Genetic Algorithm Based Improved ESTAR Nonlinear Models for Modelling Sunspot Numbers and Global Temperatures

Smooth Transition Autoregressive (STAR) models are employed to describe cyclical data. As estimation of parameters of STAR using nonlinear methods was time-consuming, Genetic algorithm (GA), a powerful optimization procedure was applied for the same. Further, optimal one step and two step ahead forecasts along with their forecast error variances are derived theoretically for fitted STAR model using conditional expectations. Given the importance of the issue of global warming, the current paper aims to model the sunspot numbers and global mean temperatures. Further, appropriate tests are carried out to see if the model employed is appropriate for the datasets.


Introduction
Linear time-series family of models viz., Autoregressive integrated moving average (ARIMA), are useful for modelling and forecasting of time-series datasets in order to extract substantial statistics and other characteristics of the data. It takes into consideration that data points taken over different time-epochs may have internal structure, like, autocorrelation, trend or seasonal variation. A time series model is linear if it can be written as a linear function of past observations, errors and other exogenous variables. These linear models have gained much popularity as it is relatively simple and further there exist a good number of computer software which has inbuilt packages for fitting such models. The linear model, however, is inadequate as it is not capable of capturing many important characteristic features like presence of non-Gaussian colored noise in the error term, cyclicity, nonlinearity, limit cycle behavior, volatility clustering, leverage effects, and chaotic behavior of time-series datasets. These types of characteristics feature of time-series datasets can be modelled efficiently using various nonlinear time series models available in the literature. Keeping this in mind, time-series analysis has moved towards the nonlinear domain over the last four decades, as it not only provides a better fit to the data but is also capable to capture different features which cannot be captured solely by linear models. Further, nonlinear models are more suitable for accurately relating dynamics of a time-series, and for making better multi-step-ahead forecasts. To this end, to capture cyclical phenomena in time-series data, Smooth Transition Autoregressive (STAR) family of parametric nonlinear time-series models was propounded by Terasvirta (1994). There are mainly two models under the STAR family, Exponential Smooth Transition Autoregressive (ESTAR) and Logistic Smooth Transition Autoregressive (LSTAR).
The advantages of using STAR models can be gauge from the fact that these types of models are of particular importance to describe those data sets that have cyclical variations along with chaotic periods, changes in economic aggregates influenced by changes in the manners of many different agents and it is highly unlikely that all agents react simultaneously to a given economic signal. In STAR models, transitions are possible along a continuous scale, making the regimeswitching process smooth which helps overcome the abrupt switch in parameter values characteristic of simpler Threshold autoregressive (TAR) models.
The importance of precise modelling of climatic variables such as temperature and sunspots numbers can be judged from the fact that researchers have used variety of models ranging from parametric to nonparametric and recently machine learning too (Citakoglu et  In this manuscript, after a brief introduction in Section 1, Section 2 describes ESTAR models in detail. In Section 3, a brief narrative on fitting of star models is given. In Section 4, optimal one step and two step ahead forecasts of ESTAR model is theoretically derived. Section 5 illustrates the procedure with practical time-series datasets. Finally, in Section 6 some concluding remarks are put up and the papers ends with delineation of some pertinent research problems for future work.

Description of ESTAR Model
Over the last few decades, nonlinear time-series models have been applied in many fields of research. One such parametric family is the State dependent models (SDM), which are the general class of nonlinear models that include the Bilinear and Threshold autoregressive as special cases, but allow much greater flexibility. SDM are essentially Autoregressive moving average (ARMA) models in which the parameters are functions of past values of time-series (Young et al., 2001). Although SDM are of a general nature, they are nevertheless amenable to statistical analysis. This approach to nonlinear time-series analysis offers two major advantages, namely (i) SDM may be used directly in connection with the problem of forecasting.
(ii) Since these can be fitted to data without any specific prior assumptions about the form of the nonlinearity, these may be used to give us an "overview"' of the nonlinearity inherent in the data, and thus indicate whether, for example, a Bilinear, Threshold autoregressive, or even a linear model, is appropriate. Toivonen (2003) derived the discrete-time models with state-dependent parameters for nonlinear systems. Linearization followed by integration over the sampling interval was used to represent these systems. The representation gave exact solution of the nonlinear system. Numerical simulations also showed that these models could be represented by a neural network approximator trained on input-output data. Models which allow for state-dependent or regimeswitching behavior have been most popular for such analysis. One of the most popular models with regime-switching behavior is Threshold Autoregressive (TAR) models. The foremost thought of TAR models is to model a given stochastic process by a piecewise linear autoregressive model, where the determination as to whether each of the sub-models is active or not is driven by the value of a known variable. Smooth Transition Autoregressive (STAR) family of parametric nonlinear time-series models is a generalization of the threshold models, avoiding discontinuities in the autoregressive parameters as the transition which takes place from one regime to the next is determined by a continuous nonlinear function.
It is to be noted that STAR family of parametric nonlinear time-series model has the potential to capture the non-Gaussian characteristics at different time-epochs of time-series datasets. The STAR family of models can be written as: where { } is a sequence of normal (0, 2 ) independent errors, = ( 0 , 1 , … , ) ′ and = There are two different transition functions in the smooth transition autoregressive models, one is And the other one  (2) is called the LSTAR model, and with transition function (3) is called the ESTAR model. A tiny difference in the two transition functions is owing to the fact that, the logistic function changes monotonically with − , while the exponential function fluctuates symmetrically at c with − . But, it is to be kept in mind that both the functions become steeper when γ is large, which means more rapid is the speed of the transition. One of the characteristic features of (1) is that the minimum value of the transition function can be equal to zero. Also heartening to note that other linear as well as nonlinear models can be transformed as special cases of the STAR specifications.

Fitting of STAR models
The

An illustration
The data are at monthly frequency for global land-ocean temperatures (GT) and sunspot numbers of static transformation of a linear Gaussian random process. Further, it was also found that the periodogram ordinates were significant, thereby signifying presence of cyclicity. Obviously, conventional ARIMA modelling approach of the given two time-series data may not be able to describe these datasets satisfactorily.
The preliminary data analysis justifies the application of ESTAR nonlinear time-series model to describe the two time-series dataset. Accordingly, several ESTAR models were fitted to the data and the best model was identified on the basis of minimum AIC criterion. The best ESTAR models for global land-ocean temperatures and sunspot numbers are respectively obtained as

Derivation of formulae for out-of-sample forecasts
One of the main objectives when dealing with time-series analysis is the forecasting of future values of the series of interest. In this section, we confine our attention only to deriving out-ofsample one-step and two-step forecast formulae in respect of ESTAR model. However, formulae for more than two-step ahead forecasts, though quite complicated, can be derived along similar lines. The optimal predictor which minimizes the mean one-step-ahead squared prediction error is the conditional expectation given by Hence, the one-step ahead forecast error variance is For out-of-sample forecasting, parameters are replaced by their corresponding estimates. By using the above procedure one-step optimal forecasts are computed. To validate whether the formula was providing reasonable forecast, the errors were tested for randomness using run test available at R software package. The test statistics for sunspot numbers and Global mean temperature was calculated as 0.148 and -1.024 respectively with 0.88 and 0.31 as the corresponding p value. So, we can infer that the error series were found to be random for both the datasets thereby implying that the model was able to explain the data in an appropriate manner.

Conclusions
In this present investigation, methodology for fitting ESTAR nonlinear time-series model through Genetic Algorithm methodology is described. It is suggested that, for modelling and forecasting cyclical time-series data, researchers should apply this model rather than the ARIMA model. Given the importance of the issue of global warming, the methodology is employed for modelling sunspot numbers and global temperature datasets, which have been captured by the model employed. Further, optimal forecast formulae have also been derived. As a future work, effort can also be required to be directed towards considering more parsimonious subset