## 5 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Size**} & \multirow{2}{*}{**Dataset**} & \multicolumn{2}{c}{**Method 1**} & \multicolumn{2}{c}{**Method 2**} & \multicolumn{2}{c}{**Proposed**} \\ \cline{3-8}  & & **Step** & **Solver** & **Step** & **Solver** & **Step** & **Solver** \\ \hline \multirow{4}{*}{Small} & S1 & 0.369 & 0.3689 & 0.316 & 0.3162 & 0.323 & 0.3232 \\  & S2 & 0.349 & 0.3486 & 0.298 & 0.2982 & 0.301 & 0.3013 \\  & S3 & 0.683 & 0.6828 & 0.546 & 0.5459 & 0.555 & 0.5383 \\  & S4 & 0.797 & 0.7970 & 0.798 & 0.7975 & 0.793 & 0.7932 \\  & S5 & 0.261 & 0.2608 & 0.213 & 0.2129 & 0.216 & 0.2161 \\ \hline \multirow{4}{*}{Medium} & M1 & 0.248 & 0.2477 & 0.228 & 0.2281 & 0.230 & 0.2301 \\  & M2 & 0.244 & 0.2437 & 0.062 & 0.0621 & 0.088 & 0.084 \\  & M3 & 0.098 & 0.0982 & 0.068 & 0.0677 & 0.042 & 0.0420 \\  & M4 & 0.237 & 0.2368 & 0.051 & 0.0507 & 0.064 & 0.0642 \\  & M5 & 0.705 & 0.7049 & 0.705 & 0.7048 & 0.704 & 0.7045 \\ \hline \multirow{4}{*}{Large} & L1 & 0.218 & 0.2177 & 0.135 & 0.1348 & 0.143 & 0.1433 \\  & L2 & 0.148 & 0.1481 & 0.148 & 0.1478 & 0.145 & 0.1447 \\ \cline{1-1}  & L3 & 0.126 & 0.1262 & 0.126 & 0.1257 & 0.122 & 0.1221 \\ \cline{1-1}  & L4 & 0.224 & 0.2238 & 0.22 & 0.2195 & 0.219 & 0.2191 \\ \cline{1-1}  & L5 & 0.181 & 0.1809 & 0.141 & 0.1415 & 0.145 & 0.1449 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The optimal values of \(\alpha\) using the initial values provided by the three methods. Figure 2: A bar chart of the optimal values of \(\alpha\) (alpha) using a step search or Solver from Microsoft Excel when setting the initial value using T1 (Method 1), T2 (Method 2), or P (the proposed method) for large, medium, and small time series datasets. and lie in a straight line, meaning that the optimal values for \(\alpha\) obtained using the step search method or Solver from Microsoft Excel using the initial value from all three methods are in good agreement. Figure 4 illustrates the actual and predicted values when setting the initial value using Method 1, Method 2, and the proposed method. For example, in S1, when setting the initial value using Method 2 and the proposed method, similar predicted values are achieved, but when setting the initial value using Method 1, the predicted values in the early period are far from those of the other methods. Additionally, they can be interpreted in the same way in the other time series. Moreover, Table 4 reports the lowest MSE values for the three initial value setting methods used to obtain the optimal value for \(\alpha\) using the step search method and Solver from Microsoft Excel. For each dataset, the lowest MSE value was obtained from a set of 1000 MSE values. For example, for S1, when setting the initial value using Method 1, Method 2, and the proposed method, the lowest MSE values were 896,212.77, 886,047.38, and 885,070.96, respectively. This trend was the same for the other datasets. It can be seen that although the results are similar, setting the initial value using the proposed method obtained lower MSE values than the other methods. Figure 5 visually supports these findings. The initial setting methods' lowest MSE frequencies were compared using Chi-squared goodness-of-fit tests (Table 5). The null hypothesis states that the number of times that each method achieves the lowest MSE is the same. For example, for S1, the Chi-squared statistic for the lowest MSE value is 159.65 with a \(p\)-value < 0.0001, which leads to the conclusion that the number of times that each method achieves the lowest MSE is significantly different. The results for the other datasets can be interpreted in the same way. Thus, the proposed method for setting the initial value achieved the lowest MSE values, and it is evident that it quite considerably outperformed the other two methods.