## 4 Case Study
Moreover, single ES model also showed unsatisfactory performance in that this model cannot accommodate the nonstationary time series encountered in many real situations. On the other hand, as confirmed in a simulation study, the two-stage EWMA model shows the best  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Case** & **EMWA Model** & \(s=1\) & \(s=2\) & \(s=3\) & \(s=4\) & \(s=5\) & \(s=6\) & **Average** \\ \hline \multirow{3}{*}{Case 1} & Single ES & 0.17 & 0.26 & 0.35 & 0.44 & 0.53 & 0.62 & 0.40 \\  & Double ES & **0.15** & 0.22 & **0.28** & 0.35 & 0.42 & 0.51 & 0.32 \\  & Two-stage EWMA & **0.15** & **0.21** & **0.28** & **0.33** & **0.39** & **0.47** & **0.31** \\ \hline \multirow{3}{*}{Case 2} & Single ES & **0.03** & **0.05** & **0.06** & **0.07** & **0.08** & **0.09** & **0.06** \\  & Double ES & **0.03** & **0.05** & 0.07 & 0.08 & 0.09 & 0.11 & 0.07 \\  & Two-stage EWMA & **0.03** & **0.05** & **0.06** & **0.07** & **0.08** & **0.09** & **0.06** \\ \hline \multirow{3}{*}{Case 3} & Single ES & 28.46 & 45.93 & 57.41 & 66.86 & 76.29 & 85.51 & 60.08 \\  & Double ES & 28.91 & 48.23 & 61.96 & 74.09 & 87.02 & 100.51 & 66.79 \\  & Two-stage EWMA & **25.48** & **45.35** & **56.37** & **65.63** & **73.43** & **82.76** & **58.17** \\ \hline \multirow{3}{*}{Case 4} & Single ES & 5.74 & 9.35 & 12.37 & 14.78 & 16.78 & 18.39 & 12.90 \\  & Double ES & 5.67 & 9.90 & 13.58 & 16.82 & 19.77 & 22.38 & 14.69 \\  & Two-stage EWMA & **5.08** & **8.34** & **11.46** & **13.77** & **15.59** & **17.54** & **11.96** \\ \hline \multirow{3}{*}{Case 5} & Single ES & 5.92 & 9.58 & 12.50 & 14.81 & 16.75 & 18.42 & 13.00 \\  & Double ES & 5.99 & 10.10 & 13.63 & 16.66 & 19.43 & 22.01 & 14.64 \\  & Two-stage EWMA & **5.33** & **8.95** & **11.86** & **14.41** & **16.09** & **17.86** & **12.42** \\ \hline \multirow{3}{*}{Case 6} & Single ES & 53.49 & 80.72 & 99.46 & 115.55 & 129.62 & 144.23 & 103.85 \\  & Double ES & 54.49 & 85.14 & 106.53 & 124.03 & 140.21 & 156.16 & 111.09 \\  & Two-stage EWMA & **49.47** & **76.97** & **93.08** & **105.16** & **114.09** & **122.17** & **93.49** \\ \hline \multirow{3}{*}{Case 7} & Single ES & 151.05 & 230.85 & 289.31 & 340.43 & 390.81 & 442.15 & 307.43 \\  & Double ES & 158.75 & 265.42 & 342.90 & 407.23 & 477.73 & 552.12 & 367.36 \\  & Two-stage EWMA & **139.70** & **222.72** & **275.33** & **321.19** & **361.35** & **402.55** & **287.14** \\ \hline \multirow{3}{*}{Case 8} & Single ES & 40.40 & 67.80 & 96.53 & 125.10 & 153.53 & 181.59 & 110.83 \\  & Double ES & 31.51 & 44.53 & 61.65 & 80.38 & 102.38 & 121.51 & 73.66 \\  & Two-stage EWMA & **29.64** & **41.23** & **56.46** & **72.36** & **91.03** & **107.80** & **66.42** \\ \hline \multirow{3}{*}{Case 9} & Single ES & 0.25 & 0.46 & 0.66 & 0.85 & 1.03 & 1.19 & 0.74 \\  & Double ES & **0.14** & 0.31 & 0.49 & 0.68 & 0.87 & 1.07 & 0.59 \\  & Two-stage EWMA & **0.14** & **0.28** & **0.44** & **0.60** & **0.76** & **0.92** & **0.52** \\ \hline \multirow{3}{*}{Case 10} & Single ES & 29.57 & 53.40 & 74.62 & 92.72 & 108.53 & 122.14 & 80.16 \\  & Double ES & 19.87 & 44.42 & 74.80 & 102.48 & 124.28 & 144.99 & 85.14 \\  & Two-stage EWMA & **17.46** & **36.79** & **60.13** & **81.02** & **98.35** & **111.90** &  
performance in most cases because the adjustment factor of the two-stage EWMA model helps to lessen the effect of noises and the bias of forecast caused by insufficient time series observations. Besides, the two-stage EWMA model yields much smaller RMSE values than the double ES model because the drift factor better models the changing patterns of time series than the trend factor in the double ES model. In other words, the drift factor is calculated as the first-order difference of two successive observations, and it is more effective to accommodate the inherent dynamics of the time series than the trend factor which is defined as the difference of two successive estimated level factors.