Time-Series Interval Forecasting with Dual-Output Monte Carlo Dropout: A Case Study on Durian Exports

Abstract

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.