Weakened fractional-order accumulation operator for ill-conditioned discrete grey system models

Abstract

In this study, the weakened fractional-order accumulation operator for alleviating the ill-condition of discrete grey system models with the aim of improving the grey system theory is proposed. It is found that the weakened fractional-order accumulation operator composed of the improved fractional-order accumulation operator and the multiplicative transformation can not only alleviate the ill-condition of the system by decreasing the differences between the elements of the columns (rows) in the coefficient matrix but also further enhance the prediction performance of the models. Therefore, the weakened fractional-order accumulation operator is an effective improvement measure. The demonstration of the unbiasedness and affine transformation property of the discrete grey forecasting models with the weakened fractional-order accumulation operator further strengthens the theoretical basis of this new system. Two real-world time series are used as cases to demonstrate the effectiveness of the discrete grey system models with the weakened fractional-order accumulation operator compared with discrete grey forecasting models based on five other different accumulation operators(1-order accumulation operation, new information accumulation operation, fractional-order accumulation operator, damping accumulative generating operator and the conformable fractional-order accumulation operator). The results of the comparative analysis show that the proposed weakened fractional order accumulation operator can not only substantially reduce the ill-condition of the models but also have good predictive performance, both of which confirm the feasibility and validity of the method.