GM(1,1)幂模型的病态性

针对GM(1,1)幂模型参数辨识过程中可能出现的病态性问题, 首先基于矩阵求逆的条件数分析灰色模型病态程度的衡量方法, 然后按照GM(1,1)幂模型的背景值和幂指数的不同取值, 分三种情形讨论了数据矩阵求逆条件数的取值范围, 在此基础上总结影响GM(1,1)幂模型病态性的主要因素, 并通过实例加以验证. 结果表明, 在部分情形下GM(1,1)幂模型的数据矩阵求逆不存在病态性, 但在部分情形下可能出现数据矩阵求逆的病态性, 其中, 背景值和幂指数是影响模型病态性的直接因素.

The morbidity of GM(1,1) power model

Aiming at the ill-conditioned problems in parameter identification of GM(1,1) power model, this paper first analyzes the measure method of the degree of morbidity of GM(1,1) power model based on the condition number of matrix inversion, then discusses the range of the condition number in three cases for different values of the background value and power index. On this basis, this paper summarizes the main factors which affect the morbidity of GM(1,1) power model and verifies the conclusions by an actual example. The results show that there is no morbidity in some cases. However, in some cases, the model is likely to be ill-conditioned. The background value and power index are main direct factors which affect the morbidity of GM(1,1) power model.