Sunspot Periodicity - Maple Help

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Sunspot Periodicity

Introduction

This application will find the periodicity of sunspots with two separate approaches:

 • A frequency domain transformation of the data
 • Using autocorrelation

Both approaches should yield the same result.

 >

International Sunspot Data

The following data table contains mean international sunspot numbers from the year 1700 to present.

Data source: Solar Influences Data Analysis Center (SIDC), RWC Belgium, World Data Center for the Sunspot Index, Royal Observatory of Belgium, 1700-2013. Data available at: http://sidc.oma.be/html/sunspot.html

The first column contains the "Year", while the second column represents the "Annual Mean Sunspot Number".

 > $\mathrm{DataReference}≔\mathrm{DataSets}:-\mathrm{Reference}\left("quandl","SIDC/SUNSPOTS_A"\right)$
 ${\mathrm{DataReference}}{≔}\left[\begin{array}{c}{\mathrm{Data set}}\\ {\mathrm{Sunspot Numbers \left(Annual\right)}}\\ {\mathrm{Quandl SIDC/SUNSPOTS_A}}\\ {\mathrm{up to 314 rows \left(annual\right), 1 column}}\\ {\mathrm{1700-12-31 - 2013-12-31}}\end{array}\right]$ (2.1)
 >

Plot the Data

 > $\mathrm{SunspotNumber}≔\mathrm{data}\left[..,2\right]:$
 > $\mathrm{ParseYear}≔x\to \mathrm{StringTools}:-\mathrm{ParseTime}\left("%Y-%m-%d",x\right):-\mathrm{year}:$
 > $\mathrm{Year}≔\mathrm{ParseYear}~\left(\mathrm{data}\left[..,1\right]\right):$
 >

Periodicity via Fourier Transformation to the Frequency Domain

Now, calculate the period using a Fast Fourier Transform (FFT) of the first 28 data points:

 > $\mathrm{fSunspots}:=\mathrm{FFT}\left(\mathrm{SunspotNumber}\left[1..{2}^{8}\right]\right):$

Plot the power spectrum:

 > $\mathrm{samplingRate}:=1:$
 > $\mathrm{psSunspots}:=\mathrm{PowerSpectrum}\left(\mathrm{fSunspots}\right):$
 >

Note the peak at a frequency of 0.09 years-1 . Try zooming in and using the point probe to confirm the value of this peak frequency.

The period is the reciprocal of the peak frequency.

 > $\mathrm{period}:=\frac{1}{0.09}$
 ${\mathrm{period}}{≔}{11.11111111}$ (4.1)

Hence, the predicted periodicity is approximately 11 years.

Periodicity via Autocorrelation

 > $\mathrm{aSunspotNumber}:=\mathrm{AutoCorrelation}\left(\mathrm{SunspotNumber}\right):$
 > $\mathrm{SignalPlot}\left(\mathrm{aSunspotNumber}\left[1..36\right],\mathrm{labels}=\left["Years",""\right],\mathrm{title}="Autocorrelation of Sunspot Data",\mathrm{titlefont}=\left[\mathrm{Arial},14\right],\mathrm{size}=\left[800,"golden"\right]\right);$

Here the first peak is at 11 years, indicating that the periodicity of sunspots is approximately 11 years. This confirms the period predicted by the Fourier Transform approach.