## 3 Simulation Study
### Simulation Results  
From Figures 3-5, x-axis is the prediction windows and y-axis is average root mean square error (RMSE) values obtained from 100 repeated experiments. In Figure 3, the RMSE curves in Pattern 1 are exhibited. In this time series pattern, the single ES yields a smaller RMSE than the other two methods because the single ES is designed for stationary patterns. In addition, the trend factors in the double ES methods tend to be overestimated, and these inappropriately estimated trend factors lead to poor forecasting results. On the other hand, compared to the double ES method, the two-stage EWMA method produced better forecasting performance than the double ES because the adjustment factor alleviates the effect of the overestimated trend factors. Figure 4 shows that the comparative results from Pattern 2 to Pattern 5 are nonstationary time series having monotonically changing parts but no seasonality pattern. When the noises are not large, the double ES model shows comparable performance with other two methods because these patterns are composed of monotonically changing parts. In addition, the single ES model produced the worst results because it cannot consider the nonstationary time series. On the other hand, the two-stage EWMA model also produced the comparable performance with double ES in that the two-stage EWMA model also basically designed the forecasting of the monotonically changing time series. In particular, the two-stage EWMA method outperformed the other two methods in Pattern 3 because this model can properly deal with a nonlinearly increasing pattern by reducing the bias based on its adjustment term. On the other hand, when the large-sized noises are added, the double ES model performs the worst because of its inaccurately estimated trend factors. That is to say, the number of initial observations is too small to properly estimate the initial trend factors in the noisy time series, and these poor trend factor estimation results sequentially influence the successive model update process of the double ES model. Conversely, the two-stage EWMA model produces better forecasting performance because the adjustment factors can lessen the risk of the inaccurate estimation of trend factors due to the noises and small number of initial time series data. In addition, as mentioned earlier, the drift factor of the two-stage EWMA model helps to accommodate the dynamics of time series in that they are computed from the difference between the two successive time series observations--not from the difference  
Figure 3: Comparison results between a single exponential smoothing (ES), double ES, and two-stage EWMA in Pattern 1.  
between the estimated level factors, as in the double ES model. These results imply that the two-stage EWMA model is robust to the noises because the adjustment factor can successfully compensate the inaccurately estimated trend factors. Figure 4: _Cont._  
In the current study, we also investigated the properties of three EWMA models when the size of noise varies over time (heteroscedastic settings). When the initial noises are small, trend factors in the double ES are accurately estimated in early phase, and they are also appropriately updated, not affected by the noise in the early update process. Although the noise is increased over time, the double ES model tends to still accurately forecast because these accurately estimated trend factors in the early phase can alleviate the effects of noises. Conversely, the two-stage EWMA model showed smaller RMSE values when the time series started with large-sized noise because the adjustment factor can successfully compensate the inaccurately estimated trend factors, as the same as the experimental results in homoscedastic settings. The simulation results in the heteroscedastic settings demonstrate that the initial condition of time series observations might significantly affect the performance of a self-starting EWMA model when the time series patterns a have monotonically increasing or decreasing part.