The probability distributions and fractal dimension of sunspot cycles associated with ENSO phenomena

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


Various methods have been used to secure the certainty of significant relations among the sunspot cycles and some of the terrestrial climate parameters such as temperature, rainfall, and ENSO. This study investigates the behavior of ENSO cycles and mean monthly sunspot cycles. Sunspot cycles range from 1755 to 2016 whereas, ENSO cycles range from 1866 to 2012. In this regard, the appropriateness of distributions is investigated with the help of Kolmogorov-Smirnov D, Anderson-Darling, and chi-square tests. It is found that most of the sunspot cycle follows generalized Pareto distribution whereas, generalized extreme value distribution was found appropriate for ENSO cycles. Probability distribution is used to analyze the behavior of each sunspot cycle and ENSO cycle separately. Probability distribution indicates the tail behavior of each cycle; tail explored correlation cycles. Furthermore, self-similar and self-affine fractal dimension methods are used to compute Hurst exponents to determine the persistency of the available data. Fractal dimension has an ability to study the complexity involved in sunspot and ENSO cycles. The fractal dimension and Hurst exponent describe persistency (smoothness) and complexity of data. Hurst exponent measures long-term behavior of time series, making it more helpful for forecasting. This is the measure of regularity or irregularity (chaos) of the time function in the form of their persistency or anti-persistency, respectively. Hurst exponents are computed using rescaled range analysis method and box counting methods. Both these methods are suitable for long-term forecasting. The results of this study confirm that during the period 1980–2000, ENSO cycles were very active. Simultaneously, ENSO was active for the periods 1982–1983, 1986–1987, 1991–1993, 1994–1995, and 1997–1998; these periods include two strongest periods of the century viz., 1982–1983 and 1997–1998. Sunspot cycles and ENSO cycles both were found to be persistent. Self-similar fractal dimensions exhibited a better persistency and a better correlation as compared to self-affine fractal dimension. This research is a part of a larger research project investigating the correlation of sunspot cycles and ENSO cycles, and the influence of ENSO cycles on variations of the local climatic parameters which in turn depends on solar activity changes.


El Niño-southern oscillation (ENSO) Hurst exponent (HE) Fractal dimension (FD) Kolmogorov-Smirnov D test (KST) Anderson-Darling test (ADT) Chi-square test (CST) 



The contents of the paper are a form of the first author’s doctoral thesis. The authors are also thankful to the World Data Centre (WDC) and National Oceanic and Atmospheric Administration (NOAA) for providing the sunspot and ENSO data.


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Copyright information

© Saudi Society for Geosciences 2018

Authors and Affiliations

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

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