## 3 Simulation Study  
### Simulation Setup  
We conducted simulation studies under nine time series patterns to evaluate the performance of four EWMA methods: (1) single ES; (2) double ES; (3) triple ES; and (4) two-stage EWMA methods. The simulation was conducted under the following nine patterns:  
* Pattern 1: \(Y_{t}=N\!\left(0,\sigma_{t}^{2}\right),\;t=1,2,\dots,\;120\);
* Pattern 2: \(Y_{t}=t/3+N\!\left(0,\sigma_{t}^{2}\right),\;t=1,2,\dots,\;120\);
* Pattern 3: \(Y_{t}=\exp(t/30)+N\!\left(0,\sigma_{t}^{2}\right),\;t=1,2,\dots,\;120\);
* Pattern 4: \(Y_{t}=\max(0,t-60)+N\!\left(0,\sigma_{t}^{2}\right),\;t=1,2,\dots,\;120\);
* Pattern 5: \(Y_{t}=\min(t,\left(120-t\right))+N\!\left(0,\sigma_{t}^{2}\right),\;t=1,2,\dots, \;120\);
* Pattern 6: \(Y_{t}=\left\{ \begin{array}{l}N\!\left(0,\sigma_{t}^{2}\right)1\leq t\leq 3 0\\ 10+N\!\left(0,\sigma_{t}^{2}\right)31\leq t\leq 60\\ 20+N\!\left(0,\sigma_{t}^{2}\right)61\leq t\leq 90\\ 30+N\!\left(0,\sigma_{t}^{2}\right)91\leq t\leq 120\end{array} \right.\);
* Pattern 7: \(Y_{t}=10\sin(t/10)+N\!\left(0,\sigma_{t}^{2}\right),\;t=1,2,\dots,\;120\);
* Pattern 8: \(Y_{t}=t+10\sin(t/10)+N\!\left(0,\sigma_{t}^{2}\right),\;t=1,2,\dots,\;120\);
* Pattern 9: \(Y_{t}=\left\{ \begin{array}{l}10\sin(t/5)+N\!\left(0,\sigma_{t}^{2}\right)1 \leq t\leq 20\\ 10\sin(t/15)+N\!\left(0,\sigma_{t}^{2}\right)21\leq t\leq 80\\ 10\sin(t/10)+N\!\left(0,\sigma_{t}^{2}\right)81\leq t\leq 120\end{array} \right..\)  
The length of each time series pattern is 120, and the noises generated from the normal distribution whose mean is zero and standard deviation is \(\sigma_{t}^{2}\) were added to the baseline. In this study, we investigated the performance of EWMA methods on both homoscedasticity and the heteroscedasticity of the noises. Hence, the standard deviation of noise is \(\sigma_{t}\), which can be defined as follows:  
* Small noise (homoscedastic): \(\sigma_{t}=0.5,\;t=1,2,\dots,\;120\);
* Medium noise (homoscedastic): \(\sigma_{t}=1,\;t=1,2,\dots,\;120\);
* Large noise (homoscedastic): \(\sigma_{t}=2\;,\;t=1,2,\dots,\;120\);
* Increasing noise (heteroscedastic): \(\sigma_{t}=0.5+t/80\;,\;t=1,2,\dots,\;120\);
* Deceasing noise (heteroscedastic): \(\sigma_{t}=2-t/80\;,\;t=1,2,\dots,\;120\). Figure 2 shows the baseline of the time series in each pattern. Pattern 1 and 2 represent the simplest time series patterns, which are the stationary and linear trend, respectively. On the other hand, Pattern 3 is a time series with an exponential trend, one of the most well known nonlinearly time-varying pattern, and Pattern 4 represents a time series that changes from stationary to non-stationary. More specifically, the series has a stationary pattern from time 1 to 60, followed by a constantly increasing trend from time 61 to 120. In addition, in Pattern 5, the direction of the trend is changed from an increasing pattern to decreasing pattern. That is to say, the trend of the simulation data constantly increased from time 1 to 60, and then, it constantly decreased from time 61 to 120. The simulated data in Pattern 6 consist of four stationary patterns with step-like increments with a period size of 30. Patten 7 considers time series that exhibit constant seasonal time series whose period size is 20, and Pattern 8 shows time series that exhibit seasonal variation (period size is 20) with a linearly increasing trend. Finally, Pattern 9 considers time series with seasonal variation with changing periods. In this scenario, the periods are 10, 20, and 30, respectively. In each simulation study, we applied the self-starting EWMA model to each simulation case. That is, we built the initial EWMA models using only the first two observations as a training time series data,and forecasted \(s\)-step ahead future observations using these initial models. Then, the EWMA models were continuously updated as a successive observation was added to the training time series data. Finally, the updated EWMA model forecasted \(s\)-step ahead future observations.