基于误差最小化的GM(1,1) 模型背景值优化方法

背景值是导致GM(1,1)模型产生系统误差的主要原因之一。对此,提出一种优化的GM(1,1)模型构建方法。首先,根据GM(1,1)模型时间响应式的函数形式,利用积分中值定理拟合真实背景值,研究发展系数与背景值之间的关系；然后,构建新的灰色微分方程,采用最小二乘法进行参数估计,并利用方程组还原原始参数,使背景值同时具备无偏性和最小误差性；最后,通过具体案例验证了所提出的优化模型能够突破高增长建模的局限,对实际问题的建模精度较高。

Optimization method of background value in GM(1,1) model based on least error

The formula of background value is one of the main factors causing systematic error of GM(1,1) model. A construction method for optimizing GM(1,1) model is proposed. According to structure characteristics of GM(1,1) time response function, the mean value theorem of integral is used to fit the real background value, and the relationship between the background value and the development rate is analyzed. Then, a new grey differential equation is constructed, and the parameter vector is evaluated by using the least square method, and the original parameters are restored by equations system. The new background satisfies the unbiased property and least error. Finally, a numerical case shows that the proposed algorithm breaks the confine of modeling high growth sequences, and the application indicates that the optimized model has an obviously high accuracy in the actual problem.