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1.
Adjustment has been based on the assumption that random errors of measurements are added to functional models. In geodetic practice, we know that accuracy formulae of modern geodetic measurements often consist of two parts: one proportional to the measured quantity and the other constant. From the statistical point of view, such measurements are of mixed multiplicative and additive random errors. However, almost no adjustment has been developed to strictly address geodetic data contaminated by mixed multiplicative and additive random errors from the statistical point of view. We systematically develop adjustment methods for geodetic data contaminated with multiplicative and additive errors. More precisely, we discuss the ordinary least squares (LS) and weighted LS methods and extend the bias-corrected weighted LS method of Xu and Shimada (Commun Stat B29:83–96, 2000) to the case of mixed multiplicative and additive random errors. The first order approximation of accuracy for all these three methods is derived. We derive the biases of weighted LS estimates. The three methods are then demonstrated and compared with a synthetic example of surface interpolation. The bias-corrected weighted LS estimate is unbiased up to the second order approximation and is of the best accuracy. Although the LS method can warrant an unbiased estimate for a linear model with multiplicative and additive errors, it is less accurate and always produces a very poor estimate of the variance of unit weight. 相似文献
2.
Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis 总被引：10，自引：4，他引：6
The weighted total least squares (TLS) method has been developed to deal with observation equations, which are functions of both unknown parameters of interest and other measured data contaminated with random errors. Such an observation model is well known as an errors-in-variables (EIV) model and almost always solved as a nonlinear equality-constrained adjustment problem. We reformulate it as a nonlinear adjustment model without constraints and further extend it to a partial EIV model, in which not all the elements of the design matrix are random. As a result, the total number of unknowns in the normal equations has been significantly reduced. We derive a set of formulae for algorithmic implementation to numerically estimate the unknown model parameters. Since little statistical results about the TLS estimator in the case of finite samples are available, we investigate the statistical consequences of nonlinearity on the nonlinear TLS estimate, including the first order approximation of accuracy, nonlinear confidence region and bias of the nonlinear TLS estimate, and use the bias-corrected residuals to estimate the variance of unit weight. 相似文献
3.
Weighted total least squares: necessary and sufficient conditions, fixed and random parameters 总被引：6，自引：1，他引：5
Xing Fang 《Journal of Geodesy》2013,87(8):733-749
A standard errors-in-variables (EIV) model refers to a Gauss–Markov model with an uncertain model matrix from a geodetic perspective. Least squares within the EIV model is usually called the total least squares (TLS) technique because of its symmetrical adjustment. However, the solutions and computational advantages of the weighted TLS problem with a general weight matrix (WTLS) are mostly unknown. In this study, the WTLS problem was solved using three different approaches: iterative methods based on the normal equation, the iteratively linearized Gauss–Helmert model with algebraic Jacobian matrices, and numerical analysis. Furthermore, sufficient conditions for WTLS optimization were investigated systematically as proposed solutions yield only necessary conditions for optimality. A WTLS solution was considered to treat random parameters within the EIV model. Last, applications to test these novel algorithms are presented. 相似文献
4.
Xing Fang 《Journal of Geodesy》2014,88(8):805-816
Observation systems known as errors-in-variables (EIV) models with model parameters estimated by total least squares (TLS) have been discussed for more than a century, though the terms EIV and TLS were coined much more recently. So far, it has only been shown that the inequality-constrained TLS (ICTLS) solution can be obtained by the combinatorial methods, assuming that the weight matrices of observations involved in the data vector and the data matrix are identity matrices. Although the previous works test all combinations of active sets or solution schemes in a clear way, some aspects have received little or no attention such as admissible weights, solution characteristics and numerical efficiency. Therefore, the aim of this study was to adjust the EIV model, subject to linear inequality constraints. In particular, (1) This work deals with a symmetrical positive-definite cofactor matrix that could otherwise be quite arbitrary. It also considers cross-correlations between cofactor matrices for the random coefficient matrix and the random observation vector. (2) From a theoretical perspective, we present first-order Karush–Kuhn–Tucker (KKT) necessary conditions and the second-order sufficient conditions of the inequality-constrained weighted TLS (ICWTLS) solution by analytical formulation. (3) From a numerical perspective, an active set method without combinatorial tests as well as a method based on sequential quadratic programming (SQP) is established. By way of applications, computational costs of the proposed algorithms are shown to be significantly lower than the currently existing ICTLS methods. It is also shown that the proposed methods can treat the ICWTLS problem in the case of more general weight matrices. Finally, we study the ICWTLS solution in terms of non-convex weighted TLS contours from a geometrical perspective. 相似文献
5.
An iterative solution of weighted total least-squares adjustment 总被引：9，自引：0，他引：9
Total least-squares (TLS) adjustment is used to estimate the parameters in the errors-in-variables (EIV) model. However, its
exact solution is rather complicated, and the accuracies of estimated parameters are too difficult to analytically compute.
Since the EIV model is essentially a non-linear model, it can be solved according to the theory of non-linear least-squares
adjustment. In this contribution, we will propose an iterative method of weighted TLS (WTLS) adjustment to solve EIV model
based on Newton–Gauss approach of non-linear weighted least-squares (WLS) adjustment. Then the WLS solution to linearly approximated
EIV model is derived and its discrepancy is investigated by comparing with WTLS solution. In addition, a numerical method
is developed to compute the unbiased variance component estimate and the covariance matrix of the WTLS estimates. Finally,
the real and simulation experiments are implemented to demonstrate the performance and efficiency of the presented iterative
method and its linearly approximated version as well as the numerical method. The results show that the proposed iterative
method can obtain such good solution as WTLS solution of Schaffrin and Wieser (J Geod 82:415–421, 2008) and the presented numerical method can be reasonably applied to evaluate the accuracy of WTLS solution. 相似文献
6.
针对EIV模型的系数矩阵同时包含固定量和随机量的情况,通过将系数矩阵中的随机量提取出来纳入平差的随机模型,从而将EIV模型表示为非线性高斯-赫尔默特(Gauss-Herlmert,GH)模型形式,推导了混合LS-TLS(least squares-total least squares,LS-TLS)算法及其精度估计公式。算法适用于系数矩阵包含固定列、固定元素和随机元素的一般情况。模拟实例结果表明,混合LS-TLS算法与已有能够解决系数矩阵同时含固定量和随机量的结构性或加权TLS算法的估计结果一致;混合LS-TLS的估计结果统计上要优于LS或TLS估计结果。 相似文献
7.
8.
通用EIV平差模型及其加权整体最小二乘估计 总被引：1，自引：1，他引：0
以平差基本理论为基础,提出了EIV(errors-in-variables)平差模型的通用形式,涵盖了间接平差、条件平差、附有参数的条件平差及附有限制条件的间接平差等基本EIV模型形式。基于整体最小二乘估计准则,研究了通用EIV模型的加权整体最小二乘算法,并推导了估计结果的近似精度公式。通用EIV模型及其整体最小二乘算法是对EIV模型估计理论的进一步完善,统一的整体最小二乘算法有利于软件的编程实现,有助于推动EIV模型估计理论的应用。 相似文献
9.
部分变量误差模型(partial EIV model)的加权整体最小二乘(weighted total least-squares,WTLS)估计不具备抵御粗差的能力。鉴于粗差可能同时出现在观测值和系数矩阵中,本文在提出部分变量误差模型WTLS估计的两步迭代解法的基础上,运用抗差M估计的等价权方法,发展了一种整体抗差最小二乘(TRLS)估计方法,并采用一致最大功效统计量确定降权因子。针对WTLS估计两步迭代解法的特点,设计了两个不同的降权方案:第1个方案是在估计系数矩阵元素时,不对观测值降权,仅对系数矩阵降权;第2个方案是在估计系数矩阵元素时,既对系数矩阵降权,同时也对观测值降权。通过对模拟2D仿射变换和线性拟合实例进行计算和分析,结果表明第1方案优于第2方案,并且优于基于残差和验后单位权方差的抗差估计和现有的变量误差模型抗差估计。 相似文献
10.
通用EIV(errors-in-variables)平差模型作为经典平差模型的一般化形式,具有同时顾及多种随机误差的优势. 在通用EIV平差模型加权总体最小二乘(WTLS)的线性化估计基础上,引入正则化准则. 正则化矩阵为单位矩阵时为岭估计,添加目标函数,通过建立拉格朗日目标函数的最小化求解,导出加权通用EIV平差模型对应的岭估计解式,给出了确定岭参数的U曲线法和L曲线法. 计算了通用EIV平差模型的线性化估计、两种岭估计及其对应的方差分量值;验证岭估计对通用EIV模型的线性化估计具有促进性,可减少迭代次数,使得参数方差分量更快趋于平稳,降低参数估计的计算量. 相似文献
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12.
A new method through Gauss–Helmert model of adjustment is presented for the solution of the similarity transformations, either 3D or 2D, in the frame of errors-in-variables (EIV) model. EIV model assumes that all the variables in the mathematical model are contaminated by random errors. Total least squares estimation technique may be used to solve the EIV model. Accounting for the heteroscedastic uncertainty both in the target and the source coordinates, that is the more common and general case in practice, leads to a more realistic estimation of the transformation parameters. The presented algorithm can handle the heteroscedastic transformation problems, i.e., positions of the both target and the source points may have full covariance matrices. Therefore, there is no limitation such as the isotropic or the homogenous accuracy for the reference point coordinates. The developed algorithm takes the advantage of the quaternion definition which uniquely represents a 3D rotation matrix. The transformation parameters: scale, translations, and the quaternion (so that the rotation matrix) along with their covariances, are iteratively estimated with rapid convergence. Moreover, prior least squares (LS) estimation of the unknown transformation parameters is not required to start the iterations. We also show that the developed method can also be used to estimate the 2D similarity transformation parameters by simply treating the problem as a 3D transformation problem with zero (0) values assigned for the z-components of both target and source points. The efficiency of the new algorithm is presented with the numerical examples and comparisons with the results of the previous studies which use the same data set. Simulation experiments for the evaluation and comparison of the proposed and the conventional weighted LS (WLS) method is also presented. 相似文献
13.
变量误差（error-in-variables,EIV）模型的系数矩阵存在结构特征的情况,并且这种结构特征可以扩展到观测向量中。首先采用变量投影法将系数矩阵的增广矩阵展开成仿射矩阵形式,提取系数矩阵和观测向量中的随机量,并将EIV模型表示为非线性高斯-赫尔默特模型,然后利用非线性最小二乘原理推导了一种结构总体最小二乘法。该算法统一了普通的结构总体最小二乘法、结构数据最小二乘法以及最小二乘法。将该算法应用到真实算例和模拟算例中,两个算例结果表明,该算法与已有能够解决EIV模型结构特征的结构或加权总体最小二乘法估计结果一致,验证了该算法的有效性。同时,该算法对结构特征的提取方式简单、规律性强且易于编程实现;且在算法设计中,把结构总体最小二乘问题转换为附有参数的条件平差问题,即将其纳入到最小二乘平差理论体系,便于其扩展应用。同时对平面拟合问题的误差估计特性进行了定性分析,由分析可知参数的相对大小对估计误差的一致性有直接影响,这说明EIV模型下系数矩阵和观测向量中随机量的估计误差与真误差的一致性关系相对复杂。 相似文献
14.
Partial EIV模型的非负最小二乘方差分量估计 总被引：2，自引：2，他引：0
Partial Errors-in-Variables(Partial EIV)模型是EIV模型的扩展形式,权阵构造简单,当系数矩阵中存在非随机元素和随机元素时,Partial EIV模型的适用性更强。针对Partial EIV模型中随机模型不准确的情况,将系数矩阵和观测向量分别作为一类数据,本文在该模型的基础上,使用最小二乘方差分量估计方法,推导相关计算公式及迭代算法,分别估计出相应的方差分量估值。并对出现的负方差使用非负最小二乘理论,增加约束条件,对随机模型进行修正,得到更加合理的参数估值。试实验结果表明,本文的方法与其他方差分量估计方法等价。 相似文献
15.
在大地测量联合反演中,方差分量估计法用于确定相对权比时并没有考虑大地测量反演的病态性,利用正则化解代替最小二乘解会引入偏差,会造成方差分量估计不准确问题。针对此提出采用偏差改正方差分量估计方法,以消除正则化解引入偏差的影响,并基于残差的偏差改正方差分量估计与方差分量估计法进行模拟实验。结果表明,进行偏差改正后的方差分量估计法能够较好地反演出滑动分布情况,所提方法针对参数进行偏差改正的方差分量估计考虑了迭代初值引入的偏差,理论更为严密。并将所提方法用于Visso地震和Norcia地震反演中,验证了该方法的可靠性、合理性。 相似文献
16.
变形体的变形量通常是一个非平稳时间序列,常常包含有趋势项和随机部分,因此,可以考虑建立GM＋AR模型。使用GM模型提取趋势项,提取了趋势项的剩余部分建立AR模型。然而,在进行模型参数的估计时,由于GM模型和AR模型的系数矩阵都含有误差,传统的最小二乘（LS）法并未顾及到这一点,因而,采用LS法得到的结果并不是最优的。为了顾及系数矩阵的误差,将整体最小二乘（TLS）法引入到GM和AR两种模型的参数求解中。AR模型系数矩阵中的每个元素都是含有误差的,可以直接采用TLS法对每个元素进行改正;然而,GM模型有一列元素是固定的,并不需要改正,直接使用TLS法进行求解是不严密的,采用LS法和TLS法相结合的方法对GM模型进行参数的求解。通过具体的变形监测实例,验证了采用组合模型的LS—TLS解法具有比LS法更高的建模和预测精度。 相似文献
17.
针对部分变量误差（partial EIV）模型的加权整体最小二乘（weighted total least squares,WTLS）估值的计算需要多次迭代且效率低下的情况,根据加权LS（least square）原理,通过改进目标函数,并运用矩阵微分运算以及矩阵反演变换,提出了一种计算partial EIV模型WTLS估值的新算法。算例计算结果表明,新算法具有迭代次数少、计算效率高等优点。 相似文献
18.
三维坐标转换的通用整体最小二乘算法 总被引：1，自引：1，他引：0
三维坐标转换模型属于非线性EIV（errors-in-variables）模型,现有整体最小二乘算法均设定了某些特殊假设条件,如仅适用于小角度或者属于非统计意义上的数值解,并且不能用于结构性的系数矩阵等,算法适用性受到极大限制。本文提出了三维坐标转换模型的通用加权整体最小二乘算法,该算法适用于任意旋转角度以及一般性的权矩阵情况下的三维坐标转换模型,并且将结构性系数矩阵、同时包含随机和非随机元素的系数矩阵等情况纳入到了统一的坐标转换模型算法。实例计算表明,本文提出的算法具有通用性,适用于实际应用中的各类三维坐标转换模型。 相似文献
19.
针对Partial EIV模型的方差分量估计中未考虑参数估值偏差所带来的影响,将Partial EIV模型视为非线性函数得到参数估值的偏差及二阶近似协方差表达式,计算得到偏差改正后的参数估值,结合方差分量估计方法,更新由参数估值影响的矩阵变量,给出了基于偏差改正的方差分量估计迭代方法。试验结果表明,参数估值及其协方差主要受参数估值偏差大小的影响,加入偏差改正能够得到更加合理的参数估值及方差分量估值,偏差改正后的方差分量估值可更加合理地评估参数估值的精度信息。 相似文献
20.
Variance Component Estimation in Linear Inverse Ill-posed Models 总被引：2，自引：4，他引：2
Regularization has been applied by implicitly assuming that the weight matrix of measurements is known. If measurements are assumed to be heteroscedastic with different unknown variance components, all regularization techniques may not be proper to apply, unless techniques of variance component estimation are directly implemented. Although variance component estimation techniques have been proposed to simultaneously estimate the variance components and provide a means of regularization, the regularization parameter is treated as if it were also an extra variance component. In this paper, we assume no prior information on the model parameters and do not treat the regularization parameter as an extra variance component. Instead, we first analyze the biases of estimated variance components due to the regularization parameter and then propose bias-corrected variance component estimators. The results have shown that they work very well. Finally, we propose and investigate through simulations an iterative scheme to simultaneously estimate the variance components and the regularization parameter, in order to eliminate the effect of regularization parameter on variance components and the effect of incorrect prior weights or initial variance components on the regularization parameter. 相似文献