842 research outputs found
Experiments in orbit determination using numerical methods
The dynamics of the observed object is written as a system of integral equations. This system is solved numerically by representing the components of the force function as linear combinations of B-splines and by applying the multigrid technique. In an outer loop the orbit determination problem is solved using Newton's method.\ud
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The method is suitable for both preliminary orbit determination and orbit improvement
A Reproducing Kernel Perspective of Smoothing Spline Estimators
Spline functions have a long history as smoothers of noisy time series data, and several equivalent kernel representations have been proposed in terms of the Green's function solving the related boundary value problem. In this study we make use of the reproducing kernel property of the Green's function to obtain an hierarchy of time-invariant spline kernels of different order. The reproducing kernels give a good representation of smoothing splines for medium and long length filters, with a better performance of the asymmetric weights in terms of signal passing, noise suppression and revisions. Empirical comparisons of time-invariant filters are made with the classical non linear ones. The former are shown to loose part of their optimal properties when we fixed the length of the filter according to the noise to signal ratio as done in nonparametric seasonal adjustment procedures.equivalent kernels, nonparametric regression, Hilbert spaces, time series filtering, spectral properties
Spline approximation of a random process with singularity
Let a continuous random process defined on be -smooth,
and have an isolated
singularity point at . In addition, let be locally like a -fold
integrated -fractional Brownian motion for all non-singular points. We
consider approximation of by piecewise Hermite interpolation splines with
free knots (i.e., a sampling design, a mesh). The approximation performance
is measured by mean errors (e.g., integrated or maximal quadratic mean errors).
We construct a sequence of sampling designs with asymptotic approximation rate
for the whole interval.Comment: 16 pages, 2 figure typos and references corrected, revised classes
definition, results unchange
Statistical unfolding of elementary particle spectra: Empirical Bayes estimation and bias-corrected uncertainty quantification
We consider the high energy physics unfolding problem where the goal is to
estimate the spectrum of elementary particles given observations distorted by
the limited resolution of a particle detector. This important statistical
inverse problem arising in data analysis at the Large Hadron Collider at CERN
consists in estimating the intensity function of an indirectly observed Poisson
point process. Unfolding typically proceeds in two steps: one first produces a
regularized point estimate of the unknown intensity and then uses the
variability of this estimator to form frequentist confidence intervals that
quantify the uncertainty of the solution. In this paper, we propose forming the
point estimate using empirical Bayes estimation which enables a data-driven
choice of the regularization strength through marginal maximum likelihood
estimation. Observing that neither Bayesian credible intervals nor standard
bootstrap confidence intervals succeed in achieving good frequentist coverage
in this problem due to the inherent bias of the regularized point estimate, we
introduce an iteratively bias-corrected bootstrap technique for constructing
improved confidence intervals. We show using simulations that this enables us
to achieve nearly nominal frequentist coverage with only a modest increase in
interval length. The proposed methodology is applied to unfolding the boson
invariant mass spectrum as measured in the CMS experiment at the Large Hadron
Collider.Comment: Published at http://dx.doi.org/10.1214/15-AOAS857 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org). arXiv admin note:
substantial text overlap with arXiv:1401.827
Lattice points in polytopes, box splines, and Todd operators
Let be a list of vectors that is totally unimodular. In a previous
article the author proved that every real-valued function on the set of
interior lattice points of the zonotope defined by can be extended to a
function on the whole zonotope of the form in a unique way, where
is a differential operator that is contained in the so-called internal
\Pcal-space. In this paper we construct an explicit solution to this
interpolation problem in terms of Todd operators. As a corollary we obtain a
slight generalisation of the Khovanskii-Pukhlikov formula that relates the
volume and the number of integer points in a smooth lattice polytope.Comment: 15 pages, 4 figure
Heuristic regularization methods for numerical differentiation
AbstractIn this paper, we use smoothing splines to deal with numerical differentiation. Some heuristic methods for choosing regularization parameters are proposed, including the L-curve method and the de Boor method. Numerical experiments are performed to illustrate the efficiency of these methods in comparison with other procedures
Unconditionality of orthogonal spline systems in
We give a simple geometric characterization of knot sequences for which the
corresponding orthonormal spline system of arbitrary order is an
unconditional basis in the atomic Hardy space .Comment: 30 page
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