5,765 research outputs found
On the Coherence of WMAP and Planck Temperature Maps
The recent data release of ESA's Planck mission together with earlier WMAP
releases provide the first opportunity to compare high resolution full sky
Cosmic Microwave Background temperature anisotropy maps. To quantify the
coherence of these maps beyond the power spectrum we introduce Generalized
Phases, unit vectors in the (2l+1) dimensional representation spaces. For a
Gaussian distribution, Generalized Phases are random and if there is
non-Gaussianity, they represent most of the non-Gaussian information. The
alignment of these unit vectors from two maps can be characterized by their
angle, 0 deg expected for full coherence, and 90 deg for random vectors. We
analyze maps from both missions with the same mask and Nside=512 resolution,
and compare both power spectra and Generalized Phases. We find excellent
agreement of the Generalize Phases of Planck Smica map with that of the WMAP
Q,V,W maps, rejecting the null hypothesis of no correlations at 5 sigma for l's
l<700, l<900 and l<1100, respectively, except perhaps for l<10. Using
foreground reduced maps for WMAP increases the phase coherence. The observed
coherence angles can be explained with a simple assumption of Gaussianity and a
WMAP noise model neglecting Planck noise, except for low-intermediate l's there
is a slight, but significant off-set, depending on WMAP band. On the same
scales WMAP power spectrum is about 2.6% higher at a very high significance,
while at higher l's there appears to be no significant bias. Using our
theoretical tools, we predict the phase alignment of Planck with a hypothetical
perfect noiseless CMB experiment, finding decoherence at l > 2900; below this
value Planck can be used most efficiently to constrain non-Gaussianity.Comment: 8 pages, 8 figures, accepted for publication in MNRAS; minor
modifications and 2 new figures adde
Coherent frequentism
By representing the range of fair betting odds according to a pair of
confidence set estimators, dual probability measures on parameter space called
frequentist posteriors secure the coherence of subjective inference without any
prior distribution. The closure of the set of expected losses corresponding to
the dual frequentist posteriors constrains decisions without arbitrarily
forcing optimization under all circumstances. This decision theory reduces to
those that maximize expected utility when the pair of frequentist posteriors is
induced by an exact or approximate confidence set estimator or when an
automatic reduction rule is applied to the pair. In such cases, the resulting
frequentist posterior is coherent in the sense that, as a probability
distribution of the parameter of interest, it satisfies the axioms of the
decision-theoretic and logic-theoretic systems typically cited in support of
the Bayesian posterior. Unlike the p-value, the confidence level of an interval
hypothesis derived from such a measure is suitable as an estimator of the
indicator of hypothesis truth since it converges in sample-space probability to
1 if the hypothesis is true or to 0 otherwise under general conditions.Comment: The confidence-measure theory of inference and decision is explicitly
extended to vector parameters of interest. The derivation of upper and lower
confidence levels from valid and nonconservative set estimators is formalize
Sampling Properties of the Spectrum and Coherency of Sequences of Action Potentials
The spectrum and coherency are useful quantities for characterizing the
temporal correlations and functional relations within and between point
processes. This paper begins with a review of these quantities, their
interpretation and how they may be estimated. A discussion of how to assess the
statistical significance of features in these measures is included. In
addition, new work is presented which builds on the framework established in
the review section. This work investigates how the estimates and their error
bars are modified by finite sample sizes. Finite sample corrections are derived
based on a doubly stochastic inhomogeneous Poisson process model in which the
rate functions are drawn from a low variance Gaussian process. It is found
that, in contrast to continuous processes, the variance of the estimators
cannot be reduced by smoothing beyond a scale which is set by the number of
point events in the interval. Alternatively, the degrees of freedom of the
estimators can be thought of as bounded from above by the expected number of
point events in the interval. Further new work describing and illustrating a
method for detecting the presence of a line in a point process spectrum is also
presented, corresponding to the detection of a periodic modulation of the
underlying rate. This work demonstrates that a known statistical test,
applicable to continuous processes, applies, with little modification, to point
process spectra, and is of utility in studying a point process driven by a
continuous stimulus. While the material discussed is of general applicability
to point processes attention will be confined to sequences of neuronal action
potentials (spike trains) which were the motivation for this work.Comment: 33 pages, 9 figure
Sparsity with sign-coherent groups of variables via the cooperative-Lasso
We consider the problems of estimation and selection of parameters endowed
with a known group structure, when the groups are assumed to be sign-coherent,
that is, gathering either nonnegative, nonpositive or null parameters. To
tackle this problem, we propose the cooperative-Lasso penalty. We derive the
optimality conditions defining the cooperative-Lasso estimate for generalized
linear models, and propose an efficient active set algorithm suited to
high-dimensional problems. We study the asymptotic consistency of the estimator
in the linear regression setup and derive its irrepresentable conditions, which
are milder than the ones of the group-Lasso regarding the matching of groups
with the sparsity pattern of the true parameters. We also address the problem
of model selection in linear regression by deriving an approximation of the
degrees of freedom of the cooperative-Lasso estimator. Simulations comparing
the proposed estimator to the group and sparse group-Lasso comply with our
theoretical results, showing consistent improvements in support recovery for
sign-coherent groups. We finally propose two examples illustrating the wide
applicability of the cooperative-Lasso: first to the processing of ordinal
variables, where the penalty acts as a monotonicity prior; second to the
processing of genomic data, where the set of differentially expressed probes is
enriched by incorporating all the probes of the microarray that are related to
the corresponding genes.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS520 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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