An iterative solution of weighted total least-squares adjustment

Shen, Yunzhong; Li, Bofeng; Yi, Chen

doi:10.1007/s00190-010-0431-1

Cited by 133 publications

(78 citation statements)

References 14 publications

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“…The algorithm does not store the whole matrix Z 1:t . Instead, it has access to the previous estimate X t−1 and to the inverse of the data covariance matrix P t−1 , where P j = R −1 j and R j is the data covariance matrix defined in (18). For computation of the covariance of X, the algorithm has access to R t−1 and Q t−1 (data covariance matrix and doubly scaled data covariance matrix defined in (18)).…”

Section: Recursive Restricted Total Least Squaresmentioning

confidence: 99%

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A Recursive Restricted Total Least-Squares Algorithm

Rhode

Usevich

Markovsky

et al. 2014

IEEE Trans. Signal Process.

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Abstract-We show that the generalized total least squares (GTLS) problem with a singular noise covariance matrix is equivalent to the restricted total least squares (RTLS) problem and propose a recursive method for its numerical solution. The method is based on the generalized inverse iteration. The estimation error covariance matrix and the estimated augmented correction are also characterized and computed recursively. The algorithm is cheap to compute and is suitable for online implementation. Simulation results in least squares (LS), data least squares (DLS), total least squares (TLS), and RTLS noise scenarios show fast convergence of the parameter estimates to their optimal values obtained by corresponding batch algorithms.

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Section: Recursive Restricted Total Least Squaresmentioning

confidence: 99%

“…In [18] the more general WTLS problem is solved with an iterative procedure based on a Newton-Gauss approach, and a solution for the computation of the uncertainty bounds is presented. However, an online implementation for this approach seems challenging.…”

Section: Introductionmentioning

confidence: 99%

A Recursive Restricted Total Least-Squares Algorithm

Rhode

Usevich

Markovsky

et al. 2014

IEEE Trans. Signal Process.

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“…For further reading, see e.g., Van Huffel and Vandewalle (1991), Schaffrin and Wieser (2008), Felus (2004), Schaffrin et al (2012a, b), Mahboub et al (2012), Fang (2013Fang ( , 2014, Snow and Schaffrin (2012), Snow (2012) and Mahboub (2014), etc. Meanwhile, some other researchers investigated this problem traditionally; see e.g., Neitzel (2010) and Shen et al (2011). The term ''total least-squares (TLS)'' was coined in the field of numerical analysis by Golub and Van Loan (1980) as one of the standard solutions of this model.…”

Section: Introductionmentioning

confidence: 99%

“…The iterative method based on the GHM is also not sensitive to initial values, because the estimates in the first iteration can be a very good approximated solution, see Shen et al (2011). 3 A total least-squares (TLS) algorithm to deal with non-typical EIV model: algorithm 2…”

Section: Introductionmentioning

confidence: 99%

Adjustment of non-typical errors-in-variables models

2015

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In this contribution, non-typical errors-in-variables (EIV) model is introduced as a more general form of the so-called EIV model which is highly relevant for geodetic data processing. Total least-squares (TLS) algorithms within a typical EIV model cannot deal with the non-typical EIV models since the first order moment/mean of the random design matrix in a non-typical EIV model is not linear. In this paper, we propose a new algorithm besides a standard solution to deal with this model. To achieve this goal, first we review a classic algorithm in order to solve this problem using traditional non-linear leastsquares method within a non-linear mixed model. Then by comparison, weighted TLS algorithm within the typical EIV model can be replaced by the proposed approach after some slight modifications which results in an excellent approximate solution. This foundation is important because there is no need for linearization in the TLS algorithms. The proposed way is not sensitive to the approximate initial values of the unknown parameters and it is applicable to curve fitting, surface reconstruction or other non-typical EIV models. Here we employ it to the curve fitting. Also two examples convincingly demonstrate that the standard TLS solution of the non-typical EIV model is not admissible when it is incorrectly considered as a typical EIV model; i.e., the non-linear relationships of the elements of the random design matrix are neglected.

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“…Most of these existing TLS algorithms using the SVD approach will involve large computations. The other algorithms based on a non-linear Lagrange approach obtain the unbiased variance component in an indirect and complex way because the bias term is too difficult to be derived (Shen et al, 2011).…”

Section: Introductionmentioning

confidence: 99%

Linear observation based total least squares

et al. 2014

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This paper presents a total least squares (TLS) method in an iterative way when the observations are linear with applications in two-dimensional linear regression and three-dimensional coordinate transformation. The second order smaller terms are preserved and the unbiased solution and the variance component estimate are both obtained rigorously from traditional non-linear least squares theory. Compared with the traditional TLS algorithm dealing with the so called errors-invariables (EIV) model, this algorithm can be used to analyse all the observations involved in the observation vector and the design matrix coequally; the non-linear adjustment with constraints or partial EIV model can also be solved using the same method. In addition, all the errors of observations can be considered in a heteroscedastic or correlated case, and the calculation of the solution and the variance component estimate are much simpler than the traditional TLS and its related improved algorithms. Experiments using statistical methods show the deviations between the designed true value of the variables and the estimated ones using this algorithm and the traditional least squares algorithm respectively, and the mean value of the posteriori variance in 1000 simulations of coordinate transformation is computed as well to test and verify the efficiency and unbiased estimation of this algorithm.

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An iterative solution of weighted total least-squares adjustment

Cited by 133 publications

References 14 publications

A Recursive Restricted Total Least-Squares Algorithm

A Recursive Restricted Total Least-Squares Algorithm

Adjustment of non-typical errors-in-variables models

Linear observation based total least squares

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