Error and its upper bound estimation between the solutions of GM(1, 1) grey forecasting models

• Then the relative error uniform upper bounds between the GM(1, 1, W) model and GM(1, 1, C) model were deduced.
• The quantitative relation between the relative error uniform upper bound and the developing coefficient was obtained.
• The result provides a criterion for whether the connotation model can be substituted reasonably by the whitenization model.

The solution of a whitenization GM(1, 1) model, called GM(1, 1, W), instead of a connotation GM(1, 1) model (GM(1, 1, C)) is usually used to forecast the trend of a grey system. This paper discusses the relative error between the solutions of these two models. First, the relative error upper bounds are derived when the developing coefficient a is located in two applicable ranges, (−2, 0] and [0, 2). Treating the relative error upper bound as a function of a and the observation time k, its properties are researched to obtain the upper bound of this relative error upper bound function, which can be called the relative error uniform upper bound. The relation between this uniform upper bound and the data sequence length n is ultimately demonstrated. The research indicates that the relative error between the solutions of GM(1, 1, W) and GM(1, 1, C) is less than 3.31% when the developing coefficient a is within the range [−1/(n + 1), 2/(n + 2)] (n ⩾ 4), and hence GM(1, 1, C) can be replaced appropriately with GM(1, 1, W). However, the relative error between the solutions of these two models may exceed 19.97% when the developing coefficient a is out of the range [−1/(n + 1), 2/(n + 2)], and in this case, GM(1, 1, C) cannot be replaced fittingly with GM(1, 1, W). Therefore, this work provides a criterion for whether the connotation model can be substituted reasonably by the whitenization model.