On The Errors-In-Variables Model With Singular Dispersion Matrices

Schaffrin, Burkhard; Snow, Kyle; Neitzel, Frank

doi:10.2478/jogs-2014-0004

Cited by 20 publications

(6 citation statements)

References 7 publications

(11 reference statements)

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“…Using the function of a cubic polynomial as an example, a brief overview of the errors-in-variables (EIV) model with regular dispersion matrix is presented in the following. In the case of the EIV model with a singular dispersion matrix, please refer to the contribution of Schaffrin et al [15]. Ezhov et al [2] elaborated a spline function constructed from ordinary cubic polynomials…”

Section: A Review Of the Errors-in-variables (Eiv) Modelmentioning

confidence: 99%

“…In order to appropriately consider this structure of the functional matrix, special TLS algorithms must be applied, as presented by Mahboub [12], Schaffrin et al [13] or Han et al [14], while this case can be solved in a standard way in the GH model. Furthermore, arbitrary and even singular dispersion matrices can be introduced into the GH model as shown by Schaffrin et al [15] and Neitzel and Schaffrin [16]. Therefore, the iteratively linearized GH model can be regarded as a generalized approach for solving constrained weighted total least squares (CWTLS) problems.…”

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confidence: 99%

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Total Least Squares Spline Approximation

Neitzel

Ezhov

2019

Mathematics

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Spline approximation, using both values y i and x i as observations, is of vital importance for engineering geodesy, e.g., for approximation of profiles measured with terrestrial laser scanners, because it enables the consideration of arbitrary dispersion matrices for the observations. In the special case of equally weighted and uncorrelated observations, the resulting error vectors are orthogonal to the graph of the spline function and hence can be utilized for deformation monitoring purposes. Based on a functional model that uses cubic polynomials and constraints for continuity, smoothness and continuous curvature, the case of spline approximation with both the values y i and x i as observations is considered. In this case, some of the columns of the functional matrix contain observations and are thus subject to random errors. In the literature on mathematics and statistics this case is known as an errors-in-variables (EIV) model for which a so-called “total least squares” (TLS) solution can be computed. If weights for the observations and additional constraints for the unknowns are introduced, a “constrained weighted total least squares” (CWTLS) problem is obtained. In this contribution, it is shown that the solution for this problem can be obtained from a rigorous solution of an iteratively linearized Gauss-Helmert (GH) model. The advantage of this model is that it does not impose any restrictions on the form of the functional relationship between the involved quantities. Furthermore, dispersion matrices can be introduced without limitations, even the consideration of singular ones is possible. Therefore, the iteratively linearized GH model can be regarded as a generalized approach for solving CWTLS problems. Using a numerical example it is demonstrated how the GH model can be applied to obtain a spline approximation with orthogonal error vectors. The error vectors are compared with those derived from two least squares (LS) approaches.

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Section: A Review Of the Errors-in-variables (Eiv) Modelmentioning

confidence: 99%

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confidence: 99%

Total Least Squares Spline Approximation

Neitzel

Ezhov

2019

Mathematics

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“…Therefore, we only give some references e.g. Zeng et al 2015, Zhang et al (2013), Neitzel (2010), Neitzel and Schaffrin (2016), Snow and Schaffrin (2012), Shen et al (2011), Schaffrin et al (2014), Schaffrin and Felus (2008), Mahboub (2012Mahboub ( , 2014Mahboub ( , 2016, Mahboub et al (2012Mahboub et al ( , 2015, Mahboub and Sharifi (2013a, b), Paláncz and Awange (2012), Amiri-simkooei and Jazaeri (2012), Fang (2011Fang ( , 2013Fang ( , a, b c, 2015, Fang et al ( , 2016, Lu et al (2014), Zhou and Fang (2015) and Fang and Wu (2015) etc. In the rest of this paper we define these two parts for a DEIV model.…”

Section: Dynamic Errors-in-variables (Deiv) Modelmentioning

confidence: 99%

A solution to dynamic errors-in-variables within system equations

2017

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We noticed that if INS data is used as system equations of a Kalman filter algorithm for integrated direct geo-referencing, one encounters with a dynamic errors-invariables (DEIV) model. Although DEIV model has been already considered for observation equations of the Kalman filter algorithm and a solution namely total Kalman filter (TKF) has been given to it, this model has not been considered for system equations (dynamic model) of the Kalman filter algorithm. Thus, in this contribution, for the first time we consider DEIV model for both observation equations and system equations of the Kalman filter algorithm and propose a least square prediction namely integrated total Kalman filter in contrast to the TKF solution of the previous approach. The variance matrix of the unknown parameters are obtained. Moreover, the residuals for all variables are predicted. In a numerical example, integrated direct geo-referencing problem is solved for a GPS-INS system. Keywords Dynamic errors-in-variables Á System equations Á Integrated total Kalman filter Á Direct geo-referencing

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“…While this paper treats the 2D similarity transformation in the framework of a nonlinear Gauss-Helmert Model by iterative linearization, it will be of major interest as well how it can be handled within an EIV-Model (''Errors-In-Variables'') by setting up nonlinear normal equations and solving them iteratively, all with singular covariance matrices for both vector and matrix observations. Two other papers on this subject have recently been published; see Schaffrin et al (2014) and Jazaeri et al (2014 Estimated dispersion matrices for the residuals and their cross-covariance matrix:…”

Section: Parametersmentioning

confidence: 99%

“…For earlier discussions of this application, see, e.g., Teunissen (1988), Bleich and Illner (1989), Koch et al (2000), Fang (2014), or Chang (2015) among many others. For an alternative approach, see Schaffrin (2003), as well as Schaffrin et al (2014).…”

Section: Introductionmentioning

confidence: 99%