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NON-LINEAR DYNAMICS OF THE SUNSPOT NUMBER CYCLE 

1. Sunspots are evanescent dark spots on the sun ranging in size from megameters to gigameters in diameter. Their life span varies from two days to two weeks and their number at any given time from a few to more than a hundred. The total number of sunspots is known to serve as a measure of the level of solar activity (Rogers, 2006) (Yan, 2013). The time series of the number of sunspots appears to form a cyclical pattern with a period of about eleven-years. However, both the period and the amplitude of this cycle are variable and irregular apparently containing long run and irregular cycles of their own (Hathaway, 1994) (Rogers, 2006) (Miletsky, 2014). Sunspot cycles have presented researchers with an enigmatic and vexing quandary for centuries because their irregular nature is not well understood (Hathaway, The Solar Cycle, 2010).
2. Sunspot counts have been recorded and studied since 1610 and all aspects of the patterns in the time series have been subjected to intense scrutiny and interpretation in terms of solar phenomena and their effects on earth (Usoskin, A solar cycle lost, 2009) (Usoskin, Grand minima and maxima of solar activity, 2007) (Hathaway, The Solar Cycle, 2010). The interest in sunspot numbers has grown in the climate change era due to advances in satellite measurements of solar activity and also because of the possible effects of changes in solar irradiance on climate (Weart, 2003) (Haigh, 2007) (Fox, 2004). However, certain issues in the utility of sunspot count time series data remain unresolved, the most prominent being the irregular nature of both the short term solar cycle and the long wave of its amplitude. In this short note we show that these issues may be addressed by separating the regular cyclical components of the solar cycle from the irregular and by describing the system as a sum of two independent phenomena – one cyclical and the other random. The random component is shown to be a non-Gaussian Hurst process with dependence and persistence, properties known to create irregular patterns from randomness (Hurst, 1951) (Koutsoyiannis D. , 2002) (Mandelbrot B. , 1972) (Kim, 2006) (Watari, 1995) (Zhou, 2014).
3. Daily counts of sunspots converted into monthly means are maintained and provided by WDC-SILSO2, Royal Observatory of Belgium, Brussels (SIDC, 2015). Although daily data are available, monthly means are expected to provide sufficient precision for this work because the unit of time represents less than one percent of the period of the average solar cycle. Currently these data are available as Version 2 of the SILSO dataset. The release of version 2 has removed certain inconsistencies between the SILSO data and the sunspot count data maintained by the NOAA (National Oceanic and Atmospheric Administration of the USA). The SILSO mean monthly sunspot counts are available from January 1749 to November 2015 as of this writing. The latter segment of the data series starting from January 1818 contains additional information including the number of different counts taken and both the mean and the standard deviation of these counts. This portion of the dataset is considered to be more reliable than the older data (Clette, 2014) (SIDC, 2015) and it therefore forms the principal source data for this investigation. The time series of monthly mean sunspot counts are decomposed into two components – a cyclical component and a random component. The cyclical component is estimated with a triangular wave generator for Microsoft Excel provided by e-circuitcenter.com (Circuit Center, 2008). The VBA code used is included in the Appendix. The function can generate an asymmetric triangular wave in time with specified low value, high value, time to reach the high value from the low value, and time to reach the low value from the high value. Two cycles are overlaid, one short wave and one long wave. The short wave cycle is used to describe the 11-year solar cycle and the long wave is used to capture the gradual waxing and waning of the amplitude of the solar cycle. The low value is set to zero for both cycles. Prediction errors are computed as the difference between observed counts and the counts predicted by the two overlaid wave functions. The optimal values of the wave parameters are then estimated by minimizing the sum of squared prediction errors with a Generalized Reduced Gradient (GRG) trial and error procedure provided in Excel’s “Solver” tool (Munshi, Methods for Estimating the Hurst Exponent of Stock Returns, 2015).
4. A hypothesis test of the correlation is used to determine whether the optimal wave functions explain a sufficient fraction of the total sum of squared deviations from the mean to represent the underlying cyclical component of the sunspot phenomenon. The hypotheses are stated as H0: ρ=0 versus HA: ρ>0, where ρ represents the correlation in the underlying population from which the sample was taken. If H0: ρ=0 is rejected, the residuals of the optimal waveform are taken to be the random component of the sunspot count series and subjected to further tests with Rescaled Range Analysis to determine whether the residuals are Gaussian or whether they contain the so called Hurst phenomenon that implies long term memory and persistence (Hurst, 1951) (Koutsoyiannis D. , 2002) (Mandelbrot-Wallis, 1969). Hypothesis tests are carried out at a maximum false positive error rate of α=0.001 per comparison in accordance with “Revised standards for statistical evidence” published by the National Academy of Sciences (Johnson, 2013). For multiple comparisons the total study-wide error rate is reported with a Bonferroni adjustment (Holm, 1979). Rescaled range analysis is carried out by taking subsamples without replacement from the residuals in eight cycles. The total sample size of the residuals is N=2375. In the first cycle, 2 subsamples of ѵ≈1187 are taken, in the second cycle 3 subsamples of ѵ≈791, and in the eighth cycle 24 subsamples of ѵ≈99 are taken. The complete sub-sampling strategy is described in Table 1 where n refers to the average sample size per cycle.
5. The value of the Hurst exponent H is computed according to the equation R/S = ѵ^H where R/S is the rescaled range, ѵ is the subsample size, and H is the Hurst exponent. The rescaled range for each subsample is computed as the range of the cumulative sums of differences from the mean that has been “rescaled” by the standard deviation of the subsample. This is the key variable that makes it possible for the exponent H to distinguish a Gaussian series with no dependence or memory from a Hurst series that exhibits long term memory and persistence. The value of H lies in the range 0<H<1. In theory, a value of H=0.5 indicates a Gaussian series with the interval 0<H<0.5 indicating anti-persistence and the interval 0.5<H<1 indicating persistence (Hurst, 1951) (Mandelbrot B. , 1972). These theoretical interpretations apply only when both N and ѵ approach infinity. In actual empirical tests, the observed values of H are affected by N, ѵ, and the sub-sampling structure used in the test (Granero, 2008). In addition, the method employed in estimating the mean unbiased value of H implied by the subsamples also affects the results (Munshi, Methods for Estimating the Hurst Exponent of Stock Returns, 2015) (Munshi, The Hurst exponent of precipitation, 2015). Therefore, it is necessary to calibrate the sampling structure and estimation method with a known Gaussian series to determine the neutral Gaussian value of H under identical test conditions. The observed value of H for the test series is then compared with the calibrated neutral value to determine whether the data contain evidence of non-Gaussian behavior. Each subsample implies a value of H. The best unbiased estimate of H from all 75 subsamples is made using four different estimation methods for both the test series and the calibration Gaussian series. They are (1) a simple mean of all 75 H values, (2) a weighted mean of the H values, (3) a simple linear regression with y=ln(R/S) and x=ln(ѵ) with the intercept set to zero, and (4) a weighted regression of the same linear model. In the weighted procedures the values taken from a subsample are weighted by the size of the subsample which varies from 1188 to 99. All four estimates are compared with the corresponding calibration values and the difference tested for statistical significance at α=0.001. Non-Gaussian behavior is implied only if the null hypothesis that ΔH=0 is rejected in all four tests.
6. Figure 1: Monthly mean sunspot counts Jan 1818 to November 2015 and the optimal regular wave function (ORW)
7. In Figure 1 the irregular wave of the monthly mean sunspot count data series is shown in red and the optimal regular wave function (ORW) in blue. The graph covers the entire study period from January 1818 to November 2015. The ORW is estimated with a Generalized Reduced Gradient numerical method that minimizes the sum of squared prediction errors (Munshi, Methods for Estimating the Hurst Exponent of Stock Returns, 2015). Details of the ORW appear in the table below Figure 1. The ORW consists of a short wave and a long wave. The optimal ORW parameters are those at which the sum of squared prediction errors is minimized. This condition yields an asymmetric short wave solar cycle with a rising period of 50 months and a falling period of 81 months; with a total period of 131 months being very close to the well-known 11-year solar cycle. The initial amplitude of the short wave solar cycle is 100 sunspots but it follows a long wave with an amplitude of 130 sunspots. The long wave is symmetrical with a period of 100 years and it implies that the amplitude of the short wave solar cycle varies from 100 to 230 sunspots in a 100-year cycle as evident from the graphical display in Figure 1 where we see a close correspondence between the data and the ORW. This visual intuition is supported by the statistics in the table below Figure 1 which show that more than 50% of the sum of squared deviations of sunspot counts from the mean are explained by the ORW model yielding a correlation coefficient of 0.745. The statistical significance of this correlation is established by the high value of the t-statistic of t=81.72 shown in the table below Figure 1. These relationships support the assumption that the ORW is the true underlying phenomenon of nature that generates sunspots and that variations from it represent the randomness of nature. The ORW residuals are shown in Figure 2 along with a corresponding random Gaussian series. The visual comparison indicates that the ORW residuals are not Gaussian because they contain patterns.
8. Patterns in residuals often imply that the model is incomplete because it has missed certain behaviors in the data. However, it can also mean that the residual patterns are the product of the Hurst phenomenon in which dependence and long term memory in random data generate faux patterns (Hurst, 1951) (Koutsoyiannis, 2002). Behavior of this kind in sunspot data has been reported in previous studies as a way of addressing the irregularity of solar cycles (Kim, 2006) (Zhou, 2014) (Watari, 1995). In this study we use rescaled range analysis to determine whether the Hurst exponent of the ORW residuals can provide a suitable explanation for the observed patterns. Four different methods are used for the estimation of H with rescaled range analysis. The table below shows that in all four cases, The Hurst exponent of the ORW residuals greatly exceeds the neutral Gaussian value in the calibration series. The least difference is 0.2266 observed in the last row of the table for the weighted OLS estimation method. The standard error of the two values of H are computed as 0.003 and 0.00275 and the standard deviation of the sampling distribution of differences is estimated to be 0.004. The corresponding t-statistic is t=56.56 indicating a strong statistical significance of the difference. We conclude that the Hurst exponent of the random component of the sunspot data is higher than the neutral Gaussian value and that therefore the ORW residuals contain the Hurst phenomenon of long term memory and persistence. Such behavior in the residuals can explain the apparent patterns in the data as shown in the charts below.
   

   

   

9. CONCLUSION: A hurdle to the analysis and understanding of the cyclical behavior of sunspot counts has been the irregular nature of these cycles. We therefore propose that the phenomenon is best described as the sum of two components – one regular and cyclical and the other irregular and random. For mean monthly sunspot counts in the sample period 1/1818-11/2015 we show that the regular and cyclical component of the phenomenon consists of two superimposed wave functions, one a short wave and the other a long wave. The short wave is identified as a 131-month asymmetric and triangular cycle of sunspot counts with a 50-month rising leg and an 81-month falling leg. The long wave is found to be a 100-year symmetric triangular wave in which the amplitude of the short wave fluctuates between 100 and 230 sunspots. This optimal regular component of sunspot number behavior is constructed by minimizing the residuals of the compound wave function. It is shown that these residuals are not random Gaussian but that they tend to form patterns.
10. Rescaled range analysis of the residuals shows that they contain the Hurst effect of memory and persistence and it is proposed that the patterns in the residuals may be explained in terms of the Hurst phenomenon of nature. This finding suggests that, at an annual time scale, the behavior of sunspot cycles may be understood in terms of its regular and cyclical components overlaid with a Hurst process. It is not likely that such persistence exists at the longer time scale of 11 years at which the sunspot cycle is studied in the context of climate change [LINK] . With thanks to @pgeerkens on twitter.

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