## 4 Case Study  
In this section, we investigated the performance of EWMA methods using 18 real time series datasets. Table 1 and Figure 6 show the characteristics and time series patterns of these real data, respectively. These real data are available in public data repositories (Federal Reserve Bank of St. Louis ([[https://research.stlouisfed.org/](https://research.stlouisfed.org/)]([https://research.stlouisfed.org/](https://research.stlouisfed.org/))); [[https://datamarket.com/data/list/?q=provider%3](https://datamarket.com/data/list/?q=provider%3) Atsdl]([https://datamarket.com/data/list/?q=provider%3](https://datamarket.com/data/list/?q=provider%3) Atsdl); [[https://datahub.io/core/global-temp#data](https://datahub.io/core/global-temp#data)]([https://datahub.io/core/global-temp#data](https://datahub.io/core/global-temp#data))). As shown in Figure 6, most time series data in real cases are nonstationary patterns, and they also have large-sized noises. In real situations, prior knowledge on periods and seasonality factors for the triple ES is not available. However, we already demonstrated that the triple ES is not effective for our problem settings in the simulation study. Hence, we only considered the single ES, double ES, and two-stage EWMA models in the real time series cases. In the case study, we applied the same experimental setting as the simulation study. At first, we constructed the initial EWMA models using only the first two observations as training time series data and forecasted \(s\)-step ahead future observations using these initial models. In the next step, we updated the EWMA models by adding the successive new observation to the training time series. Finally, the updated model forecasted the \(s\)-step ahead future observations. This continuous model update process was repeated until the (\(L\)-s) time series observations were accumulated where \(L\) was the length of the total time series data. In addition, as the same in the simulation study, we considered the smoothing parameter values between 0.1 and 0.9 with a step size of 0.1, and we reported the lowest average RMSE values among the total smoothing parameter sets. Finally, the RMSE values are computed by varying the forecasting window (\(s\)) from one to six. Table 2 shows the comparison results between three EWMA models. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **Duration of Time Series** & **Time Scale** \\ \hline Case 1: Unemployment rate in USA & 2000.01–2019.01 & Monthly \\ Case 2: USD–EUR exchange rate & 2000.01–2019.02 & Monthly \\ Case 3: USD–KRW exchange rate & 2000.01–2019.02 & Monthly \\ Case 4: WTI crude oil price & 2000.01–2019.01 & Monthly \\ Case 5: Brent crude oil price & 2000.01–2019.01 & Monthly \\ Case 6: S\&P 500 index & 2009.01–2019.01 & Monthly \\ Case 7: NASDAQ-100 index & 2000.01–2019.02 & Monthly \\ Case 8: Australian expenditure on financial services & 1969.09–1994.03 & Quarterly \\ Case 9: Chemical process temperature readings & NA & Minute \\ Case 10: Changes in the Earth’s rotation day length & 1821–1970 & Yearly \\ Case 11: Total building and construction activities in Australia & 1973.09–1995.03 & Quarterly \\ Case 12: Money supply in USA & 1890–1974 & Yearly \\ Case 13: Birth per 10,000 of 23 year old people & 1961–2019 & Yearly \\ Case 14: Numbers on unemployment benefit in USA & 1984.01–2019.01 & Monthly \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of real cases. Figure 6: Time series data of 14 real cases. In Table 2, the lowest RMSE values in each case and prediction window are highlighted as bold and underline for more effectively showing the best EWMA model. As shown in Table 2, the experimental results in most cases are the almost same as those in simulation studies. That is to say, the double ES model tends to show the worst forecasting results because the trend factors in double ES are sensitive to large-sized noises.