The nonlinear multi-variable grey Bernoulli model and its applications

This work uses the vector-valued Bernoulli equation to build a nonlinear multi-variable grey Bernoulli model, which is available to describe the nonlinear relationship between the output variables. By using Pade´ approximation, the proposed model can be implemented with high time efficiency. Additionally, the Sine Cosine Algorithm is employed to determine the Bernoulli exponent, thereby enhancing prediction accuracy. To evaluate the predictive performance of the proposed model, three case studies using three real-world data sets with different features of predicting per capita household income, fuel prices and crude oil prices are carried out. The results are compared with three existing grey multi-input multi-output models. Experimental results demonstrate that the proposed model excels in handling nonlinear relationships between variables and has strong robustness against noise, consistently delivering lower error values, demonstrating superior predictive performance.