## 5 Conclusions  
In this study, we examined the properties of the EWMA models as a base forecasting model of a self-starting forecasting process. To this end, we conducted situation studies under the situations that only small initial time series data (number of initial observations is two). In the simulation study, we generated a number of scenarios that reflected a variety of practical situations. These simulation scenarios consider different time series patterns and sizes of noises. In addition to the simulation study, we also applied the self-starting forecasting using the EMWA models on the various real case data. Through comparative experiments in such a number of cases, we can summarize the characteristics of EMWA models as base models of self-starting the forecasting process as follows:  
* Single ES performs best only in the stationary time series and yields unsatisfactory results in nonstationary patterns. Thus, this model is not proper for the base model for the self-starting forecasting process in that, in many real situations, there is no assurance that the time series data is changed with stationary patterns. * Double ES shows comparable or better performances than other EMWA models when the time series observations are monotonically increased or decreased. However, this model is vulnerable to the noises when there are no sufficient time series observations to compensate the effect of noise. In other words, the trend factor in the double ES model cannot be accurately estimated when large-sized noises are added to the insufficient time series observations, and these poorly estimated trend factors sequentially influence the successive model update process of the double ES model. Therefore, the double ES model might not be proper for a base model of the self-starting forecasting process in that the time series data in many real situations often contain large-sized noises. * The seasonal factor in the triple ES model should be carefully estimated for the sake of more accurate forecasting. However, the seasonal factors are poorly estimated when the initial time series observations are not sufficient, and thus, the triple ES shows the worst performance although this model is designed for handling seasonality patterns. In addition, prior knowledge on the true period is not available when time series observations are not sufficiently accumulated. For these reasons, the tripe ES model is not appropriate to be used for a base model of the self-starting forecasting process. * Conversely, the two-stage EWMA model tends to yield comparable or better performance than other EWMA models in all cases. In particular, this model outperforms other EWMA models as a base model for the self-starting forecasting process in the complex time series (i.e., non-stationary and noisy time series) because of the drift factor and adjustment factor. That is to say, the drift factor calculated as the first-order difference of two successive observations helps to accommodate the dynamics of the time series, and the adjustment factor helps to lessen the intrinsic bias caused by the noises and insufficient initial time series data. Finally, these appropriately estimated factors also lead to desirable EWMA model updates in the self-starting process. Based on these various comparative results, we suggest using the two-stage EMWA model as a base model of the self-starting forecasting process. For the further study, we will carefully investigate the effects of the smoothing parameters on the self-starting forecasting performance. In addition, we will also study the systematic way to select the optimal smoothing parameters of the EWMA model for the self-starting forecasting process. Finally, for the sake of exploiting these advantages of the two-stage EWMA as a base model for the self-starting approach, we will attempt to apply it to self-starting approach-based control chart techniques.