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Deep neural networks (DNNs) are prominent in predictive analytics for accurately forecasting target variables. However, inherent uncertainties necessitate constructing prediction intervals for reliability. The existing literature often lacks practical methodologies for creating predictive intervals, especially for time series with trends and seasonal patterns. This paper explicitly details a practical approach integrating dual-output Monte Carlo Dropout (MCDO) with DNNs to approximate predictive means and variances within a Bayesian framework, enabling forecast interval construction. The dual-output architecture employs a custom loss function, combining mean squared error with Softplus-derived predictive variance, ensuring non-negative variance values. Hyperparameter optimization is performed through a grid search exploring activation functions, dropout rates, epochs, and batch sizes. Empirical distributions of predictive means and variances from the MCDO demonstrate the results of the dual-output MCDO DNNs. The proposed method achieves a significant improvement in forecast accuracy, with an RMSE reduction of about 10% compared to the seasonal autoregressive integrated moving average model. Additionally, the method provides more reliable forecast intervals, as evidenced by a higher coverage proportion and narrower interval widths. A case study on Thailand's durian export data showcases the method's utility and applicability to other datasets with trends and/or seasonal components. [ABSTRACT FROM AUTHOR]

Deep neural networks (DNNs) are prominent in predictive analytics for accurately forecasting target variables. However, inherent uncertainties necessitate constructing prediction intervals for reliability. The existing literature often lacks practical methodologies for creating predictive intervals, especially for time series with trends and seasonal patterns. This paper explicitly details a practical approach integrating dual-output Monte Carlo Dropout (MCDO) with DNNs to approximate predictive means and variances within a Bayesian framework, enabling forecast interval construction. The dual-output architecture employs a custom loss function, combining mean squared error with Softplus-derived predictive variance, ensuring non-negative variance values. Hyperparameter optimization is performed through a grid search exploring activation functions, dropout rates, epochs, and batch sizes. Empirical distributions of predictive means and variances from the MCDO demonstrate the results of the dual-output MCDO DNNs. The proposed method achieves a significant improvement in forecast accuracy, with an RMSE reduction of about 10% compared to the seasonal autoregressive integrated moving average model. Additionally, the method provides more reliable forecast intervals, as evidenced by a higher coverage proportion and narrower interval widths. A case study on Thailand’s durian export data showcases the method’s utility and applicability to other datasets with trends and/or seasonal components.

[Display omitted] • A multi-input dual-output neural network is constructed to obtain the best interval. • Two constrained bi-objective optimization problems are established and solved. • An algorithm specifically solving constrained multi-objective problem is employed. • Coverage constraint is introduced to optimize the two optimal interval coefficients. • Model averaging is used to combine advantages to skillfully solve "no free lunch" The uncertainty analysis of wind speed forecasting using the Lower Upper Bound Estimation (LUBE) is an advanced interval prediction method that does not require assumptions about data distribution. However, previous studies have primarily relied on single neural network models, overlooking the benefits of model averaging. Moreover, they assumed symmetric upper and lower bounds of true values in training data, which may not hold for real data with asymmetric features. To address these issues, we propose a multi-input dual-output wind speed interval forecasting system (MDWSIFS). Utilizing neural network models, we create two different outputs for each model by scaling the output values with interval scaling coefficients 1 + γ 1 and 1 - γ 2 , respectively. Subsequently, we propose two constrained multi-objective optimization problems and introduce non-dominated sorting genetic algorithm II (NSGA-II), a method that has been proven to be highly suitable for solving constrained bi-objective optimization problems. By using NSGA-II to optimize a multi-objective problem with coverage probability constraints, the optimal coefficients γ 1 and γ 2 are determined, thereby the prediction interval is defined. Finally, through a model averaging strategy integrated with several neural network models, we use NSGA-II to optimize the weights of sub-models to achieve a more accurate final prediction interval. The test results indicate the superiority of MDWSIFS over existing models, with the metric reaching unprecedented levels across multiple datasets. These findings not only signify an advancement in wind speed forecasting but also promise improved efficiency in wind energy utilization and reduced operational costs for power systems. [ABSTRACT FROM AUTHOR]