单位权中误差约束下的最优估计理论及其精度分析

论文总字数：29504字

摘 要

在数据处理中不仅要面对偶然误差，还常常需要处理系统误差以及粗差的干扰问题。然而常用的最小二乘法并不具备抵抗粗差的能力。在数据量更大、数据类型更多元化的现代测量工作中，粗差的出现更加频繁，这使得传统最小二乘法难以很好地适应现代测量数据处理需求。为了进一步提高参数估计的精度，研究能充分利用单位权中误差信息、并具有抗差性能的参数估计方法具有重要意义。

本文主要工作内容总结如下：

（1）从理论上分析了经典最小二乘估计理论的优缺点。从Gauss-Markoff模型定义出发，推导了经典最小二乘法的相关公式，从数学上严格证明了经典最小二乘法所具有的统计特性，分析了粗差对参数估计结果的具体影响，并得出经典最小二乘法具有无偏性、有效性等优良统计特性之优点以及无法抵抗粗差影响之缺点的结论。

（2）详细介绍了稳健估计基本原理。首先简单介绍了两种主流的粗差处理策略，随后重点介绍基于方差膨胀模型处理策略的稳健M估计原理，以及其影响函数的定义，同时经过从影响函数的角度对稳健M估计经典方法的稳健性进行分析，确定经典方法在理论中以及实际数据处理中具有一些不足之处。

（3）提出了一种单位权中误差约束下的调权参数估计方法。针对经典M估计的不足，同时充分利用先验单位权中误差等信息，介绍了前人提出的基于单位权中误差约束下的概率最小二乘法，并对其具体公式进行了推导。经过对概率最小二乘法影响函数特性的分析，得出该方法在对粗差降权方面具有一定缺陷的结论，并针对该缺陷对概率最小二乘法进行了改进，提出对原方法的处取切线，形成分段函数，进而实现改正数在以外的相应观测值自动剔除。

（4）通过仿真实验验证了新方法的抗差性。基于MATLAB仿真，使用水准网平差和GPS网平差两个案例。在数据中人为地添加粗差，使用新方法和经典方法（包括概率最小二乘法）进行处理，并对其处理结果进行分析，得出新方法比经典方法更具有自适应能力，以及新方法的抗差性比概率最小二乘法有所提升的结论。

关键词：最小二乘法，参数稳健估计，单位权中误差，概率最小二乘法，分段调权

ABSTRACT

In data processing, not only accidental errors but also systematic errors and gross errors are often needed to be dealt with. However, the commonly used least square method does not have the ability to resist gross error. In modern measurement work with larger data volume and more diversified data types, gross errors occur more frequently, which makes it difficult for traditional least square method to meet the requirements of modern measurement data processing. In order to improve the accuracy of parameter estimation, it is of great significance to study the parameter estimation method which can make full use of the unit weight error information and has the tolerance property.

The main contents of this paper are summarized as follows:

(1) the advantages and disadvantages of classical least squares estimation theory are analyzed theoretically. Starting from the Gauss - George model defined, the related formula of classic least square method is deduced, proved mathematically rigorous statistical properties of the classical least squares of gross error is analyzed with the result of parameter estimation of the specific impact, and it is concluded that classic least square method with unbiasedness, effectiveness etc. With the advantages of excellent statistical properties and could not resist the drawback of the gross error influence of conclusion.

(2) the basic principle of robust estimation is introduced in detail. First introduces the two mainstream of gross error handling strategy, then focuses on the processing strategy based on variance inflation model robust M estimation principle, as well as its influence function definition, at the same time after from the Angle of the influence function of robust M estimation robustness analysis of the classic methods, determine the classic methods in theory and actual data processing has some disadvantages.

(3) a method of weight adjustment parameter estimation under the constraint of unit weight median error is proposed. Aiming at the deficiency of classical M estimation and making full use of prior information such as median error of unit weight, this paper introduces the probability least square method based on the constraint of median error of unit weight, and deduces its specific formula. Through the analysis of the characteristics of the influence function probability least-square method, it is concluded that the method has some defects on the gross error rights, and aims at the defect of probability and the least square method is improved, take the tangent of the original method is put forward, form a piecewise function, thus realize the correct number on the outside of the corresponding observations out automatically.

(4) the tolerance of the new method is verified by simulation experiment. Based on MATLAB simulation, two cases of leveling network adjustment and GPS network adjustment are used. The new method and classical method (including probabilistic least square method) are used to process the data artificially, and the results are analyzed, and the conclusion is drawn that the new method is more adaptive than the classical method, and the tolerance of the new method is improved than the probabilistic least square method.

KEY WORDS: least square method, robust estimation, unit weighted errors, probability least square method, Segment adjusted weight

目 录

摘 要 I

ABSTRACT II

第一章 绪论 1

1.1研究目的与选题意义 1

1.2国内外研究历史与现状 2

1.3论文主要内容与组织结构 3

第二章 误差理论及其处理方法 5

2.1误差的概念 5

2.2描述误差的统计量 5

2.2.1方差和方差阵 6

2.2.2协因数阵和权阵 7

2.2.3误差传播律 7

2.3最小二乘法 8

2.3.1 G-M模型 9

2.3.2 G-M模型的最小二乘解 10

2.4最小二乘解的统计特性 11

2.4.1无偏性 11

2.4.2有效性 12

2.4.3最小二乘估计与极大似然估计 14

2.5误差对改正数的影响 15

2.5.1真误差和改正数之间的关系 15

2.5.2平差因子的性质 16

2.5.3真误差对改正数的影响 17

2.6最小二乘法抗差性能分析 18

2.7本章小结 20

第三章 粗差的探测与处理 21

3.1粗差的处理策略 21

3.1.1期望漂移模型 21

3.1.2方差膨胀模型 22

3.2粗差探测 23

3.2.1单个粗差的探测（Baarda数据探测法） 23

3.2.2多个粗差的探测 24

3.3稳健估计 25

3.3.1稳健M估计基本原理 25

3.3.2相关观测的M估计 27

3.3.3 M估计的稳健性要求 28

3.4稳健估计的经典方法 30

3.4.1 Huber函数 30

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