Ensemble learning using multivariate variational mode decomposition based on the Transformer for multi-step-ahead streamflow forecasting

Reliable and accurate streamflow forecasting is critical in the domain of water resources management. However, the inherently non-stationary and stochastic nature of streamflow poses a formidable challenge to achieving accuracy in streamflow forecasting. In this study, we introduce an MVMD-ensembled Transformer model (MVMD-Transformer). This model employs the MVMD technique, which allows for simultaneous time–frequency analysis of streamflow and other potential influencing factors. The model aligns common modes in the decomposition results, ensuring that the different variables corresponding to each mode have the same center frequency. This alignment overcomes frequency mismatches and helps uncover the intrinsic patterns and essential features between streamflow and associated variables. During the forecasting phase, the Transformer component of the MVMD-Transformer model establishes connections among streamflow and other influencing factors across pairs of nodes in each mode. We tested the effectiveness of the MVMD-Transformer model on streamflow forecasting in the Shiyang River, Heihe River, and Shule River basins situated in the Hexi Corridor of Northwest China, with 1-, 3-, 5-, and 7-day forecasting horizons. The MVMD-Transformer model harnesses MVMD for the simultaneous decomposition of forecast variables (precipitation, air temperature, air pressure, soil moisture) and the response variable (streamflow). Subsequently, the resulting modes from the MVMD were fed into the Transformer, serving as the forecast analytics engine, for streamflow forecasting. Furthermore, we conducted a comprehensive performance evaluation by comparing the MVMD-Transformer model against four alternatives: the VMD-ensembled Transformer model (VMD-Transformer), CEEMDAN-ensembled Transformer model (CEEMDAN-Transformer), stand-alone Transformer model, and LSTM model. The results indicate that MVMD-Transformer outperformed all other models, achieving Nash-Sutcliffe coefficient (NSE) values exceeding 0.85 in the majority of the forecasting scenarios. This superior performance highlights the proficiency of the MVMD approach in more accurately unraveling the intricate interdependencies between streamflow and its various potential influencing factors, thus significantly improving the precision of streamflow forecasting.