FRACTAL ANALYSIS OF PERIODIC TIME SERIES OF WOLF NUMBERS (SOLAR ACTIVITY METRIC)

Abstract

Periodic time series are ubiquitous: both natural and anthropogenic processes often follow various-length periods, resonating into complex patterns. Research aiming to detect those patterns and predict future process evolution has been active for centuries, but the real breakthrough happened in with introduction of Fractal analysis techniques by B. Mandelbrought and R/S method by H. Hurst. Current research is focused on a methodology of fractal parameters calculation such as Hurst fractal dimensions, Hurst exponent, and defining a formal model of periodic time series. The domain for approbating this methodology was chosen to be data on solar activity, defined by sunspots count – Wolf numbers. The peculiarity of the time series of the number of sunspots, is that the division into cycles is carried out naturally, namely, the absence of spots or their small number are the boundaries of the cycles. In addition, their repeatability in a qualitative sense is identical, but in a quantitative sense, we have a clearly expressed difference in the duration of the cycles, in the nature of the trends present in each of them, in the chaotic dynamics of amplitudes, and therefore in the chaotic change in the dispersion of levels, the asymmetry of the branches of growth and decline, and in values of the number of spots. By their physical essence, time series characterize the dynamics of the measured indicator, which may depend on the influence of many factors, both external and internal, and which are implicitly reflected in its behavior. This, in turn, does not create a completely identical picture of the changing states of the time series data source. For this reason, forecasting methods based on time series trends have inaccuracy and a short horizon. The selection of cycles in time series based on the nature of trend behavior can provide significant results and useful information for making and making important decisions. Fractal characteristics of cycles provide new knowledge about the cyclical process of changes in solar activity. So, the fractal dimension indicates complexity (chaoticity in the dynamics of the level values), the Hurst exponent allows to establish randomness, regularity and the presence of a trend.