Statistical description of the dataset.

Accurately predicting carbon price is crucial for risk avoidance in the carbon financial market. In light of the complex characteristics of the regional carbon price in China, this paper proposes a model to forecast carbon price based on the multi-factor hybrid kernel-based extreme learning machine (HKELM) by combining secondary decomposition and ensemble learning. Variational mode decomposition (VMD) is first used to decompose the carbon price into several modes, and range entropy is then used to reconstruct these modes. The multi-factor HKELM optimized by the sparrow search algorithm is used to forecast the reconstructed subsequences, where the main external factors innovatively selected by maximum information coefficient and historical time-series data on carbon prices are both considered as input variables to the forecasting model. Following this, the improved complete ensemble-based empirical mode decomposition with adaptive noise and range entropy are respectively used to decompose and reconstruct the residual term generated by VMD. Finally, the nonlinear ensemble learning method is introduced to determine the predictions of residual term and final carbon price. In the empirical analysis of Guangzhou market, the root mean square error(RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE) of the model are 0.1716, 0.1218 and 0.0026, respectively. The proposed model outperforms other comparative models in predicting accuracy. The work here extends the research on forecasting theory and methods of predicting the carbon price.