Temporal Trend and Fluctuation Learning via Enhanced Attention Mamba for Carbon Price Interval Forecasting

Accurate carbon price forecasting is essential for transforming complex carbon trading markets into efficiently managed and stably operating systems. Existing long-term time series forecasting methods struggle to capture the nonlinear and non-stationary characteristics inherent in carbon prices. To address this limitation, we propose the Temporal Trend and Fluctuation Learning (TTFL) model for interval-valued carbon price forecasting. The model first uses wavelet decomposition to separate the forecasting task into two branches: Price Trend Learning (PTL) and Price Fluctuation Learning (PFL). The PTL branch adopts a forward-backward enhanced Mamba architecture to extract low-frequency, long-term trend features. This design facilitates price interactions across time steps. The enhanced Mamba module leverages a state space model (SSM) to preserve historical information selectively and employs a forgetting gate to recover missing information. As a result, the model captures complementary dependencies across different price points, improving prediction reliability. The PFL branch integrates an attention mechanism with the standard Mamba architecture to model high-frequency temporal dynamics. It provides fine-grained short-term volatility information essential for market participants. We also introduce an interval-valued recovery loss function. This loss quantifies the overlap between predicted and actual interval prices, emphasizes trend learning, and stabilizes model training. We evaluate the TTFL model on three real-world carbon trading markets. Comparative experiments demonstrate that TTFL achieves superior prediction accuracy and robustness relative to baseline methods. Through collaborative learning and selective state space modeling, our approach not only outperforms traditional forecasting models but also offers stakeholders a practical tool for navigating complex carbon market environments. This work contributes a novel forecasting paradigm that integrates multivariate collaborative learning with selective state space modeling. It provides actionable insights for policymaking, investment strategy development, and risk management in the energy and environmental sectors.