STATISTICALLY OPTIMAL PHENOMENOLOGICAL MODELLING OF VARIABLE STARS

Abstract

A set of algorithms and programs is reviewed, which are most effective for statistically optimal mathematical modelling of various types of variability. Although these methods were proposed and applied for photometry and polarimetry of 2000+ variable stars, as excellent laboratories to study various processes, they may be applied to signals of any nature, e.g. technical, medical or social. For "mono-periodic" stars (with high level of coherence), "global" approximations are most effective - either a trigonometric polynomial of statistically optimal order (mainly for pulsating stars or for eclipsing binaries with smooth curves (EW, EB), or for stars with elliptic variability. However, for Algol-type variables or transits of exoplanets, a "special shape/pattern" approximations superimposed onto the second-order trigonometric polynomial are recommended. For quasi-periodic variations, the wavelet analysis may be used (in some programs, with an adaptive effective width determination), or the scalegram analysis with further local weighted approximation with an optimal width. This method may be effective even in a case of strong flickering, fractal variations, red noise. Wavelet and (especially) scalegram analysis (some modifications are called “multi-resolution”, “multi time-scale”) may separate variability at different time scales. Particularly, some types suggest long-term aperiodic variability, which may be effectively modelled by an algebraic or trigonometrical polynomial. In a case of trends, the periodogram analysis is to be made using a complete mathematical model instead of super-simplified “detrending” or “pre-whitening”, which may cause wrong (biased) results. Some algorithms are illustrated by application to the observations.