Adaptive Variational Mode Decomposition via RIME Optimization and Fuzzy Entropy for PatchTST-Based Financial Time-Series Forecasting

The inherent non-linear, non-stationary, multi-scale nature of financial time-series data makes the task of financial forecasting exceptionally difficult. While deep learning models incorporating decomposition techniques have achieved significant progress in forecasting, their performance heavily relies on the signal’s quality during decomposition and the setting of parameters. To this end, this study presents a forecasting framework for the S&P 500 index using RFEO–VMD–PatchTST (RIME and Fuzzy Entropy Optimized Variational Mode Decomposition with PatchTST). We utilized the RIME optimization algorithm to autonomously obtain VMD’s optimal number of modes K and penalty term. To identify which is the best-performing model among the evaluated models under the reported test protocol, i.e., Sample Entropy (SE), Fuzzy Entropy (FE), and Permutation Entropy (PE), were discussed and used as objectives for optimization. In particular, the experimental analysis showed that Fuzzy Entropy achieves better decomposition quality with K = 2, producing two adaptive IMFs and yields the optimal decomposition that represents the primary driving forces of market trends. Subsequently, we independently model each IMF using the Patch Time Series Transformer (PatchTST) and then combine the IMF forecasts to obtain the final prediction. Experiments with various evaluation criteria were conducted on S&P 500 index data, and the results show that our framework consistently outperforms conventional methods, including statistical, machine learning, deep learning, and decomposition-based approaches. Additionally, to verify that our framework is robust, stable, and statistically significant, we provide ablation studies, multi-horizon forecasting, rolling-window analysis, random-seed experiments, and Diebold–Mariano tests.