## 5 Results and Discussion
Similarly, they produced MSE values of 31,757.83, 31,338.43, and 30,365.02, respectively, for M1, and 1,639,370.09, 1,619,541.12, and 1,616,325.69, respectively, for L1. Thus, it can be concluded that, when using the same value of \(\alpha\), the initial value provided by the proposed method provided the lowest MSE for all sizes of datasets and quite considerably outperformed the other two methods in \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Size** & **Dataset** & **Method 1** & **Method 2** & **Proposed** &
\begin{tabular}{c} **Chi-Squared** \\ **Value** \\ \end{tabular} & \(p\)**-Value** \\ \hline \multirow{5}{*}{Small} & S1 & 425 & 145 & 430 & 159.65 & \textless{}0.0001 \\ \cline{2-6}  & S2 & 300 & 97 & 603 & 389.05 & \textless{}0.0001 \\ \cline{2-6}  & S3 & 316 & 264 & 420 & 37.86 & \textless{}0.0001 \\ \cline{2-6}  & S4 & 155 & 0 & 845 & 1214.15 & \textless{}0.0001 \\ \cline{2-6}  & S5 & 248 & 223 & 529 & 173.22 & \textless{}0.0001 \\ \hline \multirow{5}{*}{Medium} & M1 & 285 & 218 & 497 & 127.27 & \textless{}0.0001 \\ \cline{2-6}  & M2 & 348 & 172 & 480 & 143.26 & \textless{}0.0001 \\ \cline{2-6}  & M3 & 146 & 295 & 559 & 262.47 & \textless{}0.0001 \\ \cline{2-6}  & M4 & 361 & 234 & 405 & 47.31 & \textless{}0.0001 \\ \cline{2-6}  & M5 & 363 & 146 & 491 & 182.50 & \textless{}0.0001 \\ \hline \multirow{5}{*}{Large} & L1 & 449 & 165 & 386 & 133.47 & \textless{}0.0001 \\ \cline{2-6}  & L2 & 4 & 56 & 940 & 1660.26 & \textless{}0.0001 \\ \cline{2-6}  & L3 & 66 & 273 & 661 & 547.42 & \textless{}0.0001 \\ \cline{2-6}  & L4 & 78 & 127 & 795 & 962.71 & \textless{}0.0001 \\ \cline{2-6}  & L5 & 302 & 221 & 447 & 102.722 & \textless{}0.0001 \\ \hline \multicolumn{6}{c}{Average} & 256.40 & 175.73 & 565.87 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Chi-squared goodness-of-fit test results for comparing the lowest MSE value frequencies of the three initial value setting methods. Figure 6: The frequencies of achieving the lowest MSE when setting the initial value using Method 1, Method 2, and the proposed method. this endeavor. Bar charts of the average MSE values obtained using the three methods for each dataset are shown in Figure 8. The results clearly show that the proposed method performed much better than Method 1 and Method 2. \begin{table}
\begin{tabular}{c c c c c} \hline
**Size** & **Dataset** & **Method 1** & **Method 2** & **Proposed** \\ \hline \multirow{6}{*}{Small} & S1 & 984,214.12 & 969,740.55 & 967,322.56 * \\ \cline{2-5}  & S2 & 1,594,382.33 & 1,569,643.61 & 1,564,879.67 * \\ \cline{2-5}  & S3 & 122,698.44 & 105,228.68 & 104,818.88 * \\ \cline{2-5}  & S4 & 460,672.15 & 461,335.23 & 456,778.22 * \\ \cline{2-5}  & S5 & 146,968.36 & 144,174.95 & 143,896.02 * \\ \hline \multirow{6}{*}{Medium} & M1 & 31,757.83 & 31,338.43 & 30,365.02 * \\ \cline{2-5}  & M2 & 1,204,852.71 & 1,064,595.04 & 1,062,339.88 * \\ \cline{2-5}  & M3 & 2,188,507.09 & 2,166,421.78 & 2,160,496.8 * \\ \cline{2-5}  & M4 & 634,169.22 & 584,823.33 & 582,716.40 * \\ \cline{2-5}  & M5 & 181,293.72 & 181,455.69 & 178,257.81 * \\ \hline \multirow{6}{*}{Large} & L1 & 1,639,370.09 & 1,619,541.12 & 1,616,325.69 * \\ \cline{2-5}  & L2 & 2,371,310.64 & 2,371,031.69 & 2,367,682.30 * \\ \cline{1-1} \cline{2-5}  & L3 & 707,504.85 & 707,387.44 & 706,683.96 * \\ \cline{1-1} \cline{2-5}  & L4 & 36,068.72 & 35,952.54 & 35,942.25 * \\ \cline{1-1} \cline{2-5}  & L5 & 69,863.02 & 69,555.56 & 69,277.16 * \\ \hline \end{tabular}  
* The best performance in terms of average MSE value.  
\end{table} 
Table 6: The average MSE values for the three initial value setting methods using predetermined \(\alpha\) values ranging from 0 to 1 (0.001, 0.002,..., and 1). Figure 7: MSE of the three initial value setting methods for 1000 sets of conditions of S1, M1, and L1.