Sunspot cycles have influenced on earth climates. In the time series, sunspot cycles in second difference have stationary nature. Autocorrelation (AC), Partial Autocorrelation (PAC) and LjungBox Q-statistics test are used for checking white noise in solar cycles. Moreover, the unit root test with Augmented Dickey Fuller (ADF) test has applied for verification of stationary. For a selection of appropriate models, diagnostic checking is used. For checking normality of sunspot cycles test of normality are used. Test of normality based on skewness, kurtosis and Jurque-Bera test. This chapter is utilizing the stochastic autoregressive and moving average (ARMA) modeling, ARGARCH (1, 1) process and ARMA-GARCH (1, 1) models and forecast evolution of sunspot cycles. Least Square Estimation is used for ARMA process. Various best fitted ARMA models estimate and forecast for each sunspot cycle. Least square method is used to calculate ARAM models and quasi maximum likelihood estimation (QMLE) are used to calculate AR-GARCH and ARMA-GARCH models. The selection of ARMA, AR-GARCH and ARMA-GARCH models are focused on smallest value of Durbin-Watson statistics test. Durbin-Watson (DW) statistics test value of each sunspot cycle is less than 2 which shows that sunspot observations are correlated to each other. GARCH (1, 1) stationary volatility model has the best forecasting model as compared with other models. Diagnostic checking is used to identify and estimation of most appropriate models and confirmation is found by forecasting evolution. The Gaussian quasi maximum likelihood estimation (QMLE) is used to calculate AR-GARCH and ARMA-GARCH models. ARCH effect is found by checking Lagrange Multiplier test, correlogram squared residuals and test of normality. Forecasting evolutions are verified by Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE) and Theil’s U-Statistics test (U test). AIC, SIC and HQC, maximum log likelihood estimation are also calculated. GARCH (1, 1) is leptokurtic in both perspective AR (p) and ARMA (p, q) for sunspot cycles. The adequate ARMA, AR-GARCH and ARMA-GARCH modeling for fractional Brownian motion of sunspot cycles will be useful to predict the dynamical variables for 24th cycle and next coming cycles in the future. The comparison of forecasting evolution of ARAM, AR-GARCH and ARMA-GARCH process are also discussed. RMSE, MAE and U test are described that ARMA model is an appropriate model for sunspot cycles. MAPE exhibits that ARAM-GARCH model is the most appropriate model of sunspot cycles. The persistency analysis of solar activities and ENSO cycles by using fractal dimension and Hurst exponent. The fractal dimension (FD) represents the roughness and complexity of the time series data of solar activities (sunspot activity) and ENSO cycles. Whereas Hurst exponent (HE) provides the smoothness of the data. This study investigates the relationship between self-similar fractal dimension (FDS) and self-affine fractal dimension (FDA). FDS calculated by Box counting method and self-affine fractal dimension (FDA) is calculated by using rescaled range method. The Hurst exponents also calculated by FD in both methods, self-similar fractal dimension (FDS) and selfaffine fractal dimension (FDA). The spectral exponent is calculated by using equation (2.13) and the autocorrelation coefficient to calculate by using equation (2.16). The table 2.1: indicate the numerical relationship between Hurst exponents, spectral exponent and autocorrelation coefficient of both fractal dimensions self-similar and self-affine. In the sunspots cycles indicate that each cycle is persistent and correlated, the values of Hurst exponent lies from 0.5 to 1 but HES values are greater than HEA. All values of the spectral exponent (αs and αA) of each sunspot cycles behave like Brownain noise which indicates the long term dependency. The autocorrelation coefficient is also lies in a significant range of both fractal dimensions. The relation of self-similar and selfaffine fractal dimensions which is described in equation (2.15) in failed in each sunspot cycles because this relation is valid if fractal dimension is lies from 1.5 to 2 but each sunspots cycles fractal dimension lies from less than 1.5. The spectral exponent value which is calculated by Higuchi’s Fractal dimension (FDH) also identify the strong correlation among Sunspots Cycles. We analyzed the complexity of each cycle of ENSO data along total cycles and active ENSO period, then compared them by estimating self-similar fractal dimension (FDS) by using Box counting method and self-affine fractal dimension (FDA) by rescaled range method, and Hurst exponents also calculated through FD which is show that self-similar fractal dimension (FDS) for ENSO cycles and Total data are less than those for self-affine fractal dimension (FDA). Similarly, self-similar fractal dimension (FDS) of ENSO cycles is also less than self-similar fractal dimension (FDs) of Sunspot cycles. This means that ENSO cycles are more persistent than Sunspot cycles data. Sunspots cycles and ENSO are correlated to each other. It is also verifying the fact if FD increases, then H decreases. The cycle will prolong sunspot activity has greater means and the tail prolongs. In the end a relation between probability distribution and Fractal Dimension establishes the persistency approach for both of the data sets. The mean- tail assessment confirm the FD-HE analysis. This study can be useful for further investigation of the impact of Sunspot and ENSO related local climatic variability. Different methods have been used to develop the certainty of significant relations among the Sunspot cycles and some of the terrestrial climate parameters such as temperature, rainfall and ENSO etc. This study explores the dependence of ENSO cycles on Mean Monthly Sunspots Cycles. Sunspot cycles range from 1755 to 2008 whereas, ENSO cycles range from 1866 to 2012. To find the above mentioned dependence probability distribution approach is utilized. In this regards 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 (GPD) whereas, Generalized Extreme Value Distribution (GEV) were found appropriate for ENSO cycles. This study confirmed that during the period 1980-2000 ENSO cycles were very active. Simultaneously, El Nino was active for the periods 1982-83, 1986- 87, 1991-1993, 1994-95, and 1997-98 these periods include two strongest periods of the century viz. 1982-83 and 1997-98. Two consecutive periods 1991-1993 and 1994-1995 were cold periods. Sunspots cycles and ENSO cycles both were found to be persistent. 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 term depend on solar activity changes. In the time series data, sunspots and ENSO cycles have stressed the Generalized Pareto Distribution (GPD) and Generalized Extreme Value Distribution (GEV). These distributions have a heavy tail on the right side. In the next section of this chapter described the analysis of heavy tail parameter for further analyze the data behavior. All the solar cycles (1-24) has a stationary nature as the differencing parameter (0 < d < 0.5) in both perspective self-similar (dS) and self-affine (dA) which represent that the dynamic is more regular. The heavy tail parameter βS as well as βA exploring that asymptotically equivalent to Pareto law which is showing that the strength of the dynamics is regular and periodic for all the solar cycles. For each sunspot cycle heavy tails are profound. The heavy tail parameter (β) and differencing parameter (d = HE-0.5) are obtained from the Hurst parameter (0.5 < HE < 1) persistent. The heavy tail parameter (β) value towards 2 depicted that the strength of heavy tail decreases. Similarly, every persistent data contains the heavy tail since for d > 0 the HT parameter β > 1. Similarly, All the ENSO cycles (1-23) also has a stationary nature as the differencing parameter (0 < d < 0.5) in self-similar (dS) and self-affine (dA) respectively. The heavy tail parameters ( βS and βA) of ENSO cycles are depicted asymptotically equivalent to Pareto law. The comparative study of solar cycle and the ENSO cycle conclude that the heavy tail parameter (βS) of the ENSO cycles, values approximately 1 which are explored that ENSO cycle data behave heavy tail increase as compared to sunspots time series data. This study concludes that ENSO cycles have more heavy tail as compare to sunspot cycles. In the time series data, the tail parameter helps to analysis the persistency and long term dependency. Statistical modeling based on two types of correlation short range correlation and long range correlation. All sunspot cycles explored the strength of long-range correlation (?). The strength of self-similar long-range correlation (1 < ?? < 3) and the self-affine strength of longrange correlation (-1<?? < 1) is persistent in the perspective of 0.5 < HES < 1 and 0.5 < HEA < 1. The novelty of this study shows that every value of sunspot cycles is strongly correlated to preceding ones in both manner self-similar as well as self-affine. Similarly, each ENSO cycle shows that each value is strongly correlated to preceding ones in both manner self-similar (??) and self-affine (??). In all aspects Self-similar technique is more appropriate as compared to selfaffine. The unit root test is used for non-stationary data. H0 is rejected when p-value is less than 5% or the critical value of absolute value of Augmented Dickey Fuller (ADF) test is greater than at 1% and 5% significance level. This study analyzed that the heavy tail parameter (βS and βA) of Sunspot and ENSO cycles are stationary. Similarly long-range correlation (?? ??? ??) of Sunspot and ENSO cycles are also stationary in time series data. The persistency of AR12192 is determined by using self-similar fractal dimensions (Box counting FDB and Correlation Dimension FDC) involving Hurst exponent. The fractal dimension (FD) expresses the complexity and roughness of the active region AR12192, whereas Hurst exponent (HE) provides the smoothness of the active region. The Hurst exponents are calculated by FD which is calculated by both techniques. The spectral exponent and the autocorrelation coefficient are also calculated by both techniques. The active region AR12192 has a stationarity nature as the differencing parameter follows the inequality 0 < d < 0.5 in both perspective Box counting (dB) and correlation dimension (dC). The heavy tail parameter (β) is less than 2 which confirms the equivalence of the asymptotic nature of heavy tail (one sided) and the Pareto law which confirms that the underlying dynamics is strong and regular. For the he active region AR12192 heavy tails are profound. The heavy tail parameter (β) and differencing parameter (d = HE-0.5) are obtained from the Hurst parameter (0.5 < HE < 1) showing persistency. The heavy tail parameter (β) tending towards 2 depicts that strength of heavy tail is decreasing. If the image is persistent, then heavy tails exist. This implies that d > 0 and heavy tail parameter β > 1 heavy tails exist. The active region AR12192 is found to be persistent, correlated and heavy tailed. The spectral exponent (αB and αC) of AR12192 behaves like Brownain noise which indicates the long term dependency. The autocorrelation coefficient is found to be significant using both the fractal dimensions FDB and FDC. Mathematical morphological operations such as erosion, dilation, closing and opening are also analysis for AR12192. The novelty of this study has delivered image segmentation of rotating sunspots with genetic algorithm. The criteria of selection of best contour points from the proposed fitness criteria and then improvements have been prepared by the process of crossover and mutation which are the most important features of Genetic algorithm (GA). Although GA is an approximated approach, but it may provide very accurate results by making a suitable selection of crossover and mutation. In the proposed algorithm we have applied crossover in a different way between the contour points of each cycle the proposed idea is unique and producing promising results. AR9114 and AR10696 of solar cycles 23rd are used as a case study to calculate the image segmentation of active contour." xml:lang="en_US