﻿ 基于改进的向后-向前选择法粗差定位与估值算法
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 大地测量与地球动力学  2019, Vol. 39 Issue (5): 538-540  DOI: 10.14075/j.jgg.2019.05.019

### 引用本文

CHAI Shuangwu, YANG Xiaoqin. Algorithm for Location and Evaluation of Gross Errors Based on Improved Backward-Forward Selection Method[J]. Journal of Geodesy and Geodynamics, 2019, 39(5): 538-540.

### Foundation support

National Natural Science Foundation of China, No. 51504159; Youth Foundation of Taiyuan University of Technology, No. 2014TD008.

### Corresponding author

YANG Xiaoqin, PhD, majors in measurement data processing and mining subsidence, E-mail: yangxiaoqin@tyut.edu.cn.

### 第一作者简介

CHAI Shuangwu, postgraduate, majors in measurement data processing and mining subsidence, E-mail:2964633881@qq.com.

### 文章历史

1. 太原理工大学矿业工程学院, 太原市新矿院路18号, 030024

1 粗差定位的基本原理 1.1 传统的向后-向前选择法

1.2 改进的向后-向前选择法

1.2.1 平差模型的整体检验

 ${\chi ^2} = \frac{{{\mathit{\boldsymbol{V}}^{\rm{T}}}\mathit{\boldsymbol{PV}}}}{{\sigma _0^2}} \sim {\chi ^2}(f)$ (1)

1.2.2 Baarda数据探测法

 ${w_i} = \frac{{\left| {{v_i}} \right|}}{{{\sigma _0}\sqrt {{q_{{v_i}}}} }} \sim N(0,1)$ (2)

1.2.3 统计量的相关系数

 ${w_i} = \frac{{\left| {{v_i}} \right|}}{{{\sigma _0}\sqrt {{q_{{v_i}}}} }} = \frac{{\mathit{\boldsymbol{e}}_i^{\rm{T}}\mathit{\boldsymbol{V}}}}{{{\sigma _0}\sqrt {{q_{{v_i}}}} }}$ (3)
 ${w_j} = \frac{{\left| {{v_j}} \right|}}{{{\sigma _0}\sqrt {{q_{{v_j}}}} }} = \frac{{\mathit{\boldsymbol{e}}_j^{\rm{T}}\mathit{\boldsymbol{V}}}}{{{\sigma _0}\sqrt {{q_{{v_j}}}} }}$ (4)

 $\begin{array}{l} {\sigma _{{w_i}{w_j}}} = \frac{{\mathit{\boldsymbol{e}}_i^{\rm{T}}}}{{{\sigma _0}\sqrt {{q_{{v_i}}}} }}D\left( \mathit{\boldsymbol{V}} \right){\left( {\frac{{\mathit{\boldsymbol{e}}_j^{\rm{T}}}}{{{\sigma _0}\sqrt {{q_{{v_j}}}} }}} \right)^{\rm{T}}} = \\ \frac{{\mathit{\boldsymbol{e}}_i^{\rm{T}}}}{{{\sigma _0}\sqrt {{q_{{v_i}}}} }}\sigma _0^2{\mathit{\boldsymbol{Q}}_{VV}}\frac{{{\mathit{\boldsymbol{e}}_j}}}{{{\sigma _0}\sqrt {{q_{{v_j}}}} }} = \frac{{\mathit{\boldsymbol{e}}_i^{\rm{T}}{\mathit{\boldsymbol{Q}}_{VV}}{\mathit{\boldsymbol{e}}_j}}}{{\sqrt {{q_{{v_i}}}{q_{{v_j}}}} }} = \frac{{{q_{{v_{ij}}}}}}{{\sqrt {{q_{{v_i}}}{q_{{v_j}}}} }} \end{array}$ (5)

 $\rho = \left| {\frac{{{q_{{v_{ij}}}}}}{{\sqrt {{q_{{v_i}}}{q_{{v_j}}}} }}} \right|$ (6)

1.2.4 偏相关系数的检验

 $F = \left| {\frac{{{T_{m - 1}} - {T_m}}}{{{T_m}/\left( {n - m} \right)}}} \right| \sim F\left( {1,n - m} \right)$ (7)

FFα(1, nm)时，观测值Li不含粗差; 否则，观测值Li含粗差。

2 多维粗差定位算法

1) 对所有观测值进行LS平差，对平差模型按式(1)进行整体检验。若检验通过，观测值中不含粗差，结束; 否则，观测值中可能含有粗差。

2) 计算标准化残差值wi，取wi最大值wmax对应的观测值Li按式(2)进行u检验。若检验通过，则继续步骤5) ~ 8)。

3) 若未通过检验，根据式(6)依次计算观测值Li对应的统计量wmax与第j(ji)个观测值Lj对应的统计量wj之间的相关系数ρ。若ρmax＞0.7，将wmax对应的观测值Li存入列表A，将ρmax对应的观测值Lj存入列表B，同时剔除观测值LiLj; 若ρmax≤0.7，仅将wmax对应的观测值Li存入列表C，同时剔除观测值Li

4) 重新对剩余的观测值进行LS平差。重复步骤2)和步骤3)，直到u检验通过。

5) 经过前面几步，可以得到一些怀疑含有粗差的观测值：观测值对应的统计量具有相关性的分别存放在列表A和B中，不具有相关性的存放在列表C中。分别对列表A和列表B中的观测值根据式(7)进行偏相关系数的F检验。若检验通过，说明该观测值不含粗差，将其恢复到“可靠”观测值中; 若未通过检验，将其添加到列表Gross中。

6) 若列表A和B中的观测值检验完成，则清空列表A和B; 若未完成，则重复步骤5)。

7) 每次从列表C中取出一个观测值Li，与前面几步确定的所有不含粗差的“可靠”观测值进行LS平差，对观测值Li进行单独的u检验。若检验通过，说明之前是误判，将该观测值恢复到不含粗差的观测值中; 若检验未通过，说明Li是粗差，将其添加到列表Gross中，直到完成对列表C中观测值的检验。

8) 对剩下的所有不含粗差的观测值重新进行整体检验，即重复步骤1) ~8)。

3 粗差估值

 ${\mathit{\boldsymbol{V}}_g} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{B}}_g}}&\mathit{\boldsymbol{E}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat X}}}\\ {{{\mathit{\boldsymbol{ \boldsymbol{\hat \varDelta} }}}_g}} \end{array}} \right] - {\mathit{\boldsymbol{l}}_g}$ (8)

 ${\mathit{\boldsymbol{V}}_r} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{B}}_r}}&{\bf{0}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat X}}}\\ {{{\mathit{\boldsymbol{ \boldsymbol{\hat \varDelta} }}}_g}} \end{array}} \right] - {\mathit{\boldsymbol{l}}_r}$ (9)
 图 1 IBFS算法流程图 Fig. 1 The flow chart of IBFS algorithm

 $\mathit{\boldsymbol{V}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{V}}_g}}\\ {{\mathit{\boldsymbol{V}}_r}} \end{array}} \right],\mathit{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{B}}_g}}&\mathit{\boldsymbol{E}}\\ {{\mathit{\boldsymbol{B}}_r}}&{\bf{0}} \end{array}} \right],\mathit{\boldsymbol{\hat K}} = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{\hat X}}}\\ {{{\mathit{\boldsymbol{ \boldsymbol{\hat \varDelta} }}}_g}} \end{array}} \right]$
 $\mathit{\boldsymbol{l}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{l}}_g}}\\ {{\mathit{\boldsymbol{l}}_r}} \end{array}} \right],\mathit{\boldsymbol{P}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{P}}_g}}&{\bf{0}}\\ {\bf{0}}&{{\mathit{\boldsymbol{P}}_r}} \end{array}} \right]$

 $\mathit{\boldsymbol{V}} = \mathit{\boldsymbol{A\hat K}} - \mathit{\boldsymbol{l}}$ (10)

VTPV =min的约束条件下，按照求函数极值的方法，可得：

 $\mathit{\boldsymbol{\hat K}} = {\left( {{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{PA}}} \right)^{ - 1}}{\mathit{\boldsymbol{A}}^{\rm{T}}}\mathit{\boldsymbol{Pl}}$ (11)

 $\mathit{\boldsymbol{\bar l}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{l}}_g}}\\ {{\mathit{\boldsymbol{l}}_r}} \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} {{{\mathit{\boldsymbol{ \boldsymbol{\hat \varDelta} }}}_g}}\\ {\bf{0}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{l}}_g} - {{\mathit{\boldsymbol{ \boldsymbol{\hat \varDelta} }}}_g}}\\ {{\mathit{\boldsymbol{l}}_r}} \end{array}} \right],\mathit{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} {{\mathit{\boldsymbol{B}}_g}}\\ {{\mathit{\boldsymbol{B}}_r}} \end{array}} \right]$ (12)

 $\mathit{\boldsymbol{\bar V}} = \mathit{\boldsymbol{B\bar X}} - \mathit{\boldsymbol{\bar l}}$ (13)

 $\mathit{\boldsymbol{\bar X = }}{\left( {{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{PB}}} \right)^{ - 1}}{\mathit{\boldsymbol{B}}^{\rm{T}}}\mathit{\boldsymbol{P\bar l}}$ (14)

 ${\hat \sigma _0} = \sqrt {\frac{{{{\mathit{\boldsymbol{\bar V}}}^{\rm{T}}}\mathit{\boldsymbol{P\bar V}}}}{{n - t}}}$ (15)
4 算例分析

 图 2 水准网示意图 Fig. 2 Diagram of the leveling network

5 结语

1) 当观测值中存在多个粗差，且构建的统计量之间相关时，传统的数据探测法和向后-向前选择法无法正确实现粗差定位，容易犯第3类错误(粗差转移)，而改进的向后-向前选择法保留了向后-向前选择法的优点，并避免弃真错误和纳伪错误。

2) 为了能够探测较小的粗差，通常需要在向后选择阶段适当增大显著性水平，以增加纳伪概率、降低弃真概率; 而在向前选择过程中可以适当降低显著性水平，来降低纳伪概率，两种方式组合可以实现很好的粗差探测效果。根据实验，在向后选择过程中显著性水平取α=0.10，向前选择过程中显著性水平取α=0.01，可以实现较好的粗差定位效果。

3) 实验表明，加入模拟的5个粗差(1个小粗差、2个较大粗差和3个大粗差)，本文算法也能很好地探测出小的粗差，说明本文算法在多个粗差同时存在的情况下，即使有大粗差可能掩盖小粗差的情况，也非常有效。推导的粗差估值公式比较严密，且粗差估值也比较准确。

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Algorithm for Location and Evaluation of Gross Errors Based on Improved Backward-Forward Selection Method
CHAI Shuangwu1     YANG Xiaoqin1
1. College of Mining Technology, Taiyuan University of Technology, 18 Xinkuangyuan Road, Taiyuan 030024, China
Abstract: In this paper, the traditional backward-forward selection method of gross error locating method is improved: testing the overall adjustment model, calculating the statistic correlation coefficient and using the partial correlation coefficient test are added to locate the statistics related to the gross error observation values. The simulation results show that this algorithm can accurately locate the multidimensional gross errors, effectively improve the gross errors location and transfer, and the estimation results of gross errors are reliable.
Key words: data snooping; statistical correlation coefficient; partial correlation coefficient test; location of gross errors; evaluation of gross errors