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1. 东华理工大学测绘工程学院, 江西南昌 330013;
2. 流域生态与地理环境监测国家测绘地理信息局重点实验室, 江西南昌 330013;
3. 江西省数字国土重点实验室, 江西南昌 330013

A Newton Algorithm for Multivariate Total Least Squares Problems
WANG Leyang1,2,3, ZHAO Yingwen1, CHEN Xiaoyong1,2,3, ZANG Deyan1,2,3
1. Faculty of Geomatics, East China University of Technology, Nanchang 330013, China;
2. Key Laboratory of Watershed Ecology and Geographical Environment Monitoring, NASG, Nanchang 330013, China;
3. Jiangxi Province Key Lab for Digital Land, Nanchang 330013, ChinaAbstract
First author: WANG Leyang (1983—),male,PhD,associate professor, majors in geodetic inversion and geodetic data processing.E-mail:wleyang@163.com
Abstract: In order to improve calculation efficiency of parameter estimation, an algorithm for multivariate weighted total least squares adjustment based on Newton method is derived. The relationship between the solution of this algorithm and that of multivariate weighted total least squares adjustment based on Lagrange multipliers method is analyzed. According to propagation of cofactor, 16 computational formulae of cofactor matrices of multivariate total least squares adjustment are also listed. The new algorithm could solve adjustment problems containing correlation between observation matrix and coefficient matrix. And it can also deal with their stochastic elements and deterministic elements with only one cofactor matrix. The results illustrate that the Newton algorithm for multivariate total least squares problems could be practiced and have higher convergence rate.
Key words: multivariate total least squares     Newton method     propagation of cofactor     affine transformation

1 多元总体最小二乘解法 1.1 多元变量EIV模型

1.2 多元总体最小二乘问题的牛顿解法

(ε为很小正值，本文取10-12)时，并通过逆拉直运算计算

1.3 牛顿法解和拉格朗日乘数法解之间的关系

2.2 协因数阵计算公式

 $L$ $\overset{\wedge }{\mathop{V}}\,$ $\overset{\wedge }{\mathop{L}}\,$ $\xi$ $L$ $Q$ $-{{\overset{\wedge }{\mathop{X}}\,}_{5}}$ $Q-{{\overset{\wedge }{\mathop{X}}\,}_{5}}$ $-Q\overset{\wedge }{\mathop{{{\sum }^{T}}}}\,\overset{\wedge }{\mathop{X_{4}^{T}}}\,$ $\overset{\wedge }{\mathop{V}}\,$ $-{{\overset{\wedge }{\mathop{X}}\,}_{5}}$ ${{\overset{\wedge }{\mathop{X}}\,}_{5}}$ $0$ $Q\overset{\wedge }{\mathop{{{\sum }^{T}}}}\,\overset{\wedge }{\mathop{X_{4}^{T}}}\,$ $\overset{\wedge }{\mathop{L}}\,$ $Q-{{\overset{\wedge }{\mathop{X}}\,}_{5}}$ $0$ $Q-{{\overset{\wedge }{\mathop{X}}\,}_{5}}$ $0$ $\xi$ $-{{\overset{\wedge }{\mathop{X}}\,}_{4}}\overset{\wedge }{\mathop{\sum }}\,Q$ ${{\overset{\wedge }{\mathop{X}}\,}_{4}}\overset{\wedge }{\mathop{\sum }}\,Q$ $0$ ${{\overset{\wedge }{\mathop{X}}\,}_{3}}$

3 算例分析 3.1 模拟算例

 名称 N法 L-N法 Fang(2011)法 Schaffrin(2009)法 WLS法 参数真值 仿射参数 ${{\xi }_{11}}$ 0.900 742 0.900 742 0.900 742 0.900 742 0.935 548 0.9 ${{\xi }_{21}}$ -0.800 035 -0.800 035 -0.800 035 -0.800 035 -0.804 088 -0.8 ${{\xi }_{31}}$ 0.998 770 0.998 770 0.998 770 0.998 770 0.954 356 1 ${{\xi }_{12}}$ 0.599 708 0.599 708 0.599 708 0.599 708 0.629 577 0.6 ${{\xi }_{22}}$ 0.700 151 0.700 151 0.700 151 0.700 151 0.680 361 0.7 ${{\xi }_{32}}$ 4.999 848 4.999 848 4.999 848 4.999 848 4.982 601 5 差值范数 $\left\| \Delta \Xi \right\|$ 0.001 482 0.001 482 0.001 482 0.001 482 0.070 192 - 平均迭代次数 i 3.02 3.01 5.372 5.974 - -

 图 1 所估参数的直方图及其正态分布拟合曲线 Fig. 1 Histogram and fitting curve of normal distribution of estimated parameters

3.2 实测算例

\begin{align} & {{P}_{XY}}=diag([9.8361,5.5357,12.7369, \\ & 12.0099,10.181,11.3661,11.147. \\ & 5.8834,9.8322,7.5678]) \\ \end{align}

 参数 ${{\xi }_{11}}$ ${{\xi }_{21}}$ ${{\xi }_{31}}$ ${{\xi }_{12}}$ ${{\xi }_{22}}$ ${{\xi }_{32}}$ 估值 0.999 827 207 612 -0.000 007 228 476 29.761 507 890 910 -0.000 175 251 547 0.999 965 497 834 19.683 772 954 162

 参数$\overset{\wedge }{\mathop{\Xi }}\,$的协因数阵 0.000 000 083 211 -0.000 000 002 016 0.000 012 130 226 -0.000 000 000 008 0.000 000 000 001 -0.000 000 002 403 -0.000 000 002 016 0.000 000 062 539 -0.000 000 155 129 0.000 000 000 001 -0.000 000 000 005 -0.000 000 000 024 0.000 012 130 226 -0.000 000 155 129 0.036 683 888 367 -0.000 000 002 010 -0.000 000 000 076 -0.000 003 465 192 -0.000 000 000 008 0.000 000 000 001 -0.000 000 002 010 0.000 000 086 273 -0.000 000 005 576 0.000 025 548 944 0.000 000 000 001 -0.000 000 000 005 -0.000 000 000 076 -0.000 000 005 576 0.000 000 102 710 -0.000 003 426 705 -0.000 000 002 403 -0.000 000 000 024 -0.000 003 465 192 0.000 025 548 944 -0.000 003 426 705 0.050 234 807 436

4 结 论

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http://dx.doi.org/10.11947/j.AGCS.2016.20150246

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#### 文章信息

WANG Leyang, ZHAO Yingwen, CHEN Xiaoyong, ZANG Deyan

A Newton Algorithm for Multivariate Total Least Squares Problems

Acta Geodaeticaet Cartographica Sinica, 2016, 45(4): 411-417.
http://dx.doi.org/10.11947/j.AGCS.2016.20150246