Heavy Tail Analysis of Sunspot Cycles Based on Stochastic Modeling

Sunspot Cycles Based on Stochastic Modeling


  • Danish Hassan Mathematical Sciences Research Centre, Federal Urdu University of Arts, Sciences and Technology, Karachi, Pakistan. Department of Applied Sciences, National Textile University, Karachi Campus


Akaike Information criterion (AIC), Bayesian-Schwarz Information criterion (BIC), Hannan-Quinn Information criterion (HIC), Log-likelihood, FARIMA & Heavy tails


Among other stochastic models, fractional auto regressive integrating moving average (FARIMA) is distinct because of its appropriateness for modeling stationary time series with long range dependence (long memory or persistence). Results obtained in this manuscript shows appropriateness of FARIMA model for the analysis of sunspot number. Analyzing for stationary, each cycle out of the 24 sunspot cycles were modeled. FARIMA can be used for modeling using different techniques. In view of the parameters obtained by maximum likelihood test two most appropriate techniques are adopted. These two are Direct Method and Whittle approximation Method. Results are obtained by applying these two types of FARIMA, the significance of these two methods were observed and compared using significance tests. For FARIMA models the fractional differencing parameter d is most decisive for the determination of persistency. In this regard four types of model (0, d,0), (1, d,0), (0, d,1) and (1, d,1) are used. The adequacy of each of the models is determined with the help of Akaike, Bayesian-Schwarz and Hannan-Quinn Information criterion. The investigations made using these models are reliable for both short and long sunspot cycles. Finally, tail analysis is performed in view of the parameter (α) it is observed that heavy tails exist for each sunspots cycle confirming long range dependence. The study is useful to examine the sunspot historical data using the FARIMA model to understand their long term behavior.


Hassan, D., S. Abbas., M.R.K. Ansari, & B. Jan, The study of sunspots and K-index data in the perspective of probability distributions. International Journal of Physical and Social Sciences, 4(1): 23-41(2014).

Ocher, D. Stationary & Nonstationary Farima Models-Model Choice, Forecasting, Aggregation & Intervention, Doctoral Dissertation, University of Konstanz, (1999), retrieved from: http://kops.unikonstanz.de/handle/123456789/733.

Wang, M. J., G.H. Tzeng, & T.D. Jane, A fuzzy ARIMA model by using quadratic programming approach for time series data, Int J Inf Syst Logist Manag, 5(1): 41-51 (2009).

Cappé, O., E. Moulines., J.C. Pesquet., A. Petropulu, & X. Yang, Long-range dependence and heavy-tail modeling for tele traffic data. Signal Processing Magazine, IEEE, 19(3): 14-27 (2002).

Feldman, R, & M. Taqqudus, M, A practical guide to heavy tails: statistical techniques and applications. Springer Science & Business Media (1998).

Rust, H.W., M. Kallache., H.J. Schellnhuber, & J.Kropp, Confidence intervals for flood return level estimates using a bootstrap approach. Springer, 61-81(2011).

Meerschaert, M. M., P. Roy, & Q. Shao, Parameter estimation for exponentially tempered power law distributions. Communications in Statistics-Theory and Methods, 41(10): 1839-1856 (2012).

Liu, J., Y. Shu., L. Zhang., F. Xue, & O.W. Yang, Traffic modeling based on FARIMA models, Electrical and Computer Engineering, 1999 IEEE Canadian Conference(IEEE).1:162-167 (1999).

Gourieroux, C., & J. Jasiaky, Truncated maximum likelihood, goodness of fit tests and tail analysis, Quantification and Simulation of Economic Processes, 36: (1998).

Barunik, J., & L. Kristoufek, On Hurst exponent estimation under heavy-tailed distributions. Physica A: Statistical Mechanics and its Applications, 389(18): 3844-3855 (2010).

Katsev, S., & I. L’Heureux, Are Hurst exponents estimated from short or irregular time series meaningful? Computers & Geosciences, 29(9):1085-1089(2003).

Sun, W., S. Rachev., F. Fabozzi, & P. Kalev, Long-range dependence and heavy tailedness in modelling trade duration. Workig Paper, University of Karlsruhe. (2005).

Reisen, V., B. Abraham, & S. Lopes, Estimation of parameters in ARFIMA processes: A simulation study. Communications in Statistics-Simulation and Computation. 30(4): 787-803 (2001).

Stanislavsky, A. A., K. Burnecki., M. Magdziarz.,A. Weron, & K. Weron, FARIMA modeling of solar flare activity from empirical time series of soft X-ray solar emission. The Astrophysical Journal, 693 (2):1877 (2009).




How to Cite

Hassan, D. . (2021). Heavy Tail Analysis of Sunspot Cycles Based on Stochastic Modeling: Sunspot Cycles Based on Stochastic Modeling. Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences, 56(1), 59–68. Retrieved from https://ppaspk.org/index.php/PPAS-A/article/view/157