261 research outputs found
Localized spherical deconvolution
We provide a new algorithm for the treatment of the deconvolution problem on
the sphere which combines the traditional SVD inversion with an appropriate
thresholding technique in a well chosen new basis. We establish upper bounds
for the behavior of our procedure for any loss. It is important
to emphasize the adaptation properties of our procedures with respect to the
regularity (sparsity) of the object to recover as well as to inhomogeneous
smoothness. We also perform a numerical study which proves that the procedure
shows very promising properties in practice as well.Comment: Published in at http://dx.doi.org/10.1214/10-AOS858 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Testing the isotropy of high energy cosmic rays using spherical needlets
For many decades, ultrahigh energy charged particles of unknown origin that
can be observed from the ground have been a puzzle for particle physicists and
astrophysicists. As an attempt to discriminate among several possible
production scenarios, astrophysicists try to test the statistical isotropy of
the directions of arrival of these cosmic rays. At the highest energies, they
are supposed to point toward their sources with good accuracy. However, the
observations are so rare that testing the distribution of such samples of
directional data on the sphere is nontrivial. In this paper, we choose a
nonparametric framework that makes weak hypotheses on the alternative
distributions and allows in turn to detect various and possibly unexpected
forms of anisotropy. We explore two particular procedures. Both are derived
from fitting the empirical distribution with wavelet expansions of densities.
We use the wavelet frame introduced by [SIAM J. Math. Anal. 38 (2006b) 574-594
(electronic)], the so-called needlets. The expansions are truncated at scale
indices no larger than some , and the distances between
those estimates and the null density are computed. One family of tests (called
Multiple) is based on the idea of testing the distance from the null for each
choice of , whereas the so-called PlugIn approach is
based on the single full expansion, but with thresholded wavelet
coefficients. We describe the practical implementation of these two procedures
and compare them to other methods in the literature. As alternatives to
isotropy, we consider both very simple toy models and more realistic
nonisotropic models based on Physics-inspired simulations. The Monte Carlo
study shows good performance of the Multiple test, even at moderate sample
size, for a wide sample of alternative hypotheses and for different choices of
the parameter . On the 69 most energetic events published by the
Pierre Auger Collaboration, the needlet-based procedures suggest statistical
evidence for anisotropy. Using several values for the parameters of the
methods, our procedures yield -values below 1%, but with uncontrolled
multiplicity issues. The flexibility of this method and the possibility to
modify it to take into account a large variety of extensions of the problem
make it an interesting option for future investigation of the origin of
ultrahigh energy cosmic rays.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS619 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
High Frequency Asymptotics for Wavelet-Based Tests for Gaussianity and Isotropy on the Torus
We prove a CLT for skewness and kurtosis of the wavelets coefficients of a
stationary field on the torus. The results are in the framework of the
fixed-domain asymptotics, i.e. we refer to observations of a single field which
is sampled at higher and higher frequencies. We consider also studentized
statistics for the case of an unknown correlation structure. The results are
motivated by the analysis of cosmological data or high-frequency financial data
sets, with a particular interest towards testing for Gaussianity and isotropyComment: 33 pages, 3 figure
Needlet algorithms for estimation in inverse problems
We provide a new algorithm for the treatment of inverse problems which
combines the traditional SVD inversion with an appropriate thresholding
technique in a well chosen new basis. Our goal is to devise an inversion
procedure which has the advantages of localization and multiscale analysis of
wavelet representations without losing the stability and computability of the
SVD decompositions. To this end we utilize the construction of localized frames
(termed "needlets") built upon the SVD bases. We consider two different
situations: the "wavelet" scenario, where the needlets are assumed to behave
similarly to true wavelets, and the "Jacobi-type" scenario, where we assume
that the properties of the frame truly depend on the SVD basis at hand (hence
on the operator). To illustrate each situation, we apply the estimation
algorithm respectively to the deconvolution problem and to the Wicksell
problem. In the latter case, where the SVD basis is a Jacobi polynomial basis,
we show that our scheme is capable of achieving rates of convergence which are
optimal in the case, we obtain interesting rates of convergence for other
norms which are new (to the best of our knowledge) in the literature, and
we also give a simulation study showing that the NEED-D estimator outperforms
other standard algorithms in almost all situations.Comment: Published at http://dx.doi.org/10.1214/07-EJS014 in the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Giant osmotic pressure in the forced wetting of hydrophobic nanopores
The forced intrusion of water in hydrophobic nanoporous pulverulent material
is of interest for quick storage of energy. With nanometric pores the energy
storage capacity is controlled by interfacial phenomena. With subnanometric
pores, we demonstrate that a breakdown occurs with the emergence of molecular
exclusion as a leading contribution. This bulk exclusion effect leads to an
osmotic contribution to the pressure that can reach levels never previously
sustained. We illustrate on various electrolytes and different microporous
materials, that a simple osmotic pressure law accounts quantitatively for the
enhancement of the intrusion and extrusion pressures governing the forced
wetting and spontaneous drying of the nanopores. Using electrolyte solutions,
energy storage and power capacities can be widely enhanced
Inversion of noisy Radon transform by SVD based needlet
A linear method for inverting noisy observations of the Radon transform is
developed based on decomposition systems (needlets) with rapidly decaying
elements induced by the Radon transform SVD basis. Upper bounds of the risk of
the estimator are established in () norms for functions
with Besov space smoothness. A practical implementation of the method is given
and several examples are discussed
NEED-VD: a second-generation wavelet algorithm for estimation in inverse problems
We provide a new algorithm for the treatment of inverse problems which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. Our goal is to devise an inversion procedure which has the advantages of localization and multiscale analysis of wavelet representations without losing the stability and computability of the SVD decompositions. To this end we utilize the construction of localized frames (termed ``needlets") built upon the SVD bases. We consider two different situations : the ``wavelet" scenario, where the needlets are assumed to behave similarly to true wavelets, and the ``Jacobi-type" scenario, where we assume that the properties of the frame truly depend on the SVD basis at hand (hence on the operator). To illustrate each situation, we apply the estimation algorithm respectively to the deconvolution problem and to the Wicksell problem. In the latter case, where the SVD basis is a Jacobi polynomial basis, we show that our scheme is capable of achieving rates of convergence which are optimal in the case, we obtain interesting rates of convergence for other norms which are new (to the best of our knowledge) in the literature, and we also give a simulation study showing that the NEED-VD estimator outperforms other standard algorithms in almost all situations
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