﻿ 灰色稳健总体最小二乘估计及高铁路基变形预测
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 大地测量与地球动力学  2018, Vol. 38 Issue (2): 141-146  DOI: 10.14075/j.jgg.2018.02.007

### 引用本文

CHEN Yang, WEN Hongyan, QIN Hui, et al. Robust Total Least Squares Estimated in GM(1, 1) for High-Speed Railway Foundation Deformation Prediction[J]. Journal of Geodesy and Geodynamics, 2018, 38(2): 141-146.

### Foundation support

National Natural Science Foundation of China, No.41461089;Guangxi Graduate Education Innovation Program, No. YCSW2017155;Open Fund of Guangxi Key Laboratory of Spatical Information and Geomatics, No.1638025-26; Guangxi Bagui Scholar Special Fund of Post and Innovation; Open Fund of Key Laboratory for Digital Land and Resources of Jiangxi Province, East China University of Technology, No. DLLJ201711.

### 第一作者简介

CHEN Yang, postgraduate, majors in deformation monitoring and data processing, E-mail:550510778@qq.com.

### 文章历史

1. 桂林理工大学测绘地理信息学院，桂林市雁山街319号，541006;
2. 武汉大学测绘学院，武汉市珞喻路129号，430079;
3. 东华理工大学江西省数字国土重点实验室，南昌市广兰大道418号，330013

1 PEIV-TLS-IGGⅢ稳健估计

GM(1, 1)白化微分方程式为：

 $\frac{{{\rm{d}}{\mathit{\boldsymbol{x}}^{\left( 1 \right)}}}}{{{\rm{d}}t}} + a{\mathit{\boldsymbol{x}}^{\left( 1 \right)}} = b$ (1)

 $\mathit{\boldsymbol{\hat a}} = {\left[ {\begin{array}{*{20}{c}} a&b \end{array}} \right]^{\rm{T}}}$ (2)

 $\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} { - 0.5\left( {{x^{\left( 1 \right)}}\left( 1 \right) + {x^{\left( 1 \right)}}\left( 2 \right)} \right)}&1\\ { - 0.5\left( {{x^{\left( 1 \right)}}\left( 2 \right) + {x^{\left( 1 \right)}}\left( 3 \right)} \right)}&1\\ \vdots&\vdots \\ { - 0.5\left( {{x^{\left( 1 \right)}}\left( {n - 1} \right) + {x^{\left( 1 \right)}}\left( n \right)} \right)}&1 \end{array}} \right]$
 $\mathit{\boldsymbol{L}} = {\left[ {\begin{array}{*{20}{c}} {{x^{\left( 0 \right)}}\left( 2 \right)}&{{x^{\left( 0 \right)}}\left( 3 \right)}& \cdots &{{x^{\left( 0 \right)}}\left( n \right)} \end{array}} \right]^{\rm{T}}}$

 $\mathit{\boldsymbol{L}} + {{\mathit{\boldsymbol{ \varepsilon }}}_y} = \mathit{\boldsymbol{B}} \cdot \mathit{\boldsymbol{\hat a}}$ (3)

 $\mathit{\boldsymbol{\hat a}} = {\left[ {\begin{array}{*{20}{c}} a&b \end{array}} \right]^{\rm{T}}} = {\left[ {{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{B}}} \right]^{ - 1}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{L}}$ (4)

 $\begin{array}{*{20}{c}} {\varphi \left( {{\mathit{\boldsymbol{e}}_y},{\mathit{\boldsymbol{e}}_B},\mathit{\boldsymbol{\hat a}}} \right) = \mathit{\boldsymbol{L}} + {\mathit{\boldsymbol{e}}_y} - \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{B}}_1} + {\mathit{\boldsymbol{e}}_B}}&{{\mathit{\boldsymbol{B}}_2}} \end{array}} \right] \cdot \mathit{\boldsymbol{\hat a}}}\\ {{\mathit{\boldsymbol{e}}_y} \in {R^{n \times 1}},{\mathit{\boldsymbol{e}}_B} \in {R^{n \times 1}}} \end{array}$ (5)

 $\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{e}}_y}}\\ {{\mathit{\boldsymbol{e}}_B}} \end{array}} \right] \propto N\left( {\left[ \begin{array}{l} 0\\ 0 \end{array} \right],\left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_L}}&0\\ 0&{{\mathit{\boldsymbol{ \boldsymbol{\varSigma} }}_B}} \end{array}} \right]} \right)$ (6)

 $\begin{array}{*{20}{c}} {f\left( {{\mathit{\boldsymbol{e}}_y},{\mathit{\boldsymbol{e}}_B},a} \right) \approx \varphi \left( {\mathit{\boldsymbol{e}}_y^0,\mathit{\boldsymbol{e}}_B^0,{{\mathit{\boldsymbol{\hat a}}}^0}} \right) + {\mathit{\boldsymbol{B}}_{10}}\left( {{\mathit{\boldsymbol{e}}_y} - \mathit{\boldsymbol{e}}_y^0} \right) + }\\ {{\mathit{\boldsymbol{B}}_{20}}\left( {{\mathit{\boldsymbol{e}}_B} - \mathit{\boldsymbol{e}}_B^0} \right) + {\mathit{\boldsymbol{A}}_0}\left( {\mathit{\boldsymbol{\hat a}} - {{\mathit{\boldsymbol{\hat a}}}^0}} \right) = 0} \end{array}$ (7)

 ${\mathit{\boldsymbol{B}}_{10}} = \frac{{\partial \varphi \left( {{\mathit{\boldsymbol{e}}_y},{\mathit{\boldsymbol{e}}_B},\mathit{\boldsymbol{\hat a}}} \right)}}{{\partial {\mathit{\boldsymbol{e}}_y}}},{\mathit{\boldsymbol{B}}_{20}} = \frac{{\partial \varphi \left( {{\mathit{\boldsymbol{e}}_y},{\mathit{\boldsymbol{e}}_B},\mathit{\boldsymbol{\hat a}}} \right)}}{{\partial {\mathit{\boldsymbol{e}}_B}}},$
 ${\mathit{\boldsymbol{A}}_0} = \frac{{\partial \varphi \left( {{\mathit{\boldsymbol{e}}_y},{\mathit{\boldsymbol{e}}_B},\mathit{\boldsymbol{\hat a}}} \right)}}{{\partial \mathit{\boldsymbol{\hat a}}}}$

 ${\mathit{\boldsymbol{W}}^0} = \varphi \left( {\mathit{\boldsymbol{e}}_y^0,\mathit{\boldsymbol{e}}_B^0,{{\mathit{\boldsymbol{\hat a}}}^0}} \right) - {\mathit{\boldsymbol{B}}_{10}}\mathit{\boldsymbol{e}}_y^0 - {\mathit{\boldsymbol{B}}_{20}}\mathit{\boldsymbol{e}}_B^0$ (8)

 ${\mathit{\boldsymbol{B}}_{10}}{\mathit{\boldsymbol{e}}_y} + {\mathit{\boldsymbol{B}}_{20}}{\mathit{\boldsymbol{e}}_B} + {\mathit{\boldsymbol{A}}_0}\left( {\mathit{\boldsymbol{\hat a}} - {{\mathit{\boldsymbol{\hat a}}}^0}} \right) + {\mathit{\boldsymbol{W}}^0} = 0$ (9)

IGGⅢ抗差方案充分考虑了实际观测数据，具有良好的抵御粗差能力[9]，其权因子为：

 $\mathit{\boldsymbol{W}}\left( u \right) = \left\{ {\begin{array}{*{20}{c}} {1,\left| u \right| < {k_0}}\\ {\frac{{{k_0}}}{{\left| u \right|}}{{\left( {\frac{{{k_1} - \left| u \right|}}{{{k_1} - {k_0}}}} \right)}^2},{k_0} \le \left| u \right| < {k_1}}\\ {0,\left| u \right| \ge {k_1}} \end{array}} \right.$ (10)

 $\begin{array}{l} {\mathit{\boldsymbol{W}}_y} = {\rm{diag}}\left( {{\mathit{\boldsymbol{W}}_{y,1}},{\mathit{\boldsymbol{W}}_{y,2}}, \cdots {\mathit{\boldsymbol{W}}_{y,n}}} \right),\\ {\mathit{\boldsymbol{W}}_B} = {\rm{diag}}\left( {{\mathit{\boldsymbol{W}}_{B,1}},{\mathit{\boldsymbol{W}}_{B,2}}, \cdots {\mathit{\boldsymbol{W}}_{B,n}}} \right) \end{array}$ (11)

 $\begin{array}{*{20}{c}} {\mathit{\Phi }\left( {{\mathit{\boldsymbol{e}}_y},{\mathit{\boldsymbol{e}}_B},\mathit{\boldsymbol{\lambda }},\mathit{\boldsymbol{\hat a}}} \right) = \mathit{\boldsymbol{e}}_y^{\rm{T}}\mathit{\boldsymbol{Q}}_1^{ - 1}{\mathit{\boldsymbol{e}}_y} + \mathit{\boldsymbol{e}}_B^{\rm{T}}\mathit{\boldsymbol{Q}}_2^{ - 1}{\mathit{\boldsymbol{e}}_B} - 2{\mathit{\boldsymbol{\lambda }}^{\rm{T}}}}\\ {\left[ {{\mathit{\boldsymbol{B}}_{10}}{\mathit{\boldsymbol{e}}_y} + {\mathit{\boldsymbol{B}}_{20}}{\mathit{\boldsymbol{e}}_B} + {\mathit{\boldsymbol{A}}_0}\left( {\mathit{\boldsymbol{\hat a}} - {{\mathit{\boldsymbol{\hat a}}}^0}} \right) + {\mathit{\boldsymbol{W}}^0}} \right]} \end{array}$ (12)

 $\frac{{\partial \mathit{\Phi }}}{{2\partial {\mathit{\boldsymbol{e}}_y}}} = \mathit{\boldsymbol{e}}_y^{\rm{T}}\mathit{\boldsymbol{Q}}_1^{ - 1} - {\mathit{\boldsymbol{\lambda }}^{\rm{T}}}{\mathit{\boldsymbol{B}}_{10}} = 0$ (13)
 $\frac{{\partial \mathit{\Phi }}}{{2\partial {\mathit{\boldsymbol{e}}_B}}} = \mathit{\boldsymbol{e}}_B^{\rm{T}}\mathit{\boldsymbol{Q}}_2^{ - 1} - {\mathit{\boldsymbol{\lambda }}^{\rm{T}}}{\mathit{\boldsymbol{B}}_{20}} = 0$ (14)
 $\frac{{\partial \mathit{\Phi }}}{{2\partial \mathit{\boldsymbol{\hat a}}}} = - {\mathit{\boldsymbol{\lambda }}^{\rm{T}}}{\mathit{\boldsymbol{A}}_0} = 0$ (15)
 $\frac{{\partial \mathit{\Phi }}}{{2\partial \mathit{\boldsymbol{\lambda }}}} = - \left( {{\mathit{\boldsymbol{B}}_{10}}{\mathit{\boldsymbol{e}}_y} + {\mathit{\boldsymbol{B}}_{20}}{\mathit{\boldsymbol{e}}_B} + {\mathit{\boldsymbol{A}}_0}\left( {\mathit{\boldsymbol{\hat a}} - {{\mathit{\boldsymbol{\hat a}}}^0}} \right) + \mathit{\boldsymbol{W}}} \right) = 0$ (16)

 $\mathit{\boldsymbol{N}} = {\mathit{\boldsymbol{B}}_{10}}{\mathit{\boldsymbol{Q}}_1}\mathit{\boldsymbol{B}}_{10}^{\rm{T}} + {\mathit{\boldsymbol{B}}_{20}}{\mathit{\boldsymbol{Q}}_2}\mathit{\boldsymbol{B}}_{20}^{\rm{T}}$ (17)

 $\mathit{\boldsymbol{\hat a}} = - {\left[ {\mathit{\boldsymbol{A}}_0^{\rm{T}}{\mathit{\boldsymbol{N}}^{ - 1}}{\mathit{\boldsymbol{A}}_0}} \right]^{ - 1}}\mathit{\boldsymbol{A}}_0^{\rm{T}}{\mathit{\boldsymbol{N}}^{ - 1}}{\mathit{\boldsymbol{W}}^0} + {{\mathit{\boldsymbol{\hat a}}}^0}$ (18)
 $\mathit{\boldsymbol{\lambda }} = - {\mathit{\boldsymbol{N}}^{ - 1}}\left[ {{\mathit{\boldsymbol{A}}_0}\left( {\mathit{\boldsymbol{\hat a}} - {{\mathit{\boldsymbol{\hat a}}}^0}} \right) + {\mathit{\boldsymbol{W}}^0}} \right]$ (19)

 $\left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{\hat e}}}_y}}\\ {{{\mathit{\boldsymbol{\hat e}}}_B}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{Q}}_1}\mathit{\boldsymbol{B}}_{10}^{\rm{T}}}\\ {{\mathit{\boldsymbol{Q}}_2}\mathit{\boldsymbol{B}}_{20}^{\rm{T}}} \end{array}} \right]\mathit{\boldsymbol{\lambda }}$ (20)

${{{\mathit{\boldsymbol{\hat{e}}}}}_{y}}$${{{\mathit{\boldsymbol{\hat{e}}}}}_{B}}$${\mathit{\boldsymbol{\hat{a}}}}$作为新的初值重新定权后，代入式(18)进行迭代计算，当$\left\| {{{\mathit{\boldsymbol{\hat{a}}}}}^{\left( i+1 \right)}}\text{-}{{{\mathit{\boldsymbol{\hat{a}}}}}^{\left( i \right)}} \right\|$时，计算结束。

 ${{\hat x}^{\left( 1 \right)}}\left( {t + 1} \right) = \left( {{x^{\left( 0 \right)}}\left( 1 \right) - \frac{b}{a}} \right){{\rm{e}}^{ - at}} + \frac{b}{a}$ (21)

 ${{\hat x}^{\left( 0 \right)}}\left( {t + 1} \right) = \left( {1 - {{\rm{e}}^a}} \right)\left( {{x^{\left( 0 \right)}}\left( 1 \right) - \frac{b}{a}} \right) \times {{\rm{e}}^{ - at}}$ (22)

2 模拟实验设计

2.1 方案一分析

 图 1 方案一发展系数a预测图像 Fig. 1 The development coefficient of alphabet a prediction images

 图 2 方案一灰作用量b预测图像 Fig. 2 The grey action of alphabet b prediction images
2.2 方案二分析

 图 3 方案二含一个粗差各模型预测图像 Fig. 3 Scheme 2 prediction images with single outlier

 图 4 方案二含两个粗差各模型预测图像 Fig. 4 Scheme 2 prediction images with two outliers
3 工程实例对比分析

 图 5 高铁路基沉降预测曲线图像 Fig. 5 High speed railway subsidence prediction image

4 结语

1) 通过模拟实验可以得出，当观测数据中含有粗差时，LS估计和PEIV-TLS估计都对粗差很敏感，不具备抗差性。PEIV-TLS因为考虑了系数矩阵的误差，其估计精度高于LS估计，但PEIV-TLS模型估计值方差较大，预测不稳定。

2) 由工程实例可知，PEIV-TLS-IGGⅢ模型考虑了系数矩阵误差，并在总体最小二乘条件下求解未知参数，预测精度高于LS-IGGⅢ模型。同时，PEIV-TLS-IGGⅢ模型具有一定的抗差性，预测值稳定性高于PEIV-TLS模型。

3) 高铁路基前期下沉量大，下沉速度快，容易产生测量误差，并且前期的测量值对后期的沉降预测影响较大。而PEIV-TLS-IGGⅢ方法使用少量数据就能实现高精度预测，为迅速准确预测工后沉降量、确定何时开始轨道工程施工提供依据。

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Robust Total Least Squares Estimated in GM(1, 1) for High-Speed Railway Foundation Deformation Prediction
CHEN Yang1     WEN Hongyan1     QIN Hui1     WANG Qingtao1     ZHOU Lü2,3
1. College of Geomatics and Geoinformation, Guilin University of Technology, 319 Yanshan Street, Guilin 541006, China;
2. School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China;
3. Key Laboratory for Digital Land and Resources of Jiangxi Province, East China University of Technology, 418 Guanglan Road, Nanchang 330013, China
Abstract: Least squares estimation and partial errors-in variables total least squares donot have the ability to resist gross errors. As gross error may also appear in the observed value and the coefficient matrix in differential equations, this paper puts forward a partial errors-invariables total least squares model based on IGGⅢ differential resistance. This paper also compares the robust least squares, partial errors-in variables total least squares with the new algorithm systematically, usingparameter estimation results, stability through simulation data, and high-speed railway observations data. The results show that the new algorithm's accuracy is high, which can be applied to the high-speed railway subsidence prediction.
Key words: robust partial errors-in variables total least squares estimation; GM(1, 1); high-speed railway subsidence prediction