112 research outputs found
The phylogenetic effective sample size and jumps
The phylogenetic effective sample size is a parameter that has as its goal
the quantification of the amount of independent signal in a phylogenetically
correlated sample. It was studied for Brownian motion and Ornstein-Uhlenbeck
models of trait evolution. Here, we study this composite parameter when the
trait is allowed to jump at speciation points of the phylogeny. Our numerical
study indicates that there is a non-trivial limit as the effect of jumps grows.
The limit depends on the value of the drift parameter of the Ornstein-Uhlenbeck
process
A Consistent Estimator of the Evolutionary Rate
We consider a branching particle system where particles reproduce according
to the pure birth Yule process with the birth rate L, conditioned on the
observed number of particles to be equal n. Particles are assumed to move
independently on the real line according to the Brownian motion with the local
variance s2. In this paper we treat particles as a sample of related
species. The spatial Brownian motion of a particle describes the development of
a trait value of interest (e.g. log-body-size). We propose an unbiased
estimator Rn2 of the evolutionary rate r2=s2/L. The estimator Rn2 is
proportional to the sample variance Sn2 computed from n trait values. We find
an approximate formula for the standard error of Rn2 based on a neat asymptotic
relation for the variance of Sn2
Critical case stochastic phylogenetic tree model via the Laplace transform
Birth-and-death models are now a common mathematical tool to describe
branching patterns observed in real-world phylogenetic trees. Liggett and
Schinazi (2009) is one such example. The authors propose a simple
birth-and-death model that is compatible with phylogenetic trees of both
influenza and HIV, depending on the birth rate parameter. An interesting
special case of this model is the critical case where the birth rate equals the
death rate. This is a non-trivial situation and to study its asymptotic
behaviour we employed the Laplace transform. With this we correct the proof of
Liggett and Schinazi (2009) in the critical case.Comment: To appear in Demonstratio Mathematic
Convergence of finite volume scheme for three dimensional Poisson's equation
We construct and analyze a finite volume scheme for numerical solution of a
three-dimensional Poisson equation. This is an extension of a two-dimensional
approach by Suli 1991. Here we derive optimal convergence rates in the discrete
H^1 norm and sub-optimal convergence in the maximum norm, where we use the
maximal available regularity of the exact solution and minimal smoothness
requirement on the source term. We also find a gap in the proof of a key
estimate in a reference in Suli 1991 for which we present a modified and
completed proof. Finally, the theoretical results derived in the paper are
justified through implementing some canonical examples in 3D
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