基于初始条件优化的GM(1,1)幂模型及其应用

针对GM（1，1）幂模型幂指数和初始条件优化问题，提出了一种基于初始条件和幂指数协同优化的方法。根据新信息优先原理，通过引入权重信息控制函数优化初始条件，表现新旧信息在初始条件构建中作用大小的变化规律，最大限度提取小样本序列中的有效信息，反应新旧信息共同对系统趋势变化的影响；以平均相对误差最小化为目标，参数间约束关系作为条件，构建非线性优化模型，实现GM（1，1）幂模型的幂指数和初始条件协同优化。最后，通过我国网络购物用户规模预测实例研究表明，优化的模型实现模型平均相对误差在理论上的最小化，其建模效果要优于其他对比模型，并将其用于2016-2020年网购用户规模预测，表明本文模型的实用性和有效性。

Construction and Application of GM(1,1) Power Model based on the Optimized Initial Condition

The GM(1,1) power model has been used in many fields, such as electricity loading forecast, industrial waste emissions, and urban water consumption, ever since it was proposed by professor Deng.Specifically, this model has wide applications, because it has a dynamic adaptive power index, which can adapt to different characteristics of various sequences. However, this model still has its drawbacks that the power exponent is difficult to obtain and the conventional initial condition does not satisfy the principle of new information priority. Therefore, large error gap exists between forecasts and original observations. To this end, approaches on collaborative optimization of power exponent and initial condition are put forward to obtain a superior grey power model. The detailed procedures for this novel can be described as follows.Initially, based on the principle of new information priority, the function of controlling weight is introduced to optimize the initial condition, which reflects the effect of new-old information on the trend of the system together with the max use of existing information and expresses the changing law. Subsequently, the non-linear optimized model is constructed to optimize the power exponent and initial condition together with the target of minimizing the average relative error. Then the optimal initial condition and power index can be obtained. Lastly, this new proposed grey power model is employed to forecast the amount of online shopper, compared with three competing models, namely GPM(1,1,x⁽¹⁾(1)), GPM(1,1,x⁽¹⁾(n)), and GPM(1,1,β). Experimental results illustrate that the novel optimized model achieves the best performance among the four models and gets the minimum value of average relative error theoretically. Therefore, this novel grey model can be used for future amount prediction of China's online shopper.For the future forecasts, the novel grey power model having optimized initial condition and optimal power exponent is utilized to estimate the amount of China's online shopper from 2016 to 2020. Additionally, the real observations from 2016 to 2018 are collected to match the forecasted values. The compared results show that the errors between forecasts and original data from 2016 to 2018 are all less than 10%, which means the great performance of this proposed model. Therefore, conclusions can be made that this newly designed grey power model has good validity and practicability and it can be further used in other fields.Moreover, according to the predicted results, the amount of China's online shoppers will reach 900 million in 2020. Many residents will shop on the internet instead of outdoor shopping.