Interpretable and probabilistic forecasting of financial time series via hybrid RBF-KAN-Transformer with Bayesian uncertainty quantification

Financial forecasting remains central to quantitative finance, yet it is persistently challenged by non-stationarity, volatility clustering, and the high demands for accuracy and reliability. We introduce the RBF-KAN-Transformer, an innovative hybrid deep architecture that integrates Radial Basis Functions (RBF) for local nonlinear encoding, Kolmogorov–Arnold Network (KAN)-inspired interpretable projections, and a Transformer for global temporal modeling. The model is evaluated on two major financial indices-the Shanghai Shenzhen China Securities Index (CSI) 300 and Nasdaq 100 Futures, demonstrates state-of-the-art (SOTA) performance. To ensure statistical reliability, we apply a bias correction method based on the residual mean estimated during walk-forward validation (yˆcorr=yˆ+μres), which eliminates systematic forecast drift and yields statistically negligible bias (paired t-test p=1.000). Uncertainty quantification is achieved through Multipass Bayesian Estimation (MPBE) and Variational Inference (VI), producing well-calibrated 95% prediction intervals (Prediction Interval Coverage Probability, PICP = 94.8% on CSI 300). On the CSI 300, our model achieves R2=0.9864, Mean Absolute Percentage Error (MAPE) = 1.19%, and Modified Directional Accuracy (MDA) = 64.18%; it maintains strong performance on the more volatile Nasdaq 100 (R2=0.9840, MAPE = 4.20%). Ablation studies confirm the necessity of each core component—RBF encoding, KAN-inspired projection, Transformer encoder, and uncertainty-aware output head with RBF-KAN-Transformer-VI variant consistently outperforming recent SOTA models across accuracy, calibration, and efficiency. Moreover, the model offers multi-level interpretability: RBF centers identify salient market regimes, KAN-style functions enable symbolic decomposition, and attention weights reveal temporal dependencies. This work establishes a new benchmark for interpretable, uncertainty-aware, and statistically robust financial time series forecasting.