Empirical mode decomposition and long-range correlation analysis of sunspot time series

Sunspots, which are the best known and most variable features of the solar surface, affect our planet in many ways. The number of sunspots during a period of time is highly variable and arouses strong research interest. When multifractal detrended fluctuation analysis (MF-DFA) is employed to study the fractal properties and long-range correlation of the sunspot series, some spurious crossover points might appear because of the periodic and quasi-periodic trends in the series. However many cycles of solar activities can be reflected by the sunspot time series. The 11-year cycle is perhaps the most famous cycle of the sunspot activity. These cycles pose problems for the investigation of the scaling behavior of sunspot time series. Using different methods to handle the 11-year cycle generally creates totally different results. Using MF-DFA, Movahed and coworkers employed Fourier truncation to deal with the 11-year cycle and found that the series is long-range anti-correlated with a Hurst exponent, H, of about 0.12. However, Hu and co-workers proposed an adaptive detrending method for the MF-DFA and discovered long-range correlation characterized by H ≈ 0.74. In an attempt to get to the bottom of the problem in the present paper, empirical mode decomposition (EMD), a data-driven adaptive method, is applied to first extract the components with different dominant frequencies. MF-DFA is then employed to study the long-range correlation of the sunspot time series under the influence of these components. On removing the effects of these periods, the natural longrange correlation of the sunspot time series can be revealed. With the removal of the 11-year cycle, a crossover point located at around 60 months is discovered to be a reasonable point separating two different time scale ranges, H ≈ 0.72 and H ≈ 1.49. And on removing all cycles longer than 11 years, we have H ≈ 0.69 and H ≈ 0.28. The three cycle-removing methods-Fourier truncation, adaptive detrending and the proposed EMD-based method-are further compared, and possible reasons for the different results are given. Two numerical experiments are designed for quantitatively evaluating the performances of these three methods in removing periodic trends with inexact/exact cycles and in detecting the possible crossover points.