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Lie geometry of 2 x 2 Markov matrices

Citation

Sumner, JG, Lie geometry of 2 x 2 Markov matrices, Journal of Theoretical Biology, 327 pp. 88-90. ISSN 0022-5193 (2013) [Refereed Article]

Copyright Statement

Copyright 2013 Elsevier Ltd

DOI: doi:10.1016/j.jtbi.2013.01.026

Abstract

In recent work discussing model choice for continuous-time Markov chains, we have argued that it is important that the Markov matrices that define the model are closed under matrix multiplication (Sumner et al., 2012a and Sumner et al., 2012b). The primary requirement is then that the associated set of rate matrices form a Lie algebra. For the generic case, this connection to Lie theory seems to have been first made by Johnson (1985), with applications for specific models given in Bashford et al. (2004) and House (2012). Here we take a different perspective: given a model that forms a Lie algebra, we apply existing Lie theory to gain additional insight into the geometry of the associated Markov matrices. In this short note, we present the simplest case possible of 2×2 Markov matrices. The main result is a novel decomposition of 2×2 Markov matrices that parameterises the general Markov model as a perturbation away from the binary-symmetric model. This alternative parameterisation provides a useful tool for visualising the binary-symmetric model as a submodel of the general Markov model.

Item Details

Item Type:Refereed Article
Research Division:Mathematical Sciences
Research Group:Applied Mathematics
Research Field:Biological Mathematics
Objective Division:Expanding Knowledge
Objective Group:Expanding Knowledge
Objective Field:Expanding Knowledge in the Mathematical Sciences
Author:Sumner, JG (Dr Jeremy Sumner)
ID Code:87059
Year Published:2013
Funding Support:Australian Research Council (DE130100423)
Web of Science® Times Cited:2
Deposited By:Mathematics and Physics
Deposited On:2013-11-06
Last Modified:2017-12-07
Downloads:0

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