MAVKA: PROGRAM OF STATISTICALLY OPTIMAL DETERMINATION OF PHENOMENOLOGICAL PARAMETERS OF EXTREMA. PARABOLIC SPLINE ALGORITHM AND ANALYSIS OF VARIABILITY OF THE SEMI-REGULAR STAR Z UMa

Advanced MAVKA software for the approximation of extrema observations is used to analyze the variability of the brightness of pulsating and eclipsing stars, but may be useful in analyzing signals of any nature. A new algorithm using a parabolic (quadratic) spline is proposed. In contrast to the traditional definition of a spline as a piecewise-defined function at fixed intervals, a spline is proposed to be divided into three intervals, but the positions of the boundaries between the intervals are additional parameters. The spline defect is 1, that is, the function and its first derivative are continuous and the second derivative can be discontinuous at the boundaries. Such a function is an enhancement of the "asymptotic parabola" (Marsakova and Andronov 1996). The dependence of the fixed signal approximation accuracy on the location of the boundaries of the interval is considered. The parameter accuracy estimates using the least squares method and the bootstrap are compared. It is recommended to use the difference between the 0.975 and 0.025 percentiles (divided by 2 . 1:96) as the accuracy estimate of a given parameter in the bootstrap method. The variability of the semi-regular pulsating star Z UMa is analyzed. The presence of multicomponent variability of an object, including four periodic oscillations (188.88(3), 197.89(4) days and halves of both) and significant variability of the amplitudes and phases of individual oscillations is shown. The approximation using the parabolic spline is only slightly better than the asymptotic parabola for our sampling of the complete interval. It is expectedly better for larger subintervals. The use of different complementary methods allows us to get a statistically optimal phenomenological approximation.