On the Determination of Cycles and Randomness of Sunspot Series Using the Periodogram

A periodogram device was applied to the sunspot series X∗ t . The analysis resulted in a period estimate of 10.8 ≈ 11 years. The periodogram was displayed graphically and the largest peak corresponded to a frequency of 0.0926. A significant test on randomness of the series was also carried out to ascertain this result. It was observed that the period 10.8 which accounted for 80.68% of the variance of Xt was statistically significant at the 0.05 level.

Sunspots have been observed since ancient times and is believed to be solar magnetic disturbances manifesting as dark spots on the surface of the sun.In 1610, shortly after viewing the sun with his new telescope, Galileo made the first European observations of sunspots.Continuous daily observations started at the Zurich observatory in 1849 and earlier observations have been used to extend the records back to 1610 (Hathaway, 2010).Schwabe (1843) collected 17 years of sunspot observations while searching for intramercurial planets.His observations revealed an 11-year periodicity in the number of visible sunspots.However, when Schuster (1906) analysed the sunspot series, the period was found to be 11.125 years.Rogers and Richards (2004), analysed archival data on sunspot numbers and sunspot areas to derive the solar activity cycles based on these variables.From their work, they were able to identify 11-year Schwabe cycle, Hale cycle, Gleissberg cycle and a cycle at approximately 10 years.
Festus (2010) also confirm the existence of cycles in sunspot numbers using Turkey's smoothing method, and this was calculated to be 10.94 years.Xu et al. (2010) applied the empirical mode description and autoregressive model to long term sunspot numbers.The method was evaluated using data of the solar 23; and the result was remarkably the predictions made by the solar dyanamo and precursor approaches for cycle 23.
The investigation of the sunspot time series has led to advances in spectrum estimation.As an example, Yule (1927) introduced the concept of a finite parameter model for a stationary random process with special references to wolfer's sunspot numbers.
One way of analyzing time series is based on the assumption that it is made up of sine and cosine waves with different frequencies.The assumption applies the concept of periodogram first introduced by Schuster (1898).Schuster proposed the word as a variable quantity which corresponds to the spectrum of a luminous radiation.He showed that the periodogram could yield information on periodic components of a time series and could be applied even when the periods are not known a priori.
In the olden era, periodogram was used to investigate hidden periodicities and estimate the amplitude of a sine component of known frequency buried in noise.However, this work seeks to extend the importance of the periodogram in identifying which, if any, periodic components explain a large enough percentage of the variance in the time series of interest, and to use the periodogram to test the significance of the result obtained.
From Fourier series model, it could be recalled that a time series of length N can be exactly reproduced by summing N/2 sinusoidal wave forms cycle lengths of N/1, N/2, . . ., N/(N/2) or two observations.The goal of the periodogram analysis is to partition the sum of squares total (S S T ) for an overall time series of length N into a set of N/2 sum of squares (S S ) component that correspond to the amount of variance accounted for by each of these cycles.
In addition to the above advantage, if we assume that the time series consist of approximately equal amount of variance due to each of the N/2 periodic components, then it will be wise to consider a periodogram as a tool for testing whether a particular series is white noise or not.

Method of Analysis
The technique to be used in this work is the periodogram device.From the plot of the raw data (see Figure 1), it would be reasonable to assumed that the time series X * t is made up of sine and cosine wave form with different frequencies ( f i ).This can be represented using the Fourier series expression: estimated by where Then, the periodogram I ( f ) consist of the q values: It should be noted here that the periodogram is simply the sum of squares associated with the pair of coefficients and the frequencies.
Thus, the proportions of each sum of squares relative to the S S T is given by:

S S p(i) = S S (i) S S T ;
that is dividing each S S by S S T .
In the above expressions, it is assume that the series X t contains a systematic sine and cosine components with amplitude, phase angle ∅ and frequency f i , so that with Rsin∅ = α and Rcos∅ = β.
If this assumption were true, then I ( f i ) would tend to be inflated because and the hypothesis that X t is white noise is rejected.
However, if X t were white noise.That is, truly random, containing no sinusoidal component; then we have:

Data Analysis and Results
The analysis was carried out using the Minitab software.
For each of the N/2 = 108/2 = 54 cyclic components, a and b coefficients were calculated.From these coefficients, the periodogram ordinates or intensities I( f i ) were obtained as shown in Table 1 below This value when compared with the critical value of 0.12334 shows that the largest peak of period 10.8 is statistically significant and so H 0 is rejected.Similarly, the second largest S S value (15028819) of period 9.818 was also tested, and was shown to be insignificant.This eventually terminates the test procedure.Therefore, the only significant periodic component of the sunspot series is 10.8 years.

Conclusion
Recently, there are strong indications that the warming and cooling of the earth might be due to the changes in the number of observed sunspots (Eyeni, 2010).In order to provide solutions to the global warming in particular, several scientists have engaged in determining the cycle of sunspot numbers which is one of the factors affecting the warming and cooling nature of the earth.Several periods have been identified by various scientists using different methods without undergoing test to ascertain their claims.This work, however, has analysed the sunspot series using the periodogram device.The period has been identified to be 10.8 years with a frequency of 0.0926.This value is not only close to the values obtained by the afformentioned scientists but is supported by the significant test.