For the case of design matrix in EIV(errors-in-variables) model containing both fixed elements and random elements, this paper proposes a mixed LS-TLS(least squares-total least squares) algorithm and deduces the precision estimator by reformulating an EIV model as a nonlinear Gauss-Helmert model, in which random elements are extracted to the random model of adjustment. This algorithm can be applied to the general design matrix including simultaneously fixed columns, fixed elements and random elements. The example illustrates that the solution of mixed LS-TLS equal the solution of structured or weighted TLS algorithms which can solve mixed LS-TLS problem. Additionally, the solution of mixed LS-TLS statistically superior to solution of LS or TLS.
FANG Xing
,
ZENG Wenxian
,
LIU Jingnan
,
YAO Yibin
,
WANG Yong
. Mixed LS-TLS Estimation Based on Nonlinear Gauss-Helmert Model[J]. Acta Geodaetica et Cartographica Sinica, 2016
, 45(3)
: 291
-296
.
DOI: 10.11947/j.AGCS.2016.20150157
[1] GOLUB GH, VAN LOAN C F. An Analysis of the Total Least Squares Problem[J]. SIAM Journal on Numerical Analysis, 1980, 17(6):883-893.
[2] XU Peiliang, LIU Jingnan, SHI Chuang. Total Least Squares Adjustment in Partial Errors-in-variables Models:Algorithm and Statistical Analysis[J]. Journal of Geodesy, 2012, 86(8):661-675.
[3] VAN HUFFEL S,VANDEWALLE J.The Total Least Squares Problem:Computational Aspects and Analysis[M]. Philadelphia:Society for Industrial and Applied Mathematics, 1991.
[4] ADCOCK R J. Note on the Method of Least Squares[J]. The Analyst, 1877, 4(6):183-184.
[5] SCHAFFRIN B, WIESER A. On Weighted Total Least-squares Adjustment for Linear Regression[J]. Journal of Geodesy, 2008, 82(7):415-421.
[6] SHEN Yunzhong, LI Bofeng, CHEN Yi. An Iterative Solution of Weighted Total Least-squares Adjustment[J]. Journal of Geodesy, 2010, 85(4):229-238.
[7] FANG Xing. Weighted Total Least Squares:Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8):733-749.
[8] FANG Xing. A Total Least Squares Solution for Geodetic Datum Transformations[J]. Acta Geodaetica et Geophysica, 2014, 49(2):189-207.
[9] NEITZELF. Generalization of Total Least-squares on Example of Unweighted and Weighted 2D Similarity Transformation[J]. Journal of Geodesy, 2010, 84(12):751-762.
[10] FANG Xing. Weighted Total Least Squares Solutions for Applications in Geodesy[D]. Germany:Leibniz University Hannover, 2011.
[11] TONG Xiaohua, JIN Yanmin, ZHANG Songlin, et al. Bias-Corrected Weighted Total Least-squares Adjustment of Condition Equations[J]. Journal of Surveying Engineering,2014, 141(2):0401-0413.
[12] 胡川,陈义. 非线性整体最小平差迭代算法[J]. 测绘学报, 2014, 43(7):668-674. DOI:10.13485/j.cnki.11-2089.2014.0111. HU Chuan, CHEN Yi. An Iterative Algorithm for Nonlinear Total Least Squares Adjustment[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(7):668-674.DOI:10.13485/j.cnki.11-2089.2014.0111.
[13] ZENG Wenxian, LIU Jingnan, YAO Yibin. On Partial Errors-in-variables Models with Inequality Constraints of Parameters and Variables[J]. Journal of Geodesy, 2015, 89(2):111-119.
[14] 曾文宪, 方兴, 刘经南, 等. 附有不等式约束的加权整体最小二乘算法[J]. 测绘学报, 2014, 43(10):1013-1018. DOI:10.13485/j.cnki.11-2089.2014.0173. ZENG Wenxian, FANG Xing, LIU Jingnan, et al. Weighted Total Least Squares Algorithm with Inequality Constraints[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(10):1013-1018. DOI:10.13485/j.cnki.11-2089.2014.0173.
[15] FANG Xing. On Non-combinatorial Weighted Total Least Squares with Inequality Constraints[J]. Journal of Geodesy, 2013, 88(8):805-816.
[16] FANG Xing. A Structured and Constrained Total Least-Squares Solution with Cross-covariances[J]. Studia Geophysica et Geodaetica, 2014, 58(1):1-16.
[17] FANG Xing. Weighted Total Least-squares with Constraints:A Universal Formula for Geodetic Symmetrical Transformations[J]. Journal of Geodesy, 2015, 89(5):459-469.DOI:10.1007/s00190-015-0790-8.
[18] ZHANG Songlin, ZHANG Kun. On a Basic Multivariate EIV Model with Linear Equality Constraints[J]. Applied Mathematics and Computation, 2014, 236:247-252.
[19] GOLUB GH, HOFFMAN A, STEWART GW. A Generalization of the Eckart-Young-Mirsky Matrix Approximation Theorem[J]. Linear Algebra and Its Applications, 1987, 88-89:317-327.
[20] DUNNE BE, WILⅡLAMSONGA. QR-based TLS and Mixed LS-TLS Algorithms with Applications to Adaptive ⅡR Filtering[J]. IEEE Transactions on Signal Processing, 2003, 51(2):386-394.
[21] YAN Shijian, FAN Jinyan. The Solution Set of the Mixed LS-TLS Problem[J]. International Journal of Computer Mathematics, 2001, 77(4):545-561.
[22] 胡川, 陈义, 彭友. 混合结构总体最小二乘参数估计[J]. 大地测量与地球动力学, 2013, 33(4):56-60. HU Chuan, CHEN Yi, PENG You.On Mixed Structured Total Least Squares for Parameters Estimation[J]. Journal of Geodesy and Geodynamics, 2013, 33(4):56-60.
[23] MAHBOUB V, SHARIFI MA. On Weighted Total Least-squares with Linear and Quadratic Constraints[J]. Journal of Geodesy, 2013, 87(3):279-286.
[24] 孔建,姚宜斌, 黄承猛. 非线性模型的一阶偏导数确定方法及其在TLS精度评定中的应用[J]. 大地测量与地球动力学, 2011, 31(3):1-5. KONG Jian, YAO Yibin, HUANG Chengmeng. Method for Determining First-order Partial Derivative of Nonlinear Model and Its Application in TLS Accuracy Assessment[J]. Journal of Geodesy and Geodynamics, 2011, 31(3):1-5.
[25] KOCH K R. Parameter Estimation and Hypothesis Testing in Linear Models[M]. Berlin:Springer, 1999.
[26] 曾文宪, 陶本藻. 三维坐标转换的非线性模型[J]. 武汉大学学报(信息科学版), 2003, 28(5):566-568. ZENG Wenxian, TAO Benzao. Non-linear Adjustment Model of Three-dimensional Coordinate Transformation[J].Geomatics and Information Science of Wuhan University, 2003, 28(5):566-568.
