Multi-step-ahead and interval carbon price forecasting using transformer-based hybrid model

Accurate and stable carbon price forecasts serve as a reference for assessing the stability of the carbon market and play a vital role in enhancing investment and operational decisions. However, realizing this goal is still a significant challenge, and researchers usually ignore multi-step-ahead and interval forecasting due to the non-linear and non-stationary characteristics of carbon price series and its complex fluctuation features. In this study, a novel hybrid model for accurately predicting carbon prices is proposed. The proposed model combines multi-step-ahead and interval carbon price forecasting based on the Hampel identifier (HI), time-varying filtering-based empirical mode decomposition (TVFEMD), and transformer model. First, HI identifies and corrects outliers in carbon price. Second, TVFEMD decomposes carbon price into several intrinsic mode functions (imfs) to reduce the non-linear and non-stationarity of carbon price to obtain more regular features in series. Next, these imfs are reconstructed by sample entropy (SE). Subsequently, the orthogonal array tuning method is used to optimize the transformer model's hyperparameters to obtain the optimal model structure. Finally, after hyperparameter optimization and quantile loss function, the transformer is used to perform multi-step-ahead and interval forecasting on each part of the reconstruction, and the final prediction result is obtained by summing them up. Five pilot carbon trading markets in China were selected as experimental objects to verify the proposed model's prediction performance. Various benchmark models and evaluation indicators were selected for comparison and analysis. Experimental results show that the proposed HI-TVFEMD-transformer hybrid model achieves an average MAE of 0.6546, 1.3992, 1.6287, and 2.2601 for one-step, three-step, five-step, and ten-step-ahead forecasting, respectively, which significantly outperforms other models. Furthermore, interval forecasts almost always have a PICI above 0.95 at a confidence interval of 0.1, thereby indicating the effectiveness of the hybrid model in describing the uncertainty in the forecasts. Therefore, the proposed hybrid model is a reliable carbon price forecasting tool that can provide a dependable reference for policymakers and investors.