病态不确定性平差模型的岭估计算法
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摘要
测量数据在获取的过程中,常存在不确定性,它们会影响参数估计结果,不确定性平差模型的解算方法可以有效提高参数估计的有效性和可靠性。当观测方程的系数矩阵存在接近零的奇异值,采用岭估计可有效抑制观测方程病态性对参数估值结果的影响。当不确定性平差模型出现病态,其受系数矩阵误差和观测值误差的影响更为严重,本文将岭估计法应用于病态不确定性平差模型,推导了迭代算法,以提高解的稳定性,并用算例验证,结果表明了新方法的有效性和可行性。
Uncertainties usually exist in the process of acquisition of measurement data, which affects the parameter estimation results. The solution method of uncertainty adjustment model can effectively improve the validity and reliability of parameter estimation. When the coefficient matrix of the observation equation has a singular value close to zero, the ridge estimation can effectively suppress the influence of the ill-posed state of the observation equation on the parameter estimation results.When the uncertainty adjustment model is ill-posed, it is more seriously affected by the error of the coefficient matrix and the observation,this paper applies ridge estimation method to ill-posed uncertainty adjustment model, derives an iterative algorithm to improve the stability and reliability of the result, and verifies it with two examples. The results show that the new method is effective and feasible.
Uncertainties usually exist in the process of acquisition of measurement data, which affects the parameter estimation results. The solution method of uncertainty adjustment model can effectively improve the validity and reliability of parameter estimation. When the coefficient matrix of the observation equation has a singular value close to zero, the ridge estimation can effectively suppress the influence of the ill-posed state of the observation equation on the parameter estimation results.When the uncertainty adjustment model is ill-posed, it is more seriously affected by the error of the coefficient matrix and the observation,this paper applies ridge estimation method to ill-posed uncertainty adjustment model, derives an iterative algorithm to improve the stability and reliability of the result, and verifies it with two examples. The results show that the new method is effective and feasible.
引文
[1] 杨元喜.卫星导航的不确定性、不确定度与精度若干注记[J].测绘学报,2012,41(5):646-650.YANG Yuanxi.Some notes on uncertainty,uncertainty measure and accuracy in satellite navigation[J].Acta Geodaetica et Cartographica Sinica,2012,41(5):646-650.
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[3] 宋迎春,谢雪梅,陈晓林.不确定性平差模型的平差准则与解算方法[J].测绘学报,2015,44(2):135-141.DOI:10.11947/j.AGCS.2015.20130213.SONG Yingchun,XIE Xuemei,CHEN Xiaolin.Adjustment criterion and algorithm in adjustment model with uncertain[J].Acta Geodaetica et Cartographica Sinica,2015,44(2):135-141.DOI:10.11947/j.AGCS.2015.20130213.
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[5] 王新洲.最小不确定度约束下的极大可能性估计[J].测绘工程,2003,12(1):5-8.WANG Xinzhou.Maximum possibility estimation restricted by least uncertainty[J].Engineering of Surveying and Mapping,2003,12(1):5-8.
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[9] 宋迎春,金昊,崔先强.带有不确定性的观测数据平差解算方法[J].武汉大学学报(信息科学版),2014,39(7):788-792.SONG Yingchun,JIN Hao,CUI Xianqiang.Adjustment algorithm about observation data with uncertain[J].Geomatics and Information Science of Wuhan University,2014,39(7):788-792.
[10] 王志忠,陈丹华,宋迎春.具有不确定性平差算法[J].测绘学报,2017,46(7):834-840.DOI:10.11947/j.AGCS.2017.20160522.WANG Zhizhong,CHEN Danhua,SONG Yingchun.An algorithm in adjustment model with uncertainty[J].Acta Geodaetica et Cartographica Sinica,2017,46(7):834-840.DOI:10.11947/j.AGCS.2017.20160522.
[11] 王志忠,宋迎春,何玲莉.系数矩阵中部分有界不确定性的混合平差算法[J].测绘学报,2018,47(9):1171-1178.DOI:10.11947/j.AGCS.2018.20170344.WANG Zhizhong,SONG Yingchun,HE Lingli.Mixed adjustment algorithm for part of the coefficient matrix with uncertainty[J].Acta Geodaetica et Cartographica Sinica,2018,47(9):1171-1178.DOI:10.11947/j.AGCS.2018.20170344.
[12] 崔希璋,於宗俦,陶本藻,等.广义测量平差[M].武汉:武汉大学出版社,2009:9.CUI Xizhang,YU Zongchou,TAO Benzao,et al.Generalized surveying adjustment[M].Wuhan:Wuhan University Press,2009:9.
[13] TIKHONOV A.Solution of incorrectly formulated problems and the regularization method[J].Soviet Math Dokl,1963,5(4):1035-1038.
[14] TIKHONOV A N,ARSENIN V Y.Solutions of ill-posed problems[M].New York:John Wiley & Sons,1977.
[15] HOERL A E,KENNARD R W.Ridge regression:biased estimation for nonorthogonal problems[J].Technometrics,1970,12(1):55-67.
[16] HOERL A E,KENNARD R W.Ridge regression:applications to nonorthogonal problems[J].Technometrics,1970,12(1):69-82.
[17] 鲁铁定.总体最小二乘平差理论及其在测绘数据处理中的应用[D].武汉:武汉大学,2010.LU Tieding.Research on the total least squares and its applications in surveying data processing[D].Wuhan:Wuhan University,2010.
[18] 葛旭明,伍吉仓.病态总体最小二乘问题的广义正则化[J].测绘学报,2012,41(3):372-377.GE Xuming,WU Jicang.Generalized regularization to ill-posed total least squares problem[J].Acta Geodaetica et Cartographica Sinica,2012,41(3):372-377.
[19] 袁振超,沈云中,周泽波.病态总体最小二乘模型的正则化算法[J].大地测量与地球动力学,2009,29(2):131-134.YUAN Zhenchao,SHEN Yunzhong,ZHOU Zebo.Regularized total least-squares solution to ill-posed error-in-variable model[J].Journal of Geodesy and Geodynamics,2009,29(2):131-134.
[20] 葛旭明,伍吉仓.误差限的病态总体最小二乘解算[J].测绘学报,2013,42(2):196-202.GE Xuming,WU Jicang.A regularization method to ill-posed total least squares with error limits[J].Acta Geodaetica et Cartographica Sinica,2013,42(2):196-202.
[21] 王乐洋,许才军,鲁铁定.病态加权总体最小二乘平差的岭估计解法[J].武汉大学学报(信息科学版),2010,35(11):1346-1350.WANG Leyang,XU Caijun,LU Tieding.Ridge estimation method in ill-posed weighted total least squares adjustment[J].Geomatics and Information Science of Wuhan University,2010,35(11):1346-1350.
[22] 王振杰,欧吉坤.用L-曲线法确定岭估计中的岭参数[J].武汉大学学报(信息科学版),2004,29(3):235-238.WANG Zhenjie,OU Jikun.Determining the ridge parameter in a ridge estimation using L-curve method[J].Geomatics and Information Science of Wuhan University,2004,29(3):235-238.
[23] 郭建锋.测量平差系统病态性的诊断与处理[D].郑州:信息工程大学,2002.GUO Jianfeng.The diagnosis and process of ill-conditioned adjustment system[D].Zhengzhou:Information Engineering University,2002.
[2] 陶本藻.GIS质量控制中不确定度理论[J].测绘学院学报,2000,17(4):235-238.TAO Benzao.Basic theory of uncertainty of quality control in GIS[J].Journal of Institute of Surveying and Mapping,2000,17(4):235-238.
[3] 宋迎春,谢雪梅,陈晓林.不确定性平差模型的平差准则与解算方法[J].测绘学报,2015,44(2):135-141.DOI:10.11947/j.AGCS.2015.20130213.SONG Yingchun,XIE Xuemei,CHEN Xiaolin.Adjustment criterion and algorithm in adjustment model with uncertain[J].Acta Geodaetica et Cartographica Sinica,2015,44(2):135-141.DOI:10.11947/j.AGCS.2015.20130213.
[4] 杨元喜.关于“新的点位误差度量”的讨论[J].测绘学报,2009,38(3):280-282.YANG Yuanxi.Discussion on “a new measure of positional error”[J].Acta Geodaetica et Cartographica Sinica,2009,38(3):280-282.
[5] 王新洲.最小不确定度约束下的极大可能性估计[J].测绘工程,2003,12(1):5-8.WANG Xinzhou.Maximum possibility estimation restricted by least uncertainty[J].Engineering of Surveying and Mapping,2003,12(1):5-8.
[6] 陶本藻.精确度和不确定度估计及应用[J].勘察科学技术,2003(5):24-27.TAO Benzao.Estimation of accuracy and uncertainty and its application[J].Site Investigation Science and Technology,2003(5):24-27.
[7] 陈伟,张践.最小不确定度估计及其在测量数据处理中的应用[J].测绘通报,2013(1):16-18,32.CHEN Wei,ZHANG Jian.Least uncertainty estimation theory and its applications in survey data processing[J].Bulletin of Surveying and Mapping,2013(1):16-18,32.
[8] 邹永刚,翟京生,刘雁春,等.利用不确定度的海底数字高程模型构建[J].武汉大学学报(信息科学版),2011,36(8):964-968.ZOU Yonggang,ZHAI Jingsheng,LIU Yanchun,et al.Seabed DEM construction based on uncertainty[J].Geomatics and Information Science of Wuhan University,2011,36(8):964-968.
[9] 宋迎春,金昊,崔先强.带有不确定性的观测数据平差解算方法[J].武汉大学学报(信息科学版),2014,39(7):788-792.SONG Yingchun,JIN Hao,CUI Xianqiang.Adjustment algorithm about observation data with uncertain[J].Geomatics and Information Science of Wuhan University,2014,39(7):788-792.
[10] 王志忠,陈丹华,宋迎春.具有不确定性平差算法[J].测绘学报,2017,46(7):834-840.DOI:10.11947/j.AGCS.2017.20160522.WANG Zhizhong,CHEN Danhua,SONG Yingchun.An algorithm in adjustment model with uncertainty[J].Acta Geodaetica et Cartographica Sinica,2017,46(7):834-840.DOI:10.11947/j.AGCS.2017.20160522.
[11] 王志忠,宋迎春,何玲莉.系数矩阵中部分有界不确定性的混合平差算法[J].测绘学报,2018,47(9):1171-1178.DOI:10.11947/j.AGCS.2018.20170344.WANG Zhizhong,SONG Yingchun,HE Lingli.Mixed adjustment algorithm for part of the coefficient matrix with uncertainty[J].Acta Geodaetica et Cartographica Sinica,2018,47(9):1171-1178.DOI:10.11947/j.AGCS.2018.20170344.
[12] 崔希璋,於宗俦,陶本藻,等.广义测量平差[M].武汉:武汉大学出版社,2009:9.CUI Xizhang,YU Zongchou,TAO Benzao,et al.Generalized surveying adjustment[M].Wuhan:Wuhan University Press,2009:9.
[13] TIKHONOV A.Solution of incorrectly formulated problems and the regularization method[J].Soviet Math Dokl,1963,5(4):1035-1038.
[14] TIKHONOV A N,ARSENIN V Y.Solutions of ill-posed problems[M].New York:John Wiley & Sons,1977.
[15] HOERL A E,KENNARD R W.Ridge regression:biased estimation for nonorthogonal problems[J].Technometrics,1970,12(1):55-67.
[16] HOERL A E,KENNARD R W.Ridge regression:applications to nonorthogonal problems[J].Technometrics,1970,12(1):69-82.
[17] 鲁铁定.总体最小二乘平差理论及其在测绘数据处理中的应用[D].武汉:武汉大学,2010.LU Tieding.Research on the total least squares and its applications in surveying data processing[D].Wuhan:Wuhan University,2010.
[18] 葛旭明,伍吉仓.病态总体最小二乘问题的广义正则化[J].测绘学报,2012,41(3):372-377.GE Xuming,WU Jicang.Generalized regularization to ill-posed total least squares problem[J].Acta Geodaetica et Cartographica Sinica,2012,41(3):372-377.
[19] 袁振超,沈云中,周泽波.病态总体最小二乘模型的正则化算法[J].大地测量与地球动力学,2009,29(2):131-134.YUAN Zhenchao,SHEN Yunzhong,ZHOU Zebo.Regularized total least-squares solution to ill-posed error-in-variable model[J].Journal of Geodesy and Geodynamics,2009,29(2):131-134.
[20] 葛旭明,伍吉仓.误差限的病态总体最小二乘解算[J].测绘学报,2013,42(2):196-202.GE Xuming,WU Jicang.A regularization method to ill-posed total least squares with error limits[J].Acta Geodaetica et Cartographica Sinica,2013,42(2):196-202.
[21] 王乐洋,许才军,鲁铁定.病态加权总体最小二乘平差的岭估计解法[J].武汉大学学报(信息科学版),2010,35(11):1346-1350.WANG Leyang,XU Caijun,LU Tieding.Ridge estimation method in ill-posed weighted total least squares adjustment[J].Geomatics and Information Science of Wuhan University,2010,35(11):1346-1350.
[22] 王振杰,欧吉坤.用L-曲线法确定岭估计中的岭参数[J].武汉大学学报(信息科学版),2004,29(3):235-238.WANG Zhenjie,OU Jikun.Determining the ridge parameter in a ridge estimation using L-curve method[J].Geomatics and Information Science of Wuhan University,2004,29(3):235-238.
[23] 郭建锋.测量平差系统病态性的诊断与处理[D].郑州:信息工程大学,2002.GUO Jianfeng.The diagnosis and process of ill-conditioned adjustment system[D].Zhengzhou:Information Engineering University,2002.
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