Uniformization Stable Markov Models and Their Jordan Algebraic Structure

IF 1.5 2区 数学 Q2 MATHEMATICS, APPLIED
Luke Cooper, Jeremy Sumner
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引用次数: 1

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1822-1851, December 2023.
Abstract. We provide a characterization of the continuous-time Markov models where the Markov matrices from the model can be parameterized directly in terms of the associated rate matrices (generators). That is, each Markov matrix can be expressed as the sum of the identity matrix and a rate matrix from the model. We show that the existence of an underlying Jordan algebra provides a sufficient condition, which becomes necessary for (so-called) linear models. We connect this property to the well-known uniformization procedure for continuous-time Markov chains by demonstrating that the property is equivalent to all Markov matrices from the model taking the same form as the corresponding discrete-time Markov matrices in the uniformized process. We apply our results to analyze two model hierarchies practically important to phylogenetic inference, obtained by assuming (i) time reversibility and (ii) permutation symmetry, respectively.
均匀化稳定马尔可夫模型及其Jordan代数结构
SIAM矩阵分析与应用杂志,第44卷,第4期,1822-1851页,2023年12月。摘要。我们提供了连续时间马尔可夫模型的表征,其中模型中的马尔可夫矩阵可以直接根据相关的速率矩阵(生成器)参数化。也就是说,每个马尔可夫矩阵都可以表示为来自模型的单位矩阵和速率矩阵的和。我们证明了一个潜在的约旦代数的存在提供了一个充分条件,这成为(所谓的)线性模型的必要条件。我们将这一性质与众所周知的连续时间马尔可夫链的均匀化过程联系起来,证明了该性质等价于模型中的所有马尔可夫矩阵在均匀化过程中具有与相应的离散时间马尔可夫矩阵相同的形式。我们应用我们的结果来分析两个模型层次实际上对系统发育推理很重要,分别通过假设(i)时间可逆性和(ii)排列对称性获得。
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来源期刊
CiteScore
2.90
自引率
6.70%
发文量
61
审稿时长
6-12 weeks
期刊介绍: The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.
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