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Register a new userAstroinformatics: Statistically Optimal Approximations of Near-Extremal Parts with Application to Variable Stars
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The software MAVKA is described, which was elaborated for statistically optimal determination of the characteristics of the extrema of 1000+ variable stars of different types, mainly eclipsing and pulsating. The approximations are phenomenological, but not physical. As often, the discovery of a new variable star is made on time series of a single-filter (single-channel) data, and there is no possibility to determine parameters needed for physical modelling (e.g. temperature, radial velocities, mass ratio of binaries). Besides classical polynomial approximation AP (we limited the degree of the polynomial from 2 to 9), there are realized symmetrical approximations (symmetrical polynomials SP, wall-supported horizontal line WSL and parabola WSP, restricted polynomials of non-integer order based on approximations of the functions proposed by Andronov (2012) and Mikulasek (2015) and generally asymmetric functions (asymptotic parabola AP, parabolic spline PS, generalized hyperbolic secant function SECH and log-normal-like BSK). This software is a successor of the Observation Obscurer with some features for the variable star research, including a block for running parabola RP scalegram and approximation. Whereas the RP is oriented on approximation of the complete data set. MAVKA is pointed to parts of the light curve close to extrema (including total eclipses and transits of stars and exoplanets). The functions for wider intervals, covering the eclipse totally, were discussed in 2017Ap.....60...57A . Global and local approximations are reviewed in 2020kdbd.book..191A . The software is available at http://uavso.org.ua/mavka and https://katerynaandrych.wixsite.com/mavka. We have analyzed the data from own observations, as well as from monitoring obtained at ground-based and space (currently, mainly, TESS) observatories. It may be used for signals of any nature.
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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.
The expert system for time series analysis of irregularly spaced signals is reviewed. It consists of a number of complementary algorithms and programs, which may be effective for different types of variability. Obviously, for a pure sine signal, all the methods should produce the same results. However, for irregularly spaced signals with a complicated structure, e.g. a sum of different components, different methods may produce significantly different results. The basic approach is based on classical method of the least squares (1994OAP.....7...49A). However, contrary to common step-by-step methods of removal important components (e.g. mean, trend (detrending), sine wave (prewhitening), where covariations between different components are ignored, i.e. erroneously assumed to be zero, we use complete mathematical models. Some of the methods are illustrated on the observations of the semi-regular pulsating variable RY UMa. The star shows a drastic cyclic change of semi-amplitude of pulsations between 0.01 to 0.37mag, which is interpreted as a bias between the waves with close periods and a beat period of 4000d (11yr). The dominating period has changed from 307.35(8)d before 1993 to 285.26(6)d after 1993. The initial epoch of the maximum brightness for the recent interval is 2454008.8(5). It is suggested that the apparent period switch is due to variability of amplitudes of these two waves and an occasional swap of the dominating wave.
The Kepler space telescope has revolutionised our knowledge about exoplanets and stars and is continuing to do so in the K2 mission. The exquisite photometric precision, together with the long, uninterrupted observations opened up a new way to investigate the structure and evolution of stars. Asteroseismology, the study of stellar oscillations, allowed us to investigate solar-like stars and to peer into the insides of red giants and massive stars. But many discoveries have been made about classical variable stars too, ranging from pulsators like Cepheids and RR Lyraes to eclipsing binary stars and cataclysmic variables, and even supernovae. In this review, which is far from an exhaustive summary of all results obtained with Kepler, we collected some of the most interesting discoveries, and ponder on the role for amateur observers in this golden era of stellar astrophysics.
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 $3$.
We present a comparison of the Gaia DR1 samples of pulsating variable stars - Cepheids and RR Lyrae type - with the OGLE Collection of Variable Stars aiming at the characterization of the Gaia mission performance in the stellar variability domain. Out of 575 Cepheids and 2322 RR Lyrae candidates from the Gaia DR1 samples located in the OGLE footprint in the sky, 559 Cepheids and 2302 RR Lyrae stars are genuine pulsators of these types. The number of misclassified stars is low indicating reliable performance of the Gaia data pipeline. The completeness of the Gaia DR1 samples of Cepheids and RR Lyrae stars is at the level of 60-75% as compared to the OGLE Collection dataset. This level of completeness is moderate and may limit the applicability of the Gaia data in many projects.
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