Sciences, University of Tasmania, Hobart TAS 7001, Australia.
BMC Evol Biol. 2014 Dec 4;14:236. doi: 10.1186/s12862-014-0236-6.
Hadamard conjugation is part of the standard mathematical armoury in the analysis of molecular phylogenetic methods. For group-based models, the approach provides a one-to-one correspondence between the so-called "edge length" and "sequence" spectrum on a phylogenetic tree. The Hadamard conjugation has been used in diverse phylogenetic applications not only for inference but also as an important conceptual tool for thinking about molecular data leading to generalizations beyond strictly tree-like evolutionary modelling.
For general group-based models of phylogenetic branching processes, we reformulate the problem of constructing a one-one correspondence between pattern probabilities and edge parameters. This takes a classic result previously shown through use of Fourier analysis and presents it in the language of tensors and group representation theory. This derivation makes it clear why the inversion is possible, because, under their usual definition, group-based models are defined for abelian groups only.
We provide an inversion of group-based phylogenetic models that can implemented using matrix multiplication between rectangular matrices indexed by ordered-partitions of varying sizes. Our approach provides additional context for the construction of phylogenetic probability distributions on network structures, and highlights the potential limitations of restricting to group-based models in this setting.
Hadamard 共轭是分子系统发育方法分析中标准数学工具的一部分。对于基于群组的模型，该方法在系统发育树上提供了所谓的“边长度”和“序列”谱之间的一一对应关系。Hadamard 共轭不仅在推断中，而且在作为思考分子数据的重要概念工具方面，在各种系统发育应用中得到了广泛应用，从而导致超越严格树状进化建模的概括。
对于一般的基于群组的系统发育分支过程模型，我们重新制定了构建模式概率和边缘参数之间一一对应关系的问题。这采用了以前通过傅里叶分析显示的经典结果，并以张量和群表示理论的语言呈现。该推导清楚地表明了为什么反转是可能的，因为，根据它们的通常定义，基于群组的模型仅针对交换群定义。
我们提供了基于群组的系统发育模型的反转，可以使用大小不同的有序分区索引的矩形矩阵之间的矩阵乘法来实现。我们的方法为网络结构上的系统发育概率分布的构建提供了额外的背景，并突出了在此设置中限制基于群组的模型的潜在局限性。
