9,618 research outputs found
Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 1: The Over-Sampled Case
In \cite{Abramovich04}, it was demonstrated that the likelihood ratio (LR) for multivariate complex Gaussian distribution has the invariance property that can be exploited in many applications. Specifically, the probability density function (p.d.f.) of this LR for the (unknown) actual covariance matrix does not depend on this matrix and is fully specified by the matrix dimension and the number of independent training samples . Since this p.d.f. could therefore be pre-calculated for any a priori known , one gets a possibility to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically ``as likely'' as the a priori unknown actual covariance matrix. This ``expected likelihood'' (EL) quality assessment allows for significant improvement of MUSIC DOA estimation performance in the so-called ``threshold area'' \cite{Abramovich04,Abramovich07d}, and for diagonal loading and TVAR model order selection in adaptive detectors \cite{Abramovich07,Abramovich07b}. Recently, a broad class of the so-called complex elliptically symmetric (CES) distributions has been introduced for description of highly in-homogeneous clutter returns. The aim of this series of two papers is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative of CES, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix \mSigma_{0}. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario () while Part 2 deals with the under-sampled scenario ()
Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 2: The Under-Sampled Case
In the first part of this series of two papers, we extended the expected likelihood approach originally developed in the Gaussian case, to the broader class of complex elliptically symmetric (CES) distributions and complex angular central Gaussian (ACG) distributions. More precisely, we demonstrated that the probability density function (p.d.f.) of the likelihood ratio (LR) for the (unknown) actual scatter matrix \mSigma_{0} does not depend on the latter: it only depends on the density generator for the CES distribution and is distribution-free in the case of ACG distributed data, i.e., it only depends on the matrix dimension and the number of independent training samples , assuming that . Additionally, regularized scatter matrix estimates based on the EL methodology were derived. In this second part, we consider the under-sampled scenario () which deserves a specific treatment since conventional maximum likelihood estimates do not exist. Indeed, inference about the scatter matrix can only be made in the -dimensional subspace spanned by the columns of the data matrix. We extend the results derived under the Gaussian assumption to the CES and ACG class of distributions. Invariance properties of the under-sampled likelihood ratio evaluated at \mSigma_{0} are presented. Remarkably enough, in the ACG case, the p.d.f. of this LR can be written in a rather simple form as a product of beta distributed random variables. The regularized schemes derived in the first part, based on the EL principle, are extended to the under-sampled scenario and assessed through numerical simulations
Non-Gaussianity in the Weak Lensing Correlation Function Likelihood -- Implications for Cosmological Parameter Biases
We study the significance of non-Gaussianity in the likelihood of weak
lensing shear two-point correlation functions, detecting significantly non-zero
skewness and kurtosis in one-dimensional marginal distributions of shear
two-point correlation functions in simulated weak lensing data. We examine the
implications in the context of future surveys, in particular LSST, with
derivations of how the non-Gaussianity scales with survey area. We show that
there is no significant bias in one-dimensional posteriors of
and due to the non-Gaussian likelihood distributions of shear
correlations functions using the mock data ( deg). We also present a
systematic approach to constructing approximate multivariate likelihoods with
one-dimensional parametric functions by assuming independence or more flexible
non-parametric multivariate methods after decorrelating the data points using
principal component analysis (PCA). While the use of PCA does not modify the
non-Gaussianity of the multivariate likelihood, we find empirically that the
one-dimensional marginal sampling distributions of the PCA components exhibit
less skewness and kurtosis than the original shear correlation
functions.Modeling the likelihood with marginal parametric functions based on
the assumption of independence between PCA components thus gives a lower limit
for the biases. We further demonstrate that the difference in cosmological
parameter constraints between the multivariate Gaussian likelihood model and
more complex non-Gaussian likelihood models would be even smaller for an
LSST-like survey. In addition, the PCA approach automatically serves as a data
compression method, enabling the retention of the majority of the cosmological
information while reducing the dimensionality of the data vector by a factor of
5.Comment: 16 pages, 10 figures, published MNRA
The non-Gaussianity of the cosmic shear likelihood - or: How odd is the Chandra Deep Field South?
(abridged) We study the validity of the approximation of a Gaussian cosmic
shear likelihood. We estimate the true likelihood for a fiducial cosmological
model from a large set of ray-tracing simulations and investigate the impact of
non-Gaussianity on cosmological parameter estimation. We investigate how odd
the recently reported very low value of really is as derived from
the \textit{Chandra} Deep Field South (CDFS) using cosmic shear by taking the
non-Gaussianity of the likelihood into account as well as the possibility of
biases coming from the way the CDFS was selected.
We find that the cosmic shear likelihood is significantly non-Gaussian. This
leads to both a shift of the maximum of the posterior distribution and a
significantly smaller credible region compared to the Gaussian case. We
re-analyse the CDFS cosmic shear data using the non-Gaussian likelihood.
Assuming that the CDFS is a random pointing, we find
for fixed . In a
WMAP5-like cosmology, a value equal to or lower than this would be expected in
of the times. Taking biases into account arising from the way the
CDFS was selected, which we model as being dependent on the number of haloes in
the CDFS, we obtain . Combining the CDFS data
with the parameter constraints from WMAP5 yields and for a flat
universe.Comment: 18 pages, 16 figures, accepted for publication in A&A; New Bayesian
treatment of field selection bia
KL Estimation of the Power Spectrum Parameters from the Angular Distribution of Galaxies in Early SDSS Data
We present measurements of parameters of the 3-dimensional power spectrum of
galaxy clustering from 222 square degrees of early imaging data in the Sloan
Digital Sky Survey. The projected galaxy distribution on the sky is expanded
over a set of Karhunen-Loeve eigenfunctions, which optimize the signal-to-noise
ratio in our analysis. A maximum likelihood analysis is used to estimate
parameters that set the shape and amplitude of the 3-dimensional power
spectrum. Our best estimates are Gamma=0.188 +/- 0.04 and sigma_8L = 0.915 +/-
0.06 (statistical errors only), for a flat Universe with a cosmological
constant. We demonstrate that our measurements contain signal from scales at or
beyond the peak of the 3D power spectrum. We discuss how the results scale with
systematic uncertainties, like the radial selection function. We find that the
central values satisfy the analytically estimated scaling relation. We have
also explored the effects of evolutionary corrections, various truncations of
the KL basis, seeing, sample size and limiting magnitude. We find that the
impact of most of these uncertainties stay within the 2-sigma uncertainties of
our fiducial result.Comment: Fig 1 postscript problem correcte
A re-analysis of the three-year WMAP temperature power spectrum and likelihood
We analyze the three-year WMAP temperature anisotropy data seeking to confirm
the power spectrum and likelihoods published by the WMAP team. We apply five
independent implementations of four algorithms to the power spectrum estimation
and two implementations to the parameter estimation. Our single most important
result is that we broadly confirm the WMAP power spectrum and analysis. Still,
we do find two small but potentially important discrepancies: On large angular
scales there is a small power excess in the WMAP spectrum (5-10% at l<~30)
primarily due to likelihood approximation issues between 13 <= l <~30. On small
angular scales there is a systematic difference between the V- and W-band
spectra (few percent at l>~300). Recently, the latter discrepancy was explained
by Huffenberger et al. (2006) in terms of over-subtraction of unresolved point
sources. As far as the low-l bias is concerned, most parameters are affected by
a few tenths of a sigma. The most important effect is seen in n_s. For the
combination of WMAP, Acbar and BOOMERanG, the significance of n_s =/ 1 drops
from ~2.7 sigma to ~2.3 sigma when correcting for this bias. We propose a few
simple improvements to the low-l WMAP likelihood code, and introduce two
important extensions to the Gibbs sampling method that allows for proper
sampling of the low signal-to-noise regime. Finally, we make the products from
the Gibbs sampling analysis publically available, thereby providing a fast and
simple route to the exact likelihood without the need of expensive matrix
inversions.Comment: 14 pages, 7 figures. Accepted for publication in ApJ. Numerical
results unchanged, but interpretation sharpened: Likelihood approximation
issues at l=13-30 far more important than potential foreground issues at l <=
12. Gibbs products (spectrum and sky samples, and "easy-to-use" likelihood
module) available from http://www.astro.uio.no/~hke/ under "Research
Estimating the large-scale angular power spectrum in the presence of systematics: a case study of Sloan Digital Sky Survey quasars
The angular power spectrum is a powerful statistic for analysing cosmological
signals imprinted in the clustering of matter. However, current galaxy and
quasar surveys cover limited portions of the sky, and are contaminated by
systematics that can mimic cosmological signatures and jeopardise the
interpretation of the measured power spectra. We provide a framework for
obtaining unbiased estimates of the angular power spectra of large-scale
structure surveys at the largest scales using quadratic estimators. The method
is tested by analysing the 600 CMASS mock catalogues constructed by Manera et
al. (2013) for the Baryon Oscillation Spectroscopic Survey (BOSS). We then
consider the Richards et al. (2009) catalogue of photometric quasars from the
Sixth Data Release (DR6) of the Sloan Digital Sky Survey (SDSS), which is known
to include significant stellar contamination and systematic uncertainties.
Focusing on the sample of ultraviolet-excess (UVX) sources, we show that the
excess clustering power present on the largest-scales can be largely mitigated
by making use of improved sky masks and projecting out the modes corresponding
to the principal systematics. In particular, we find that the sample of objects
with photometric redshift exhibits no evidence of
contamination when using our most conservative mask and mode projection. This
indicates that any residual systematics are well within the statistical
uncertainties. We conclude that, using our approach, this sample can be used
for cosmological studies.Comment: 18 pages, 18 figures. Version accepted by MNRA
Hierarchical Cosmic Shear Power Spectrum Inference
We develop a Bayesian hierarchical modelling approach for cosmic shear power
spectrum inference, jointly sampling from the posterior distribution of the
cosmic shear field and its (tomographic) power spectra. Inference of the shear
power spectrum is a powerful intermediate product for a cosmic shear analysis,
since it requires very few model assumptions and can be used to perform
inference on a wide range of cosmological models \emph{a posteriori} without
loss of information. We show that joint posterior for the shear map and power
spectrum can be sampled effectively by Gibbs sampling, iteratively drawing
samples from the map and power spectrum, each conditional on the other. This
approach neatly circumvents difficulties associated with complicated survey
geometry and masks that plague frequentist power spectrum estimators, since the
power spectrum inference provides prior information about the field in masked
regions at every sampling step. We demonstrate this approach for inference of
tomographic shear -mode, -mode and -cross power spectra from a
simulated galaxy shear catalogue with a number of important features; galaxies
distributed on the sky and in redshift with photometric redshift uncertainties,
realistic random ellipticity noise for every galaxy and a complicated survey
mask. The obtained posterior distributions for the tomographic power spectrum
coefficients recover the underlying simulated power spectra for both - and
-modes.Comment: 16 pages, 8 figures, accepted by MNRA
A Measurement of the Angular Power Spectrum of the CMB Temperature Anisotropy from the 2003 Flight of Boomerang
We report on observations of the Cosmic Microwave Background (CMB) obtained
during the January 2003 flight of Boomerang . These results are derived from
195 hours of observation with four 145 GHz Polarization Sensitive Bolometer
(PSB) pairs, identical in design to the four 143 GHz Planck HFI polarized
pixels. The data include 75 hours of observations distributed over 1.84% of the
sky with an additional 120 hours concentrated on the central portion of the
field, itself representing 0.22% of the full sky. From these data we derive an
estimate of the angular power spectrum of temperature fluctuations of the CMB
in 24 bands over the multipole range (50 < l < 1500). A series of features,
consistent with those expected from acoustic oscillations in the primordial
photon-baryon fluid, are clearly evident in the power spectrum, as is the
exponential damping of power on scales smaller than the photon mean free path
at the epoch of last scattering (l > 900). As a consistency check, the
collaboration has performed two fully independent analyses of the time ordered
data, which are found to be in excellent agreement.Comment: 11 pages, 7 figures, 3 tables. High resolution figures and data are
available at http://cmb.phys.cwru.edu/boomerang/ and
http://oberon.roma1.infn.it/boomerang/b2
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