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基于卡尔曼滤波的MGM-多维~AR($p$) 模型的构建及其应用

熊萍萍1,2,檀成伟2,3,闫书丽1,姚天祥1   

  1. 1. 南京信息工程大学管理工程学院, 南京 210044; 2. 南京信息工程大学江苏省统 计科学研究基地,南京 210044; 3. 南京信息工程大学数学与统计学院, 南京 210044
  • 出版日期:2018-04-25 发布日期:2021-06-29

熊萍萍, 檀成伟, 闫书丽, 姚天祥. 基于卡尔曼滤波的MGM-多维~AR($p$) 模型的构建及其应用[J]. 系统科学与数学, 2021, 41(4): 1131-1149.

XIONG Pingping , TAN Chengwei , YAN Shuli , YAO Tianxiang. Construction and Application of MGM-Multidimensional AR ($p$) Model Based on Kalman Filter[J]. Journal of Systems Science and Mathematical Sciences, 2021, 41(4): 1131-1149.

Construction and Application of MGM-Multidimensional AR ($p$) Model Based on Kalman Filter

XIONG Pingping1,2 , TAN Chengwei2,3 , YAN Shuli1 , YAO Tianxiang1   

  1. 1.  School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing210044; 2. Jiangsu Statistical Science Research Base, Nanjing University of Information Science and Technology, Nanjing 210044;3. College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044
  • Online:2018-04-25 Published:2021-06-29
由于受到外界不确定性因素的干扰, 导致实际数据偏离 模拟的趋势, 使得灰色多变量~MGM($1, m$) 模型预测效果不佳, 而多维平稳序列自回归模型~(AR($p$)) 能够有效反应具体数据与整体趋势之间产生的偏差, 从而可以掌握外界环境对目标数据发展趋势带来的影响. 由此文章首先利用卡尔曼滤波对给定的小样本数据做平滑处理, 消除数据观测时产生的噪声误差, 然后根据~MGM($1, m$) 模型对处理后的数据建模, 将得到的模拟预测值作为样本数据的趋势项, 并将残差作为样本数据的随机项, 再通过多维~AR($p$) 模型对随机项进行分析, 最后将~MGM($1, m$) 模型的趋势项与多维~AR($p$) 模型模拟的随机项相加得到基于卡尔曼滤波的~MGM- 多维~AR($p$) 模型的模拟预测值. 将该模型和~MGM($1, m$) 模型, 多维~AR($p$) 模型和~GM-AR 组合模型分别应用于衡量杭州市雾霾程度的相关指标中建模分析, 结果表明: 文中提出的组合优化模型相比其他3个模型, 拟合效果更佳, 预测结果更精确.
In this paper, a new hybrid model is proposed to improve the prediction accuracy of multivariate grey model (MGM($1, m$)) when it is used to simulate oscillation data. Firstly, Kalman filter is used to eliminate the noise error in the measurement of observation data. Secondly, according to MGM($1, m$) model, the overall development trend of oscillation data can be well reflected, and the residual error between the simulated value and the real data is close to stable distribution. Therefore, the Multidimensional Stationary Sequence Autoregressive (AR($p$)) model is introduced to analyze the residuals, and the variation rules of the residuals are obtained. Finally, the simulated predicted values of MGM($1, m$) model and the residual values calculated by multidimensional AR($p$) model are integrated to obtain the final results of the MGM-multidimensional AR($p$) model. The new combination model and other three models are applied to an example analysis, and the accuracy and feasibility of the new model are verified by comparative analysis. The results show that the new hybrid model can effectively improve the simulation and prediction accuracy of MGM($1, m$) for oscillation data, and broaden the application range of the model.}
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