Performance analysis of the Least-Squares estimator in Astrometry

We characterize the performance of the widely-used least-squares estimator in astrometry in terms of a comparison with the Cramer-Rao lower variance bound. In this inference context the performance of the least-squares estimator does not offer a closed-form expression, but a new result is presented (Theorem 1) where both the bias and the mean-square-error of the least-squares estimator are bounded and approximated analytically, in the latter case in terms of a nominal value and an interval around it. From the predicted nominal value we analyze how efficient is the least-squares estimator in comparison with the minimum variance Cramer-Rao bound. Based on our results, we show that, for the high signal-to-noise ratio regime, the performance of the least-squares estimator is significantly poorer than the Cramer-Rao bound, and we characterize this gap analytically. On the positive side, we show that for the challenging low signal-to-noise regime (attributed to either a weak astronomical signal or a noise-dominated condition) the least-squares estimator is near optimal, as its performance asymptotically approaches the Cramer-Rao bound. However, we also demonstrate that, in general, there is no unbiased estimator for the astrometric position that can precisely reach the Cramer-Rao bound. We validate our theoretical analysis through simulated digital-detector observations under typical observing conditions. We show that the nominal value for the mean-square-error of the least-squares estimator (obtained from our theorem) can be used as a benchmark indicator of the expected statistical performance of the least-squares method under a wide range of conditions. Our results are valid for an idealized linear (one-dimensional) array detector where intra-pixel response changes are neglected, and where flat-fielding is achieved with very high accuracy.Comment: 35 pages, 8 figures. Accepted for publication by PAS

Finite-temperature properties of hard-core bosons confined on one-dimensional optical lattices

We present an exact study of the finite-temperature properties of hard-core bosons (HCB's) confined on one-dimensional optical lattices. Our solution of the HCB problem is based on the Jordan-Wigner transformation and properties of Slater determinants. We analyze the effects of the temperature on the behavior of the one-particle correlations, the momentum distribution function, and the lowest natural orbitals. In addition, we compare results obtained using the grand-canonical and canonical descriptions for systems like the ones recently achieved experimentally. We show that even for such small systems, as small as 10 HCB's in 50 lattice sites, there are only minor differences between the energies and momentum distributions obtained within both ensembles.Comment: RevTex file, 12 pages, 16 figures, published versio

Fluctuations of a surface relaxation model in interacting scale free networks

Isolated complex networks have been studied deeply in the last decades due to the fact that many real systems can be modeled using these types of structures. However, it is well known that the behavior of a system not only depends on itself, but usually also depends on the dynamics of other structures. For this reason, interacting complex networks and the processes developed on them have been the focus of study of many researches in the last years. One of the most studied subjects in this type of structures is the Synchronization problem, which is important in a wide variety of processes in real systems. In this manuscript we study the synchronization of two interacting scale-free networks, in which each node has $ke$ dependency links with different nodes in the other network. We map the synchronization problem with an interface growth, by studying the fluctuations in the steady state of a scalar field defined in both networks. We find that as $ke$ slightly increases from $ke=0$, there is a really significant decreasing in the fluctuations of the system. However, this considerable improvement takes place mainly for small values of $ke$, when the interaction between networks becomes stronger there is only a slight change in the fluctuations. We characterize how the dispersion of the scalar field depends on the internal degree, and we show that a combination between the decreasing of this dispersion and the integer nature of our growth model are the responsible for the behavior of the fluctuations with $ke$.Comment: 11 pages, 4 figures and 1 tabl

Analysis of the Bayesian Cramer-Rao lower bound in astrometry: Studying the impact of prior information in the location of an object

Context. The best precision that can be achieved to estimate the location of a stellar-like object is a topic of permanent interest in the astrometric community. Aims. We analyse bounds for the best position estimation of a stellar-like object on a CCD detector array in a Bayesian setting where the position is unknown, but where we have access to a prior distribution. In contrast to a parametric setting where we estimate a parameter from observations, the Bayesian approach estimates a random object (i.e., the position is a random variable) from observations that are statistically dependent on the position. Methods. We characterize the Bayesian Cramer-Rao (CR) that bounds the minimum mean square error (MMSE) of the best estimator of the position of a point source on a linear CCD-like detector, as a function of the properties of detector, the source, and the background. Results. We quantify and analyse the increase in astrometric performance from the use of a prior distribution of the object position, which is not available in the classical parametric setting. This gain is shown to be significant for various observational regimes, in particular in the case of faint objects or when the observations are taken under poor conditions. Furthermore, we present numerical evidence that the MMSE estimator of this problem tightly achieves the Bayesian CR bound. This is a remarkable result, demonstrating that all the performance gains presented in our analysis can be achieved with the MMSE estimator. Conclusions The Bayesian CR bound can be used as a benchmark indicator of the expected maximum positional precision of a set of astrometric measurements in which prior information can be incorporated. This bound can be achieved through the conditional mean estimator, in contrast to the parametric case where no unbiased estimator precisely reaches the CR bound.Comment: 17 pages, 12 figures. Accepted for publication on Astronomy & Astrophysic

Force distribution in a randomly perturbed lattice of identical particles with 1/r21/r^2 pair interaction

We study the statistics of the force felt by a particle in the class of spatially correlated distribution of identical point-like particles, interacting via a $1/r^2$ pair force (i.e. gravitational or Coulomb), and obtained by randomly perturbing an infinite perfect lattice. In the first part we specify the conditions under which the force on a particle is a well defined stochastic quantity. We then study the small displacements approximation, giving both the limitations of its validity, and, when it is valid, an expression for the force variance. In the second part of the paper we extend to this class of particle distributions the method introduced by Chandrasekhar to study the force probability density function in the homogeneous Poisson particle distribution. In this way we can derive an approximate expression for the probability distribution of the force over the full range of perturbations of the lattice, i.e., from very small (compared to the lattice spacing) to very large where the Poisson limit is recovered. We show in particular the qualitative change in the large-force tail of the force distribution between these two limits. Excellent accuracy of our analytic results is found on detailed comparison with results from numerical simulations. These results provide basic statistical information about the fluctuations of the interactions (i) of the masses in self-gravitating systems like those encountered in the context of cosmological N-body simulations, and (ii) of the charges in the ordered phase of the One Component Plasma.Comment: 23 pages, 10 figure

Orbits for eighteen visual binaries and two double-line spectroscopic binaries observed with HRCAM on the CTIO SOAR 4m telescope, using a new Bayesian orbit code based on Markov Chain Monte Carlo

We present orbital elements and mass sums for eighteen visual binary stars of spectral types B to K (five of which are new orbits) with periods ranging from 20 to more than 500 yr. For two double-line spectroscopic binaries with no previous orbits, the individual component masses, using combined astrometric and radial velocity data, have a formal uncertainty of ~0.1 MSun. Adopting published photometry, and trigonometric parallaxes, plus our own measurements, we place these objects on an H-R diagram, and discuss their evolutionary status. These objects are part of a survey to characterize the binary population of stars in the Southern Hemisphere, using the SOAR 4m telescope+HRCAM at CTIO. Orbital elements are computed using a newly developed Markov Chain Monte Carlo algorithm that delivers maximum likelihood estimates of the parameters, as well as posterior probability density functions that allow us to evaluate the uncertainty of our derived parameters in a robust way. For spectroscopic binaries, using our approach, it is possible to derive a self-consistent parallax for the system from the combined astrometric plus radial velocity data ("orbital parallax"), which compares well with the trigonometric parallaxes. We also present a mathematical formalism that allows a dimensionality reduction of the feature space from seven to three search parameters (or from ten to seven dimensions - including parallax - in the case of spectroscopic binaries with astrometric data), which makes it possible to explore a smaller number of parameters in each case, improving the computational efficiency of our Markov Chain Monte Carlo code.Comment: 32 pages, 9 figures, 6 tables. Detailed Appendix with methodology. Accepted by The Astronomical Journa