On weighted total least-squares adjustment for linear regression

By Burkhard Schaffrin, Andreas Wieser. 2008

271 citations recorded by the source corpus.

Subfields: Applied Mathematics, Artificial Intelligence, Environmental Engineering

View Paper: https://www.semanticscholar.org/paper/2ab423ace5111a72aabd915ab28e36702c8af4c0

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Citations in the corpus

Total least squares adjustment in partial errors-in-variables models: algorithm and statistical analysis (2012, methodology): “…step, the approach to using the method of Lagrange multipliers to solve EIV models, as developed by Deming (1931), has been followed by almost all the works on weighted TLS (see e.g. Gerhold 1969; Schaffrin and Wieser 2008; Felus and Burtch 2009; Schaffrin and Felus 2009; Shen et al. 2011).”
A total least squares solution for geodetic datum transformations (2014, methodology): “Two fundamental algorithms for adjusting the GMM are the Procrustes analysis (e.g., Grafarend and Awange 2003; Grafarend 2006; Awange et al. 2010; Grafarend and Awange 2012) and the unit quaternion-based approach (e.g., Shen et al. 2006).”
Generalization of total least-squares on example of unweighted and weighted 2D similarity transformation (2010, background): “In order to solve this adjustment problem, Akyilmaz (2007) elaborates on a so-called “generalized TLS procedure” (GTLS) which is known to not necessarily furnish the weighted TLS solution according to Schaffrin and Wieser (2008) .”
Iterative algorithm for weighted total least squares adjustment (2014, background): “In the absence of such correlation between y and A, the weighted total least squares (WTLS) problem seeks to solve the following optimization problem [20]”
On least-squares solution to 3D similarity transformation problem under Gauss–Helmert model (2015, methodology): “It is noted that more general weighting strategies, e.g., those by Schaffrin and Felus (2008), Schaffrin and Wieser (2008) and Teunissen (1985, 1988), may disturb the equivalence, maybe even the existence of closed-form solution.”
A structured and constrained Total Least-Squares solution with cross-covariances (2013, background): “Schaffrin and Felus (2005) and Schaffrin (2006) presented the iterative linear fixed and stochastic constrained TLS solutions based on a fixed form solution, which is separately detailed in Schaffrin and Wieser (2008) specifically for their unconstrained TLS solution.”
On weighted total least-squares for geodetic transformations (2012, methodology): “…the following algorithm was found: By defining, Py = Q−1y , [N , c] = AT Py[A, y], ξ (0) = N−1c, ν(0) = 0 and QA = Q0 ⊗ QX, the following iterative method has been proposed by Schaffrin and Wieser (2008) consisting of three steps: First step: ξ̂ (1) := ( AT ( Qy + (( ξ (0) )T Q0ξ (0) ) .”
Weighted total least squares formulated by standard least squares theory (2012, methodology): “We use the data presented in Neri et al. (1989) and later used by Schaffrin and Wieser (2008) (see Table 1).”
Weighted total least squares: necessary and sufficient conditions, fixed and random parameters (2013, methodology): “One frequently used TLS approach is based on the singular value decomposition (SVD) procedure (Felus 2004; Akyilmaz 2007; Schaffrin and Felus 2008), and another commonly explored option is to utilize auxiliary Lagrange multipliers to rearrange the specially weighted TLS problem as a constrained mini”
On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms (2008, background): “This subject was not presented here for brevity; the interested reader is referred to Schaffrin and Wieser (2008) .”
Data-snooping procedure applied to errors-in-variables models (2013, background): “(3) Schaffrin and Wieser (2008) introduced the “weighted TLS” to the geodetic community based on the inverse of the covariance matrix.”
Application of least squares variance component estimation to errors-in-variables models (2013, methodology): “In the studies by Neri et al. (1989) and Schaffrin and Wieser (2008), only estimates for the unknown parameters x = [a, b]T were provided.”
An iterative solution of weighted total least-squares adjustment (2011, methodology): “18 as ( ATQ−1l A ) ξ̂ WLS = ATQ−1l y (19) The WTLS solution derived by Schaffrin and Wieser (2008) is ξ̂ = ( ATQ−1l A − v̂Q0 )−1 ATQ−1l y (20) with the scalar v̂ being v̂ = ( y − Aξ̂ )T Q−1l QxQ −1 l ( y − Aξ̂ ) (21) where Q0 and Qx are defined in Eq.”
On non-combinatorial weighted total least squares with inequality constraints (2014, methodology): “From the point of view of model formulations, quite a few algorithms have been developed to adjust the univariate EIV model (see e.g., Markovsky et al. 2006; Schuermans et al. 2007; Schaffrin and Wieser 2008; Neitzel 2010; Fang 2011, 2013a; Shen et al. 2011; Amiri-Simkooei and Jazaeri 2012; Grafaren”
Variance components in errors-in-variables models: estimability, stability and bias analysis (2014, methodology): “…more than one century and found a wide variety of applications, either from the point of view of approximation (see, e.g., Golub and Loan 1980; Huffel and Vandewalle 1991) or from the statistical point of view (see, e.g., Pearson 1901; Deming 1931, 1934; Schaffrin and Wieser 2008; Xu et al. 2012).”