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 大地测量与地球动力学  2021, Vol. 41 Issue (11): 1106-1110  DOI: 10.14075/j.jgg.2021.11.002

### 引用本文

ZOU Shilin, WU Xing, WANG Fengwei. Regularized Robust Solution for Ill-Posed Weighted Total Least Squares Model[J]. Journal of Geodesy and Geodynamics, 2021, 41(11): 1106-1110.

### 第一作者简介

ZOU Shilin, associate professor, majors in survey data processing and geographic information application, E-mail: slzou@ecut.edu.cn.

### 文章历史

1. 东华理工大学测绘工程学院，南昌市广兰大道418号，330013;
2. 东华理工大学勘察设计研究院，江西省抚州市学府路56号，344000;
3. 同济大学测绘与地理信息学院，上海市四平路1239号，200092

1 等权病态总体最小二乘模型的正则化解

 ${\boldsymbol{y}} - {{\boldsymbol{e}}_y} = ({\boldsymbol{A}} - {{\boldsymbol{E}}_A}){\boldsymbol{x}}$ (1)

 $\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{e}}_y}}\\ {{{\boldsymbol{e}}_A}} \end{array}} \right]\sim \left( {\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} 0\\ 0 \end{array}} \right]}&{\sigma _{\rm{0}}^{\rm{2}}\left[ {\begin{array}{*{20}{c}} {{{\boldsymbol{Q}}_y}}&{}\\ {}&{{{\boldsymbol{Q}}_A}} \end{array}} \right]} \end{array}} \right)$ (2)

 ${\boldsymbol{e}}_y^{\rm{T}}{{\boldsymbol{e}}_y} + {\boldsymbol{e}}_A^{\rm{T}}{{\boldsymbol{e}}_A} + \alpha {{\boldsymbol{x}}^{\rm{T}}}{\boldsymbol{x}} = \min$ (3)

 ${\boldsymbol{\hat x}} = {({{\boldsymbol{A}}^{\rm{T}}}{\boldsymbol{A}} + b{{\boldsymbol{I}}_n})^{ - 1}}{{\boldsymbol{A}}^{\rm{T}}}{\boldsymbol{y}}$ (4)

2 病态加权总体最小二乘模型的正则化抗差解 2.1 正则化解

 ${{\boldsymbol{Q}}_y}{\boldsymbol{ = P}}_y^{ - 1}{\boldsymbol{, }}{{\boldsymbol{Q}}_A} = {{\boldsymbol{Q}}_0} \otimes {{\boldsymbol{Q}}_{\boldsymbol{x}}}$ (5)

 ${\boldsymbol{e}}_y^{\rm{T}}{{\boldsymbol{P}}_y}{{\boldsymbol{e}}_y} + {\boldsymbol{e}}_A^{\rm{T}}({{\boldsymbol{P}}_0} \otimes {{\boldsymbol{P}}_{\boldsymbol{x}}}){{\boldsymbol{e}}_A} + \alpha {{\boldsymbol{x}}^{\rm{T}}}{\boldsymbol{x}} = \min$ (6)

 $\begin{array}{l} \varPhi ({{\boldsymbol{e}}_y}, {{\boldsymbol{e}}_A}, {\boldsymbol{\lambda }}, {\boldsymbol{x}}) = {\boldsymbol{e}}_y^{\rm{T}}{{\boldsymbol{P}}_y}{{\boldsymbol{e}}_y} + {\boldsymbol{e}}_A^{\rm{T}}({{\boldsymbol{P}}_0} \otimes {{\boldsymbol{P}}_{\boldsymbol{x}}}){{\boldsymbol{e}}_A} +\\ \alpha {{\boldsymbol{x}}^{\rm{T}}}{\boldsymbol{x}} + 2{{\boldsymbol{\lambda }}^{\rm{T}}}({\boldsymbol{y}} - {\boldsymbol{Ax}} - {{\boldsymbol{e}}_y} + ({{\boldsymbol{x}}^{\rm{T}}} \otimes {{\boldsymbol{I}}_n}) \cdot {{\boldsymbol{e}}_A}) \end{array}$ (7)

 $\frac{1}{2}\frac{{\partial \varPhi }}{{\partial {{\boldsymbol{e}}_y}}} = {\boldsymbol{Q}}_y^{ - 1}{{\boldsymbol{\tilde e}}_y} - {\boldsymbol{\hat \lambda }} = {\rm{0}}$ (8a)
 $\frac{1}{2}\frac{{\partial \varPhi }}{{\partial {{\boldsymbol{e}}_A}}} = ({\boldsymbol{Q}}_0^{ - 1} \otimes {\boldsymbol{Q}}_{\boldsymbol{x}}^{ - 1}){{\boldsymbol{\tilde e}}_A} + ({\boldsymbol{\hat x}} \otimes {{\boldsymbol{I}}_n}){\boldsymbol{\hat \lambda }} = {\rm{0}}$ (8b)
 $\frac{1}{2}\frac{{\partial \varPhi }}{{\partial {\boldsymbol{\hat \lambda }}}} = {\boldsymbol{y}} - {\boldsymbol{A\hat x}} - {{\boldsymbol{\tilde e}}_y} + ({{\boldsymbol{\hat x}}^{\rm{T}}} \otimes {{\boldsymbol{I}}_n}){{\boldsymbol{\tilde e}}_A} = {\rm{0}}$ (8c)
 $\frac{1}{2}\frac{{\partial \varPhi }}{{\partial {\boldsymbol{\hat x}}}} = \alpha {\boldsymbol{\hat x}} - {{\boldsymbol{A}}^{\rm{T}}}{\boldsymbol{\hat \lambda }} + {\boldsymbol{\tilde E}}_A^{\rm{T}}{\boldsymbol{\hat \lambda }} = {\rm{0}}$ (8d)

 ${{\boldsymbol{\tilde e}}_y} = {{\boldsymbol{Q}}_y}{\boldsymbol{\hat \lambda }}$ (9a)
 ${{\boldsymbol{\tilde e}}_A} = - ({{\boldsymbol{Q}}_0}{\boldsymbol{\hat x}} \otimes {{\boldsymbol{Q}}_{\boldsymbol{x}}}){\boldsymbol{\hat \lambda }}$ (9b)
 ${{\boldsymbol{\tilde E}}_A} = - {{\boldsymbol{Q}}_x}{\boldsymbol{\hat \lambda }}{{\boldsymbol{\hat x}}^{\rm{T}}}{{\boldsymbol{Q}}_0}$ (9c)

 ${\boldsymbol{\hat \lambda }} = {\left( {{{\boldsymbol{Q}}_y} + ({{{\boldsymbol{\hat x}}}^{\rm{T}}}{{\boldsymbol{Q}}_0}{\boldsymbol{\hat x}}){{\boldsymbol{Q}}_{\boldsymbol{x}}}} \right)^{ - 1}}({\boldsymbol{y}} - {\boldsymbol{A\hat x}})$ (10)

 $\begin{array}{l} - {{\boldsymbol{A}}^{\rm{T}}}{\boldsymbol{\hat \lambda }} = {{\boldsymbol{A}}^{\rm{T}}}{\left( {{{\boldsymbol{Q}}_y} + ({{{\boldsymbol{\hat x}}}^{\rm{T}}}{{\boldsymbol{Q}}_0}{\boldsymbol{\hat x}}){{\boldsymbol{Q}}_{\boldsymbol{x}}}} \right)^{ - 1}}({\boldsymbol{A\hat x}} - {\boldsymbol{y}})=\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\boldsymbol{\tilde E}}_A^{\rm{T}}{\boldsymbol{\hat \lambda }} - \alpha {\boldsymbol{\hat x}} \end{array}$ (11)

 ${{\boldsymbol{A}}^{\rm{T}}}{\left( {{{\boldsymbol{Q}}_y} + ({{{\boldsymbol{\hat x}}}^{\rm{T}}}{{\boldsymbol{Q}}_0}{\boldsymbol{\hat x}}){{\boldsymbol{Q}}_{\boldsymbol{x}}}} \right)^{ - 1}}({\boldsymbol{A\hat x}} - {\boldsymbol{y}}) = {{\boldsymbol{Q}}_0}{\boldsymbol{\hat x}}\hat v - \alpha {\boldsymbol{\hat x}}$ (12)

 ${\boldsymbol{\hat x}} = {({{\boldsymbol{A}}^{\rm{T}}}{{\boldsymbol{Q}}_{\boldsymbol{l}}}{\boldsymbol{A}} - \hat v{{\boldsymbol{Q}}_0} + \alpha {{\boldsymbol{I}}_n})^{ - 1}}{{\boldsymbol{A}}^{\rm{T}}}{{\boldsymbol{Q}}_{\boldsymbol{l}}}{\boldsymbol{y}}$ (13)

 $\hat \sigma = \frac{{{\boldsymbol{\tilde e}}_y^{\rm{T}}{{\boldsymbol{Q}}_y}{{{\boldsymbol{\tilde e}}}_y} + {\boldsymbol{\tilde e}}_A^{\rm{T}}{{\boldsymbol{Q}}_A}{{{\boldsymbol{\tilde e}}}_A}}}{{m - n}}$ (14)

 $\begin{array}{c} {{\boldsymbol{Q}}_{\boldsymbol{l}}} = {(1 + {{{\boldsymbol{\hat x}}}^{\rm{T}}}{\boldsymbol{x}})^{ - 1}}{{\boldsymbol{I}}_m}\\ {\boldsymbol{\hat \lambda }} = {(1 + {{{\boldsymbol{\hat x}}}^{\rm{T}}}{\boldsymbol{x}})^{ - 1}}({\boldsymbol{y}} - {\boldsymbol{A\hat x}})\\ \hat v = {(1 + {{{\boldsymbol{\hat x}}}^{\rm{T}}}{\boldsymbol{x}})^{ - 2}}{({\boldsymbol{y}} - {\boldsymbol{A\hat x}})^{\rm{T}}}({\boldsymbol{y}} - {\boldsymbol{A\hat x}}) \end{array}$ (15)

 ${\boldsymbol{\hat x}} = {\left( {{{\boldsymbol{A}}^{\rm{T}}}{\boldsymbol{A}} + (\alpha - \hat v)(1 + {{{\boldsymbol{\hat x}}}^{\rm{T}}}{\boldsymbol{\hat x}}){{\boldsymbol{I}}_n}} \right)^{ - 1}}{{\boldsymbol{A}}^{\rm{T}}}{\boldsymbol{y}}$ (16)

2.2 基于中位数法的正则化抗差解

 ${{\bar P}_i} = \left\{ \begin{array}{l} {P_i}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| {{{\hat e}_i} \le a} \right|\\ {P_i}a{\rm{/}}\left| {{{\hat v}_i}} \right|\;\;\;\;\;\;\;\;\;a < \left| {{{\hat e}_i}} \right| \le b\\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} b < \left| {{{\hat e}_i}} \right| \end{array} \right.$ (17)

 ${\boldsymbol{\hat X}} = \left[ {\begin{array}{*{20}{c}} {{{{\boldsymbol{\hat X}}}_1}}& \cdots &{{{{\boldsymbol{\hat X}}}_{C_m^n}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\hat x_1^1}& \cdots &{\hat x_{C_m^n}^1}\\ \vdots &{}& \vdots \\ {\hat x_1^n}& \cdots &{\hat x_{C_m^n}^n} \end{array}} \right]$ (18)

3 算例分析 3.1 数值算例

 $z(y) = \int_a^b {K(x, y)f(x){\rm{d}}x}$ (21)

 $\begin{array}{l} K(x, y) = \frac{{1.0}}{{1.0 + 100{{(y - x)}^2}}}\\ f(x) = \frac{{\exp ({\beta _1}) + \exp ({\beta _2})}}{{10.9550408}} - 0.052130913 \end{array}$ (22)

 $\begin{array}{l} z({y_i}) = \frac{1}{2}\Delta x\sum\limits_{j = 1}^{50} {[K({x_j},{y_i})f({x_j}) + } \\ \;\;\;\;\;\;\;\;\;K({x_{j + 1}},{y_i})f({x_{j + 1}})] \end{array}$ (23)

 $RMSE = \sqrt {\frac{{{{({\boldsymbol{\hat X}} - {\boldsymbol{\bar X}})}^{\rm{T}}}({\boldsymbol{\hat X}} - {\boldsymbol{\bar X}})}}{n}}$ (24)

 图 1 最小二乘解和总体最小二乘解 Fig. 1 Least squares solution and total least squares solution

 图 2 正则化解、正则化抗差解与真值对比 Fig. 2 Comparison between regularized solution, regularized robust solution and true values

 图 3 不同算法500次实验获得估值的RMSE Fig. 3 The root mean square error of parameter estimation of four algorithms for 500 experiments
3.2 病态测边网算例

 图 4 空间测边网平面点位分布 Fig. 4 The point position distribution of the space net in XY plane

4 结语

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Regularized Robust Solution for Ill-Posed Weighted Total Least Squares Model
ZOU Shilin1,2     WU Xing2     WANG Fengwei3
1. Faculty of Geomatics, East China University of Technology, 418 Guanglan Road, Nanchang 330013, China;
2. Institute of Survey and Design, East China University of Technology, 56 Xuefu Road, Fuzhou 344000, China;
3. College of Surveying and Geo-Informatics, Tongji University, 1239 Siping Road, Shanghai 200092, China
Abstract: The coefficient matrix of the error-in-variables (EIV) model is ill-posed, and the precision of the coefficient matrix and the observation are not equal, so we derive the regularized solution of the ill-posed weighted total least squares model using the Lagrangian multiplier method. We prove the existing regularized solution of the ill-posed total least squares model to be a special case of the new model. On this basis, we propose the regularized robust solution based on the median method for ill-posed weighted total least squares model, and the effectiveness of the new algorithm is verified by two examples of the first kind Fredholm integral equation and ill-posed trilateration network. The results show that the least squares solution and the total least squares solution have poor accuracies and seriously deviate from the true value due to the influence of ill-posedness and the outliers of the model, while the accuracy of the regularized solution has been improved for weakening the ill-posedness of the model taking into account the errors of coefficient matrix and the observations. On the basis of regularized solution, the regularized robust solution reconstructs the weight matrix using the equivalent weight function, which can effectively resist the influence of gross error and has the highest accuracy.
Key words: ill-posed model; total least squares; regularized method; robust estimation