Solar System Research

, Volume 53, Issue 5, pp 399–409 | Cite as

An Analysis of Heavy Tail and Long-Range Correlation of Sunspot and El Nino-Southern Oscillation (ENSO) Cycles

  • Asma ZaffarEmail author
  • Shaheen Abbas
  • Muhammad Rashid Kamal Ansari


Heavy tail analysis of Sunspot and El Nino-southern oscillation (ENSO) cycles is performed. Also, their significant behavior is investigated and the long-range correlation (persistency) is also determined. In a time series data, the heavy tail analysis helps to determine the persistency and long term dependency. All Sunspot Cycles and El Nino-southern oscillation (ENSO) Cycles are stationary in nature and each data value is strongly correlated to previous values. The purpose of the modeling performed is to evaluate the strength of long- range correlation as analyzed and to quantify the uncertainty which is hidden in Sunspots and El Nino-southern oscillation (ENSO) Cycles. All Sunspot Cycles and ENSO Cycles have the differencing parameter in the range 0 < d < 0.5 in both self-similar (dS) and self-affine (dA) cases. This means that the background dynamics are more regular. The heavy tail parameter βS (self-similar) as well as βA (self-affine) asymptotically follow the Pareto law which shows that the dynamics for all the Sunspot Cycles and El Nino-southern oscillation (ENSO) Cycles is regular and periodic. Heavy tail parameter (β) and differencing parameter (d = H – 0.5) is obtained from the Hurst Exponent ranging between 0.5 and 1 (persistent data). This study concludes that El Nino-southern oscillation (ENSO) Cycle data behave heavy tail and it is more elongated as compared to Sunspots time series data. All Sunspot Cycles and El Nino-southern oscillation (ENSO) Cycles demonstrate a strong long-range correlation (\(\gamma \)). The strength of self-similar long-range correlation (1 < \(~{{\gamma }_{{\text{S}}}} < 3\)) and the self-affine strength of long-range correlation (–1 < \({{\gamma }_{{\text{A}}}} < 1\)) demonstrate persistency in the perspective that 0.5 < HS < 1 and 0.5 < HA < 1. This study shows that every value of El Nino-southern oscillation (ENSO) Cycles and Sunspot Cycles are strongly correlated to preceding values in both the self-similar and self-affine cases. Unit root test is applied to the tail parameter and the strength of long range-correlation of El Nino-southern oscillation (ENSO) and Sunspot Cycles confirms stationary behavior of the parameters. The variation of earth climatic has a strong influence in Sunspots Cycles and El Nino-southern oscillation (ENSO) Cycles. Sunspots and El Nino-southern oscillation (ENSO) have strong correlation with each other (Asma et  al. 2018). The El Nino-southern oscillation (ENSO) cycles influence on the variation of the parameter of local climate which depends on the changes in solar activity.


ENSO Heavy Tail Analysis Long-range correlation Persistent Hurst Exponent (H) 



The authors are grateful to the World Data Centre (WDC) and the National Oceanic and Atmospheric Administration (NOAA) for providing the Sunspots and ENSO data.


  1. 1.
    Anderson, R.Y., Solar-cycle modulation of ENSO: a mechanism for Pacific and global climate change, Proc. Sixth Annual Pacific Climate (PACLIM) Workshop, Asilomar, California, March 5–8, 1989, Betan Court, J.L and Mackey, A.M., Eds., Sacramento, CA: Calif. Dep. Water Resour., 1990, pp. 27–81.Google Scholar
  2. 2.
    Anderson, R.Y., Possible connection between surface winds, solar activity and the Earth’s magnetic field, Nature, 1992, vol. 358, pp. 51–53.ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Anderson, J.D., Lieberman, M., and Stewart, R.N., Ethylene production by apple protoplasts, Plant Physiol., 1979, vol. 63, pp. 931–935.CrossRefGoogle Scholar
  4. 4.
    Arif, M., Askari, H. and Rashid, A., Influence of El Nino on Pakistan Rainfall: Research Study Report, Karachi: Pak. Meteorol. Dep., 1993, p. 25.Google Scholar
  5. 5.
    Asma, Z., Shaheen, A., and Rashid, A., The probability distributions and fractal dimension of sunspot cycles associated with ENSO phenomena, Arab. J. Geosci., 2018, vol. 11, p. 63.CrossRefGoogle Scholar
  6. 6.
    Beran, J., Statistics for Long-Memory Processes, Boca Raton, FL: CRC Press, 1994.zbMATHGoogle Scholar
  7. 7.
    Bhalme, H.N. and Jadhav, S.K., The southern oscillation and its relation to the monsoon rainfall, J. Climatol., 1984, vol. 4, no. 5, pp. 509–520.CrossRefGoogle Scholar
  8. 8.
    Box, G.E.P., Jenkins, G.M., and Reinsel, G.C., Time Series Analysis: Forecasting and Control, Englewood Cliffs: Prentice Hall, 1994, 3rd ed.zbMATHGoogle Scholar
  9. 9.
    Burrough, P.A., Fractal dimensions of landscapes and other environmental data, Nature, 1981, vol. 294, pp. 240–242.ADSCrossRefGoogle Scholar
  10. 10.
    Burrough, P.A., Multiscale sources of spatial variation in soil. I. The application of fractal concepts to nested levels of soil variation, J. Soil Sci., 1983, vol. 34, pp. 577–597.CrossRefGoogle Scholar
  11. 11.
    Chambers, M.J., Testing for unit roots with flow data and varying sampling frequency, J. Econometrics, 2004, vol. 119, pp. 1–18.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chan, N.H. and Wei, C.Z., Asymptotic inference for nearly nonstationary AR(1) processes, Ann. Stat., 1987, vol. 15, pp. 1050–1063.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chaudhary, Q.Z., Farooqi, A.B. and Rasul, G., Tendency of Pakistan’s Current Climate and Climate Change Scenarios: Climate Change Impact Assessment and Adaptation Strategies in Pakistan, Nairobi: U.N. Environ. Program, 1998, p. 31.Google Scholar
  14. 14.
    Cole, J.E., Dunbar, R.B., McClanahan, T.R., and Muthiga, N.A., Tropical Pacific forcing of decadal SST variability in the western Indian Ocean over the past two centuries, Science, 2000, vol. 287, no. 5453, pp. 617–619.ADSCrossRefGoogle Scholar
  15. 15.
    da Silva Lopes, A.C., Deterministic seasonality in Dickey–Fuller tests: should we care? Empirical Econ., 2006, vol. 31, no. 1, pp. 165–182.CrossRefGoogle Scholar
  16. 16.
    Demon, P.E., Production and decay of radiocarbon and its modulation by geomagnetic field-solar activity changes with possible implications for global environment, in Secular, Solar An Geomagnetic Variation in the Last 10 000 Years, Stephenson, F.R. and Wolfendala, A.W., Eds., New York: Springer-Verlag, 1988, pp. 267–285.Google Scholar
  17. 17.
    Feldman, R. and Taqqu, M., A Practical Guide to Heavy Tails: Statistical Techniques and Applications, New York: Springer-Verlag, 1998, Gouriéroux, C. and Jasiaky, J., Truncated Maximum Likelihood and Nonparametric Tail Analysis: Working Papers No. 98-25, Paris: Center Res. Econ. Stat., 1998.Google Scholar
  18. 18.
    Gourieroux C., and Jasiaky J.: 1998, Truncated Maximum Likelihood, Goodness of Fit Tests Aand Tail Analysis (No. 1998, 36). Discussion Papers, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.Google Scholar
  19. 19.
    Hanslmeier, A., Denkmayr, K., and Weiss, P., Long-term prediction of solar activity using the combined method, Sol. Phys., 1999, vol. 184, no. 1, pp. 213–218.ADSCrossRefGoogle Scholar
  20. 20.
    Hurst, H.E., The long-term storage capacity of reservoir, Trans. Am. Soc. Civ. Eng., 1951, vol. 116, no. 2447, p. 770.Google Scholar
  21. 21.
    Malamud, B.D. and Turcotte, D.L., Self-affine time series: measures of weak and strong persistence, J. Stat. Plan. Influence, 1999, vol. 80, pp. 173–196.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mark, A.D. and Aronson, P.B., Scale-dependent fractal dimensions of topographic surfaces: An empirical investigation, with applications in geomorphology and computer mapping, J. Int. Assoc. Math. Geol., 1984, vol. 16, pp. 671–683.CrossRefGoogle Scholar
  23. 23.
    McCarthy, D., An introduction to testing for unit roots using SAS®: the case of U.S. national health expenditures, Proc. SAS Global Forum 2015, Dallas, TX, 2015, no. 3294.Google Scholar
  24. 24.
    Priestly, M.B., Spectral Analysis and Time Series, London: Academic, 1981.Google Scholar
  25. 25.
    Ruzmaikin, A., Can El Nińo amplify the solar forcing of climate? Geophys. Res. Lett., 1999, vol. 26, no. 15, pp. 2255–2258.ADSCrossRefGoogle Scholar
  26. 26.
    Salakhutdinova, I.I., A fractal structure of the time series of global indices of solar activity, Sol. Phys., 1998, vol. 181, no. 1, pp. 221–235.ADSCrossRefGoogle Scholar
  27. 27.
    Taqqu, M.S. and Samorodnitsky, G., Linear models with long-range dependence and with finite or infinite variance, in New Directions in Time Series Analysis, New York: Springer-Verlag, 1992, part 2, pp. 325–340.Google Scholar
  28. 28.
    Sun, W., Rachev, S., Fabozzi, F., and Kalev, P., Fractals in trade duration: capturing long-range dependence and heavy tailedness in modeling trade duration, Ann. Fin., 2008, vol. 4, no. 2, pp. 217–241.CrossRefGoogle Scholar
  29. 29.
    Tinsley, B. and Deen, G., Apparent tropospheric response to MeV–GeV particle flux variations: A connection via electrofreezing of supercooled water in high-level clouds? J. Geophys. Res.: Atmos., 1991, vol. 96, no. 12, pp. 22 283–22 296. ADSCrossRefGoogle Scholar
  30. 30.
    Turcotte, D.L., Fractal and Chaos in Geology and Geophysics, Cambridge: Cambridge Univ. Press, 1992.zbMATHGoogle Scholar
  31. 31.
    Turcotte, D.L., Fractals and Chaos in Geology and Geophysics, Cambridge: Cambridge Univ. Press, 1997, 2nd ed.CrossRefGoogle Scholar
  32. 32.
    Webster, P.J. and Yang, S., Monsoon and ENSO: selectively interactive systems, Q. J. R. Meteorol. Soc., 1992, vol. 118, pp. 877–926.ADSCrossRefGoogle Scholar
  33. 33.
    Witt, A. and Malamud, B.D., Quantification of long-range persistence in geophysical time series: conventional and benchmark-based improvement techniques, Surv. Geophys., 2013, vol. 34, no. 5, pp. 541–651. ADSCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  • Asma Zaffar
    • 1
    Email author
  • Shaheen Abbas
    • 2
  • Muhammad Rashid Kamal Ansari
    • 1
  1. 1.Department of Mathematics, Sir Syed University of Engineering and TechnologyKarachiPakistan
  2. 2.Mathematical Sciences Research Centre, Federal Urdu University of Arts, Sciences and TechnologyKarachiPakistan

Personalised recommendations