A Newton algorithm for weighted total least-squares solution to a specific errors-in-variables model with correlated measurements

Zhou, Yuegui; Kou, Xin-jian; Zhu, Jianjun; Li, Jonathan

doi:10.1007/s11200-013-0254-7

Cited by 17 publications

(3 citation statements)

References 30 publications

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“…Snow [12] and Zhou et al [13] provided a solution for the case of a full weight matrix W. Later, Zhou and Fang [14] also provided a solution for the mixed case (WLS-WTLS), in which some columns of A are assumed to be free of errors.…”

Section: Cases With Numerical Solutions Onlymentioning

confidence: 99%

A universal and fast method to solve linear systems with correlated coefficients using weighted total least squares

Wurm

2021

Meas. Sci. Technol.

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Especially in metrology and geodesy, but also in many other disciplines, the solution of overdetermined linear systems of the form Ax≈b with individual uncertainties not only in b but also in A is an important task. The problem is known in literature as weighted total least squares. In the most general case, correlations between the elements of [A,b] exist as well. The problem becomes more complicated and can—except for special cases—only be solved numerically. While the formulation of this problem and even its solution is straightforward, its implementation—when the focus is on reliability and computational costs—is not. In this paper, a robust, fast, and universal method for computing the solution of such linear systems as well as their covariance matrix is presented. The results were confirmed by applying the method to several special cases for which an analytical or numerical solution is available. If individual coefficients can be considered to be free of errors, this can be taken into account in a simple way. An implementation of the code in MATLAB is provided.

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Section: Cases With Numerical Solutions Onlymentioning

confidence: 99%

A universal and fast method to solve linear systems with correlated coefficients using weighted total least squares

Wurm

2021

Meas. Sci. Technol.

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“…The formulation of Eq. (28) can be viewed as the conditional adjustment in least squares theory (Zhou et al 2014). Therefore, the estimation of error vectorsŵ kÀ1 ,ê a k , and u k can be estimated aŝ…”

Section: Formulation Of Generalized Total Kalman Filter Algorithmmentioning

confidence: 99%

Generalized total Kalman filter algorithm of nonlinear dynamic errors-in-variables model with application on indoor mobile robot positioning

Jian

Wang

et al. 2017

Acta Geod Geophys

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In this paper, a nonlinear dynamic errors-in-variables (DEIV) model which considers all of the random errors in both system equations and observation equations is presented. The nonlinear DEIV model is more general in the structure, which is an extension of the existing DEIV model. A generalized total Kalman filter (GTKF) algorithm that is capable of handling all of random errors in the respective equations of the nonlinear DEIV model is proposed based on the Gauss-Newton method. In addition, an approximate precision estimator of the posteriori state vector is derived. A two dimensional simulation experiment of indoor mobile robot positioning shows that the GTKF algorithm is statistically superior to the extended Kalman filter algorithm and the iterative Kalman filter (IKF) algorithm in terms of state estimation. Under the experimental conditions, the improvement rates of state variables of positions x, y and azimuth w of the GTKF algorithm are about 14, 29, and 66%, respectively, compared with the IKF algorithm.

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“…The first set of real-life data was used for the experiment, which is adopted from Neri et al (1989), which was later used by TLS research literature (Schaffrin and Wieser, 2008;Shen et al, 2011;Tong et al, 2011;Amiri-Simkooei and Jazaeri, 2012;Mahboub, 2012;Fang, 2013;Gong and Li, 2014;Zhou et al, 2014). The data including the coordinates of 10 observed points and corresponding weights are shown in Table 1.…”

Section: Linear Regressionmentioning

confidence: 99%

A robust weighted total least-squares solution with Lagrange multipliers

Gong

2016

Survey Review

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Weighted total least-squares (WTLS) is becoming popular for parameter estimations in geodesy and surveying. However, it does not take into consideration the possible gross errors in observations, which may lead to a reduction in the robustness and reliability of parameter estimations. In order to solve this problem, in this study, Lagrange multipliers (LM) are employed to make WTLS solution rigorous and the IGG (Institute of Geodesy and Geophysics) weight function is employed to make WTLS solution more robust and reliable, resulting in a new robust WTLS solution (RWTLS-LM-IGG). A comparison with existing WTLS and robust WTLS solutions is conducted for linear regression and coordinate transformation, through experimental evaluation with simulation data sets (with different numbers and magnitudes of gross errors) and two sets of real-life data. The results of simulation experiments show that the variance component and the mean square error of estimated parameter vector obtained by using the existing methods increase almost linearly with an increase in the numbers and magnitudes of gross errors, but these values obtained by using the proposed method are almost stable, which means an effective reduction of the influence of the gross errors by the proposed method as compared with the existing methods. It is also found that the larger the numbers and magnitudes of gross errors, the more obvious such a reduction. Furthermore, the experiment results with two sets of real-life data are consistent with the results of simulation experiments.

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A Newton algorithm for weighted total least-squares solution to a specific errors-in-variables model with correlated measurements

Cited by 17 publications

References 30 publications

A universal and fast method to solve linear systems with correlated coefficients using weighted total least squares

A universal and fast method to solve linear systems with correlated coefficients using weighted total least squares

Generalized total Kalman filter algorithm of nonlinear dynamic errors-in-variables model with application on indoor mobile robot positioning

A robust weighted total least-squares solution with Lagrange multipliers

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