Weighted total least squares formulated by standard least squares theory
Weighted total least squares formulated by standard least squares theory
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DOI：
10.2478/v10156-011-0036-5
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发表时间：
2012-06-01
影响因子：
1.3
通讯作者：
Jazaeri, S.
中科院分区：
文献类型：
--
作者：
Amiri-Simkooei, A.;Jazaeri, S.
本文提出了一种加权总体最小二乘（WTLS）问题的简单、有吸引力且灵活的公式。它之所以简单，是因为它基于著名的标准最小二乘理论；它有吸引力是因为它允许人们直接利用最小二乘理论的现有知识体系；它灵活是因为它可用于变量含误差（EIV）模型的广泛应用领域。文中给出了两个使用真实数据和模拟数据的实证例子。第一个例子是线性回归模型，将系数矩阵的协方差矩阵设为\(Q(A)=Q(n)\otimes Q(m)\)，而第二个例子是二维仿射变换，采用协方差矩阵\(Q(A)\)的一般结构。针对这两个例子，得到了未知参数的估计值以及估计值的标准差。结果表明与基于非线性高斯 - 赫尔默特模型（GHM）得到的结果相同。我们旨在对WTLS和GHM进行公正的评估。我们进一步探索了所提出公式的高潜在能力。人们可以简单地得到WTLS估计值的协方差矩阵。此外，可以推广标准最小二乘的正交投影器，由此可以直接导出残差和观测值（以及它们的协方差矩阵）的估计值以及单位权方差。而且，约束WTLS、EIV模型的方差分量估计以及可靠性和数据探测理论可以很容易地建立起来，这些内容将在未来的出版物中呈现。
This contribution presents a simple, attractive, and flexible formulation for the weighted total least squares (WTLS) problem. It is simple because it is based on the well-known standard least squares theory; it is attractive because it allows one to directly use the existing body of knowledge of the least squares theory; and it is flexible because it can be used to a broad field of applications in the error-invariable (EIV) models. Two empirical examples using real and simulated data are presented. The first example, a linear regression model, takes the covariance matrix of the coefficient matrix as Q(A) = Q(n) circle times Q(m), while the second example, a 2-D affine transformation, takes a general structure of the covariance matrix Q(A). The estimates for the unknown parameters along with their standard deviations of the estimates are obtained for the two examples. The results are shown to be identical to those obtained based on the nonlinear Gauss-Helmert model (GHM). We aim to have an impartial evaluation of WTLS and GHM. We further explore the high potential capability of the presented formulation. One can simply obtain the covariance matrix of the WTLS estimates. In addition, one can generalize the orthogonal projectors of the standard least squares from which estimates for the residuals and observations (along with their covariance matrix), and the variance of the unit weight can directly be derived. Also, the constrained WTLS, variance component estimation for an EIV model, and the theory of reliability and data snooping can easily be established, which are in progress for future publications.