Improved Phenomenological Modelling of Multi-Component Signals: Applications to Variable Stars
Ivan L. Andronov
Department of “Mathematics, Physics and Astronomy”
Odessa National Maritime University, Ukraine
We review improvements of methods of time series analysis for signals with generally irregular distribution in an argument. Astronomical photometrical surveys are typically irregular in time, obtained in space (Hipparcos/Tycho. Kepler, GAIA, WISE etc.) or using ground-based telescopes with CCD (NSVS, ASAS, CRTS, SuperWASP etc.) or photographic (Harvard, Sonneberg, Odessa etc.) equipment. This leads to the orthogonality of the basic functions, and thus the simplified methods give biased parameters of the approximations. In the commonly used methods, there is often a "matrix-phoebia".
We have elaborated a series of algorithms and programs for statistically correct analysis, and have applied them to 2000+ variable stars of different types. The data were obtained from an international collaboration in a frame of the "Inter-Longitude Astronomy" (ILA) campaign, as well as from the international databases. Some highlights of our studies are presented.
The main improvements were done:
1) for the periodogram analysis - the parameters are determined from a complete set of equations containing the (algebraic polynomial) trend superimposed on the (multi-) harmonic wave, so no "detrending", no "prewhitening" are used;
2) for the approximations - we use additional (multi-) harmonic waves, and also "special shapes" (patterns) for parts of the light curve, which correspond to relatively fast changes (minima of the eclipsing binaries, minima and maxima for the pulsating variables);
3) "Autocorrelation analysis" (ACF) - taking into account the bias due to a trend removal (previously - only a subtraction of the sample mean was taken into account); ACF for the irregularly spaced data;
4) for the signals with bad coherence, the "scalegram" analysis is proposed, which allows to estimate a characteristic cycle length and the amplitude, as well as to provide a realistic approximation;
5) the extension of the Morlet-type wavelet for more periodic signals;
6) "running" (parabola, sine) approximations for aperiodic and “nearly periodic” variations, respectively.
Although the methods were initially elaborated for the analysis of variable stars, some of them were applied by others in geo- sciences and even in a cardiology.
Ivan L. Andronov was born in 1960 in Odessa, Ukraine. After graduation from the Department of Astronomy of the Odessa National University, he got his PhD at the Leningrad State University in 1984 under a supervision of Prof. Vladimir P. Tsesevich, the famous researcher of variable stars. The Second scientific degree of Doctor of Science was awarded in Kiev in 1995. Professor of Astronomy since 2001. In 2006, has moved to the Odessa National Maritime University, where currently is a Chair of the Department of “Mathematics, Physics and Astronomy”. An author of 275 publications listed in the “Astronomy Data System” and many popular papers and brochures. Main interest: Mathematical modeling of physical processes in the astronomical objects, including the data analysis with applications to variable stars of different types. The asteroid 11003 was named “Andronov”.