The Relationship of Discrete Grey Forecasting Model DGM and GM(1,1) Model

This paper studies theoretically the relative error between the discrete grey forecasting model abbreviated as DGM and the GM(1,1) model. Firstly, in the two applicable ranges (-2,0] and [0,2) of the developing coefficient a, the upper bound of each relative error at time k between DGM and GM(1,1) is calculated Then, by using the properties of a bivariate differentiable function, we study the relevant properties of the relative error upper bounds r(1)(a,k,n) and r(2)(a,k,n). Further; we derive the quantitative relation between the relative error and the developing coefficient a. The results show that when a is within the range [-0.2, 1] the relative error between DGM and GM(1,1) at time k is less than 5.21%, and hence DGM and GM(1,1) can substitute appositely each other; However, when a is in the range (-2,-0.4] or [1,2) the relative error may exceed 12.46%. Therefore, this work provides a quantitative criterion for whether DGM and GM(1,1) model can reasonably substitute each other.