Empirical mode decomposition and long-range correlation analysis of sunspot timeseries

Abstract

Sunspots, which are the best known and most variable features of the solar surface, affectour planet in many ways. The number of sunspots during a period of time is highly variableand arouses strong research interest. When multifractal detrended fluctuation analysis(MF-DFA) is employed to study the fractal properties and long-range correlation of thesunspot series, some spurious crossover points might appear because of the periodicand quasi-periodic trends in the series. However many cycles of solar activitiescan be reflected by the sunspot time series. The 11-year cycle is perhaps themost famous cycle of the sunspot activity. These cycles pose problems for theinvestigation of the scaling behavior of sunspot time series. Using different methods tohandle the 11-year cycle generally creates totally different results. Using MF-DFA,Movahed and co-workers employed Fourier truncation to deal with the 11-year cycleand 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 detrendingmethod for the MF-DFA and discovered long-range correlation characterized byH≈0.74. In an attempt to get to the bottom of the problem in the present paper, empiricalmode decomposition (EMD), a data-driven adaptive method, is applied to firstextract the components with different dominant frequencies. MF-DFA is thenemployed to study the long-range correlation of the sunspot time series underthe influence of these components. On removing the effects of these periods, thenatural long-range correlation of the sunspot time series can be revealed. With theremoval of the 11-year cycle, a crossover point located at around 60 months isdiscovered to be a reasonable point separating two different time scale ranges,H≈0.72 andH≈1.49. And on removing all cycles longer than 11 years, we haveH≈0.69 andH≈0.28. The three cycle-removing methods—Fourier truncation, adaptive detrending andthe proposed EMD-based method—are further compared, and possible reasonsfor the different results are given. Two numerical experiments are designed forquantitatively evaluating the performances of these three methods in removingperiodic trends with inexact/exact cycles and in detecting the possible crossoverpoints.