Generalised total least squares solution based on pseudo-observation method

Hu, Chengping; Chen, Y.; Zhu, Weidong

doi:10.1179/1752270614y.0000000155

Cited by 3 publications

(3 citation statements)

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“…(3) Specify the PDFs to the input quantities, i.e., it should be provided the PDFs of the control points coordinates in both reference frames, e.g., a joint normal distribution given by the expression (7) and another one by (15). (4) Generate vectors, by drawing randomly from the PDFs assigned to the input quantities according to step (3).…”

Section: Implementation Of the Least-squares Methods Based On The Monte Carlo Methods For Coordinate Transformationmentioning

confidence: 99%

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Monte-Carlo-based uncertainty propagation in the context of Gauss–Markov model: a case study in coordinate transformation

Rofatto¹,

Matsuoka

Klein

et al. 2019

Sci. Plena

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One of the main tasks of professionals in the Earth sciences is to convert coordinates from one reference frame into another. The coordinates transformation is widely needed and applied in all branches of modern geospatial activities. To convert from one reference frame to another, it is necessary initially to determine the parameters of a coordinate transformation model. A common way to estimate the transformation parameters is using the least-squares theory within a linearized Gauss-Markov Model (GMM). Another approach arises from a numerical method. Here, a Monte Carlo Method (MCM) is used to infer the uncertainty of estimators. This method is based on: (i) the assignment of probability distributions to the coordinates in the two reference frames, (ii) the determination of a discrete representation of the probability distribution for the transformation parameters, and (iii) the determination of the associated uncertainties from this discrete representation of the estimates of the transformation parameters. In this contribution, we compare the weighted least-squares within the GMM (WLSE-GM) and the proposed method regarding the transformation problem of computing 2D similarity transformation parameters. The results show that, the transformation parameters uncertainties are higher for the LS-MC than the WLSE-GM. This is due to the fact that the WLSE-GM solution does not take into account the uncertainties associated with the system matrix. In future studies the Monte Carlo method should be applied to the nonlinear least-squares solution.

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Section: Implementation Of the Least-squares Methods Based On The Monte Carlo Methods For Coordinate Transformationmentioning

confidence: 99%

“…Here, the random error vectors and the random noise matrix A of the functional matrix A have been synthetically generated based on a multivariate normal distribution according to Eq. ( 7) and (15), respectively. The design matrix A * for the similarity transformation is given by (16).…”

Section: Generation Of Observation Errors and Number Of Monte Carlo Trialsmentioning

confidence: 99%

Monte-Carlo-based uncertainty propagation in the context of Gauss–Markov model: a case study in coordinate transformation

Rofatto¹,

Matsuoka

Klein

et al. 2019

Sci. Plena

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show abstract

“…Other authors have also dealt with the WTLS problem (e.g. Shen et al, 2011;Tong et al, 2011;Mahboub, 2012;Xu et al 2012Xu et al , 2014Fang, 2013Fang, , 2014aFang, , 2014bFang, , 2014cFang, , 2015Fang and Wu, 2015;Hu et al 2015), all of them relying on the fundamental paper by Schafrin and Wieser (2008).…”

Section: Introductionmentioning

confidence: 98%

Weighted coordinate transformation formulated by standard least-squares theory

Mihajlović

Cvijetinović

2016

Survey Review

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This paper presents a universal model of weighted coordinate transformation, i.e. transformation considering the errors of coordinates in both coordinate systems. It is intrinsically one of the typical examples of 'error-in-variables' (EIV) models. The proposed method of LS theory application on weighted coordinate transformation does not impose any constraints on the form of functional relationship among stochastic variables. Since the basic idea is to generalise Gauss-Markov model (GMM) by introduction of so-called 'total residuals', the proposed procedure is named 'Generalised Gauss-Markov model'. Formulation of expressions for estimation of unknown transformation parameters is theoretically confirmed using the GaussHelmert model (GHM) and three different modifications of the GMM. The proposed procedure is in its essence a strict solution to total least-squares (unweighted) and weighted total leastsquares problem in coordinate transformation. This thesis is experimentally confirmed by comparison of its results with those found in four characteristic examples from the literature.

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