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A Covariance Matrix Inversion Problem arising from the Construction of Phylogenetic Trees

Published online by Cambridge University Press:  01 February 2010

Tom M. W. Nye ,
Brad J. C. Baxter  and
Walter R. Gilks
Tom M. W. Nye
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle Upon Tyne NE1 7RU, United Kingdom, [email protected], http://www.mas.ncl.ac.uk/~ntmwn/
Brad J. C. Baxter
Affiliation:
Birkbeck, University of London, Malet Street, London WC1E 7HX, United Kingdom, [email protected]
Walter R. Gilks
Affiliation:
Department of Statistics, University of Leeds, Leeds LS2 9JT, United Kingdom, [email protected]

Abstract

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We describe an efficient algorithm for the inversion of covariance matrices that arise in the context of phylogenetic tree construction. Phylogenetic trees describe the evolutionary relationships between species, and their construction is computationally demanding. Many approaches involve the symmetric matrix of evolutionary distances between species. Regarding these distances as random variables, the corresponding set of variances and covariances form a rank-4 tensor, and the inner-product defined by its inverse can be used to assign statistical scores to candidate trees. We describe a natural set of assumptions for the phylogenetic tree under construction, and show how under these assumptions the covariance tensor for a tree with n leaves can be inverted in O(n2) operations. In addition to presenting the inversion algorithm, we hope this article will open algebraic and computational problems from the field of phylogeny to a wider audience.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007 , pp. 119 - 131
Copyright
Copyright © London Mathematical Society 2007

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