An extensive conformable fractional grey model and its application

In order to expand the applicability of the conventional conformable fractional grey model, an extensive conformable fractional grey model, abbreviated as ECFGM(1,1), is proposed by the introduction of the extensive conformable fractional accumulation. Specifically, the improvements of the proposed model can be outlined as follows. First, the extensive fractional accumulation and difference are designed for accumulated generating operation and its reverse calculation, respectively. Second, based on the extensive form, the parameter estimation and time response function of the ECFGM(1,1) model are deduced, thereinto, the particle swarm optimization algorithm is employed to determine the optimal fractional order for the newly-designed model. It is worthy noting that an algorithm framework by the Monte Carlo simulation and parameter sensitivity analysis is conducted to assess the robustness of the proposed model. To validate this model’s efficacy, the novel technique is adopted to forecast China’s primary energy consumption compared with a series of competitive models. The numerical results indicate the newly-proposed model is superior to all competitors in terms of MAPE and RMSE values, thus, the proposed ECFGM(1,1) model is considered a powerful and promising method for enhancing the existing fractional grey models.