Purpose This study proposes a hybrid forecasting framework integrating grey system models (GM) with Gaussian process regression (GPR) to enhance prediction accuracy under small-sample conditions. While GM is effective in capturing trend components under small-sample conditions, it is less suited for modeling nonlinear fluctuations and providing probabilistic uncertainty quantification. To complement this, the proposed framework employs GM to model the trend sub-series, while GPR is applied to the residuals in a probabilistic manner, thereby enabling both accurate forecasts and robust nonlinear interval estimation. Design/methodology/approach The proposed framework first smooths the trend using the moving average method and then extracts seasonal and residual components. The GM model is employed to capture the trend, while the GPR model generates point forecasts and interval estimates for residuals. Particle swarm optimization is applied to optimize GM’snonlinear parameters, improving overall accuracy and robustness. Findings Experiments show that the nonlinear grey Bernoulli model with fractional-order accumulation-GPR framework outperforms the benchmarks in predictive accuracy. Case studies confirm that combining time series decomposition, grey system modeling and GPR enhances both accuracy and robustness in energy consumption forecasting. Originality/value The primary innovation of this study is a hybrid framework combining trend forecasting with probabilistic residual modeling, using GPR for nonlinear interval estimation. Comparisons with six typical hybrid grey models demonstrate superior predictive performance, offering a novel approach for complex time series modeling and energy forecasting.
Ma et al. (2026) studied this question.