## 5 Results
### Performance of MCDO NNs  
Table 1 presents a comparative analysis of the forecasting performance of MCDO and SARIMA models across six sinusoidal trend scenarios. The results suggest that MCDO generally achieves better performance than SARIMA in terms of forecast accuracy (RMSE and MAPE) and interval width reduction. The RMSE of MCDO models ranges from 1.58 to 2.82, whereas SARIMA's RMSE varies between 1.79 and 8.84, indicating that MCDO tends to produce lower errors across most models. Similarly, MCDO achieves lower MAPE values in the majority of cases. For instance, in Model 2 (varying amplitude with trend), MCDO's MAPE is 6.14%, substantially lower than SARIMA's 15.97%. Additionally, MCDO often produces narrower forecast intervals, helping reduce uncertainty while maintaining comparable or higher coverage probabilities. For example, in Model 2, MCDO achieves an interval width of 21.62 compared to 28.56 for SARIMA, suggesting more concentrated forecast bounds. Figures 9-14 visualize these results for each model and are discussed individually. For Model 1 (constant amplitude with trend), the MCDO model outperformed SARIMA(1,1,2)(0,0,0)\({}_{12}\), which was selected based on its lowest AIC value of 776.248. The MCDO model achieved a lower RMSE (2.50 vs. 4.57) and MAPE (4.30% vs. 7.44%). Additionally, the MCDO model produced significantly narrower WIF (22.94 vs. 33.58) while maintaining a 100% CIF for both models, as shown in Figure 9.  
For Model 2 (varying amplitude with trend), the MCDO model demonstrated superior accuracy, achieving a lower RMSE (3.40 vs. 8.84) and MAPE (6.14% vs. 15.97%) compared to SARIMA. Additionally, MCDO produced a narrower prediction interval width (21.62 vs. 28.56) while maintaining a slightly higher CIF of 100% compared to SARIMA's 96.67%, as shown in Figure 10. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & \multicolumn{2}{c}{**RMSE**} & \multicolumn{2}{c}{**MAPE**} & \multicolumn{1}{c}{**WIF**} \\
**Model** & **MCDO** & **SARIMA** & **MCDO** & **SARIMA** & **(MCDO vs. SARIMA)** \\ \hline
1. Constant Amplitude & 2.50 & 4.57 & 4.30\% & 7.44\% & 22.94 vs. 33.58 \\
2. Varying Amplitude & 3.40 & 8.84 & 6.14\% & 15.97\% & 21.62 vs. 28.56 \\
3. Mixed Frequency & 1.78 & 1.79 & 2.93\% & 2.70\% & 8.95 vs. 10.07 \\
4. Phase Shift & 2.00 & 12.56 & 3.12\% & 21.78\% & 18.56 vs. 36.88 \\
5. Nonlinear Modulation & 1.79 & 3.68 & 2.90\% & 6.13\% & 28.85 vs. 48.86 \\
6. Noise Variability & 2.51 & 2.74 & 4.05\% & 4.56\% & 13.90 vs. 31.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance comparison of MCDO and SARIMA across models (test data). Figure 8: Lowest 15 RMSEs for MCDO configurations for Model 6. Figure 11: Interval forecast comparison for Model 3 (last 12 months). Pink and deep pink lines show MCDO forecasts; blue and navy lines show SARIMA forecasts. Green line denotes actual data. Figure 10: Interval forecast comparison for Model 2 (last 12 months). Pink and deep pink lines show MCDO forecasts; blue and navy lines show SARIMA forecasts. Green line denotes actual data. Figure 9: Interval forecast comparison for Model 1 (last 12 months). Pink and deep pink lines show MCDO forecasts; blue and navy lines show SARIMA forecasts. Green line denotes actual data. For Model 3 (mixed frequency with trend), the SARIMA model selected was ARIMA(0,1,1)(2,0,1)\({}_{12}\), which achieved the lowest AIC of 896.358. Both MCDO and SARIMA demonstrated similar accuracy, with RMSE values of 1.78 and 1.79, respectively. However, MCDO slightly outperformed SARIMA in terms of MAPE (2.93% vs. 2.70%)  
Figure 14: Interval forecast comparison for Model 6 (last 12 months). Pink and deep pink lines show MCDO forecasts; blue and navy lines show SARIMA forecasts. Green line denotes actual data. Figure 12: Interval forecast comparison for Model 4 (last 12 months). Pink and deep pink lines show MCDO forecasts; blue and navy lines show SARIMA forecasts. Green line denotes actual data. Figure 13: Interval forecast comparison for Model 5 (last 12 months).