A generalized Fellner-Schall method for smoothing parameter estimation with application to Tweedie location, scale and shape models

We consider the estimation of smoothing parameters and variance components in models with a regular log likelihood subject to quadratic penalization of the model coefficients, via a generalization of the method of Fellner (1986) and Schall (1991). In particular: (i) we generalize the original method to the case of penalties that are linear in several smoothing parameters, thereby covering the important cases of tensor product and adaptive smoothers; (ii) we show why the method's steps increase the restricted marginal likelihood of the model, that it tends to converge faster than the EM algorithm, or obvious accelerations of this, and investigate its relation to Newton optimization; (iii) we generalize the method to any Fisher regular likelihood. The method represents a considerable simplification over existing methods of estimating smoothing parameters in the context of regular likelihoods, without sacrificing generality: for example, it is only necessary to compute with the same first and second derivatives of the log-likelihood required for coefficient estimation, and not with the third or fourth order derivatives required by alternative approaches. Examples are provided which would have been impossible or impractical with pre-existing Fellner-Schall methods, along with an example of a Tweedie location, scale and shape model which would be a challenge for alternative methods

X-ray Lighthouses of the High-Redshift Universe. II. Further Snapshot Observations of the Most Luminous z>4 Quasars with Chandra

We report on Chandra observations of a sample of 11 optically luminous (Mb<-28.5) quasars at z=3.96-4.55 selected from the Palomar Digital Sky Survey and the Automatic Plate Measuring Facility Survey. These are among the most luminous z>4 quasars known and hence represent ideal witnesses of the end of the "dark age ''. Nine quasars are detected by Chandra, with ~2-57 counts in the observed 0.5-8 keV band. These detections increase the number of X-ray detected AGN at z>4 to ~90; overall, Chandra has detected ~85% of the high-redshift quasars observed with snapshot (few kilosecond) observations. PSS 1506+5220, one of the two X-ray undetected quasars, displays a number of notable features in its rest-frame ultraviolet spectrum, the most prominent being broad, deep SiIV and CIV absorption lines. The average optical-to-X-ray spectral index for the present sample (=-1.88+/-0.05) is steeper than that typically found for z>4 quasars but consistent with the expected value from the known dependence of this spectral index on quasar luminosity. We present joint X-ray spectral fitting for a sample of 48 radio-quiet quasars in the redshift range 3.99-6.28 for which Chandra observations are available. The X-ray spectrum (~870 counts) is well parameterized by a power law with Gamma=1.93+0.10/-0.09 in the rest-frame ~2-40 keV band, and a tight upper limit of N_H~5x10^21 cm^-2 is obtained on any average intrinsic X-ray absorption. There is no indication of any significant evolution in the X-ray properties of quasars between redshifts zero and six, suggesting that the physical processes of accretion onto massive black holes have not changed over the bulk of cosmic time.Comment: 15 pages, 7 figures, accepted for publication in A

Designing a Belief Function-Based Accessibility Indicator to Improve Web Browsing for Disabled People

The purpose of this study is to provide an accessibility measure of web-pages, in order to draw disabled users to the pages that have been designed to be ac-cessible to them. Our approach is based on the theory of belief functions, using data which are supplied by reports produced by automatic web content assessors that test the validity of criteria defined by the WCAG 2.0 guidelines proposed by the World Wide Web Consortium (W3C) organization. These tools detect errors with gradual degrees of certainty and their results do not always converge. For these reasons, to fuse information coming from the reports, we choose to use an information fusion framework which can take into account the uncertainty and imprecision of infor-mation as well as divergences between sources. Our accessibility indicator covers four categories of deficiencies. To validate the theoretical approach in this context, we propose an evaluation completed on a corpus of 100 most visited French news websites, and 2 evaluation tools. The results obtained illustrate the interest of our accessibility indicator

Application of Monte Carlo Algorithms to the Bayesian Analysis of the Cosmic Microwave Background

Power spectrum estimation and evaluation of associated errors in the presence of incomplete sky coverage; non-homogeneous, correlated instrumental noise; and foreground emission is a problem of central importance for the extraction of cosmological information from the cosmic microwave background. We develop a Monte Carlo approach for the maximum likelihood estimation of the power spectrum. The method is based on an identity for the Bayesian posterior as a marginalization over unknowns. Maximization of the posterior involves the computation of expectation values as a sample average from maps of the cosmic microwave background and foregrounds given some current estimate of the power spectrum or cosmological model, and some assumed statistical characterization of the foregrounds. Maps of the CMB are sampled by a linear transform of a Gaussian white noise process, implemented numerically with conjugate gradient descent. For time series data with N_{t} samples, and N pixels on the sphere, the method has a computational expense $KO[N^{2} +- N_{t} +AFw-log N_{t}], where K is a prefactor determined by the convergence rate of conjugate gradient descent. Preconditioners for conjugate gradient descent are given for scans close to great circle paths, and the method allows partial sky coverage for these cases by numerically marginalizing over the unobserved, or removed, region.Comment: submitted to Ap

Toward the estimation of background fluctuations under newly-observed signals in particle physics

When the number of events associated with a signal process is estimated in particle physics, it is common practice to extrapolate background distributions from control regions to a predefined signal window. This allows accurate estimation of the expected, or average, number of background events under the signal. However, in general, the actual number of background events can deviate from the average due to fluctuations in the data. Such a difference can be sizable when compared to the number of signal events in the early stages of data analysis following the observation of a new particle, as well as in the analysis of rare decay channels. We report on the development of a data-driven technique that aims to estimate the actual, as opposed to the expected, number of background events in a predefined signal window. We discuss results on toy Monte Carlo data and provide a preliminary estimate of systematic uncertainty

Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting

In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix $X$ formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose $X$ into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance. The third problem is a basic geometric question: given points $v_1,v_2,...,v_n\in \R^k$ (where $n > k$) determine whether there is a centered ellipsoid passing \emph{exactly} through all of the points. We show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a subspace $U$ that ensures any positive semidefinite matrix $L$ with column space $U$ can be recovered from $D+L$ for any diagonal matrix $D$ using a convex optimization-based heuristic known as minimum trace factor analysis. This result leads to a new understanding of the structure of rank-deficient correlation matrices and a simple condition on a set of points that ensures there is a centered ellipsoid passing through them.Comment: 20 page

A Meaner King uses Biased Bases

The mean king problem is a quantum mechanical retrodiction problem, in which Alice has to name the outcome of an ideal measurement on a d-dimensional quantum system, made in one of (d+1) orthonormal bases, unknown to Alice at the time of the measurement. Alice has to make this retrodiction on the basis of the classical outcomes of a suitable control measurement including an entangled copy. We show that the existence of a strategy for Alice is equivalent to the existence of an overall joint probability distribution for (d+1) random variables, whose marginal pair distributions are fixed as the transition probability matrices of the given bases. In particular, for d=2 the problem is decided by John Bell's classic inequality for three dichotomic variables. For mutually unbiased bases in any dimension Alice has a strategy, but for randomly chosen bases the probability for that goes rapidly to zero with increasing d.Comment: 5 pages, 1 figur

Principal Component Analysis with Noisy and/or Missing Data

We present a method for performing Principal Component Analysis (PCA) on noisy datasets with missing values. Estimates of the measurement error are used to weight the input data such that compared to classic PCA, the resulting eigenvectors are more sensitive to the true underlying signal variations rather than being pulled by heteroskedastic measurement noise. Missing data is simply the limiting case of weight=0. The underlying algorithm is a noise weighted Expectation Maximization (EM) PCA, which has additional benefits of implementation speed and flexibility for smoothing eigenvectors to reduce the noise contribution. We present applications of this method on simulated data and QSO spectra from the Sloan Digital Sky Survey.Comment: Accepted for publication in PASP; v2 with minor updates, mostly to bibliograph

The Ages and Abundances of the M87 Globular Clusters

A subset of 150 globular clusters in M87 has been selected on the basis of S/N ratio for abundance and age determinations from the sample of Paper I. Indices measuring the strength of the strongest spectral features were determined for the M87 GCs and from new data for twelve galactic GCs. Combining the new and existing data for the galactic GCs and comparing the $(U-R)$ colors and the line indices gives qualitative indications for the ages and abundances of the GCs. Quantitative results are obtained by applying the Worthey (1994) models for the integrated light of stellar systems of a single age, calibrated by observations of galactic GCs, to deduce abundances and ages for the objects in our sample. We find that the M87 GCs span a wide range in metallicity, from very metal poor to somewhat above solar metallicity. The mean [Fe/H] of -0.95 dex is higher than that of the galactic GC system, and there is a metal rich tail that reaches to higher [Fe/H] than one finds among the galactic GCs. The mean metallicity of the M87 GC system is about a factor of four lower than that of the M87 stellar halo at a fixed projected radius $R$. The metallicity inferred from the X-ray studies is similar to that of the M87 stellar halo, not to that of GCs. We infer the relative abundances of Na, Mg, and Fe in the M87 GCs from the strength of their spectral features. The behavior of these elements between the metal rich and metal poor M87 GCs is similar to that shown by the galactic GCs and by halo stars in the Galaxy. The pattern of chemical evolution in these disparate old stellar systems is indistinguishable. We obtain a median age for the M87 GC system of 13 Gyr, similar to that of the galactic GCs, with a small dispersion about this value.Comment: 56 pages with included postscript figures; added derived M87 GC metallicities to Table 2, a statistical analysis of possible bimodality, an appendix on the metallicity calibration of U-R and the Washington system, and other smaller changes. Accepted for publication in ApJ. (See paper for complete version of the Abstract.

Toward particle-level filtering of individual collision events at the Large Hadron Collider and beyond

Low-energy strong interactions are a major source of background at hadron colliders, and methods of subtracting the associated energy flow are well established in the field. Traditional approaches treat the contamination as diffuse, and estimate background energy levels either by averaging over large data sets or by restricting to given kinematic regions inside individual collision events. On the other hand, more recent techniques take into account the discrete nature of background, most notably by exploiting the presence of substructure inside hard jets, i.e. inside collections of particles originating from scattered hard quarks and gluons. However, none of the existing methods subtract background at the level of individual particles inside events. We illustrate the use of an algorithm that will allow particle-by-particle background discrimination at the Large Hadron Collider, and we envisage this as the basis for a novel event filtering procedure upstream of the official reconstruction chains. Our hope is that this new technique will improve physics analysis when used in combination with state-of-the-art algorithms in high-luminosity hadron collider environments