## 7 Conclusions and Discussions  
This study demonstrates that MCDO neural networks perform more effectively than traditional models such as SARIMA across several forecasting criteria. Results from both simulation studies and real-world applications highlight MCDO's strengths in predictive accuracy, uncertainty quantification, and adaptability to complex periodic behavior. These findings are summarized through the following key aspects:  
* Forecasting Accuracy: Across all six sinusoidal models, MCDO consistently yielded lower RMSE and MAPE than SARIMA. In the phase shift model, RMSE dropped to 2.00 and MAPE to 3.12%, compared to SARIMA's 12.56 and 21.78%, respectively. Likewise, in the nonlinear modulation model, MCDO achieved an RMSE of 1.79 versus 3.68. * Uncertainty Quantification: MCDO produced narrower and more reliable prediction intervals while maintaining high coverage. For instance, in the constant amplitude model, its WIF was 22.94 versus SARIMA's 33.58. Even in the challenging mixed frequency model, MCDO maintained 100% CIF while reducing WIF to 8.95 from SARIMA's 10.07, demonstrating improved precision and efficiency. * Adaptability to Sinusoidal Structures: Unlike SARIMA, which relies on predefined seasonal components and assumptions of stationarity, MCDO can learn periodic patterns directly from data. This adaptability enables the modeling of complex sinusoidal signals, including variations in amplitude, phase, and frequency, while capturing residual dynamics. * Generalization Beyond Ideal Sinusoidal Patterns: While the simulated models assume ideal periodicity, many real-world time series deviate from such structure. MCDO remains effective in these settings, as it does not rely on strict model assumptions. Suitability of sinusoidal modeling can be assessed using tools such as decomposition, autocorrelation, or spectral analysis. * Real-World Performance: Applied to the AirPassengers dataset, MCDO offered better forecast accuracy (RMSE: 32.42 vs. 37.59; MAPE: 5.79% vs. 6.21%) and better-calibrated intervals, achieving 91.67% CIF compared to SARIMA's 66.67%, despite a wider prediction interval. * Computational Efficiency: Compared to SARIMA, the MCDO model requires greater computational resources due to its neural network architecture and the use of multiple stochastic forward passes during inference. For the real-world AirPassengers dataset, inference with 20,000 passes on a consumer-grade laptop (Intel Core i9-13980 HX processor, 64 GB RAM) took approximately 49.36 s, whereas SARIMA fitting and prediction completed in about 3.39 s. Despite the increased computational cost, MCDO provides additional benefits, including improved accuracy and more informative uncertainty quantification. Its efficiency can be further enhanced by reducing the number of forward passes or by leveraging parallel hardware such as GPUs. * Future Applications and Federated Learning Potential: While this study focused on centralized learning, a promising direction for future research is the integration of MCDO into federated learning (FL) frameworks. Federated MCDO could enable decentralized forecasting across edge devices or distributed sensors while preserving data privacy. Recent studies have shown that ensemble-based strategies can enhance generalization and uncertainty quantification in FL settings [44,45]. Additionally, adaptive sampling and model fusion techniques have been proposed to improve performance under non-independent and identically distributed data conditions and communication constraints [46]. Ensemble learning has also demonstrated effectiveness in environmental forecasting contexts [47], suggesting further opportunities to apply MCDO in federated or distributed scenarios. Integrating MCDO into theseapproaches may offer a scalable and interpretable solution for privacy-preserving time series forecasting. Conceptualization, P.S. and U.K.; Methodology, P.S. ; Software, P.S. and U.K.; Validation, P.S. and U.K.; Formal analysis, U.K.; Investigation, P.S. ; Resources, P.S.