Estimating parameters of fluctuations in the cosmic microwave background

We address questions that arise in statistical analyses of recently detected fluctuations in the Cosmic Microwave Background (CMB). Estimators of the quadrupole amplitude, Q, and spectral index, n, of the CMB angular fluctuation power spectrum are considered. Families of unbiased estimators of Q2 and existence conditions for minimum variance estimators of n are given. We find that the common practice of excluding the quadrupole is not recommended if one is interested in unbiased estimators. We explain previousiy reported correlations of the estimators and show how they depend on the multiple used to normalize the spectrum. We show that a finite beam resolution does not justify the use of truncated least-squares to estimate harmonic coefficients of CMB data

Optimal low-rank approximations of Bayesian linear inverse problems

In the Bayesian approach to inverse problems, data are often informative, relative to the prior, only on a low-dimensional subspace of the parameter space. Significant computational savings can be achieved by using this subspace to characterize and approximate the posterior distribution of the parameters. We first investigate approximation of the posterior covariance matrix as a low-rank update of the prior covariance matrix. We prove optimality of a particular update, based on the leading eigendirections of the matrix pencil defined by the Hessian of the negative log-likelihood and the prior precision, for a broad class of loss functions. This class includes the F\"{o}rstner metric for symmetric positive definite matrices, as well as the Kullback-Leibler divergence and the Hellinger distance between the associated distributions. We also propose two fast approximations of the posterior mean and prove their optimality with respect to a weighted Bayes risk under squared-error loss. These approximations are deployed in an offline-online manner, where a more costly but data-independent offline calculation is followed by fast online evaluations. As a result, these approximations are particularly useful when repeated posterior mean evaluations are required for multiple data sets. We demonstrate our theoretical results with several numerical examples, including high-dimensional X-ray tomography and an inverse heat conduction problem. In both of these examples, the intrinsic low-dimensional structure of the inference problem can be exploited while producing results that are essentially indistinguishable from solutions computed in the full space

Estimating a potential without the agony of the partition function

Estimating a Gibbs density function given a sample is an important problem in computational statistics and statistical learning. Although the well established maximum likelihood method is commonly used, it requires the computation of the partition function (i.e., the normalization of the density). This function can be easily calculated for simple low-dimensional problems but its computation is difficult or even intractable for general densities and high-dimensional problems. In this paper we propose an alternative approach based on Maximum A-Posteriori (MAP) estimators, we name Maximum Recovery MAP (MR-MAP), to derive estimators that do not require the computation of the partition function, and reformulate the problem as an optimization problem. We further propose a least-action type potential that allows us to quickly solve the optimization problem as a feed-forward hyperbolic neural network. We demonstrate the effectiveness of our methods on some standard data sets

Estudio de la fracción insaponificable obtenida del aceite extraído de la semilla de la luffa cylindrica

Se obtuvo y caracterizo la materia insaponificable del aceite de las semillas de la Luffa Cylindrica procedentes de Risaralda suministrada por la empresa ESTROCOL. La materia prima se sometio a un proceso de secado para la posterior extraccion del aceite por el metodo soxhlet, empleando Hexano como solvente. La saponificacion del aceite obtenido y la cuantificacion de la materia insaponificable se determino segun Norma Tecnica Colombiana NTC 235-2. El fraccionamiento se realizo empleando separacion en fase solida y dos sistemas de solventes: Hexano-Acetato de Etilo y Hexano-Diclorometano; donde las fracciones obtenidas fueron analizadas por cromatografia en capa delgada para observar la separacion de los compuestos esterolicos para su posterior analisis. La caracterizacion de la materia insaponificable se hizo por cromatografia de gases acoplado a masas, dando como resultado la posible presencia de fitoesteroles (Lanosterol (10.96%) y Ergosta-7,22-dien-3-ol,(3.beta.,5.alpha) (15.06%)) como componentes mayoritarios. Adicionalmente se determino el contenido de vitamina E y ¿À-Caroteno en el aceite de las semillas de la Luffa Cylindrica, empleando cromatografia liquida de alta eficiencia como metodo analitico; obteniendose un valor de 621 ppm para Vitamina E y de 58 ppm para ¿À-Caroteno. Los resultados obtenidos fueron comparados con aceites de origen vegetal y animal

Radioactive impact in sediments from an estuarine system affected by industrial wastes releases

A big fertilizer industrial complex and a vast extension of phosphogypsum piles (12 km2), sited in the estuary formed by the Odiel and Tinto river mouths (southwest of Spain), are producing an unambiguous radioactive impact in their surrounding aquatic environment through radionuclides from the U-series. The levels and distribution of radionuclides in sediments from this estuarine system have been determined. The analyses of radionuclide concentrations and activity ratios have provided us with an interesting information to evaluate the extension, degree and routes of the radioactive impact, as well as for the knowledge of the different pathways followed for the radioactive contamination to disturb this natural system. The obtained results indicate that the main pathway of radioactive contamination of the estuary is through the dissolution in its waters of the radionuclides released by the industrial activities and their later fixation on the particulate materials. Tidal activity also plays an important role in the transport and homogenization along the estuary of the radioactivity released from the fertilizer plants. D 2002 Elsevier Science Ltd. All rights reserved.Junta de Andalucia (España) project 1FD97-0900-C02-02 (AMB

Minimum local distance density estimation

We present a local density estimator based on first-order statistics. To estimate the density at a point, x, the original sample is divided into subsets and the average minimum sample distance to x over all such subsets is used to define the density estimate at x. The tuning parameter is thus the number of subsets instead of the typical bandwidth of kernel or histogram-based density estimators. The proposed method is similar to nearest-neighbor density estimators but it provides smoother estimates. We derive the asymptotic distribution of this minimum sample distance statistic to study globally optimal values for the number and size of the subsets. Simulations are used to illustrate and compare the convergence properties of the estimator. The results show that the method provides good estimates of a wide variety of densities without changes of the tuning parameter, and that it offers competitive convergence performance.United States. Department of Energy. Applied Mathematical Sciences Program (Award DE-FG02-08ER2585)United States. Department of Energy. Applied Mathematical Sciences Program (Award de-sc0009297