基于最小二乘法的灰色系统应用

Application of Grey System Based on Least Square Method

本文主要研究了长沙空气质量与多种影响因素之间的关系,利用各季度AQI的平均值作为空气质量好坏的衡量指标。首先,利用灰色关联分析的方法,对各个影响因素做了定性的分析,计算出各个影响因素对各季度AQI平均值的相对关联度。根据灰色关联分析的结果做优势分析,并得出影响各季度AQI平均值的五个主要影响因素,分别为日均最高温、降水量、绿化面积、日均最低温和大风级天数。然后,根据优势分析的结论,将五个主要影响因素作为自变量,各季度AQI平均值作为因变量,并假设空气质量与主要影响因素的关系为线性关系,做了相应的回归分析。根据所收集的数据,利用最小二乘法给出了各个参数的无偏估计,从而建立了主要影响因素与各季度AQI平均值之间的数学表达式,用于制定空气质量的改善方案。最后,建立了灰色系统下的GM(1,1)模型,并利用最小二乘原理对模型进行了白化求解,然后对2017第四季度的AQI平均值和2018第一季度的AQI平均值进行了预测。并对预测结果进行了残差检验,发现预测效果并不理想,在文章结尾处给出了改进的GM(1,1)模型,并重新对后两个季度的AQI平均值进行预测,预测结果为:2017年第四季度AQI平均值的预测值为67.3718,2018年第一季度AQI平均值的预测值为84.9393,与实际结果差别不大,说明了模型的有效性。

This paper mainly studies the relationship between air quality and various influencing factors in Changsha, using the average AQI of each quarter as a measure of air quality. First of all, using the method of grey correlation analysis, the paper makes a qualitative analysis of each influencing factor, and calculates the relative correlation degree of each influencing factor to the average AQI of each quarter. According to the results of the grey correlation analysis, the superiority analysis is made, and five main factors influencing the average AQI of each quarter are obtained, which are the highest daily temperature, precipitation, green area, lowest daily temperature and gale days. Then, according to the conclusion of advantage analysis, five main influencing factors are taken as independent variables, the average AQI of each quarter is taken as dependent variable, and the relationship between air quality and main influencing factors is assumed to be linear, and the corresponding regression analysis is made. According to the collected data, the unbiased estimation of each parameter is given by using the least square method;thus the mathematical expression between the main influencing factors and the average AQI of each quarter is established, which is used to formulate the air quality improvement plan. Finally, the GM (1,1) model under the grey system is established, and the model is whitened by using the least square principle. And the AQI average in the fourth quarter of 2017 and the first quarter of 2018 are predicted. Then the residual test is carried out on the prediction results, and it is found that the prediction effect is not ideal. At the end of the article, an improved GM (1,1) model is given, and the AQI average value of the last two quarters is predicted again. The prediction result is: the AQI average value in the fourth quarter of 2017 is 67.3718, and the AQI average value in the first quarter of 2018 is 84.9393, which is not much different from the actual result, indicating the effectiveness of the model.