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.

本研究提出了一种弱化分数阶累积算子来缓解离散灰色系统模型的病态，旨在改进灰色系统理论。发现由改进的分数阶累加算子和乘法变换组成的弱化分数阶累加算子不仅可以通过减小列（行）元素之间的差异来缓解系统的病态。系数矩阵还能进一步提升模型的预测性能。因此，弱化分数阶累加算子是一种有效的改进措施。用弱化分数阶累积算子证明离散灰色预测模型的无偏性和仿射变换特性，进一步加强了这一新系统的理论基础。以两个真实世界时间序列为例，对比基于其他五种不同累积算子（一阶累积操作、新信息累加运算，分数阶累加运算符，阻尼累积生成算子和一致分数阶累积算子）。对比分析结果表明，提出的削弱分数阶累加算子不仅可以大幅度降低模型的病态，而且具有良好的预测性能，这两者都证实了该方法的可行性和有效性。