利用最小一乘法改进的灰色模型的导航卫星钟差预报
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摘要
在卫星钟差波动较大的情况下,为了克服基于最小二乘法估计灰色模型参数对卫星钟差预报精度的不足,本文利用最小一乘法对传统灰色模型进行了改进。在建模的过程中,采用以误差绝对值之和最小为优化原则,针对目标函数不可微的特点,运用线性规划的方法对灰色预报模型的模型参数进行了估计。此外,将改进后的预报模型应用到卫星钟差波动较大情况下钟差的预报中,并将预报结果与传统灰色模型的预报结果进行了对比分析。结果表明:在卫星钟差波动较大的情况下,该方法相比传统灰色模型的预报结果有显著改善,为高精度的卫星钟差预报提供了一种新思路。
In order to overcome the shortcomings of satellite clock bias( SCB) forecasting accuracy based on the grey model( GM( 1,1)) parameters of least squares estimation in the case of large SCB fluctuations,this paper uses the least absolute deviations( LAD) to improve the GM( 1,1). During the modeling process,the principle of minimizing the absolute sum of errors is used as the optimization principle. Aiming at the non-differentiable characteristics of the objective function,the linear programming method is used to estimate the model parameters of the GM( 1,1). In addition,the improved forecasting model is applied to the forecast of the SCB when the SCB fluctuates greatly,and the forecast results are compared with the results predicted by GM( 1,1). The results show that the prediction results of the GM( 1,1)-LAD are significantly improved compared with the GM( 1,1) in the case of large fluctuations in the SCB,which provides a new idea for high-precision SCB prediction.
In order to overcome the shortcomings of satellite clock bias( SCB) forecasting accuracy based on the grey model( GM( 1,1)) parameters of least squares estimation in the case of large SCB fluctuations,this paper uses the least absolute deviations( LAD) to improve the GM( 1,1). During the modeling process,the principle of minimizing the absolute sum of errors is used as the optimization principle. Aiming at the non-differentiable characteristics of the objective function,the linear programming method is used to estimate the model parameters of the GM( 1,1). In addition,the improved forecasting model is applied to the forecast of the SCB when the SCB fluctuates greatly,and the forecast results are compared with the results predicted by GM( 1,1). The results show that the prediction results of the GM( 1,1)-LAD are significantly improved compared with the GM( 1,1) in the case of large fluctuations in the SCB,which provides a new idea for high-precision SCB prediction.
引文
[1]于烨,张慧君,李孝辉.顾及钟差随机项的GPS卫星钟差预报[J].测绘通报,2018(6):1-6.
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[3]于烨,张慧君,李孝辉.组合模型在卫星钟差中长期预报中的应用和比较[J].电子测量技术,2017,40(11):7-11.
[4]HUANG G W,ZHANG Q.Real-time estimation of satellite clock offset using adaptively robust Kalman filter with classified adaptive factors[J].GPS Solutions,2012,16(4):531-539.
[5]于烨,张慧君,李孝辉.基于一阶差分修正指数曲线法的GPS卫星钟差预报[J].探测与控制学报,2018,40(4):94-98.
[6]LI M,QU L,ZHAO Q,et al.Precise point positioning with the Bei Dou Navigation Satellite System[J].Sensors,2014,14(1):927-943.
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[8]李晓宇,杨洋,胡晓粉,等.基于改进灰色ARMA模型的卫星钟差短期预报研究[J].大地测量与地球动力学,2013,33(1):59-63.
[9]LIANG Y,REN C,YANG X,et al.A grey model based on first differences in the application of satellite clock bias prediction[J].Chinese Astronomy and Astrophysics,2016,40(1):79-93.
[10]YU Y,ZHANG H,LI X.Three kinds of algorithm and its performance analysis in the prediction of satellite clocks bias[C]∥Proceedings of the 13th IEEE International Conference on Electronic Measurement and Instruments.[S.l.]:IEEE,2017:536-543.
[11]梁月吉,任超,杨秀发,等.结合双树复小波和广义回归神经网络的钟差预报方法研究[J].测绘通报,2016(1):6-10.
[12]路晓峰,杨志强,贾小林,等.灰色系统理论的优化方法及其在卫星钟差预报中的应用[J].武汉大学学报(信息科学版),2008,33(5):492-495.
[13]于烨.存在钟跳情况下的卫星钟差预报[C]∥中国天文学会2017年学术年会摘要集.乌鲁木齐:中国天文学会,2017:52-52.
[14]崔先强,焦文海.灰色模型在卫星钟差预报中的应用[J].武汉大学学报(信息科学版),2005,30(5):447-450.
[15]于烨,张慧君,李孝辉,等.基于Vondrak滤波1阶差分的灰色模型的卫星钟差预报方法[J].天文学报,2018,59(2):75-84.
[16]刘明.最小一乘法与最小二乘法:基于例证的比较[J].统计与决策,2012(20):12-15.
[17]顾乐民.曲线拟合的最小一乘法[J].同济大学学报(自然科学版),2011,39(9):1377-1382.
[18]谢开贵,宋乾坤,周家启.最小一乘线性回归模型研究[J].系统仿真学报,2002,14(2):189-192.
[2]HUANG G W,ZHANG Q,XU G C.Real-time clock offset prediction with an improved model[J].GPS Solutions,2014,18(1):95-104.
[3]于烨,张慧君,李孝辉.组合模型在卫星钟差中长期预报中的应用和比较[J].电子测量技术,2017,40(11):7-11.
[4]HUANG G W,ZHANG Q.Real-time estimation of satellite clock offset using adaptively robust Kalman filter with classified adaptive factors[J].GPS Solutions,2012,16(4):531-539.
[5]于烨,张慧君,李孝辉.基于一阶差分修正指数曲线法的GPS卫星钟差预报[J].探测与控制学报,2018,40(4):94-98.
[6]LI M,QU L,ZHAO Q,et al.Precise point positioning with the Bei Dou Navigation Satellite System[J].Sensors,2014,14(1):927-943.
[7]于烨,张慧君,李孝辉,等.基于GM(1,1)和MECM组合模型的GPS卫星钟差中短期预报[J].天文学报,2018,59(3):19-30.
[8]李晓宇,杨洋,胡晓粉,等.基于改进灰色ARMA模型的卫星钟差短期预报研究[J].大地测量与地球动力学,2013,33(1):59-63.
[9]LIANG Y,REN C,YANG X,et al.A grey model based on first differences in the application of satellite clock bias prediction[J].Chinese Astronomy and Astrophysics,2016,40(1):79-93.
[10]YU Y,ZHANG H,LI X.Three kinds of algorithm and its performance analysis in the prediction of satellite clocks bias[C]∥Proceedings of the 13th IEEE International Conference on Electronic Measurement and Instruments.[S.l.]:IEEE,2017:536-543.
[11]梁月吉,任超,杨秀发,等.结合双树复小波和广义回归神经网络的钟差预报方法研究[J].测绘通报,2016(1):6-10.
[12]路晓峰,杨志强,贾小林,等.灰色系统理论的优化方法及其在卫星钟差预报中的应用[J].武汉大学学报(信息科学版),2008,33(5):492-495.
[13]于烨.存在钟跳情况下的卫星钟差预报[C]∥中国天文学会2017年学术年会摘要集.乌鲁木齐:中国天文学会,2017:52-52.
[14]崔先强,焦文海.灰色模型在卫星钟差预报中的应用[J].武汉大学学报(信息科学版),2005,30(5):447-450.
[15]于烨,张慧君,李孝辉,等.基于Vondrak滤波1阶差分的灰色模型的卫星钟差预报方法[J].天文学报,2018,59(2):75-84.
[16]刘明.最小一乘法与最小二乘法:基于例证的比较[J].统计与决策,2012(20):12-15.
[17]顾乐民.曲线拟合的最小一乘法[J].同济大学学报(自然科学版),2011,39(9):1377-1382.
[18]谢开贵,宋乾坤,周家启.最小一乘线性回归模型研究[J].系统仿真学报,2002,14(2):189-192.