Total least-squares adjustment of condition equations

This paper proposes a method for determining the vertical deformations treated as random fields. It is assumed that the monitored surfaces are subject not only to deterministic deformations, but also to random fluctuations. Furthermore, the existence of random noise coming from surface’s vibrations is also assumed. Such noise disturbs the deformation’s functional models. Surface monitoring with the use of the geodetic levelling network of a free control network class is carried out. Assuming that, in some cases, the control networks are insufficient in surface’s deformation analysis, additional and non–measurable reference points have been provided. The prediction of these points’ displacements and estimation of the free control network points’ displacement are carried out using the collocation method applying the total least squares adjustment. The proposed theoretical solutions were verified by the simulation methods and on the example of a real control network.

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Total least squares adjustment in inequality constrained partial errors-in-variables models: optimality conditions and algorithms

Survey Review ◽

10.1080/00396265.2021.1911509 ◽

2021 ◽

pp. 1-14

Author(s):

Jian Xie ◽

Dongfang Lin ◽

Sichun Long

Keyword(s):

Least Squares ◽

Optimality Conditions ◽

Total Least Squares ◽

Errors In Variables ◽

Least Squares Adjustment

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Scaled weighted total least-squares adjustment for partial errors-in-variables model

Journal of Geodetic Science ◽

10.1515/jogs-2016-0010 ◽

2016 ◽

Vol 6 (1) ◽

Cited By ~ 1

Author(s):

J. Zhao

Keyword(s):

Least Squares ◽

Variance Component ◽

Maximum Likelihood Method ◽

Coefficient Matrix ◽

Total Least Squares ◽

Likelihood Method ◽

Errors In Variables ◽

Weighted Total Least Squares ◽

Least Squares Adjustment ◽

Observation Vector

AbstractScaled total least-squares (STLS) unify LS, Data LS, and TLS with a different choice of scaled parameter. The function of the scaled parameter is to balance the effect of random error of coefficient matrix and observation vector for the estimate of unknown parameter. Unfortunately, there are no discussions about how to determine the scaled parameter. Consequently, the STLS solution cannot be obtained because the scaled parameter is unknown. In addition, the STLS method cannot be applied to the structured EIV casewhere the coefficient matrix contains the fixed element and the repeated random elements in different locations or both. To circumvent the shortcomings above, the study generalize it to a scaledweighted TLS (SWTLS) problem based on partial errors-in-variable (EIV) model. And the maximum likelihood method is employed to derive the variance component of observations and coefficient matrix. Then the ratio of variance component is proposed to get the scaled parameter. The existing STLS method and WTLS method is just a special example of the SWTLS method. The numerical results show that the proposed method proves to bemore effective in some aspects.

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