## 1 Introduction  
Time series data play a crucial role in various fields, including economics, finance, environmental sciences, and engineering, as they provide valuable insights into temporal patterns and trends. Among these, sinusoidal time series are particularly significant due to their ability to model periodic behaviors observed in both natural and artificial systems [1, 2]. For instance, in astronomy, periodic variations in solar time and astronomical measurements arise from Earth's rotation and orbit around the Sun, which can be effectively captured using sinusoidal models [3]. Similarly, in electrical engineering, sinusoidal waveforms form the foundation for understanding alternating current behaviors, as they accurately represent the periodic fluctuations in voltage and current in real-world circuits [4]. In management, sinusoidal functions assist in predicting and managing demand variations in inventory systems, as demonstrated by [PERSON] [5] in forecasting dynamic demand shifts over time. In biological sciences, sinusoidal models are instrumental in analyzing circadian rhythms, which regulate sleep cycles, hormone secretion, and metabolism [6]. Oceanography relies on sinusoidal patterns to predict tidal movements [7], while quantum mechanics and wave theory employ sinusoidal functions to describe particle behaviors and wave propagation, driving advancements in modern physics [8]. Recent advancements in machine learning (ML), particularly in neural networks, have shown significant promise in addressing the challenges of time series forecasting [9; 10]. Neural networks excel at capturing nonlinear patterns and dependencies, making them well-suited for modeling sinusoidal time series. However, traditional neural networks primarily provide point estimates, often failing to quantify the uncertainty inherent in their predictions. To address this limitation, [PERSON] and [PERSON] [11] introduced the use of dropout in neural networks as a method for approximate Bayesian inference, allowing for uncertainty estimation in predictions. Building upon this, [PERSON] and [PERSON] [12] extended the application of Monte Carlo dropout (MCDO) to explicitly distinguish between aleatoric uncertainty, which arises from inherent data noise, and epistemic uncertainty, which stems from limited model knowledge. Additionally, [PERSON] et al. [13] originally proposed dropout as a regularization technique, which later became a foundational tool for uncertainty quantification in deep learning models. Further advancements in uncertainty quantification have explored various techniques to improve reliability in forecasting. [PERSON] et al. [14] applied Bayesian networks for short-term traffic flow prediction, demonstrating their potential in time series modeling. [PERSON] et al. [15] combined Bayesian neural networks (BNNs) with ensemble methods to improve uncertainty estimation in time series forecasting. Similarly, [PERSON] et al. [16] employed variational inference for neural network weights, supporting dropout as an effective tool for uncertainty quantification. In the domain of medical image analysis, [PERSON] et al. [17] showed that MCDO enhances model consistency and repeatability, particularly in classification and regression tasks. These contributions underscore the importance of integrating uncertainty estimation into forecasting models to improve reliability across diverse applications. To address computational challenges, [PERSON] et al. [18] integrated echo state networks (ESNs) with MCDO, enabling uncertainty estimation with minimal computational overhead. Similarly, [PERSON] et al. [19] proposed the bootstrapping reservoir computing network ensemble (BRCNE), which blends Bayesian linear regression with ensemble learning to generate robust prediction intervals for noisy, nonlinear time series. In a practical application, [PERSON] and [PERSON] [10] demonstrated the effectiveness of MCDO neural networks (NNs) in interval forecasting for Thailand's durian export data.