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Phylogenetic confidence intervals for the optimal trait value

Part of: Mathematical biology in general Stochastic analysis Applications Inference from stochastic processes

Published online by Cambridge University Press:  30 March 2016

Krzysztof Bartoszek  and
Serik Sagitov
Krzysztof Bartoszek*
Affiliation:
Uppsala University
Serik Sagitov*
Affiliation:
Chalmers University of Technology and the University of Gothenburg
*
∗Postal address: Department of Mathematics, Uppsala University, 751 06 Uppsala, Sweden. Email address: [email protected]
∗∗Postal address: Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, 412 96 Göteborg, Sweden.
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Abstract

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We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

Keywords

Central limit theoremconditioned Yule processmacroevolutionmartingalesOrnstein-Uhlenbeck processphylogenetics

MSC classification

Primary: 62M05: Markov processes 62P10: Applications to biology and medical sciences
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.) 92B99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1115 - 1132
Copyright
Copyright © Applied Probability Trust 2015 

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