4,912 research outputs found
Elementary vectors and conformal sums in polyhedral geometry and their relevance for metabolic pathway analysis
A fundamental result in metabolic pathway analysis states that every flux
mode can be decomposed into a sum of elementary modes. However, only a
decomposition without cancelations is biochemically meaningful, since a
reversible reaction cannot have different directions in the contributing
elementary modes. This essential requirement has been largely overlooked by the
metabolic pathway community.
Indeed, every flux mode can be decomposed into elementary modes without
cancelations. The result is an immediate consequence of a theorem by
Rockafellar which states that every element of a linear subspace is a conformal
sum (a sum without cancelations) of elementary vectors (support-minimal
vectors). In this work, we extend the theorem, first to "subspace cones" and
then to general polyhedral cones and polyhedra. Thereby, we refine Minkowski's
and Carath\'eodory's theorems, two fundamental results in polyhedral geometry.
We note that, in general, elementary vectors need not be support-minimal, in
fact, they are conformally non-decomposable and form a unique minimal set of
conformal generators.
Our treatment is mathematically rigorous, but suitable for systems
biologists, since we give self-contained proofs for our results and use
concepts motivated by metabolic pathway analysis. In particular, we study cones
defined by linear subspaces and nonnegativity conditions - like the flux cone -
and use them to analyze general polyhedral cones and polyhedra.
Finally, we review applications of elementary vectors and conformal sums in
metabolic pathway analysis
Inference for covariate adjusted regression via varying coefficient models
We consider covariate adjusted regression (CAR), a regression method for
situations where predictors and response are observed after being distorted by
a multiplicative factor. The distorting factors are unknown functions of an
observable covariate, where one specific distorting function is associated with
each predictor or response. The dependence of both response and predictors on
the same confounding covariate may alter the underlying regression relation
between undistorted but unobserved predictors and response. We consider a class
of highly flexible adjustment methods for parameter estimation in the
underlying regression model, which is the model of interest. Asymptotic
normality of the estimates is obtained by establishing a connection to varying
coefficient models. These distribution results combined with proposed
consistent estimates of the asymptotic variance are used for the construction
of asymptotic confidence intervals for the regression coefficients. The
proposed approach is illustrated with data on serum creatinine, and finite
sample properties of the proposed procedures are investigated through a
simulation study.Comment: Published at http://dx.doi.org/10.1214/009053606000000083 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres
Functional data analysis on nonlinear manifolds has drawn recent interest.
Sphere-valued functional data, which are encountered for example as movement
trajectories on the surface of the earth, are an important special case. We
consider an intrinsic principal component analysis for smooth Riemannian
manifold-valued functional data and study its asymptotic properties. Riemannian
functional principal component analysis (RFPCA) is carried out by first mapping
the manifold-valued data through Riemannian logarithm maps to tangent spaces
around the time-varying Fr\'echet mean function, and then performing a
classical multivariate functional principal component analysis on the linear
tangent spaces. Representations of the Riemannian manifold-valued functions and
the eigenfunctions on the original manifold are then obtained with exponential
maps. The tangent-space approximation through functional principal component
analysis is shown to be well-behaved in terms of controlling the residual
variation if the Riemannian manifold has nonnegative curvature. Specifically,
we derive a central limit theorem for the mean function, as well as root-
uniform convergence rates for other model components, including the covariance
function, eigenfunctions, and functional principal component scores. Our
applications include a novel framework for the analysis of longitudinal
compositional data, achieved by mapping longitudinal compositional data to
trajectories on the sphere, illustrated with longitudinal fruit fly behavior
patterns. RFPCA is shown to be superior in terms of trajectory recovery in
comparison to an unrestricted functional principal component analysis in
applications and simulations and is also found to produce principal component
scores that are better predictors for classification compared to traditional
functional functional principal component scores
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