﻿ 加权EIV模型的经典最小二乘算法
 文章快速检索 高级检索
 大地测量与地球动力学  2019, Vol. 39 Issue (5): 541-546, 550  DOI: 10.14075/j.jgg.2019.05.020

### 引用本文

XIE Jian, LONG Sichun. Classic Least Squares Method to the Weighted EIV Model[J]. Journal of Geodesy and Geodynamics, 2019, 39(5): 541-546, 550.

### Foundation support

National Natural Science Foundation of China, No. 41704007, 41877283; Scientific Research Project of the Education Department of Hunan Province, No. 16C0632; Doctors' Scientific Research Fund of Hunan University of Science and Technology, No. E51673; Open Fund of Hunan Province Key Laboratory of Coal Resources Clean-Utilization and Mine Environment Protection, No. E21610.

### 第一作者简介

XIE Jian, PhD, lecturer, majors in surveying adjustment and surveying data processing, E-mail: xiejian@csu.edu.cn.

### 文章历史

1. 湖南科技大学煤炭资源清洁利用与矿山环境保护湖南省重点实验室，湖南省湘潭市桃园路，411201

1 经典最小二乘原理概述

 $\mathit{\boldsymbol{y}} = \mathit{\boldsymbol{A\xi }} + \mathit{\boldsymbol{e}}$ (1)

 $\mathit{\boldsymbol{\hat \xi }} = {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{Q}}_y^{ - 1}\mathit{\boldsymbol{A}}} \right)^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{Q}}_y^{ - 1}\mathit{\boldsymbol{y}}$ (2)

 $\left\{ {\begin{array}{*{20}{l}} {\mathit{\boldsymbol{\widehat y}} = {\mathit{\boldsymbol{P}}_A}\mathit{\boldsymbol{y}}}\\ {\mathit{\boldsymbol{v}} = \mathit{\boldsymbol{\widehat e}} = \mathit{\boldsymbol{y}} - \mathit{\boldsymbol{\widehat y}} = {\mathit{\boldsymbol{P}}_{\frac{1}{A}}}\mathit{\boldsymbol{y}}} \end{array}} \right.$ (3)

 $\boldsymbol{A}^{\mathrm{T}} \hat{\boldsymbol{\lambda}}-\boldsymbol{E}_{A}^{\mathrm{T}} \hat{\boldsymbol{\lambda}}=0$ (8)
 $\hat{\boldsymbol{e}}=-\boldsymbol{Q}_{l l} \hat{\boldsymbol{B}}^{\mathrm{T}} \hat{\lambda}$ (9)
 $\mathit{\boldsymbol{\hat \lambda }} = {\left( {\mathit{\boldsymbol{\widehat B}}{\mathit{\boldsymbol{Q}}_{ll}}{{\mathit{\boldsymbol{\widehat B}}}^{\rm{T}}}} \right)^{ - 1}}(\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{A\hat \xi }})$ (10)

 $\mathit{\boldsymbol{\widehat e}} = - {\mathit{\boldsymbol{Q}}_{ll}}{\mathit{\boldsymbol{\widehat B}}^{\rm{T}}}{\left( {\mathit{\boldsymbol{\widehat B}}{\mathit{\boldsymbol{Q}}_{ll}}{{\mathit{\boldsymbol{\widehat B}}}^{\rm{T}}}} \right)^{ - 1}}(\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{A}}\mathit{\boldsymbol{\widehat \xi }})$ (11)

 $f\left( {\mathit{\boldsymbol{l}} + \mathit{\boldsymbol{e}},\mathit{\boldsymbol{\xi }}} \right) = \left( {\mathit{\boldsymbol{A}} - {\mathit{\boldsymbol{E}}_A}} \right)\mathit{\boldsymbol{\xi }} - \mathit{\boldsymbol{y}} + {\mathit{\boldsymbol{e}}_y}$ (15)

 $\begin{array}{*{20}{c}} {f\left( {\mathit{\boldsymbol{l}} + \mathit{\boldsymbol{e}},\mathit{\boldsymbol{\xi }}} \right) = \frac{{\partial f}}{{\partial {\mathit{\boldsymbol{\xi }}^{\rm{T}}}}}\left| {_{l + {\mathit{\boldsymbol{e}}^{\left( i \right)}},{\xi ^{\left( i \right)}}}} \right.{\rm{d}}{\xi ^{i + 1}} + \frac{{\partial f}}{{\partial {\mathit{\boldsymbol{l}}^{\rm{T}}}}}\left| {_{l + {e^{\left( i \right)}},{\xi ^{\left( i \right)}}}} \right. \cdot }\\ {{{\left( {\mathit{\boldsymbol{e}} - \mathit{\boldsymbol{e}}} \right)}^{\left( i \right)}} + f\left( {\mathit{\boldsymbol{l}} + {\mathit{\boldsymbol{e}}^{\left( i \right)}},{\mathit{\boldsymbol{\xi }}^{\left( i \right)}}} \right) = }\\ {\frac{{\partial f}}{{\partial {\mathit{\boldsymbol{\xi }}^{\rm{T}}}}}\left| {_{l + {e^{\left( i \right)}},{\xi ^{\left( i \right)}}}} \right.{\rm{d}}{\xi ^{i + 1}} + \frac{{\partial f}}{{\partial {\mathit{\boldsymbol{l}}^{\rm{T}}}}}\left| {_{l + {e^{\left( i \right)}},{\xi ^{\left( i \right)}}}} \right. \cdot }\\ {\mathit{\boldsymbol{e}} + f\left( {\mathit{\boldsymbol{l}},{\mathit{\boldsymbol{\xi }}^{\left( i \right)}}} \right) = \left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{E}}_A^{\left( i \right)}} \right){\rm{d}}{\mathit{\boldsymbol{\xi }}^{\left( {i + 1} \right)}} + }\\ {\left[ {\begin{array}{*{20}{c}} {{{\left( {{\mathit{\boldsymbol{\xi }}^{\left( i \right)}}} \right)}^{\rm{T}}} \otimes {\mathit{\boldsymbol{l}}_n}}&{ - {\mathit{\boldsymbol{l}}_n}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{e}}_A^{\left( {i + 1} \right)}}\\ {\mathit{\boldsymbol{e}}_y^{\left( {i + 1} \right)}} \end{array}} \right] + \mathit{\boldsymbol{A}}{\mathit{\boldsymbol{\xi }}^{\left( i \right)}} - \mathit{\boldsymbol{y}}} \end{array}$ (16)

 $\begin{array}{l} {\rm{d}}{{\mathit{\boldsymbol{\hat \xi }}}^{\left( {i + 1} \right)}} = \\ {\left( {{{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{\hat E}}_A^{\left( {i + 1} \right)}} \right)}^{\rm{T}}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}{\mathit{\boldsymbol{Q}}_{ll}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}} \right)}^{\rm{T}}}} \right)}^{ - 1}}\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{\hat E}}_A^{\left( {i + 1} \right)}} \right)} \right)^{ - 1}} \cdot \\ {\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{\hat E}}_A^{\left( {i + 1} \right)}} \right)^{\rm{T}}}{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}{\mathit{\boldsymbol{Q}}_{ll}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}} \right)}^{\rm{T}}}} \right)^{ - 1}}\left( {\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat \xi }}}^{\left( i \right)}}} \right) \end{array}$ (17)
 $\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat \xi }}}^{\left( {i + 1} \right)}} = {{\mathit{\boldsymbol{\hat \xi }}}^{\left( i \right)}} + {\rm{d}}{{\mathit{\boldsymbol{\hat \xi }}}^{\left( {i + 1} \right)}} = }\\ {{{\left( {{{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{\hat E}}_A^{\left( {i + 1} \right)}} \right)}^{\rm{T}}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}{\mathit{\boldsymbol{Q}}_{ll}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}} \right)}^{\rm{T}}}} \right)}^{ - 1}}\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{\hat E}}_A^{\left( {i + 1} \right)}} \right)} \right)}^{ - 1}} \cdot }\\ {{{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{\hat E}}_A^{\left( {i + 1} \right)}} \right)}^{\rm{T}}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}{\mathit{\boldsymbol{Q}}_{ll}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}} \right)}^{\rm{T}}}} \right)}^{ - 1}}\left( {\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{\hat E}}_A^{\left( {i + 1} \right)}{{\mathit{\boldsymbol{\hat \xi }}}^{\left( i \right)}}} \right)} \end{array}$ (18)

 $\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{Q}}_{\hat \xi \hat \xi }} = \left( {{{\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{\hat E}}_A^{\left( {i + 1} \right)}} \right)}^{\rm{T}}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}{\mathit{\boldsymbol{Q}}_{ll}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}} \right)}^{\rm{T}}}} \right)}^{ - 1}}} \right. \cdot }\\ {{{\left. {\left( {\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{\hat E}}_A^{\left( {i + 1} \right)}} \right)} \right)}^{ - 1}}} \end{array}$ (19)
 $\begin{array}{*{20}{c}} {\sigma _0^2 = \frac{{{{\mathit{\boldsymbol{\hat e}}}^{\rm{T}}}\mathit{\boldsymbol{P\hat e}}}}{{n - u}} = }\\ {\frac{{{{\left( {\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat \xi }}}^{\left( {i + 1} \right)}}} \right)}^{\rm{T}}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}{\mathit{\boldsymbol{Q}}_{ll}}{{\left( {{{\mathit{\boldsymbol{\hat B}}}^{\left( i \right)}}} \right)}^{\rm{T}}}} \right)}^{ - 1}}\left( {\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{A}}{{\mathit{\boldsymbol{\hat \xi }}}^{\left( {i + 1} \right)}}} \right)}}{{n - u}}} \end{array}$ (20)

3 基于经典LS原理的WTLS方法

 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{y}} - {\mathit{\boldsymbol{E}}_A}\mathit{\boldsymbol{\xi }} = \left( {\mathit{\boldsymbol{A}} - {\mathit{\boldsymbol{E}}_A}} \right)\mathit{\boldsymbol{\xi }} + \left( {{\mathit{\boldsymbol{e}}_y} - {\mathit{\boldsymbol{E}}_A}\mathit{\boldsymbol{\xi }}} \right) = }\\ {\left( {\mathit{\boldsymbol{A}} - {\mathit{\boldsymbol{E}}_A}} \right)\mathit{\boldsymbol{\xi }} - \mathit{\boldsymbol{Be}}} \end{array}$ (21)

$\mathit{\boldsymbol{\tilde y}} = {\rm{ }}\mathit{\boldsymbol{y}}{\rm{ }} - \mathit{\boldsymbol{E}}{_A}\mathit{\boldsymbol{\xi }}, \mathit{\boldsymbol{\tilde A}} = {\rm{ }}\mathit{\boldsymbol{A}}{\rm{ }} - {\rm{ }}\mathit{\boldsymbol{E}}{_A}, {\rm{ }}\mathit{\boldsymbol{\tilde e}} = - {\rm{ }}\mathit{\boldsymbol{Be}}$，则式(21)可写成类似式(1)的形式：

 $\mathit{\boldsymbol{\tilde y}} = \mathit{\boldsymbol{\tilde A\xi }} + \mathit{\boldsymbol{\tilde e}}$ (22)

 ${\mathit{\boldsymbol{Q}}_{\tilde y}} = {\mathit{\boldsymbol{Q}}_{\tilde e}} = \mathit{\boldsymbol{B}}{\mathit{\boldsymbol{Q}}_{ll}}{\mathit{\boldsymbol{B}}^{\rm{T}}}$ (23)

 $\mathit{\boldsymbol{\hat \xi }} = {\left( {{{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}}\mathit{\boldsymbol{Q}}_{\tilde y}^{ - 1}\mathit{\boldsymbol{\tilde A}}} \right)^{ - 1}}{\mathit{\boldsymbol{\tilde A}}^{\rm{T}}}\mathit{\boldsymbol{Q}}_{\tilde y}^{ - 1}\mathit{\boldsymbol{\tilde y}}$ (24)

 ${\mathit{\boldsymbol{Q}}_{\hat \xi \hat \xi }} = {\left( {{{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}}\mathit{\boldsymbol{Q}}_{\tilde y}^{ - 1}\mathit{\boldsymbol{\tilde A}}} \right)^{ - 1}}$ (25)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{\hat y}} = {\mathit{\boldsymbol{P}}_{\tilde A}}\mathit{\boldsymbol{\tilde y}} = \mathit{\boldsymbol{\tilde A\hat \xi }} = \left( {\mathit{\boldsymbol{A}} - {{\mathit{\boldsymbol{\tilde e}}}_A}} \right)\mathit{\boldsymbol{\hat \xi }}\\ \mathit{\boldsymbol{\hat e}} = P_{\tilde A}^ \bot \mathit{\boldsymbol{\tilde y}} = \mathit{\boldsymbol{\tilde y}} - \mathit{\boldsymbol{\hat y}} = \mathit{\boldsymbol{y}} - \mathit{\boldsymbol{A\hat \xi }} = {\mathit{\boldsymbol{Q}}_{\tilde y}}\mathit{\boldsymbol{\hat \lambda }} \end{array} \right.$ (26)

 ${\mathit{\boldsymbol{Q}}_{\mathit{\boldsymbol{\hat y}}}} = {\mathit{\boldsymbol{P}}_{\tilde A}}{\mathit{\boldsymbol{Q}}_{\tilde y}} = \mathit{\boldsymbol{\tilde A}}{\mathit{\boldsymbol{Q}}_{\hat \xi \hat \xi }}{{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}}$ (27)
 ${\mathit{\boldsymbol{Q}}_v} = {\mathit{\boldsymbol{Q}}_{\mathit{\boldsymbol{\hat e}}}} = \mathit{\boldsymbol{P}}_{\tilde A}^ \bot {\mathit{\boldsymbol{Q}}_{\tilde y}} = {\mathit{\boldsymbol{Q}}_{\tilde y}} - {\mathit{\boldsymbol{Q}}_{\mathit{\boldsymbol{\hat y}}}}$ (28)

 $\sigma _0^2 = \frac{{{{\mathit{\boldsymbol{\hat e}}}^{\rm{T}}}\mathit{\boldsymbol{Q}}_{\tilde y}^{ - 1}\mathit{\boldsymbol{\hat e}}}}{{n - u}} = \frac{{{{\left( {\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{A\hat \xi }}} \right)}^{\rm{T}}}\mathit{\boldsymbol{Q}}_{\tilde y}^{ - 1}\left( {\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{A\hat \xi }}} \right)}}{{n - u}}$ (29)

1) 给定系数阵A、观测向量y及协因数矩阵Qll，计算初始值${{\mathit{\boldsymbol{\hat \xi }}}^0} = {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{Q}}_{\tilde y}^{ - 1}\mathit{\boldsymbol{A}}} \right)^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{Q}}_{\tilde y}^{ - 1}\mathit{\boldsymbol{y}}$;

2) 令i=0, 1, …kk为迭代次数，依次计算${{\mathit{\boldsymbol{\hat B}}}^{(i + 1)}} = [{({{\mathit{\boldsymbol{\hat \xi }}}^{\left( i \right)}})^{\rm{T}}} \otimes {\rm{ }}\mathit{\boldsymbol{I}}{_n} - {\rm{ }}\mathit{\boldsymbol{I}}{_n}], \mathit{\boldsymbol{\hat \lambda }}{^{(i + 1)}} = {(\mathit{\boldsymbol{\hat B}}{^{(i + 1)}}\mathit{\boldsymbol{Q}}{_{ll}}\cdot{(\mathit{\boldsymbol{\hat B}}{^{(i + 1)}})^{\rm{T}}})^{ - 1}}\left( {{\rm{ }}\mathit{\boldsymbol{y}} - {\rm{ }}\mathit{\boldsymbol{A\hat \xi }}{^{(i)}}} \right),$ $\mathit{\boldsymbol{\hat e}}_A^{\left( {i + 1} \right)} = - \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{Q}}{_{AA}}}&{\mathit{\boldsymbol{Q}}{_{Ay}}} \end{array}} \right]\cdot{({{\mathit{\boldsymbol{\hat B}}}^{(i + 1)}})^{\rm{T}}}{{\mathit{\boldsymbol{\hat \lambda }}}^{(i + 1)}}, \mathit{\boldsymbol{\hat e}}_A^{\left( {i + 1} \right)} = {\rm{ivec}}\left( {\mathit{\boldsymbol{\hat e}}_A^{\left( {i + 1} \right)}} \right), {{\mathit{\boldsymbol{\tilde A}}}^{(i + 1)}} = {\rm{ }}\mathit{\boldsymbol{A}} - \mathit{\boldsymbol{\hat e}}_A^{\left( {i + 1} \right)},$ ${({\rm{ }}\mathit{\boldsymbol{Q}}_{\tilde y}^{ - 1})^{(i + 1)}} = {({\rm{ }}\mathit{\boldsymbol{B}}{^{(i + 1)}}\mathit{\boldsymbol{Q}}{\mathit{\boldsymbol{}}_{ll}}{({\rm{ }}\mathit{\boldsymbol{B}}{^{(i + 1)}})^{\rm{T}}})^{ - 1}}, {{\mathit{\boldsymbol{\tilde y}}}^{(i + 1)}} = {\rm{ }}\mathit{\boldsymbol{y}} - \mathit{\boldsymbol{\hat e}}_A^{\left( {i + 1} \right)}\mathit{\boldsymbol{\xi }}{^{(i)}}, \mathit{\boldsymbol{Q}}_{\hat \xi \hat \xi }^{(i + 1)} = {({({{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}})^{(i + 1)}}{(\mathit{\boldsymbol{Q}}_{\tilde y}^{ - 1})^{(i + 1)}}{{\mathit{\boldsymbol{\tilde A}}}^{(i + 1)}})^{1}},$ $\mathit{\boldsymbol{\hat \xi }}{^{(i + 1)}} = {\rm{ }}\mathit{\boldsymbol{Q}}_{\hat \xi \hat \xi }^{(i + 1)}{{\mathit{\boldsymbol{\tilde A}}}^{\rm{T}}}^{\left( {i + 1} \right)}{\rm{ }}{(Q_{\tilde y}^{ - 1})^{(i + 1)}};$

3) 给定一个足够小的正常数ε，若$\parallel {{\mathit{\boldsymbol{\hat \xi }}}^{\left( {i + 1} \right)}} - {{\mathit{\boldsymbol{\hat \xi }}}^{\left( i \right)}}\parallel < \varepsilon$，停止迭代，$\mathit{\boldsymbol{\hat \xi }} = {{\mathit{\boldsymbol{\hat \xi }}}^{\left( {i + 1} \right)}}$

4 算例分析

 ${\mathit{\boldsymbol{Q}}_{\hat \xi \hat \xi }} = {10^{ - 2}} \times \left( {\begin{array}{*{20}{c}} {8.700\;8}&{ - 1.6473}\\ { - 1.6473}&{0.3362} \end{array}} \right)$

 ${\mathit{\boldsymbol{D}}_{\hat \xi \hat \xi }} = \sigma _0^2{\mathit{\boldsymbol{Q}}_{\hat \xi \hat \xi }} = \left( {\begin{array}{*{20}{c}} {0.129\;1}&{ - 0.024\;4}\\ { - 0.024\;4}&{0.005\;0} \end{array}} \right)$

5 结语

 [1] 武汉大学测绘学院测量平差学科组. 误差理论与测量平差基础[M]. 武汉: 武汉大学出版社, 2014 (Surveying Adjustment Group in School of Geodesy and Geomatics of Wuhan University. Error Theory and Foundation of Surveying Adjustment[M]. Wuhan: Wuhan University Press, 2014) (0) [2] Neitzel F. Generalization of Total Least-squares on Example of Unweighted and Weighted 2D Similarity Transformation[J]. Journal of Geodesy, 2010, 84(12): 751-762 DOI:10.1007/s00190-010-0408-0 (0) [3] Shen Y Z, Li B F, Chen Y. An Iterative Solution of Weighted Total-Least Squares Adjustment[J]. Journal of Geodesy, 2011, 85(4): 229-238 DOI:10.1007/s00190-010-0431-1 (0) [4] Xu C J, Wang L Y, Wen Y M, et al. Strain Rates in the Sichuan-Yunnan Region Based upon the Total Least Squares Heterogeneous Strain Model from GPS Data[J]. Terrestrial Atmospheric and Oceanic Sciences, 2011, 22(2): 133-147 DOI:10.3319/TAO.2010.07.26.02(TibXS) (0) [5] Huffel S V, Lemmerling P. Total Least Squares and Errors-in-Variables Modeling, Analysis, Algorithms and Applications[M]. Alphen: Kluwer Academic Publishers, 2001 (0) [6] Golub G H, Loan C F V. An Analysis of the Total Least Squares Problem[J]. SIAM, 1980, 17(6): 883-893 (0) [7] Mahboub V. On Weighted Total Least Squares for Geodetic Transformations[J]. Journal of Geodesy, 2012, 86(5): 359-367 DOI:10.1007/s00190-011-0524-5 (0) [8] Amiri-Simkooei A, Jazaeri S. Weighted Total Least Squares Formulated by Standard Least Squares[J]. Journal of Geodetic Sciences, 2012, 2(2): 113-124 (0) [9] Fang X. Weighted Total Least Squares: Necessary and Sufficient Conditions, Fixed and Random Parameters[J]. Journal of Geodesy, 2013, 87(8): 733-749 DOI:10.1007/s00190-013-0643-2 (0)
Classic Least Squares Method to the Weighted EIV Model
XIE Jian1     LONG Sichun1
1. Hunan Province Key Laboratory of Coal Resources Clean-Utilization and Mine Environment Protection, Hunan University of Science and Technology, Taoyuan Road, Xiangtan 411201, China
Abstract: First, the EIV model is linearized at the optimal solution through the GHM method and the approximate variance matrix is derived. Then, the EIV model is reformulated in the form of the Gauss-Markov model. The solution to EIV model and its approximate dispersion matrix are derived using the standard least squares theory, which is equivalent to the existing results. Finally, the statistical properties of the estimation of observations and residuals are derived and the system of parameter estimation and accuracy assessment of EIV model are established.
Key words: classic least squares; error-in-variables (EIV) model; weighted total least squares (WTLS); line fitting