L1范数探测粗差失效的观测量识别方法

Abstract

粗差发生时，L1范数估计求得的条件方程闭合差较最小二乘估计（LS）的残差更能集中反映粗差，从而有助于粗差的发现与定位。然而，存在一类观测值，虽然其具有粗差发现和定位能力，但在采用L1范数估计解决粗差探测问题时，无论含有多大量级粗差都不能准确定位，为叙述方便，称其为L1抗差性失效点（robustness failpoint in L1-norm estimation，RFP-L1）。显然，只有判定测量系统不存在RFP-L1，或存在时能够准确判断其是否含有粗差，才能保证基于L1的粗差探测结果的准确、可靠，此过程中，RFP-L1的识别是问题解决的基础。本文由条件方程，推导出观测值粗差对条件方程闭合差绝对值和的影响系数计算式，得到了最小影响系数大小与观测值是否为RFP-L1的判别关系，并探讨了存在RFP-L1的测量系统设计矩阵数值特点，提出了判断RFP-L1观测值的方法。仿真试验表明，最小影响系数反映了观测值粗差对L1范数估计目标函数的影响大小，非RFP-L1和RFP-L1的最小影响系数具有分别等于1和小于1的规律性，同时得出，若观测方程中系数矩阵只有±1和0，对应的观测量均不属于RFP-L1。

Alternate 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.