10,840 research outputs found
Principal components analysis in the space of phylogenetic trees
Phylogenetic analysis of DNA or other data commonly gives rise to a
collection or sample of inferred evolutionary trees. Principal Components
Analysis (PCA) cannot be applied directly to collections of trees since the
space of evolutionary trees on a fixed set of taxa is not a vector space. This
paper describes a novel geometrical approach to PCA in tree-space that
constructs the first principal path in an analogous way to standard linear
Euclidean PCA. Given a data set of phylogenetic trees, a geodesic principal
path is sought that maximizes the variance of the data under a form of
projection onto the path. Due to the high dimensionality of tree-space and the
nonlinear nature of this problem, the computational complexity is potentially
very high, so approximate optimization algorithms are used to search for the
optimal path. Principal paths identified in this way reveal and quantify the
main sources of variation in the original collection of trees in terms of both
topology and branch lengths. The approach is illustrated by application to
simulated sets of trees and to a set of gene trees from metazoan (animal)
species.Comment: Published in at http://dx.doi.org/10.1214/11-AOS915 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Osculating orbits in Schwarzschild spacetime, with an application to extreme mass-ratio inspirals
We present a method to integrate the equations of motion that govern bound,
accelerated orbits in Schwarzschild spacetime. At each instant the true
worldline is assumed to lie tangent to a reference geodesic, called an
osculating orbit, such that the worldline evolves smoothly from one such
geodesic to the next. Because a geodesic is uniquely identified by a set of
constant orbital elements, the transition between osculating orbits corresponds
to an evolution of the elements. In this paper we derive the evolution
equations for a convenient set of orbital elements, assuming that the force
acts only within the orbital plane; this is the only restriction that we impose
on the formalism, and we do not assume that the force must be small. As an
application of our method, we analyze the relative motion of two massive
bodies, assuming that one body is much smaller than the other. Using the hybrid
Schwarzschild/post-Newtonian equations of motion formulated by Kidder, Will,
and Wiseman, we treat the unperturbed motion as geodesic in a Schwarzschild
spacetime whose mass parameter is equal to the system's total mass. The force
then consists of terms that depend on the system's reduced mass. We highlight
the importance of conservative terms in this force, which cause significant
long-term changes in the time-dependence and phase of the relative orbit. From
our results we infer some general limitations of the radiative approximation to
the gravitational self-force, which uses only the dissipative terms in the
force.Comment: 18 pages, 6 figures, final version to be published in Physical Review
Geodesic Equations on Diffeomorphism Groups
We bring together those systems of hydrodynamical type that can be written as
geodesic equations on diffeomorphism groups or on extensions of diffeomorphism
groups with right invariant or metrics. We present their formal
derivation starting from Euler's equation, the first order equation satisfied
by the right logarithmic derivative of a geodesic in Lie groups with right
invariant metrics.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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