21,613 research outputs found
On Khachiyan's algorithm for the computation of minimum-volume enclosing ellipsoids
Cataloged from PDF version of article.Given A := {a(1),..., a(m)} subset of R(d) whose affine hull is R(d), we study the problems of computing an approximate rounding of the convex hull of A and an approximation to the minimum-volume enclosing ellipsoid of V. In the case of centrally symmetric sets, we first establish that Khachiyan's barycentric coordinate descent (BCD) method is exactly the polar of the deepest cut ellipsoid method using two-sided symmetric cuts. This observation gives further insight into the efficient implementation of the BCD method. We then propose a variant algorithm which computes an approximate rounding of the convex hull of,91, and which can also be used to compute an approximation to the minimum-volume enclosing ellipsoid of A.. Our algorithm is a modification of the algorithm of Kumar and Yildirim, which combines Khachiyan's BCD method with a simple initialization scheme to achieve a slightly improved polynomial complexity result, and which returns a small "core set." We establish that our algorithm computes an approximate solution to the dual optimization formulation of the minimum-volume enclosing ellipsoid problem that satisfies a more complete set of approximate optimality conditions than either of the two previous algorithms. Furthermore, this added benefit is achieved without any increase in the improved asymptotic complexity bound of the algorithm of Kumar and Yildirim or any increase in the bound on the size of the computed core set. In addition, the "dropping idea" used in our algorithm has the potential of computing smaller core sets in practice. We also discuss several possible variants of this dropping technique. (C) 2007 Elsevier B.V. All rights reserved
Multimodal nested sampling: an efficient and robust alternative to MCMC methods for astronomical data analysis
In performing a Bayesian analysis of astronomical data, two difficult
problems often emerge. First, in estimating the parameters of some model for
the data, the resulting posterior distribution may be multimodal or exhibit
pronounced (curving) degeneracies, which can cause problems for traditional
MCMC sampling methods. Second, in selecting between a set of competing models,
calculation of the Bayesian evidence for each model is computationally
expensive. The nested sampling method introduced by Skilling (2004), has
greatly reduced the computational expense of calculating evidences and also
produces posterior inferences as a by-product. This method has been applied
successfully in cosmological applications by Mukherjee et al. (2006), but their
implementation was efficient only for unimodal distributions without pronounced
degeneracies. Shaw et al. (2007), recently introduced a clustered nested
sampling method which is significantly more efficient in sampling from
multimodal posteriors and also determines the expectation and variance of the
final evidence from a single run of the algorithm, hence providing a further
increase in efficiency. In this paper, we build on the work of Shaw et al. and
present three new methods for sampling and evidence evaluation from
distributions that may contain multiple modes and significant degeneracies; we
also present an even more efficient technique for estimating the uncertainty on
the evaluated evidence. These methods lead to a further substantial improvement
in sampling efficiency and robustness, and are applied to toy problems to
demonstrate the accuracy and economy of the evidence calculation and parameter
estimation. Finally, we discuss the use of these methods in performing Bayesian
object detection in astronomical datasets.Comment: 14 pages, 11 figures, submitted to MNRAS, some major additions to the
previous version in response to the referee's comment
Uniform sampling of steady states in metabolic networks: heterogeneous scales and rounding
The uniform sampling of convex polytopes is an interesting computational
problem with many applications in inference from linear constraints, but the
performances of sampling algorithms can be affected by ill-conditioning. This
is the case of inferring the feasible steady states in models of metabolic
networks, since they can show heterogeneous time scales . In this work we focus
on rounding procedures based on building an ellipsoid that closely matches the
sampling space, that can be used to define an efficient hit-and-run (HR) Markov
Chain Monte Carlo. In this way the uniformity of the sampling of the convex
space of interest is rigorously guaranteed, at odds with non markovian methods.
We analyze and compare three rounding methods in order to sample the feasible
steady states of metabolic networks of three models of growing size up to
genomic scale. The first is based on principal component analysis (PCA), the
second on linear programming (LP) and finally we employ the lovasz ellipsoid
method (LEM). Our results show that a rounding procedure is mandatory for the
application of the HR in these inference problem and suggest that a combination
of LEM or LP with a subsequent PCA perform the best. We finally compare the
distributions of the HR with that of two heuristics based on the Artificially
Centered hit-and-run (ACHR), gpSampler and optGpSampler. They show a good
agreement with the results of the HR for the small network, while on genome
scale models present inconsistencies.Comment: Replacement with major revision
Using an Ellipsoid Model to Track and Predict the Evolution and Propagation of Coronal Mass Ejections
We present a method for tracking and predicting the propagation and evolution
of coronal mass ejections (CMEs) using the imagers on the STEREO and SOHO
satellites. By empirically modeling the material between the inner core and
leading edge of a CME as an expanding, outward propagating ellipsoid, we track
its evolution in three-dimensional space. Though more complex empirical CME
models have been developed, we examine the accuracy of this relatively simple
geometric model, which incorporates relatively few physical assumptions,
including i) a constant propagation angle and ii) an azimuthally symmetric
structure. Testing our ellipsoid model developed herein on three separate CMEs,
we find that it is an effective tool for predicting the arrival of density
enhancements and the duration of each event near 1 AU. For each CME studied,
the trends in the trajectory, as well as the radial and transverse expansion
are studied from 0 to ~.3 AU to create predictions at 1 AU with an average
accuracy of 2.9 hours.Comment: 18 pages, 11 figure
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