An algorithmic approach to the total least-squares problem with linear and quadratic constraints

Schaffrin, Burkhard; Felus, Yaron A.

doi:10.1007/s11200-009-0001-2

Cited by 59 publications

(20 citation statements)

References 18 publications

(33 reference statements)

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“…The results of Case 1 which contain one inequality constraint correspond with the TLS solution without any constraints as presented in Schaffrin and Felus (2009). This indicates that the inequality constraint refers to an inactive constraint, which does not influence the parameter estimates at all.…”

Section: Applicationmentioning

confidence: 57%

“…In the first example, data of a geodetic application presented in Schaffrin and Felus (2009) shows a simplified ''geodetic resection'' problem. In the second application, we use the data presented in Peng et al (2006) and Zhang et al (2013) to compare results.…”

Section: Applicationsmentioning

confidence: 99%

“…Regarding prior information representing constraints of the parameters, Schaffrin and Felus (2009), Mahboub and Sharifi (2013) and Fang (2014aFang ( , 2015 investigated the equality constrained EIV model. The EIV model with linear inequality constraints was firstly adjusted by Zhang et al (2013) with an active set method based on exhaustive tests, under the condition that the weight matrix (including all random errors of the coefficient matrix and the observation vector) is the identity matrix.…”

Section: Introductionmentioning

confidence: 99%

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On the errors-in-variables model with equality and inequality constraints for selected numerical examples

Fang

2015

Acta Geod Geophys

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It is well known that the errors-in-variables (EIV) model has been treated as a special case of the traditional geodetic model, the nonlinear Gauss-Helmert model (GHM), for more than a century. In this contribution, an adjustment of the EIV model with equality and inequality constraints is investigated based on the nonlinear GHM. In each iteration, the constrained EIV model is linearized to form a quadratic program. Furthermore, the precision description is investigated for the mixed constrained problem. The demonstrated results from the numerical examples show that this approach avoids the large computational expenses of the existing combinatorial solution that normally accompany the number of inequality constraints.

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Section: Applicationmentioning

confidence: 57%

Section: Applicationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the errors-in-variables model with equality and inequality constraints for selected numerical examples

Fang

2015

Acta Geod Geophys

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“…Linear regression is the simplest experiment of multivariate EIV regression analysis and is most often used to examine the result of different kinds of TLS algorithms (Neri et al, 1989;Schaffrin et al, 2006;Schaffrin and Felus, 2009;Shen et al, 2011). The 2D linear regression model is as follows…”

Section: Example I: 2d Linear Regressionmentioning

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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“…Also in several contributions particularly in geodetic literature, its applications have been investigated. Linear regression (Schaffrin and Wieser 2008;Fang 2011), geodetic resection (Schaffrin and Felus 2008), transformation (Mahboub 2012) and rapid satellite positioning (Mahboub and Sharifi 2013) are some examples. Nevertheless, there are some other problems such as curve fitting which all the variables are subject to errors but the model is not similar to GM model.…”

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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An algorithmic approach to the total least-squares problem with linear and quadratic constraints

Cited by 59 publications

References 18 publications

On the errors-in-variables model with equality and inequality constraints for selected numerical examples

On the errors-in-variables model with equality and inequality constraints for selected numerical examples

Linear observation based total least squares

Adjustment of non-typical errors-in-variables models

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