系数矩阵误差对EIV模型平差结果的影响研究

发布时间：2018-06-15 09:09

本文选题：EIV模型 + 估计偏差　；参考：《武汉大学》2013年博士论文

【摘要】：经典的Gauss-Markov模型只假定观测向量包含随机误差,系数矩阵是非随机的固定值,当模型为线性形式时,采用最小二乘估计方法(LS:least squers)可得到模型参数的最优解。但实际应用中,许多情况下观测向量和系数矩阵均包含随机误差,这类平差模型称为EIV (errors-in-variables)模型。 EIV模型于19世纪末就已提出,20世纪80年代前主要是统计领域开展了少量研究工作。1980年,Golub和van Loan发表了著名的奇异值分解算法(该方法实质上与1901年Pearson提出的正交回归解法相同),之后EIV模型引起了各领域的广泛关注。到目前为止,EIV模型已成为基本的数学模型之一,广泛应用于信号处理、通信工程、计算机视觉等众多科学研究和工程应用领域。 EIV模型最简单的算法是忽略系数矩阵误差,采用最小二乘方法求解,但其解为近似解,不再具有最优统计特性(Xu等2012)。为了得到EIV模型的最优解,通过对最小二乘准则进行扩展,得到的能同时顾及观测向量和系数矩阵误差的整体最小二乘估计方法(TLS:total least squares)是EIV模型的严密估计方法。然而,TLS的计算量远大于LS方法,在观测值和参数数量大的情况下,TLS甚至无法求解。同时,有些参考文献实例结果表明模型的LS解和TLS解几乎没有差别(如大地坐标转换模型)。因此,我们认为EIV模型估计的一个基本问题为：究竟在什么情况下可以采用LS方法代替TLS方法?采用数学语言描述,即系数矩阵误差如何影响EIV模型的LS估计结果。遗憾的是,测绘领域尚没有文献从理论上进行研究,仅统计学领域两篇文献(Hodges和Moore1972、Davies和Hutton1975)在非常简单的假设下进行了探讨,且推导过程存在错误。因此,论文的主要研究内容之一是全面系统地研究系数矩阵误差对EIV模型LS估计结果的各种影响。平差模型的可靠性度量是平差的基本问题之一。尽管建立在Gauss-Markov模型基础上的经典可靠性理论成果丰富,但到口前为止,仅两篇文献讨论了EIV模型的可靠性理论。Schaffrin和Uzun (2011/2012)在极为特殊的权阵条件下推导了EIV模型的可靠性度量,由于公式中包括无粗差情况下的TLS解,不能用于实际计算。Proszynski (2013)直接简单套用经典可靠性理论,且只讨论了观测向量的可靠性,不能视为真正意义上的EIV模型的可靠性度量。针对EIV模型可靠性理论的缺陷,论文的主要研究内容之二是推导了系数矩阵误差对经典可靠性度量的影响,并且系统地发展了一般情况下EIV模型的可靠性理论和方法。论文的主要内容和贡献如下： (1)从EIV模型的一般情况出发,全面系统地推导了系数矩阵误差引起的LS参数估计值及其方差协方差阵的偏差、观测向最残差偏差的严密计算公式。研究结果表明,系数矩阵误差对平差结果的影响与系数矩阵量级及其方差、参数的大小有关。若定义系数矩阵的信噪比为系数矩阵量级与其中误差之比,则参数估计值的相对偏差随系数矩阵信噪比二次方的增大而迅速减小,参数估计值的相对中误差随系数矩阵信噪比的增长而减小。通常情况下,系数矩阵误差对LS参数估计值精度的影响大于对参数估计值偏差的影响。 (2)论文推导了系数矩阵误差引起的单位权方差偏差的计算公式。公式表明,单位权方差的偏差随参数二次方的增长而迅速增长,随系数矩阵方差协方差阵的增大而增长。公式从理论上完美地解释了测绘领域有关文献报道的EIV模型经典LS单位权方差估计结果异常且显著偏大的情况。 (3)在以上研究成果的基础上,通过对LS估计结果进行偏差改正,构造了EIV模型偏差改正的LS参数估计值、参数的方差协方差估计值以及单位权方差估计公式。 (4)论文研究了系数矩阵误差对经典可靠性度量的影响,导出了系数矩阵误差引起的内部可靠性和外部可靠性偏差的计算公式。公式反映了偏差随系数矩阵信噪比二次方的增长而减小,随模型本身可靠性的增大而减小。利用偏差公式,论文构造了EIV模型观测向量偏差改正的的可靠性度最公式。 (5)以partial-EIV模型为基础,建立了一般权矩阵条件下EIV模型的可靠性理论和方法,推导了观测向量和系数矩阵的内部可靠性和外部可靠性的计算公式。研究结果表明,EIV模型的多余观测数根据观测向量和系数矩阵的方差以及参数大小在观测的量和系数矩阵之间进行分配。当观测向量方差很小(系数矩阵或参数很小)时,由于观测向最(系数矩阵)分配的多余观测数很少,若出现粗差将难以发现。
[Abstract]:The classical Gauss-Markov model only assumes that the observation vector contains random error, the coefficient matrix is a non random fixed value. When the model is linear, the least square estimation method (LS:least squers) can obtain the optimal solution of the model parameters. However, in practical applications, the observation vector and the coefficient matrix all contain random errors in many situations. The class adjustment model is called EIV (errors-in-variables) model.
The EIV model has been put forward in the late nineteenth Century, and a small amount of research work was carried out in the field of statistics before 1980s. Golub and van Loan published a famous singular value decomposition algorithm (this method is essentially the same as the orthogonal regression method proposed in 1901). Then the EIV model has aroused wide attention in various fields. The EIV model has become one of the basic mathematical models, and is widely used in many scientific research and engineering applications such as signal processing, communication engineering, computer vision, etc.
The simplest algorithm of the EIV model is to ignore the error of the coefficient matrix and use the least square method to solve it, but the solution is an approximate solution and no longer has the optimal statistical property (Xu et al. 2012). In order to obtain the optimal solution of the EIV model, the least second total of the error of the observation vector and the coefficient matrix can be considered at the same time by extending the least square criterion. The multiplicative estimation method (TLS:total least squares) is the strict estimation method of the EIV model. However, the calculation of TLS is far greater than the LS method. In the case of large observation and parameter, TLS can not be solved even. At the same time, some reference examples show that the LS solution of the model and the TLS solution are almost no difference (such as the geodetic coordinate transformation model). We think one of the basic problems of EIV model estimation is: in what case can we use the LS method instead of the TLS method? The mathematical language is used to describe how the coefficient matrix error affects the LS estimation results of the EIV model. Unfortunately, there is no literature in the field of Surveying and mapping, but only two literature in the field of Statistics (Hod) Ges and Moore1972, Davies and Hutton1975) are discussed under very simple assumptions and there are errors in the derivation process. Therefore, one of the main contents of this paper is to systematically study the influence of the coefficient matrix error on the LS estimation results of the EIV model.
The reliability measurement of the adjustment model is one of the basic problems of the adjustment. Although the classical reliability theory based on the Gauss-Markov model is rich, only two papers discuss the reliability theory.Schaffrin and Uzun (2011/2012) of the EIV model so far. The reliability of the model is derived from the EIV model under the extremely special weight matrix condition. Because the formula includes the TLS solution in the case of no gross error, it can not be used to calculate the reliability theory of.Proszynski (2013) directly and simply, and only discusses the reliability of the observation vector. It can not be regarded as the reliability measure of the real EIV model. The main research of the thesis is the defect of the reliability theory of the EIV model. The two content is to deduce the influence of the coefficient matrix error on the classical reliability measurement, and systematically develop the reliability theory and method of the EIV model in general.
The main contents and contributions of the paper are as follows:
(1) from the general situation of the EIV model, the error of the estimation value of LS parameters and the covariance covariance matrix caused by the coefficient matrix error and the rigorous calculation formula of the observation to the most residual deviation are derived. The results show that the magnitude and variance of the coefficient matrix error to the result of the coefficient matrix error and the coefficient matrix are the size of the parameter. If the signal-to-noise ratio of the coefficient matrix is defined as the ratio of the coefficient matrix to the error, the relative deviation of the parameter estimation decreases rapidly with the increase of the signal to noise ratio of the coefficient matrix, and the relative error of the parameter estimation decreases with the increase of the signal to noise ratio of the coefficient matrix. In the general case, the coefficient matrix error is estimated for the LS parameter. The effect of value accuracy is greater than that on parameter estimation bias.
(2) the paper derives the calculation formula of the deviation of unit weight variance caused by the coefficient matrix error. The formula shows that the deviation of the unit weight variance increases rapidly with the increase of the parameter of the two order square, and increases with the increase of the covariance matrix of variance covariance. The formula explains perfectly the classical EIV model L in the field of Surveying and mapping in the field of Surveying and mapping. S unit weight variance estimation results are abnormal and significantly larger.
(3) on the basis of the above research results, by correcting the deviation of the LS estimation results, the LS parameter estimation value of the EIV model deviation correction, the variance covariance estimation value of the parameter and the formula of the unit weight variance estimation are constructed.
(4) the paper studies the influence of the coefficient matrix error on the classical reliability measurement, derives the calculation formula of the internal reliability and the external reliability deviation caused by the coefficient matrix error. The formula reflects the decrease of the deviation with the increase of the signal to noise ratio of the coefficient matrix and the increase of the reliability of the model with the increase of the reliability of the model, and the use of the deviation formula. The most reliable formula for the deviation correction of the EIV model is constructed.
(5) based on the partial-EIV model, the reliability theory and method of the EIV model under the condition of the general weight matrix are established. The calculation formulas for the internal reliability and external reliability of the observation vector and the coefficient matrix are derived. The results show that the superfluous observations of the EIV model are based on the variance of the observation vector and the coefficient matrix and the size of the parameters. When the observation vector is very small (the coefficient matrix or the parameter is very small), the number of superfluous observations allocated to the most (coefficient matrix) is very few, and it is difficult to find out if the gross error occurs, when the observation vector is very small (the coefficient matrix or the parameter is very small).
【学位授予单位】：武汉大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：P207

【参考文献】