A novel optimized grey model with quadratic polynomials term and its application

The grey prediction model has been widely used in various fields and demonstrated good performance. However, when the data shows non-homogeneous exponential characteristic, the effect of the grey prediction model performs poorly. Therefore, a grey prediction model with a quadratic polynomial term (denoted as NGM(1,1,$k^2$) is developed. The NGM(1,1,$k^2$) model is generalized, the GM(1,1) model, the GM(1,1,k) model, the SAIGM model and the GM(1,1,$k^2$) model are the special forms of it. Moreover, the parameter characteristics of the NGM(1,1,$k^2$) model and the effect on the modeling precision are evaluated under the multiplication transformation. To make the NGM(1,1,$k^2$) model more precise, we further analyze the error of the NGM(1,1,$k^2$) model and propose a new model, named BNGM(1,1,$k^2$) model, of which the background value is reconstructed based on the Simpson formula. Subsequently, the effectiveness of the new model is verified through four cases. The result shows that the prediction accuracy of the BNGM(1,1,$k^2$) model is significantly improved. Finally, the BNGM(1,1,$k^2$) model is applied to analyse and predict the Gross Domestic Product (GDP) of Chongqing’s primary industry, the total power of Chongqing’s agricultural machinery and the GDP of Chongqing’s wholesale and retail trades, which shows the prediction performance of the new model is superior to other models.