The traditional grey Bernoulli model often faces limitations when applied to pollutant concentration series, which may exhibit complex seasonal trends and varying data types. To address these challenges, we propose a structural extension of the traditional grey Bernoulli model by integrating a binomial equation. This extension allows for a more flexible framework suitable for diverse datasets, especially those related to environmental pollution.
First, the pollutant concentration time series is decomposed into four relatively stable seasonal sub-sequences. Binomial and nonlinear grey Bernoulli models are then integrated to predict these sub-sequences. The prediction formula of the proposed model is derived directly from the definition equation rather than from the solutions of the grey differential equation, thereby minimizing systematic errors. The particle swarm optimization algorithm is used to estimate the nonlinear parameters, while the least squares method is used to estimate the linear parameters of the model.
The BNGBM(1,1) model is used to forecast the air quality index (AQI), sulfur dioxide (SO2) concentration and particulate matter (PM2.5) concentration for seven major regions in China. The prediction results show that BNGBM(1,1) has superior accuracy compared to four competing models. The model predicts the seasonal variations of these three air pollution indicators in the selected regions for the period 2023–2024. The results show that the concentrations of all three pollution indices will decrease at different rates.
The grey Bernoulli model is well suited to sequences exhibiting quasi-exponential growth, whereas the polynomial model is more appropriate for sequences characterized by saturated growth. The integration of these two models extends their applicability. In the empirical study, despite the different development trends of the three air quality indicators in different regions of China, the proposed forecasting method demonstrates effective prediction performance for these indicators.
