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

Abstract

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.