Skip to main content Accessibility help
×
Home
Hostname: page-component-59b7f5684b-j4fss Total loading time: 0.258 Render date: 2022-09-29T13:39:42.075Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

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
Affiliation:
School of Mathematics and Statistics, Newcastle University, Newcastle Upon Tyne NE1 7RU, United Kingdom, tom.nye@ncl.ac.uk, http://www.mas.ncl.ac.uk/~ntmwn/
Brad J. C. Baxter
Affiliation:
Birkbeck, University of London, Malet Street, London WC1E 7HX, United Kingdom, b.baxter@bbk.ac.uk
Walter R. Gilks
Affiliation:
Department of Statistics, University of Leeds, Leeds LS2 9JT, United Kingdom, wally@maths.leeds.ac.uk

Abstract

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Copyright
Copyright © London Mathematical Society 2007

References

1.Baxter, B. J. C., ‘Conditionally positive functions and p-norm distance matrices’, Constr. Approx. 7 (1991) 427–440.CrossRefGoogle Scholar
2.Bruno, W. J., Socci, N. D. and Halpern, A. L., ‘Weighted neighbour-joining: A likelihood-based approach to distance-based phylogeny reconstruction’, Molecular Biology and Evolution 17 (2000) 189197.CrossRefGoogle Scholar
3.Cavalli-Sforza, L. L. and Edwards, A. W. F., ‘Phylogenetic analysis: Models and estimation procedures’, American Journal of Human Genetics 19 (1967) 233257.Google Scholar
4.Delsuc, F., Brinkmann, H. and Philippe, H., ‘Phylogenomics and the reconstruction of the tree of life’, Nature Reviews Genetics 6 (2005) 361375.CrossRefGoogle Scholar
5.Felsenstein, J., Inferring phytogenies (Sinauer, 2004).Google Scholar
6.Gascuel, O., ‘Bionj: An improved version of the neighbour-joining algorithm based on a simple model of sequence data’, Molecular Biology and Evolution 14 (1997) 685695.CrossRefGoogle Scholar
7.Gascuel, O. (ed.). Mathematics of evolution and phylogeny (Oxford University Press. 2005).Google Scholar
8.Golub, G. H. and Van Loan, C. F., Matrix computations (Johns Hopkins University Press, 1996), 3rd edn.Google Scholar
9.Satou, N. and Nei, M., ‘The neighbour-joining method: A new method for reconstructing phylogenetic trees’, Molecular Biology and Evolution 4 (1987) 406425.Google Scholar
10.Schoenberg, I. J., ‘Remarks to Maurice Fréchet's article “Sur la définition axiomatique d'une classe d'espace distanciés vectoriellement applicable sur l'espace d'Hilbert”’, Ann. of Math. 36 (1935) 724736.CrossRefGoogle Scholar
11.Sneath, P. H. A. and Sokal, R. R., Numerical taxonomy (Freeman, W. K.. 1973).Google Scholar
You have Access

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

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

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *