A VMD-Based Four-Stage Hybrid Forecasting Model with Error Correction for Complex Coal Price Series

This study proposes a four-module “decomposition–forecasting–ensemble–correction” framework to improve the accuracy of complex coal price forecasts. The framework combines Variational Mode Decomposition (VMD), adaptive Autoregressive Integrated Moving Average (ARIMA) and Gated Recurrent Unit (GRU)-Attention forecasting models, a data-driven weighted ensemble strategy, and an innovative error correction mechanism. Empirical analysis using the Bohai-Rim Steam–Coal Price Index (BSPI) shows that the framework significantly outperforms benchmark models, as validated by the Diebold–Mariano test. It reduces the Mean Absolute Percentage Error (MAPE) by 30.8% compared to a standalone GRU-Attention model, with the error correction module alone contributing a 25.1% MAPE reduction. This modular and transferable framework provides a promising approach for improving forecasting accuracy in complex and volatile economic time series.