Fractional-order grey forecasting framework for cross-regional air quality early warning

Purpose This study develops an interpretable grey forecasting framework under limited information conditions. The framework supports air quality early warning across regions. It focuses on disturbed annual Air Quality Composite Index (AQCI) sequences and heterogeneous warning pressure across cities. Design/methodology/approach This study proposes S-HFGM(1,N), a Hausdorff fractional multivariable grey model with a single indicator buffer. The model integrates target sequence buffering, Hausdorff fractional accumulation and multivariable grey modelling. The framework uses AQCI association pressure for city screening and selects socioeconomic drivers. After validation, Monte Carlo simulations based on residuals quantify uncertainty. Warnings are then interpreted for specific pollutants. The application covers the Beijing–Tianjin–Hebei region. It uses observations from 2013–2022 for training and 2023–2024 for validation. The validated model produces projections for 2025–2030. Findings The model achieves an average validation mean absolute percentage error of 3.6861%, outperforming selected grey forecasting benchmarks. The baseline projections show different trends under gradual structural evolution. Beijing faces slight upward pressure. By contrast, Shijiazhuang and Tangshan continue improving. Handan and Xingtai follow a relatively stable trajectory despite persistent pressure. The 80% and 95% prediction intervals widen with the forecast horizon, with Handan and Xingtai showing the widest 95% intervals in 2030. Future warning pressure is mainly associated with O3 and particulate matter. Practical implications The framework suggests that regional governance should move beyond uniform control measures. Instead, policies should reflect the pollutant profile of each city. Coordinated monitoring should also be strengthened for nonadjacent priority cities. Originality/value The unified grey framework combines disturbance weakening, nonlinear time scale memory reconstruction and structural external driving. It extends grey forecasting beyond single city prediction. This extension supports warning interpretation across regions for environmental policymakers and regional governance bodies.