Sun is the main source of energy for the earth and other planets. Its activity in one or the other way influences the terrestrial climate. Particularly, the solar activity manifested in the form of sunspots is found to be more influential on the earth’s climate and its magnetosphere. Links of the variability in terrestrial climate, sunspot cycles and associated magnetic cycles have been the concern of many recent studies. The role of the sun and its activities to understand the space weather and the earth’s environment interaction has been the unique importance in all eras. In this dissertation, we have fitted some adequate probability distributions and stochastic modeling on solar activity (particularly sunspots and solar flares) cycles and terrestrialmagnetic (K-index) activity data comparatively. The 24 cycles (1749-2014) of sunspots including 24th cycle that is in progress, last 4 cycles (20, 21, 22 and 23) of solar flares (1966-2008) and terrestrial K-index activity data (1932-2014) are used in the research work. We have compared the solar activity cycles and K-index activity cycles (associated with solar activity cycle) in the perspective of probability distributions. Comparing both the data we have distributed the time series (1932-2014) among 22-year cycle (2 solar cycles) of each. This kind of distribution is based on the period of one magnetic cycle of sun in which polarity is changed after each 11 years. The magnetosphere’s and magnetic field’s variation of earth can be detected and analyzed by the change in K-index data on which earth climate is depends. The geomagnetic activity is the one of the best recorded sign on earth of solar activity variations. It is basically showing a relationship between space weather and earth''s climate.Results obtained in this dissertation show thequasi-regular (persistent) dynamics of solar activity and K-index activity cycles along with the total time series data from the perspective of fractal dimension. Long-range dependence for each activity cycle is also calculated in terms of Hurst exponent. Theoretical instrument is developed between solar and K-index activity cycles to understand their long term relationship. Stochastic modeling is also fitted on the solar activity and K-index cyclic as well as on the total time series data. The result shows the heavy tail for the sunspots and K-index activity time series data used in this dissertation. The stochastic model FARIMA (Fractional Auto Regressive Integrated Moving Average) is applied on the cycles along with their total time series data as used time series are long ringing dependence (LRD). FARIMA has a capability to use on short and long term conditions. Fractional differencing parameter and heavy tails parameter are calculated to understand the strength and peak of each cycle. The parameters of FARIMA model are obtained by MLE (Maximum Likelihood Estimator). Goodness-of-fit (AIC, BIC and HIC) are used to select the best fitted model among FARIMA (0, d, 0), (1, d, 0), (0, d, 1) and (1, d, 1). The log - likelihood is also estimated for further verification of significant model. Any time series that have heavy tail, fitting FARIMA modeling for them can be more useful to understand their expected behavior in future. The underlying physics of solar activity and K-index activity cycles is modeled by FARIMA (p, d, q) in this dissertation. Finally, we have analyzed and verified that the sunspots and K-index activities are followed Markov process. Transition matrices for both the activities are estimated to understand their physical behavior in 4 different selected states. Stationarity for stochastic matrices is observed in this dissertation to understand similar physical behavior in the used activity data. 2-dimensional correlation between stochastic matrices of sunspots and K-index activity cycles are calculated to understand how much relationship strong between them. In this connection 2-dimensional correlation is also obtained between sunspots and ENSO data to observe the sunspots effects on the earth’s climate. Bayesian posterior and prior are also observed in the estimated stochastic matrices as Bayesian approach is more adequate to understand the complex in the models. By the results obtained we can say that all the activities used in this dissertation are correlated and predictable. We can use probabilistic and stochastic approach to model them. The topic is wide that we could not cover by single dissertation, the same can be done with other solar, geomagnetic and global indices that we did not use in this research work to understand the space weather and earth climate interaction more intensely.
Arabic language is one of the most developed languages of the world. It has a number of grammatical phenomenon, Omissions is one of them. Omission of any part of a sentence creates ambiguity to fully understand its meanings. Due to this phenomenon Arabic has a specific cause when viewed in the light of Semantic analysis. This study deals with the phenomenon of dropping or omission of the part of a speech. In article under review, I have explained as to how an omission becomes requirement of the text to reflect a particular meaning. I have chosen semantic study of three basic parts of verbal sentence that is Verb, Subject and Object to unveil this phenomenon in Sahih Al Bukhari. This clearly explains the significance of omission of words in the sayings of Holy Prophet Muhammad (Peace Be Upon Him).
In this dissertation, generalized simple and exponential type estimators have been developed using the information of single and two auxiliary variables for the estimation of rare and clustered population mean in adaptive cluster sampling designs. The proposed estimators are specifically developed for different situations of clustered populations in simple adaptive cluster sampling, stratified adaptive cluster sampling and systematic adaptive cluster sampling designs. In Chapter 1, the discussion has been made about the situations of rare and clustered population in which the conventional sampling designs may not be appropriate in order to achieve even moderate precision. The use of adaptive cluster sampling design along with the process in the presence of auxiliary information is also discussed. Comparison of adaptive cluster sampling with conventional sampling design and some advantages and disadvantages has also been given. Furthermore, stratified adaptive cluster sampling and systematic adaptive cluster sampling has been illustrated in the same Chapter with the detail sampling process. In Chapter 2, the literature regarding the use of auxiliary information in conventional sampling designs, adaptive cluster sampling, stratified adaptive cluster sampling and systematic adaptive cluster sampling have been discussed whereas Chapter 3 contains some basic estimators that already developed in conventional sampling designs, adaptive cluster sampling, stratified adaptive cluster sampling and systematic adaptive cluster sampling designs. The major contribution of this dissertation appears from Chapter 4 by proposing modified ratio and regression-cum-modified ratio estimators using the information of single auxiliary variable in adaptive cluster sampling by utilizing the average values of the networks with simple random sampling without replacement. The expressions of approximate bias and mean square error for the proposed estimators have been derived. The generalized form for the proposed estimators has been suggested by introduction the unknown constants. The expressions of approximate bias and mean square error have been derived for the generalized form and optimum properties have been discussed. Many conventional and non-conventional parameters of the auxiliary variable have been used as special cases of the proposed estimators. The efficiency issues in adaptive cluster sampling have also been discussed. Theoretical comparisons have been made of the proposed estimators with existing estimators. An extensive numerical study is conducted by using real and artificial population data sets for all the estimators to evaluate their performance. In Chapter 5, weighted exponential ratio-product type estimator have been developed using single auxiliary variable in adaptive cluster sampling for the situations in which the relationship between the survey variable and the auxiliary variable is non-linear. The expressions of approximate bias and mean square error have been derived. A simulation study is conducted to evaluate the performance of the proposed estimator with existing exponential type estimators. In Chapter 6, a generalized semi-exponential type estimator has been suggested based on two auxiliary variables in adaptive cluster sampling. Some exponential and non-exponential type estimators have been discussed, as the special cases of the proposed estimator. The expressions of estimated bias and minimum mean square error have been derived. A simulation study is conducted on simulated populations generated by Poisson cluster process and Ecodist Package in R, to examining the performance of proposed estimator in adaptive cluster sampling design. In Chapter 7, modified ratio and regression-cum-modified ratio estimators have been developed using the information of single auxiliary variable in stratified adaptive cluster sampling. The generalized form for the proposed estimators has been suggested by introduction the unknown constants. The expressions of approximate bias and mean square error have been derived and optimum properties have been discussed. Theoretical comparisons have been made of the proposed estimators with existing estimators. An extensive numerical study is conducted by using real and artificial population data sets for all the estimators to evaluate their performance. In Chapter 8, a generalized semi-exponential type estimator has been suggested based on two auxiliary variables by utilizing the average values of the networks in stratified adaptive cluster sampling. Some exponential and nonexponential type estimators have been discussed, as the special cases of the proposed estimator. The expressions of approximate bias and minimum mean square error have also been derived. A simulation study is conducted using the simulated populations generated by Poisson cluster process at different level of rarity and aggregation to examining the performance of proposed estimator in stratified adaptive cluster sampling design. In Chapter 9, modified ratio and regression-cum-modified ratio estimators have been developed using the information of single auxiliary variable in systematic adaptive cluster sampling. The generalized form for the proposed estimators has been suggested by introducing the unknown constants. The expressions of approximate bias and mean square error have been derived and optimum properties have been discussed. Theoretical comparison has been made of the proposed estimators with existing estimators. A numerical study is conducted by using artificial population data sets taken from Thompson (2012) for all the estimators to evaluate their performance.