稳健主成分回归及粗差探测
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摘要
针对测量平差模型中法方程系数矩阵病态和观测值存在粗差的问题,该文将主成分估计与稳健估计方法相结合,建立了稳健主成分回归方法。详细介绍了稳健主成分回归方法的步骤,即先将系数矩阵进行稳健主成分分析,得到稳健主成分,利用稳健的LTS估计法求解参数向量,同时构造了一种稳健的主成分选择统计量,便于选择最佳的主成分数目。文中构造了3种异常点检验统计量,给出了异常点和粗差的探测步骤。算例研究表明,与传统方法相比,本文方法有一定的优越性。
To solve the problem of both ill-condition and gross errors,the robust principal components regression,based on principal components estimation and robust estimation theory,is established.Firstly,to get robust principal components,the robust principal component analysis is applied to coefficient matrix.Then,the expression of the parameter estimation is deduced by LTS and a robust component selection statistic is proposed to choose the components correctly.Next,three outliers diagnostic statistics are constructed to distinguish between regular observations and the outliers.After that,the procedure of checking out outliers and gross errors is provided.The paper gives a test in the end.Test results demonstrate the effectiveness of the method.
To solve the problem of both ill-condition and gross errors,the robust principal components regression,based on principal components estimation and robust estimation theory,is established.Firstly,to get robust principal components,the robust principal component analysis is applied to coefficient matrix.Then,the expression of the parameter estimation is deduced by LTS and a robust component selection statistic is proposed to choose the components correctly.Next,three outliers diagnostic statistics are constructed to distinguish between regular observations and the outliers.After that,the procedure of checking out outliers and gross errors is provided.The paper gives a test in the end.Test results demonstrate the effectiveness of the method.
引文
[1]王振杰.测量中不适定问题的正则化解法[M].北京:科学出版社,2006:66-80.(WANG Zhenjie.The regularization solutions of ill-posed problems in geodesy[M].Beijing:Science Press,2006:66-80.)
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[3]邱卫宁,陶本藻,姚宜斌,等.测量数据处理理论与方法[M].武汉:武汉大学出版社,2008:108-150.(QIU Weining,TAO Benzao,YAO Yibin,et al.The theory and method of surveying data processing[J].Wuhan:Wuhan University Press,2008:108-150.)
[4]MASSY W F.Principal components regression in exploratory statistical research[J].Amer.Statist.Asso,1965(60):234-266.
[5]宋力杰.主成分估计与附加条件的参数平差[J].测绘工程,1997,6(1):24-27.(SONG Lijie.Principal components estimate and parameter adjustment with constraints[J].Engineering of Surveying and Mapping,1997,6(1):24-27.)
[6]归庆明,郭海涛,郭建锋,等.主成分估计用于解算卫星线阵CCD影像外方位元素[J].遥感技术与应用,2003,18(1):14-18.(GUI Qingming,GUO Haitao,GUO Jianfeng,et al.Computing the exterior orientation elements of one-line scanner satellitic imagery by using principal component estimation[J].Remote Sensing Technology and Application,2003,18(1):14-18.)
[7]冯光财,陈正阳.基于主成分回归的GPS高程曲面拟合[J].测绘科学,2007,32(1):51-52.(FENG Guangcai,CHEN Zhengyang.A new method of GPS elevation interpolation based on principal component regression[J].Science of Surveying and Mapping,2007,32(1):51-52.)
[8]岳东杰,李立瑞,郭培闪.主成分分析在GPS高程拟合中的应用[J].河海大学学报(自然科学版),2011,39(3):323-326.(YUE Dongjie,LI Lirui,GUO Peishan.Application of principal component analysis in GPS height fitting[J].Journal of Hohai University(Natural Sciences),2011,39(3):323-326.)
[9]ROUSSEEUW P J,VAN AELST S,VAN DRIESSEN K,et al.Robust multivariate regression[J].Technometrics,2004(46):293-305.
[10]王彤,何大卫.LTS回归与M估计稳健性的比较[J].中国卫生统计,1999,16(2):79-81.(WANG Tong,HE Dawei.Comparision of M-estimator and LTS estimation in regression[J].Chinese Journal of Health Statistics,1999,16(2):79-81.)
[11]杨飚,张曾科,孙政顺.基于LTS稳健初值的选权迭代法[J].科学技术与工程,2005,5(22):1720-1723.(YANG Biao,ZHANG Zengke,SUN Zhengshun.Selecting weight iteration method with initial value by LTS[J].Science Technology and Engineering,2005,5(22):1720-1723.)
[12]HUBERT M.ROUSSEEUW P G.ROBPCA:a new approach to robust principal components analysis[J].Technometrics,2005(47):64-79.
[13]ENGELEN S,HUBERT M.Fast model selection for robust calibration methods[J].Analytica Chimica Acta,2005(544):219-228.
[14]ROUSSEEUW P J.Least median of squares regression[J].Journal of the American Statistical Association,1984(79):871-880.
[15]HUBERT M,VERBOVEN S.A robust PCR method for high-dimensional regressors[J].Journal of Chemometrics,2003(17):438-452.
[2]潘雄,吕玉婷,汪耀,等.基于半参数平差模型的粗差定位与定值研究[J].武汉大学学报(信息科学版),2016,41(11):1-7.(PAN Xiong,LV Yuting,WANG Yao,et al.Research of the location and valuation of gross error based on semi-parametric adjustment model[J].Geomatics and Information Science of Wuhan University,2016,41(11):1-7.)
[3]邱卫宁,陶本藻,姚宜斌,等.测量数据处理理论与方法[M].武汉:武汉大学出版社,2008:108-150.(QIU Weining,TAO Benzao,YAO Yibin,et al.The theory and method of surveying data processing[J].Wuhan:Wuhan University Press,2008:108-150.)
[4]MASSY W F.Principal components regression in exploratory statistical research[J].Amer.Statist.Asso,1965(60):234-266.
[5]宋力杰.主成分估计与附加条件的参数平差[J].测绘工程,1997,6(1):24-27.(SONG Lijie.Principal components estimate and parameter adjustment with constraints[J].Engineering of Surveying and Mapping,1997,6(1):24-27.)
[6]归庆明,郭海涛,郭建锋,等.主成分估计用于解算卫星线阵CCD影像外方位元素[J].遥感技术与应用,2003,18(1):14-18.(GUI Qingming,GUO Haitao,GUO Jianfeng,et al.Computing the exterior orientation elements of one-line scanner satellitic imagery by using principal component estimation[J].Remote Sensing Technology and Application,2003,18(1):14-18.)
[7]冯光财,陈正阳.基于主成分回归的GPS高程曲面拟合[J].测绘科学,2007,32(1):51-52.(FENG Guangcai,CHEN Zhengyang.A new method of GPS elevation interpolation based on principal component regression[J].Science of Surveying and Mapping,2007,32(1):51-52.)
[8]岳东杰,李立瑞,郭培闪.主成分分析在GPS高程拟合中的应用[J].河海大学学报(自然科学版),2011,39(3):323-326.(YUE Dongjie,LI Lirui,GUO Peishan.Application of principal component analysis in GPS height fitting[J].Journal of Hohai University(Natural Sciences),2011,39(3):323-326.)
[9]ROUSSEEUW P J,VAN AELST S,VAN DRIESSEN K,et al.Robust multivariate regression[J].Technometrics,2004(46):293-305.
[10]王彤,何大卫.LTS回归与M估计稳健性的比较[J].中国卫生统计,1999,16(2):79-81.(WANG Tong,HE Dawei.Comparision of M-estimator and LTS estimation in regression[J].Chinese Journal of Health Statistics,1999,16(2):79-81.)
[11]杨飚,张曾科,孙政顺.基于LTS稳健初值的选权迭代法[J].科学技术与工程,2005,5(22):1720-1723.(YANG Biao,ZHANG Zengke,SUN Zhengshun.Selecting weight iteration method with initial value by LTS[J].Science Technology and Engineering,2005,5(22):1720-1723.)
[12]HUBERT M.ROUSSEEUW P G.ROBPCA:a new approach to robust principal components analysis[J].Technometrics,2005(47):64-79.
[13]ENGELEN S,HUBERT M.Fast model selection for robust calibration methods[J].Analytica Chimica Acta,2005(544):219-228.
[14]ROUSSEEUW P J.Least median of squares regression[J].Journal of the American Statistical Association,1984(79):871-880.
[15]HUBERT M,VERBOVEN S.A robust PCR method for high-dimensional regressors[J].Journal of Chemometrics,2003(17):438-452.