整体小二乘理论及其在变形监测中的应用研究
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摘要
最小二乘法(Least-Squares,LS)作为最基本的、应用最广泛的数据处理方法之一,一直受到测量学者的重视。但它应用在测量数据处理上有一个前提,假设只有观测向量包含误差,不考虑系数矩阵受误差干扰,即为高斯马尔科夫模型(G-M模型)。然而,在大多数测量问题中由于人员、模型、环境、仪器等因素造成系数矩阵中的某些元素受到误差干扰,使得系数矩阵不完全精确,出现了系数矩阵和观测向量同时包含误差的情况,称之为所有随机变量包含误差模型(Error-In-Variables,EIV)。从统计学观点来看,利用LS法解算EIV模型不够合理,不能得到参数的最优估值。因此,急需引入一种更为合理的处理方法来解决此类问题,弥补最小二乘法的不足。
直到十九世纪八十年代,才由Golub和Van Loan;总结前人针对EIV模型的研究思路,提出了整体最小二乘法(Total Least-Squares,TLS)。TLS以所有随机变量的改正数平方和达到最小为准则,对全部随机变量作最小化约束,我们可以利用该方法建立更为合理的数据处理模型,解算出精度更高的结果。三十多年来,TLS作为最小二乘法的延伸,迅速发展成为了一种新的数据处理方法。目前,自动控制、图像处理、系统辨识、信号处理等相关领域已经成功将整体最小二乘法引入。与此同时,整体最小二乘法自身也得到了巨大发展,其各种改进模型和解算方法相继被提出。基于整体最小二乘法的优势,将其引入到测量领域具有重要的理论意义和应用价值。
利用整体最小二乘法解决测量数据处理问题,有人做了一些应用探索,但是比较系统的研究还不多。本文首先系统探讨了最小二乘和整体最小二乘的基本思想及原理,从基本思想和原理层面分析了整体最小二乘法相比最小二乘法的优缺点。然后详细地推导了目前最为常用的几种整体最小二乘模型及其解算方法,将各模型进行比较,并针对测量数据处理中EIV模型的特点,在原有模型和计算方法的基础之上改进了加权整体最小二乘法。最后,分别选用不同的方法对测量数据处理中的直线拟合、二维直角坐标转换、三维小角度基准转换问题进行解算并比较；将附有约束条件的加权整体最小二乘法、多元整体最小二乘法应用于三维任意旋转角度基准转换；将TLS法应用于整体变形监测数据处理：将基于整体最小二乘法的三维任意旋转角度基准转换思想应用于变形监测数据处理,并用C#、Matlab编写相关程序,用实例验证整体最小二乘法解决测量数据处理问题的效果。
The Least-Squares (LS) method is one of the most basic and widely method for data processing. But the application has a premise that the random error exists only in the observation vector, without considering the coefficient matrix interfered by error (the Gauss-Markov model, G-M model). However, most measurement problems are called variable containing the error model (error-in-variables, EIV),because some of the elements of the coefficient matrix interfered by the personnel model, the environment, equipment, and other factors, so that the coefficient matrix is not completely accurate, the coefficient matrix and observation vector also contains the error. From statistical view, it is not reasonable to solve the EIV model with LS, and can not get the optimal solution. Therefore, there is an urgent need for a more reasonable method to solve such problems.
Golub and Van Loan advanced the Total Least-Squares (TLS) to solve the EIV model in the eighties of last century. TLS minimizes all variables which need to be fixed, so we can use this method to establish a more reasonable data processing model and get more accurate results. In the past three decades, the TLS as a new data processing method has become a hot topic. At present, this method has been successfully applied to spectral analysis, parameter estimation, automatic control, image processing, system identification, signal processing, and other related fields. At the same time, the TLS has made a big advance. The improved models have been proposed. Based on the advantages of the TLS, it is theoretical and worth to introduce this method into the measurement field.
Someone did some exploration research in Utilization of TLS in the problem of measurement data processing, but there were few systematic studies. This paper systematically expounded the basic ideas and principles of TLS, compared to the LS, a dissection of the advantages and disadvantages of the TLS was made in the basic ideas and principles level. Then we deduced in detail the most commonly TLS model and compared each model, and derived a new solution method-improved WTLS in connection with the characteristics of the EIV model of measurement data processing. Finally, different methods were selected to solute the line fitting, the two-dimensional Cartesian coordinate transformation and the three-dimensional small-angle datum transformation problem. CWTLS method and MTLS method were applied to three-dimensional coordinate transformation adapted to arbitrary rotation in the first time. Proposed overall deformation monitoring method based on the TLS. Applied three-dimensional small-angle datum transformation based on the TLS to the deformation monitoring data processing. Used C#and Matlab wrote interrelated procedures and verified the effect of TLS to solving the problem of measurement data processing.
直到十九世纪八十年代,才由Golub和Van Loan;总结前人针对EIV模型的研究思路,提出了整体最小二乘法(Total Least-Squares,TLS)。TLS以所有随机变量的改正数平方和达到最小为准则,对全部随机变量作最小化约束,我们可以利用该方法建立更为合理的数据处理模型,解算出精度更高的结果。三十多年来,TLS作为最小二乘法的延伸,迅速发展成为了一种新的数据处理方法。目前,自动控制、图像处理、系统辨识、信号处理等相关领域已经成功将整体最小二乘法引入。与此同时,整体最小二乘法自身也得到了巨大发展,其各种改进模型和解算方法相继被提出。基于整体最小二乘法的优势,将其引入到测量领域具有重要的理论意义和应用价值。
利用整体最小二乘法解决测量数据处理问题,有人做了一些应用探索,但是比较系统的研究还不多。本文首先系统探讨了最小二乘和整体最小二乘的基本思想及原理,从基本思想和原理层面分析了整体最小二乘法相比最小二乘法的优缺点。然后详细地推导了目前最为常用的几种整体最小二乘模型及其解算方法,将各模型进行比较,并针对测量数据处理中EIV模型的特点,在原有模型和计算方法的基础之上改进了加权整体最小二乘法。最后,分别选用不同的方法对测量数据处理中的直线拟合、二维直角坐标转换、三维小角度基准转换问题进行解算并比较；将附有约束条件的加权整体最小二乘法、多元整体最小二乘法应用于三维任意旋转角度基准转换；将TLS法应用于整体变形监测数据处理：将基于整体最小二乘法的三维任意旋转角度基准转换思想应用于变形监测数据处理,并用C#、Matlab编写相关程序,用实例验证整体最小二乘法解决测量数据处理问题的效果。
The Least-Squares (LS) method is one of the most basic and widely method for data processing. But the application has a premise that the random error exists only in the observation vector, without considering the coefficient matrix interfered by error (the Gauss-Markov model, G-M model). However, most measurement problems are called variable containing the error model (error-in-variables, EIV),because some of the elements of the coefficient matrix interfered by the personnel model, the environment, equipment, and other factors, so that the coefficient matrix is not completely accurate, the coefficient matrix and observation vector also contains the error. From statistical view, it is not reasonable to solve the EIV model with LS, and can not get the optimal solution. Therefore, there is an urgent need for a more reasonable method to solve such problems.
Golub and Van Loan advanced the Total Least-Squares (TLS) to solve the EIV model in the eighties of last century. TLS minimizes all variables which need to be fixed, so we can use this method to establish a more reasonable data processing model and get more accurate results. In the past three decades, the TLS as a new data processing method has become a hot topic. At present, this method has been successfully applied to spectral analysis, parameter estimation, automatic control, image processing, system identification, signal processing, and other related fields. At the same time, the TLS has made a big advance. The improved models have been proposed. Based on the advantages of the TLS, it is theoretical and worth to introduce this method into the measurement field.
Someone did some exploration research in Utilization of TLS in the problem of measurement data processing, but there were few systematic studies. This paper systematically expounded the basic ideas and principles of TLS, compared to the LS, a dissection of the advantages and disadvantages of the TLS was made in the basic ideas and principles level. Then we deduced in detail the most commonly TLS model and compared each model, and derived a new solution method-improved WTLS in connection with the characteristics of the EIV model of measurement data processing. Finally, different methods were selected to solute the line fitting, the two-dimensional Cartesian coordinate transformation and the three-dimensional small-angle datum transformation problem. CWTLS method and MTLS method were applied to three-dimensional coordinate transformation adapted to arbitrary rotation in the first time. Proposed overall deformation monitoring method based on the TLS. Applied three-dimensional small-angle datum transformation based on the TLS to the deformation monitoring data processing. Used C#and Matlab wrote interrelated procedures and verified the effect of TLS to solving the problem of measurement data processing.
引文
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[8]Schaffrin B.,Wieser A.. On weighted total least-squares adjustment for linear regression[J]. J Geodes,2008,82(7):415-421.
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[2]Golub G. H.,Van Loan C. F.. An analysis of the Total Least Squares problem[J]. SIAM J. Numer. Anal..1980,17(6):883-893.
[3]丁克良.整体最小二乘法及其在测量数据处理中的若干应用研究[D].中国科学院测量与地球物理研究所,湖北,武汉,2006.
[4]丁克良,欧吉坤,陈义.整体最小二乘法及其在测量数据处理中的应用[A].中国测绘学会.中国测绘学会第九次全国会员代表大会暨学会成立50周年纪念大会论文集[c].中国测绘学会：,2009：7.
[5]Van Huffel S.,Vandewalle J.. The Total Least Squares Problem Computational Aspects and Analysis[C]. SIAM, Philadelphia,1991.
[6]Van Huffel S.,Philippe Lemmerling. Total Least Squares and Errors-in-Variables Modeling analysis, Algorithms and Application[M]. Kluwer Academic Publishers,Dordrecht,2002.
[7]Schaffrin B.,Felus Y. A.. On the multivariate total least squares approach to empirical coordinate transformation [J]. J. Geodes,2008,82(6):373-383.
[8]Schaffrin B.,Wieser A.. On weighted total least-squares adjustment for linear regression[J]. J Geodes,2008,82(7):415-421.
[9]Shen Yunzhong,Li Bofeng,Chen Yi. An Iterative Solution of Weighted Total Least-squares Adjustment[J]. Journal of Geodesy,2010,85(4):229-238.
[10]S. Van Huffel, J. Vandewalle. Analysis and properties of the generalized total least squares problem AX≈B when some or all columns in A are subject to error[J]. SIAM J. Matrix Anal.,1989,10(3):294-315.
[11]Schaffrin B., Snow K.. Total least-squares regularization of Tykhonov type and an ancient racetrack in Corinth[J]. Linear Algebra Appl.,2010,432(8),2061-2076.
[12]Burkhard Schaffrin. A note on Constrained Total Least-Squares estimation[J].Linear Algebra and its Applications,2006,417(1):245-258.
[13]Schaffrin B., Felus Y. A.. An algorithmic approach to the total least-squares problem with linear and quadratic constraints [J]. Studia geophysical et geodetics,2009,53(1),1-16.
[14]魏木生.广义最小二乘问题的理论和计算[M].北京：科学出版社.2006.
[15]刘永辉,魏木生.TLS和LS问题的比较[J].计算数学,2003,(04)：479-492.
[16]董校洪.整体最小二乘法在工程测量上的应用[D].上海：同济大学.2009.
[17]Van Huffel S.. Orthogonal least squares fitting by conic sections. Recent advances in total least squares techniques and errors-in-variables modeling[C]. SIAM,Philadelphia,1997: 259-264.
[18]魏木生.加权广义逆、加权最小二乘和约束最小二乘问题[J].计算数学,1995,(2)：196-209.
[19]魏木生.关于TLS和LS解的扰动性分析[J].计算数学,1998,20(3)：268-273
[20]刘新国.关于整体最小二乘的可解性及扰动分析[J].应用所学学报,1996,19(2)：254-262.
[21]黄开斌,俞锦成.整体最小二乘问题的解集与极小范数解[J].南京师范大学学,1997,20(4)：1-5.
[22]罗武安.广义完全最小二乘问题可解的充分必要条件[J].北京大学学报(自然科学),2000,36(1)：1-7.
[23]何英姿,张兵,吴宏鑫.递推的TLS参数估计方法及其在飞轮FDD中的应用[J].航天控制,1998,3：17-22.
[24]周忠来,施聚生,栗苹,周勇.基于SVD-TLS的AR谱估计方法在声目标识别中的应用[J].探测与控制学报,2000,(01)：56-60.
[25]杨磊,赵拥军,王志刚.基于功率和相位联合估计TLS-ESPRIT算法的极化干涉SAR数据分析[J].测绘学报,2007,(02)：163-168.
[26]王鼎,吴瑛,田建春.基于总体最小二乘算法的多站无源定位[J].信号处理,2007,(04)：611-614.
[27]张静,徐政,王峰,常勇TLS-ESPRIT算法在低频振荡分析中的应用[J].电力系统自动化,2007,(20)：84-88.
[28]Davis T. G..Total least squares curve fitting[D]. University of South Florida,Tampa,Fla.,1998.
[29]Felus Y. A.. Application of Total Least Squares for Spatial Point Process Analysis[J]. Journal of Surveying Engineering. ASCE 2004,130:126-133.
[30]Richard L., Branham Jr.. Astronomical data reduction with total least squares[J]. New Astronomy Reviews,2001,45:649-661.
[31]Yaron A. Felus, Burkhard Schaffrin. PERFORMING SIMILARITY TRANSFORMATIONS USING THE ERROR-IN-VARIABLES MODEL[C]. ASPRS 2005 Annual Conference Baltimore, Maryland,2005.
[32]Frank Neitzel. Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation. J. Geod,2010,84:751-762.
[33]Yavuz E., Arikan F., Arikan O.. A Hybrid Reconstruction Algorithm for Computerized Ionospheric Tomography[C]. Recent Advances in Space Technologies,2005,782-787.
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