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Register a new userStatistically optimal fitting of astrometric signals
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A general purpose fitting model for one-dimensional astrometric signals is developed, building on a maximum likelihood framework, and its performance is evaluated by simulation over a set of realistic image instances. The fit quality is analysed as a function of the number of terms used for signal expansion, and of astrometric error, rather than RMS discrepancy with respect to the input signal. The tuning of the function basis to the statistical characteristics of the signal ensemble is discussed. The fit sensitivity to a priori knowledge of the source spectra is addressed. Some implications of the current results on calibration and data reduction aspects are discussed, in particular with respect to Gaia.
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A tool for representation of the one-dimensional astrometric signal of Gaia is described and investigated in terms of fit discrepancy and astrometric performance with respect to number of parameters required. The proposed basis function is based on the aberration free response of the ideal telescope and its derivatives, weighted by the source spectral distribution. The influence of relative position of the detector pixel array with respect to the optical image is analysed, as well as the variation induced by the source spectral emission. The number of parameters required for micro-arcsec level consistency of the reconstructed function with the detected signal is found to be 11. Some considerations are devoted to the issue of calibration of the instrument response representation, taking into account the relevant aspects of source spectrum and focal plane sampling. Additional investigations and other applications are also suggested.
We provide a basis to select the optimal algorithm according to the specific observational conditions in ground-based astrometry, and clarify the loss of precision in the case of not achieving optimum. The principle of the centering algorithms based on model fitting is analyzed by the method of maximum likelihood. The effective point spread function (ePSF) algorithm, which can construct an accurate model to fit the star image, and the most widely used Gaussian centering algorithm are chosen to investigate the effect of different factors on centering precision. A series of synthetic star images with different backgrounds, full width at half maximums (FWHMs) and profiles are processed by these algorithms. The profiles include the actual profiles extracted from observations and the theoretical profiles, the spatial variation of the PSF across the detector is also taken into account. Each algorithm is applied to the observations obtained from Yunnan observatory to verify the simulation results. The simulations show that ePSF fitting is obviously more precise than Gaussian fitting for a Gaussian profile star with high signal-to-noise ratio (SNR). When the center of star profile becomes sharp, or the SNR of the star decreases, the advantage of ePSF fitting will gradually decrease. The high precision of ePSF fitting is due to its appropriate weight in the weighted least squares fitting. However, a similar method using the same weight, the weighted Gaussian fitting, turned out to be poor under some conditions. The reduction results of practical observations show good agree with the simulations. For a frame of CCD image with enough stars to construct accurate ePSFs, ePSF fitting can approach the Cramer-Rao (CR) bound. Other centering algorithms may achieve the same precision under suitable conditions, but will show poor precision when not used properly.
We present the Exoplanet Simple Orbit Fitting Toolbox (ExoSOFT), a new, open-source suite to fit the orbital elements of planetary or stellar mass companions to any combination of radial velocity and astrometric data. To explore the parameter space of Keplerian models, ExoSOFT may be operated with its own multi-stage sampling approach, or interfaced with third-party tools such as emcee. In addition, ExoSOFT is packaged with a collection of post-processing tools to analyze and summarize the results. Although only a few systems have been observed with both the radial velocity and direct imaging techniques, this number will increase thanks to upcoming spacecraft and ground based surveys. Providing both forms of data enables simultaneous fitting that can help break degeneracies in the orbital elements that arise when only one data type is available. The dynamical mass estimates this approach can produce are important when investigating the formation mechanisms and subsequent evolution of substellar companions. ExoSOFT was verified through fitting to artificial data and was implemented using the Python and Cython programming languages; available for public download at https://github.com/kylemede/ExoSOFT under the GNU General Public License v3.
We introduce the program MAVKA for determination of characteristics of extrema using observations in the adjacent data intervals, with intended applications to variable stars, but it may be used for signals of arbitrary nature. We have used a dozen of basic functions, some of them use the interval near extremum without splitting the interval (algebraic polynomial in general form, Symmetrical algebraic polynomial using only even degrees of time (phase) deviation from the position of symmetry argument), others split the interval into 2 subintervals (a Taylor series of the New Algol Variable, the function of Prof. Z. Mikulav{s}ek), or even 3 parts (Asymptotic Parabola, Wall-Supported Parabola, Wall-Supported Line, Wall-Supported Asymptotic Parabola, Parabolic Spline of defect 1). The variety of methods allows to choose the best (statistically optimal) approximation for a given data sample. As the criterion, we use the accuracy of determination of the extremum. For all parameters, the statistical errors are determined. The methods are illustrated by applications to observations of pulsating and eclipsing variable stars, as well as to the exoplanet transits. They are used for the international campaigns Inter-Longitude Astronomy, Virtual Observatory and AstroInformatics. The program may be used for studies of individual objects, also using ground-based (NSVS, ASAS, WASP, CRTS et al.) and space (GAIA, KEPLER, HIPPARCOS/TYCHO, WISE et al.) surveys.
Hypothesis Selection is a fundamental distribution learning problem where given a comparator-class $Q={q_1,ldots, q_n}$ of distributions, and a sampling access to an unknown target distribution $p$, the goal is to output a distribution $q$ such that $mathsf{TV}(p,q)$ is close to $opt$, where $opt = min_i{mathsf{TV}(p,q_i)}$ and $mathsf{TV}(cdot, cdot)$ denotes the total-variation distance. Despite the fact that this problem has been studied since the 19th century, its complexity in terms of basic resources, such as number of samples and approximation guarantees, remains unsettled (this is discussed, e.g., in the charming book by Devroye and Lugosi `00). This is in stark contrast with other (younger) learning settings, such as PAC learning, for which these complexities are well understood. We derive an optimal $2$-approximation learning strategy for the Hypothesis Selection problem, outputting $q$ such that $mathsf{TV}(p,q) leq2 cdot opt + eps$, with a (nearly) optimal sample complexity of~$tilde O(log n/epsilon^2)$. This is the first algorithm that simultaneously achieves the best approximation factor and sample complexity: previously, Bousquet, Kane, and Moran (COLT `19) gave a learner achieving the optimal $2$-approximation, but with an exponentially worse sample complexity of $tilde O(sqrt{n}/epsilon^{2.5})$, and Yatracos~(Annals of Statistics `85) gave a learner with optimal sample complexity of $O(log n /epsilon^2)$ but with a sub-optimal approximation factor of $3$.
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