Monte Carlo Dropout Neural Networks for Forecasting Sinusoidal Time Series: Performance Evaluation and Uncertainty Quantification

Abstract

Accurately forecasting sinusoidal time series is essential in various scientific and engineering applications. However, traditional models such as the seasonal autoregressive integrated moving average (SARIMA) rely on assumptions of linearity and stationarity, which may not adequately capture the complex periodic behaviors of sinusoidal data, including varying amplitudes, phase shifts, and nonlinear trends. This study investigates Monte Carlo dropout neural networks (MCDO NNs) as an alternative approach for both forecasting and uncertainty quantification. The performance of MCDO NNs is evaluated across six sinusoidal time series models, each exhibiting different dynamic characteristics. Results indicate that MCDO NNs consistently outperform SARIMA in terms of root mean square error, mean absolute percentage error, and the coefficient of determination, while also producing more reliable prediction intervals. To assess real-world applicability, the airline passenger dataset is used, demonstrating MCDO’s ability to effectively capture periodic structures. These findings suggest that MCDO NNs provide a robust alternative to SARIMA for sinusoidal time series forecasting, offering both improved accuracy and well-calibrated uncertainty estimates.

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 Tawadros [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, Gal and Ghahramani [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, Kendall and Gal [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, Srivastava 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. Zhang et al. [14] applied Bayesian networks for short-term traffic flow prediction, demonstrating their potential in time series modeling. Pearce et al. [15] combined Bayesian neural networks (BNNs) with ensemble methods to improve uncertainty estimation in time series forecasting. Similarly, Blundell 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, Lemay 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, Atencia et al. [18] integrated echo state networks (ESNs) with MCDO, enabling uncertainty estimation with minimal computational overhead. Similarly, Sheng 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, Srisuradetchai and Phaphan [10] demonstrated the effectiveness of MCDO neural networks (NNs) in interval forecasting for Thailand’s durian export data. Their study found that MCDO-NN outperformed the seasonal autoregressive integrated moving average (SARIMA) model, achieving a lower root mean squared error (RMSE) and producing more reliable uncertainty estimates. Additionally, Khosravi et al. [20] emphasized the significance of prediction intervals in uncertain forecasting, leveraging Bayesian and delta methods to assess interval reliability through coverage probability and width. Recent work has also explored ensemble-based regression approaches, such as random kernel k-nearest neighbors, to improve prediction robustness and model reliability in nonparametric forecasting contexts [21].

Although MCDO has achieved success in various areas such as computer vision, medical imaging, and general time series forecasting, its application to sinusoidal time series has not been thoroughly investigated. Two main reasons likely explain this gap in literature. First, the adoption of MCDO in deep learning is relatively recent, with initial research focusing on broader applications before narrowing to specific forecasting problems. Second, while sinusoidal time series are common in scientific and engineering disciplines, they have only recently been recognized as a distinct category for MCDO-based forecasting, prompting the need for targeted investigations.

This study aims to bridge this gap by applying MCDO-NNs to sinusoidal time series and rigorously evaluating their performance in generating accurate forecast intervals under diverse conditions. The study compares MCDO-NNs with SARIMA, a widely used benchmark model for handling autocorrelated errors in time series analysis. The evaluation focuses on forecast accuracy, interval reliability, and forecast interval width, offering a comprehensive assessment of MCDO’s utility for sinusoidal time series forecasting.

2. Monte Carlo Dropout Neural Networks

2.1. Monte Carlo Dropout

MCDO extends dropout, originally designed as a regularization technique, into a method for approximate Bayesian inference, enabling neural networks to quantify prediction uncertainty. MCDO captures two key types of uncertainty:

• Aleatoric uncertainty, caused by inherent data noise such as measurement errors, is irreducible and can be modeled through predictive distributions obtained via multiple forward passes [11].
• Epistemic uncertainty, on the other hand, arises from limited training data and represents uncertainty in the model’s parameters.

MCDO addresses epistemic uncertainty by applying dropout at inference time, effectively creating a committee of sub-networks, where each forward pass samples from a distribution over model weights. The variance of these predictions serves as a confidence estimate, making MCDO a crucial tool in financial forecasting, medical diagnosis, and other high-risk applications where interpretable uncertainty is essential for decision-making [22]. Recent studies have advanced uncertainty-aware deep learning approaches for time series forecasting across a range of domains, including industrial systems, smart cities, and hybrid statistical frameworks [23,24,25].

2.2. Neural Network Model

A neural network model consists of mathematical transformations that map input features to output predictions. Consider a feedforward neural network with $L$ layers, where each layer l performs a transformation on its input to produce an output that serves as the input to the next layer.

Let $x^{\left(0\right)}$denote the input feature vector. The operation of each layer is expressed as:

$$x^{\left(l\right)}=f^{\left(l\right)}\left(W^{\left(l\right)}{x}^{\left(l-1\right)}+b^{\left(l\right)}\right),$$

(1)

where $x^{\left(l\right)}$ is the output of layer l, $W^{\left(l\right)}$ and $b^{\left(l\right)}$ represent the weight matrix and bias vector, respectively, and $f^{\left(l\right)}$ is the activation function, commonly chosen from ReLU, sigmoid, or tanh functions.

2.3. Dropout Mechanism

A dropout layer is applied after the activation function of each hidden layer during both training and inference. In training, neurons are randomly “dropped” with a dropout rate p, which refers to the probability of a neuron being deactivated. This simulates a sparse network and helps prevent overfitting. Mathematically, this is represented as:

$$x_{\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{u}\mathrm{t}}^{\left(l\right)}=d^{\left(l\right)}⨀x^{\left(l\right)},$$

(2)

where $d^{\left(l\right)}$is a binary mask vector where each element is drawn independently from a Bernoulli distribution with parameter $1-p,$ and $⨀$ denotes element-wise multiplication. The original and dropout networks are depicted in Figure 1, where neurons are randomly removed during each forward pass based on the dropout rate. This stochastic process can be interpreted as sampling from a variational distribution, forming the basis for Monte Carlo inference [26].

2.4. MCDO as Approximation

A Bayesian neural network (BNN) extends traditional neural networks by placing a probability distribution over weights, enabling a comprehensive quantification of uncertainty in predictions. Using Bayes’ theorem, the posterior distribution over the weights $W$, given the dataset $D={\left\{(x_i,y_i)\right\}}_{i=1}^N$ is computed as:

$$P(W|D)=P(W)P(D|W)/P(D).$$

(3)

However, directly computing $P(W|D)$ is computationally intractable for deep neural networks due to the high dimensionality of W and the complexity of $P(D|W)$ [27].

To approximate this, MCDO applies dropout during both training and inference, randomly deactivating a subset of activations. This effectively samples from a distribution of thinned networks, which can be viewed as drawing from an approximate posterior distribution $q(W)$. The predictive distribution for a new input $x^{\ast }$ is then obtained by marginalizing over this approximate posterior [28]:

$$p\left(y^{\ast }\left|x^{\ast },D\right.\right)\propto {\displaystyle \int p\left(y^{\ast }\left|x^{\ast },W\right.\right)}q(W)dW.$$

(4)

Since evaluating this integral is computationally expensive, Monte Carlo integration is used to approximate it. By performing $M$ stochastic forward passes with dropout, MCDO generates a set of predictions $\{y_1^{\ast },y_2^{\ast },\dots ,y_M^{\ast }\}.$ The empirical mean and variance of these predictions serve as approximations for the predictive mean and uncertainty, respectively [29]:

$${\overline{y}}^{\ast }={\displaystyle \frac{1}{M}}{\displaystyle {\sum }_{m=1}^M{y}_m^{\ast }}, {\sigma }_{y^{\ast }}^2={\displaystyle \frac{1}{M}}{\displaystyle {\sum }_{m=1}^M{\left(y_m^{\ast }-{\overline{y}}^{\ast }\right)}^2}.$$

(5)

The 95% confidence intervals for predictions are then computed as:

$${\overline{y}}^{\ast }\pm 1.96\sqrt{{\sigma }_{y^{\ast }}^2}$$

(6)

2.5. Network Input and Implementation Details

In our time series forecasting framework, the input to the neural network at time step $t$ consists of the twelve most recent lagged observations, defined as

$$x^{(0)}=[y_{t-1},y_{t-2},\dots ,y_{t-12}],$$

(7)

and the network is trained to predict the next value, $y_t$. This one-step-ahead prediction structure is used during training. To forecast multiple future steps, such as the final 12 months of the series, the model is applied recursively: after predicting one value, the output is appended to the input window, and the oldest lag is removed, allowing the model to generate the next input for the following time step.

The input features are standardized using z-score normalization, while the target values remain in their original scale. At inference time, MCDO is applied by performing $M=20,000$ stochastic forward passes with dropout layers active, yielding a distribution of predictions:

$$\left\{y_t^{\ast (1)},y_t^{\ast (2)},\dots ,y_t^{\ast (M)}\right\},$$

(8)

from which the predictive mean and standard deviation are estimated.

3. Sinusoidal Models in Simulation Studies

This section explores six sinusoidal time series models, highlighting their structures, applications, and integration of trend components [30,31].

3.1. Constant Amplitude with Trend

The constant amplitude model represents a fundamental form of sinusoidal time series, characterized by regular oscillations of a fixed amplitude around a central level:

$$y_t=A\sin (\omega t+f)+C+\beta t+{\epsilon }_t,$$

(9)

where A is the amplitude, $\omega$ is the angular frequency, $f$ is the phase shift, $C$ is the central level, $\beta t$ introduces a linear trend, and ${\epsilon }_t$ is a random error component. The inclusion of $\beta t$ allows the model to capture long-term directional movements, making it applicable to phenomena exhibiting both cyclic behavior and overall growth or decline trends.

3.2. Varying Amplitude with Trend

To model changing intensity in periodic behavior, the varying amplitude model integrates a time-dependent amplitude function $M(t)$:

$$y_t=(A+\delta t)\sin (\omega t+f)+C+\beta t+{\epsilon }_t.$$

(10)

This model is particularly useful for capturing seasonal effects, where periodic fluctuations strengthen or weaken over time.

3.3. Mixed Frequency with Trend

The mixed frequency model accounts for scenarios where multiple cyclical components overlap within a single time series:

$$y_t=A_1\sin ({\omega }_{1}t+f_1)+A_2\sin ({\omega }_{2}t+f_2)+C+\beta t+{\epsilon }_t.$$

(11)

By incorporating sine waves of different frequencies $({\omega }_1,{\omega }_2)$ and amplitudes $(A_1,A_2)$, this model captures complex cyclical behaviors, such as those observed in economic data with overlapping business and seasonal cycles.

3.4. Phase Shift with Trend

The phase shift model simulates temporal changes in periodic patterns:

$$y_t=A\sin \left(\omega t+f(t)\right)+C+\beta t+{\epsilon }_t.$$

(12)

Here, the phase shift $f(t)$ evolves over time, enabling the model to represent phenomena such as biological rhythm shifts or adjustments in seasonal patterns.

3.5. Nonlinear Modulation with Trend

The nonlinear modulation model addresses cases where the amplitude of oscillations is influenced by additional cyclical factors, such as modulated signal strength in telecommunications or seasonal businesses with varying peak demand. The model is defined as:

$$y_t=\left[A\sin (\omega t+f)+C\right]\times \left[1+B\sin (\theta t+\psi )\right]+\beta t+{\epsilon }_t.$$

(13)

3.6. Noise Variability with Trend

This model extends phase shift dynamics by incorporating stochastic fluctuations in signal strength, often observed in telecommunications and environmental systems:

$$y_t=A\sin \left(\omega t+f(t)\right)+C+\beta t+\sigma (t){\epsilon }_t,$$

(14)

where $\sigma (t)$ modulates the variance of the noise component over time.

Figure 2 illustrates the six sinusoidal models over a five-year period, each comprising 300 data points (one observation every six days). This resolution captures finer seasonal variations, allowing for a detailed analysis of trend and cyclic patterns. In practical applications, datasets are often aggregated to monthly intervals, aligning them with real-world use cases while preserving key patterns and trends.

4. Studied Settings

4.1. Simulation Models

In our simulation study, we examine six distinct sinusoidal time series models, allowing for a detailed analysis of their behaviors under a consistent upward trend (β = 0.75). The model settings are as follows:

• Model 1: Constant amplitude with trend, using a fixed amplitude of 5 units, representing a stable cyclic pattern.
• Model 2: Varying amplitude with trend, starting with an amplitude of 3 units, increasing linearly by 0.05 units per month, simulating intensifying seasonal effects.
• Model 3: Mixed frequency with trend, combining a primary cycle amplitude of 5 units with a secondary cycle amplitude of 3 units, capturing overlapping periodic patterns.
• Model 4: Phase shift with trend, initially set to zero for simplicity, but adaptable to simulate shifts in periodic events.
• Model 5: Nonlinear modulation with trend, incorporating a modulation depth of 0.5 to reflect additional cyclical influences.
• Model 6: Noise variability with trend, using a fixed amplitude of 5 units with stochastic noise modulation, modeling fluctuations in predictability over time.

Across all models, the angular frequency is set to $2\pi /12$ to reflect a yearly cycle in monthly measurements. The mean level remains at 10 units, ensuring a standardized basis for comparison.

4.2. Hyperparameter Grid

The simulation begins with a basic neural network architecture consisting of two layers with 60 and 30 neurons, establishing a baseline for model behavior. The architecture is then expanded incrementally to assess the effect of increasing complexity. A three-layer network consists of 120 neurons, while a four-layer network contains 180 neurons.

Activation functions such as ReLU, Sigmoid, and Tanh are tested to evaluate their effectiveness in capturing nonlinear relationships. To mitigate overfitting, dropout regularization is applied during training. Initial models use dropout rates between 0.2 and 0.3, with deeper networks employing rates as high as 0.7.

Training is conducted over 25, 50, or 100 epochs. Batch sizes of 16, 32, 64, and 128 are tested to examine their impact on computational efficiency and learning stability. By systematically adjusting these parameters, the study provides insights into how network depth, activation functions, and training configurations influence forecasting performance.

4.3. Model to Be Compared with MCDO NN

To assess forecast uncertainty, MCDO-NN predictions are compared with those of the SARIMA model, defined as [32,33]:

$$\left(1-{\displaystyle {\sum }_{i=1}^p{f}_i}{L}^i\right)\left(1-{\displaystyle {\sum }_{i=1}^P{\Phi }_i}{L}^{is}\right){\left(1-L\right)}^d{\left(1-L^s\right)}^D{y}_t=\left(1+{\displaystyle {\sum }_{i=1}^q{\theta }_i}{L}^i\right)\left(1-{\displaystyle {\sum }_{i=1}^Q{\Theta }_i}{L}^{is}\right){\epsilon }_t,$$

(15)

where p is the number of autoregressive terms (AR part), d is the number of nonseasonal differences needed for stationarity (integrated part), q is the number of lagged forecast errors in the prediction equation (moving average part), P is the number of seasonal autoregressive terms, D is the number of seasonal differences, Q is the number of seasonal moving average terms, and $s$ is the number of time steps for a single seasonal period. A SARIMA model is typically denoted as SARIMA(p, d, q)(P, D, Q)_s.

4.4. Comparison Criteria

In this study, the evaluation of forecasting models considers both point prediction accuracy and uncertainty quantification. For MCDO, each forecast is obtained by performing 20,000 stochastic forward passes with dropout enabled at inference. This yields a distribution of predictions at each time step, from which the predictive mean and standard deviation are calculated. The resulting statistics are then used to compute forecast intervals and assess model reliability. To evaluate the forecasting models, four key criteria are considered:

• Root Mean Square Error (RMSE): RMSE measures the average magnitude of the errors between predicted and actual values [34]. It is defined as

$$RMSE=\sqrt{{\displaystyle {\sum }_{t=1}^M{\left(y_t-{\widehat{y}}_t\right)}^2}/M},$$

(16)

where ${\widehat{y}}_t$ represents the forecasted value at time t, $y_t$ is the actual observed value, and $M$ is the total number of forecasts. A lower RMSE indicates higher prediction accuracy.

• Mean Absolute Percentage Error (MAPE): MAPE measures the average absolute percentage difference between the predicted and actual values, expressed as a percentage. It is defined as

$$MAPE={\displaystyle {\sum }_{t=1}^M\left|\left(y_t-{\widehat{y}}_t\right)/y_t\right|}/M\times 100.$$

(17)

• This metric provides a scale-independent measure of forecast accuracy [35]. Lower MAPE values reflect better model performance.

• Coverage of Interval Forecasts (CIF): CIF refers to the proportion of times the actual values fall within the model’s predicted confidence intervals. For instance, a 95% interval should ideally contain the true values 95% of the time. This metric assesses how well the model quantifies uncertainty [36].
• Width of Forecast Intervals (WIF): WIF quantifies the average width of the prediction intervals. Narrower intervals suggest more precise forecasts, while wider intervals may indicate greater uncertainty [37]. When interpreted alongside CIF, WIF helps evaluate the sharpness and reliability of the interval predictions.

All performance metrics are computed on the test set, which consists of the final segment of the time series excluded from training. This setup ensures that forecast quality reflects generalization to unseen data. Due to the temporal dependency in time series, standard cross-validation is not applicable. Therefore, a fixed train-test split is adopted to maintain the chronological order of observations [38,39].

In the simulation study, the time series is sampled every six days, producing 300 data points across a five-year span. The last 12 months, corresponding to approximately 60 observations, are held out as a test set to evaluate model performance.

5. Results

5.1. Optimal Parameters for MCDO NNs

A grid search was conducted to determine the optimal configurations for MCDO neural networks, evaluating the number of hidden layers, activation functions, dropout rates, epochs, and batch sizes.

For Model 1 (constant amplitude with trend), the best-performing neural network consists of five hidden layers with 240, 180, 120, 60, and 30 nodes, using the ReLU activation function and dropout rates of 0.6, 0.5, 0.4, 0.3, and 0.2. It was trained for 50 epochs with a batch size of 16. This configuration achieved an RMSE of 1.5049 and an $R^2$ of 0.6331, as shown in Figure 3.

For Model 2 (varying amplitude with trend), the top-performing architecture follows the same structure as Model 1. This configuration resulted in an RMSE of 2.0204 and an $R^2$ of 0.9528, demonstrating its ability to capture the increasing amplitude variations effectively. The results for Model 2 are depicted in Figure 4.

For Model 3 (mixed frequency with trend), the optimal configuration features three hidden layers with 120, 60, and 30 nodes, trained for 100 epochs with dropout rates of 0.4, 0.3, and 0.2. The batch size is set to 16, and this setup achieves an RMSE of 1.8352 and an $R^2$ of 0.5340, indicating a reasonable performance in capturing multiple overlapping periodic components. The results for this configuration are presented in Figure 5.

For Model 4 (phase shift with trend), the network shares the same architecture as Model 3 but undergoes training for 50 epochs instead of 100. This model achieves an RMSE of 1.7906 and an $R^2$ of 0.8562, indicating improved accuracy in forecasting time series with evolving phase shifts. The results are shown in Figure 6.

For Model 5 (nonlinear modulation with trend), the best-performing neural network consists of four hidden layers, utilizing the ReLU activation function and trained for 100 epochs with a batch size of 64. This configuration achieves an RMSE of 1.4387 and an $R^2$ of 0.8512, suggesting strong predictive performance in scenarios with nonlinear amplitude modulations. The results for this setup are illustrated in Figure 7.

For Model 6 (noise variability with trend), the optimal neural network consists of two hidden layers with 60 and 30 nodes, trained for 100 epochs with a batch size of 16, using the ReLU activation function. This model attains an RMSE of 1.8831 and an $R^2$ of 0.5969, reflecting moderate predictive accuracy given the added uncertainty from noise fluctuations. The results for this model are displayed in Figure 8.

The range of optimal configurations shows how well MCDO neural networks can adapt to different data characteristics. The depth and size of the networks, along with choices like dropout rates, batch sizes, and training epochs, are fine-tuned to handle varying levels of trend complexity, amplitude changes, frequency shifts, phase adjustments, nonlinearity, and noise. This flexibility ensures the networks are capable of capturing the unique patterns in each dataset. The grid search process is key here, as it identifies the best setup for each model.

5.2. Performance of MCDO NNs

Table 1. Performance comparison of MCDO and SARIMA across models (test data).

	RMSE		MAPE		WIF (MCDO vs. SARIMA)
Model	MCDO	SARIMA	MCDO	SARIMA
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

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. Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 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.

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%) while producing narrower WIF (8.95 vs. 10.07) without compromising CIF, which remained at 100% for both models, as depicted in Figure 11.

For Model 4 (phase shift with trend), the MCDO model outperformed SARIMA(1,1,2)(0,0,0)₁₂, which had an AIC of 786.576. MCDO achieved a lower RMSE (2.009 vs. 12.56) and MAPE (3.12% vs. 21.78%). It also produced a narrower WIF (18.56 vs. 36.88) and a higher CIF (100% vs. 91.67%), as illustrated in Figure 12.

For Model 5 (nonlinear modulation with trend), the MCDO model demonstrated notable advantages over SARIMA(1,1,2)(0,0,1)₁₂, which had an AIC of 857.898. While both models achieved a 100% CIF, MCDO yielded a significantly lower RMSE (1.79 vs. 3.68) and MAPE (2.90% vs. 6.13%), indicating superior forecasting accuracy. Additionally, MCDO produced a much narrower WIF (28.85 vs. 48.86), further enhancing the precision of its predictions, as shown in Figure 13.

For Model 6 (noise variability with trend), the SARIMA model selected was ARIMA(2,1,3)(2,0,1)₁₂, which achieved the lowest AIC of 987.396. The MCDO model outperformed SARIMA, achieving a lower RMSE (2.51 vs. 2.74) and MAPE (4.05% vs. 4.56%). Additionally, MCDO produced a significantly narrower WIF (13.90 vs. 31.23) while maintaining a comparable CIF, as shown in Figure 14.

While MCDO consistently produced narrower prediction intervals than SARIMA, its coverage probabilities remained comparable or higher. This combination indicates that MCDO’s intervals are not only sharper but also more reliable. In practical terms, such intervals provide more actionable forecasts, offering users greater confidence in the expected range of future values. This is particularly useful in real-world scenarios where planning or decision-making depends not just on point predictions, but also on understanding the range of likely outcomes

6. Application to a Real Dataset

This section evaluates the performance of MCDO and SARIMA models on the widely recognized AirPassengers dataset, a benchmark in time series forecasting. The dataset contains monthly totals of international airline passengers from 1949 to 1960, originally introduced by Box et al. [40], and is available in many statistical software libraries as well as public repositories such as Kaggle. Due to its well-documented structure and periodicity, it has become a standard resource for evaluating forecasting models. Previous studies, including those by Himakireeti & Vishnu [41], Ohri [42], and Castaño [43], have extensively analyzed this dataset, confirming its relevance in both traditional and machine learning-based forecasting frameworks.

6.1. Fit the Six Models to the AirPassengers Dataset

The six sinusoidal models were applied to the AirPassengers dataset to evaluate their predictive performance. Each model was optimized using curve-fitting techniques, and their effectiveness was assessed based on RMSE and $R^2$ metrics, as summarized in

Table 2. Forecasting performance of MCDO neural networks for each sinusoidal model using the AirPassengers dataset (original scale).

Model	RMSE (Original Scale)	R2 (Original Scale)
Model 1: constant amplitude	25.27	0.943
Model 2: varying amplitude	24.49	0.947
Model 3: mixed frequency	17.59	0.973
Model 4: phase shift	25.27	0.943
Model 5: nonlinear modulation	25.22	0.944
Model 6: noise variability	24.73	0.946

. Among the models, the mixed frequency with trend model demonstrated the best performance, achieving the lowest RMSE (17.59) and the highest $R^2$ (0.973). The fitted results for all models are visually depicted in Figure 15. The corresponding best-fit model for the mixed frequency with trend, which demonstrated the best performance on the training AirPassengers dataset, is:

$$\widehat{y_t}=0.148\sin (0.521t-1.189)+0.084\sin (1.051t+0.448)+0.01t+4.809$$

(18)

6.2. Hyperparameter Optimization

For the MCDO model, a systematic grid search was conducted to optimize key hyperparameters, including the number of layers, neurons, dropout rates, epochs, and batch sizes. The best-performing configuration consisted of a neural network with layers [30,60,120,180], using ReLU activation, dropout rates of [0.5, 0.4, 0.3, 0.2], trained for 50 epochs, with a batch size of 16. This setup achieved a robust performance with an RMSE of 18.54 and an $R^2$ of 0.9379, as summarized in Figure 16.

In comparison, the SARIMA model was optimized using a stepwise search strategy, selecting the configuration that minimized the Akaike Information Criterion (AIC). The optimal SARIMA configuration was ARIMA(3,0,0)(0,1,0)₁₂, which resulted in an RMSE of 18.57 and an $R^2$ of 0.9377. Both models demonstrated strong predictive capabilities, with the MCDO model slightly outperforming SARIMA in accuracy metrics.

6.3. Performance Comparison

The MCDO model exhibited superior forecasting performance compared to SARIMA, achieving a lower RMSE (32.42 vs. 37.59) and MAPE (5.79% vs. 6.21%), indicating higher accuracy in point predictions. Figure 17 shows the interval forecast comparison for this model. While MCDO produced a wider prediction interval (96.92 vs. 83.26), it maintained a significantly higher coverage probability of 91.67% compared to SARIMA’s 66.67%. This suggests that MCDO’s intervals, though broader, are more reliable in capturing actual values, making them more informative in practice. In real-world applications, such intervals offer users greater confidence in forecasted ranges, which is particularly valuable for decision-making under uncertainty. Rather than relying solely on narrow intervals, practitioners can interpret these as indicators of both predictive accuracy and model trustworthiness.

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 i9-13980HX 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 these approaches may offer a scalable and interpretable solution for privacy-preserving time series forecasting.