Sunspots Data Analysis Using Time Series

Abstract

The record of the sunspot number visible on the sun is regularly collected over the centuries by various observatories for studying the different factors influencing the sunspot cycle and solar activity. Sunspots appear in cycles, and last several years. These cycles follow a certain pattern which is well known. We analyzed monthly and yearly averages of sunspot data observed from year 1818 to 2002 using rescaled range analysis. The Hurst exponent calculated for monthly data sets are 0.8899, 0.8800 and 0.8597 and for yearly data set is 0.7187. Fractal dimensions1 calculated are 1.1100, 1.1200, 1.1403 and 1.2813. From the study of Hurst exponent and fractal dimension, we conclude that time series of sunspots show persistent behavior. The fundamental tool of signal processing is the fast Fourier transform technique (FFT). The sunspot data is also analyzed using FFT. The power spectrum of monthly and yearly averages of sunspot shows distinct peaks at 11 years confirming the well known 11-year cycle. The monthly sunspot data is also analyzed using FFT to filter the noise in the data.

On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points

“…A Taylor expansion of ρ(k) from Eq. (3) gives ρ(k) H(2H − 1)|k| 2H−2 → 1 (4) for k → ∞. It follows that for H > 1/2, the autocorrelation ρ(k) behaves like |k| 2−2H , thus x(t) has LRD.…”

“…• the last 2 11 values of the monthly mean of the total sunspot number (SSN), obtained from http://www.sidc.be/silso/datafiles (source: WDC-SILSO, Royal Observatory of Belgium, Brussels; accessed on October 31, . The Hurst exponent is widely used to examine whether the Solar activity variations might be caused by a random (i.e., white-noise) process [4,5];…”

“…Long-range dependence (LRD) is a feature of a time series that has found applications in a variety of fields, such as condensed matter [1,2], physiology [3], Solar physics [4,5], financial analyses [6,7], and astrophysical phenomena Email address: mariusz.tarnopolski@uj.edu.pl (Mariusz Tarnopolski) Preprint submitted to Physica A June 28, 2016 arXiv:1512.02928v4 [physics.data-an] 26 Jun 2016 [8,9], among others. In general, a process has the LRD property if the autocorrelations ρ(k) decay to zero so slowly that their sum does not converge [10].…”

“…The fBm branch extends below T /µ T nearly vertically until it reaches zero. The red crosses mark the locations of the time series examined in Sect 4…”

Cusum tests for changes in the Hurst exponent and volatility of fractional Brownian motion

“…For a real data example, we use the daily total sunspot number 4 from 1848/12/23 to 2019/03/31 having n = 62, 190 daily observations. Time series of sunspots are often analyzed using fBm models, see, for instance, Shaikh et al (2008). Estimator (3) and the second-order increments ratio estimator from Bardet and Surgailis (2011) applied to all data both yield H ≈ 0.469.…”