The identification method of gross error detection failpoint in L1-norm estimation

doi:10.11947/j.AGCS.2019.20180395

Abstract

Abstract: The gross errors reflected in the corresponding closure errors of conditional equations obtained by L₁-norm estimation are more significant than those in LS residuals, and thereby, the former estimation is helpful for the gross error detection and location. Unfortunately, there is a kind of observations which have the ability to detect and locate gross errors, no matter how large the gross error is, but it cannot be accurately located in L₁-norm estimation. For convenience of discussion, such observation is called as Robustness Failpoint in L₁-norm estimation, RFP-L₁ for short. Obviously, only to meet such a premise, the results of gross error detection based on L₁-norm estimation can be accurate and reliable, i.e., it is ensured that there is no RFP-L₁ in the surveying system, or whether the RFP-L₁ contains gross error can be judged accurately. And in this process, the accurate identification of RFP-L₁ is the basis for solving the problem. From conditional equation, the calculation formula of influence coefficient which reflect the extent of the influence of gross error on corresponding closure errors of conditional equations is derived, and the distinguish relation of judging whether an observation is RFP-L₁ or not according to the value of least influence coefficient is formulated. Furthermore, the numerical characteristic of design matrix that might contain RFP-L₁ is explored. And at the end, a method of identifying RFP-L₁ is put forward. The simulation results show that the least influence coefficient reflects the influence of gross errors on the objective function of L₁-norm estimation, and the least influence coefficient of normal observation and RFP-L₁ have a significant regularity equal to one and less than one respectively. In addition, it can be concluded that there is no RFP-L₁ in the surveying system with design matrix containing only±1 and 0.

Key words: L₁-norm estimation, gross errors detection, conditional equation, influence coefficient

CLC Number:

P207

YAN Guangfeng, CEN Minyi. The identification method of gross error detection failpoint in L₁-norm estimation[J]. Acta Geodaetica et Cartographica Sinica, 2019, 48(11): 1430-1438.

References 25

[1]	BAARDA W. A testing procedure for use in geodetic networks[M]. Delft:Netherlands Geodetic Commission, 1968:27-55.
[2]	於宗俦, 李明峰. 多维粗差的同时定位与定值[J]. 武汉测绘科技大学学报, 1996, 21(4):323-329. YU Zongchou, LI Mingfeng. Simultaneous location and evaluation of multidimensional gross errors[J]. Journal of Wuhan Technical University of Surveying and Mapping, 1996, 21(4):323-329.
[3]	欧吉坤. 粗差的拟准检定法(QUAD法)[J]. 测绘学报, 1999, 28(1):15-20. DOI:10.3321/j.issn:1001-1595.1999.01.004. OU Jikun. Quasi-accurate detection of gross errors (QUAD)[J]. Acta Geodaetica et Cartographica Sinica, 1999, 28(1):15-20. DOI:10.3321/j.issn:1001-1595.1999.01.004.
[4]	李德仁. 利用选择权迭代法进行粗差定位[J]. 武汉测绘科技大学学报, 1984, 9(1):46-68. LI Deren. Gross error location by means of the iteration method with variable weights[J]. Geomatics and Information Science of Wuhan University, 1984, 9(1):46-68.
[5]	RANGELOVA E, FOTOPOULOS G, SIDERIS M G. On the use of iterative re-weighting least-squares and outlier detection for empirically modelling rates of vertical displacement[J]. Journal of Geodesy, 2009, 83(6):523-535.
[6]	YANG Y, SONG L, XU T. Robust estimator for correlated observations based on bifactor equivalent weights[J]. Journal of Geodesy, 2002, 76(6-7):353-358.
[7]	杨元喜. 抗差估计理论及其应用[M]. 北京:解放军出版社, 1993. YANG Yuanxi. Robust estimation theory and its applications[M]. Beijing:PLA Press, 1993.
[8]	BASELGA S. Global optimization solution of robust estimation[J]. Journal of Surveying Engineering, 2007, 133(3):123-128.
[9]	GAO Yang, KRAKIWSKY E J, CZOMPO J. Robust testing procedure for detection of multiple blunders[J]. Journal of Surveying Engineering, 1992, 118(1):11-23.
[10]	岑敏仪. 平差前粗差的发现和定位及其若干规律[D]. 成都:西南交通大学, 1999. CEN Minyi. Gross errors detection and location before adjustment of observed values as well as some theorems for judging gross errors[D]. Chengdu:Southwest Jiaotong University, 1999.
[11]	彭军还, 陶本藻, 黄珹, 等. Lq估计的渐近方差-协方差矩阵及其特点[J]. 测绘学报, 2004, 33(4):293-298. PENG Junhuan, TAO Benzao, HUANG Cheng, et al. The asymptotic variance-covariance matrix in Lq-norm estimate and its characteristic[J]. Acta Geodaetica et Cartographica Sinica, 2004, 33(4):293-298.
[12]	彭军还. L1范数估计的巴尔达型检验及其可靠性[J]. 测绘学报, 2005, 34(3):208-212. DOI:10.3321/j.issn:1001-1595.2005.03.004. PENG Junhuan. Baarda test and its reliability in L1-norm estimation[J]. Acta Geodaetica et Cartographica Sinica, 2005, 34(3):208-212. DOI:10.3321/j.issn:1001-1595.2005.03.004.
[13]	郑肇葆. 线性规划在摄影测量粗差检测中应用的尝试[J]. 测绘学报, 1985, 14(4):241-252. ZHENG Zhaobao. Application of linear programming in the detection of blunders in photogrammetry[J]. Acta Geodaetica et Cartographica Sinica, 1985, 14(4):241-252.
[14]	MARSHALL J, BETHEL J. Basic concepts of L1 norm minimization for surveying applications[J]. Journal of Surveying Engineering, 1996, 122(4):168-179.
[15]	AMIRI-SIMKOOEI A. Formulation of L1 norm minimization in Gauss-Markov models[J]. Journal of Surveying Engineering, 2003, 129(1):37-43.
[16]	KHODABANDEH A, AMIRI-SIMKOOEI A R. Recursive algorithm for L1 norm estimation in linear models[J]. Journal of Surveying Engineering, 2011, 137(1):1-8.
[17]	YETKIN M, INAL C. L1 norm minimization in GPS networks[J]. Survey Review, 2011, 43(323):523-532.
[18]	ROUSSEUW P J, LEROY A M. Robust regression and outlier detection[M]. New York:Wiley, 1987.
[19]	CEN M, LI Z, DING Xiaoli, et al. Gross error diagnostics before least squares adjustment of observations[J]. Journal of Geodesy, 2003, 77(9):503-513.
[20]	岑敏仪, 卓健成, 李志林, 等. 判断观测值粗差能否发现和定位的一种验前方法[J]. 测绘学报, 2003, 32(2):134-138. DOI:10.3321/j.issn:1001-1595.2003.02.008. CEN Minyi, ZHUO Jiancheng, LI Zhilin, et al. A method of judging whether gross errors detectable and locatable in observations before least squares adjustment[J]. Acta Geodaetica et Cartographica Sinica, 2003, 32(2):134-138. DOI:10.3321/j.issn:1001-1595.2003.02.008.
[21]	郑肇葆. 数学规划在测绘学中应用[M]. 北京:测绘出版社, 2003. ZHENG Zhaobao. Application of mathematical programming in surveying and mapping[M]. Beijing:Surveying and Mapping Press, 2003.
[22]	束金龙, 闻人凯. 线性规划理论与模型应用[M]. 北京:科学出版社, 2003. SHU Jinlong, WEN Renkai. Linear programming theory and model applications[M]. Beijing:Science Press, 2003.
[23]	LUENBERGER D G, YE Yinyu. Linear and nonlinear programming[M]. Berlin:Springer, 2008.
[24]	范东明. 任意平面网坐标自动解算的非线性最小二乘平差算法[D]. 成都:西南交大通大学, 2000. FAN Dongming. The algorithm of nonlinear least squares adjustment of any plane networks with coordinates computed automatically[D]. Chengdu:Southwest Jiaotong University, 2000.
[25]	黄维彬. 近代平差理论及其应用[M]. 北京:解放军出版社, 1992. HUANG Weibin. Modern adjustment theory and its application[M]. Beijing:PLA Press, 1992.

The identification method of gross error detection failpoint in L₁-norm estimation

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