TDTimexer: A Carbon Price Forecasting Model Integrating Time–Frequency Enhancement and Cross-Variable Dependency Modeling

Carbon price series exhibit nonlinear fluctuations, multi-scale temporal patterns, and complex dependencies among influencing variables, making accurate forecasting challenging. To tackle these challenges, we introduce CPO-TDTimexer as a framework for forecasting carbon prices. First, the maximal information coefficient (MIC) and SHapley Additive exPlanations (SHAP) are combined to identify informative input variables. Next, a Temporal Enhancement and Spectral Adaptation (TESA) module is developed to adjust frequency responses through learnable spectral weights, extract trend and seasonal components, and preserve original sequence information through residual fusion. Differential cross-attention (DCA) is then introduced into Timexer to strengthen cross-variable dependency modeling and reduce the interference of redundant information. Finally, key model hyperparameters are tuned with the crested porcupine optimizer (CPO). The framework’s effectiveness is demonstrated through experiments in the carbon markets of Guangdong, Hubei, and Shanghai. Compared with the original Timexer, CPO-TDTimexer yields average reductions of 24.48% in MAE, 22.35% in RMSE, and 25.02% in MAPE. Ablation studies further confirm the complementary contributions of TESA, DCA, and CPO. Diebold–Mariano tests support the statistical significance of the forecasting improvements achieved by CPO-TDTimexer over the benchmark models.