Weighted total least squares formulated by standard least squares theory
By Alireza Amiri-Simkooei, S. Jazaeri. 2012
147 citations recorded by the source corpus.
Weighted total least squares formulated by standard least squares theory This contribution presents a simple, attractive, and flexible formulation for the weighted total least squares (WTLS) problem. It is simple because it is based on the well-known standard least squares theory; it is attractive because it allows one to directly use the existing body of knowledge of the least squares theory; and it is flexible because it can be used to a broad field of applications in the error-invariable (EIV) models. Two empirical examples using real and simulated data are presented. The first example, a linear regression model, takes the covariance matrix of the coefficient matrix as QA = Qn ⊗ Qm, while the second example, a 2-D affine transformation, takes a general structure of the covariance matrix QA. The estimates for the unknown parameters along with their standard deviations of the estimates are obtained for the two examples. The results are shown to be identical to those obtained based on the nonlinear Gauss-Helmert model (GHM). We aim to have an impartial evaluation of WTLS and GHM. We further explore the high potential capability of the presented formulation. One can simply obtain the covariance matrix of the WTLS estimates. In addition, one can generalize the orthogonal projectors of the standard least squares from which estimates for the residuals and observations (along with their covariance matrix), and the variance of the unit weight can directly be derived. Also, the constrained WTLS, variance component estimation for an EIV model, and the theory of reliability and data snooping can easily be established, which are in progress for future publications.
Subfields: Applied Mathematics, Computer Vision and Pattern Recognition
View Paper: https://www.semanticscholar.org/paper/f373309569557577831b41d2b43de95f3e0f4dad
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In Applied Mathematics
In Computer Vision and Pattern Recognition
Citations in the corpus
- Iterative algorithm for weighted total least squares adjustment (2014, methodology): “Amiri-Simkooei and Jazaeri [2] have recently formulated the WTLS problem based on the standard least-squares theory in which a linearly structured Gauss-Markov model (GMM) is iteratively solved instead of a nonlinear GHM.”
- Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations (2015, methodology): “2010; Amiri-Simkooei and Jazaeri 2012, 2013; Mahboub 2012; Snow 2012; Xu et al. 2012; Amiri-Simkooei 2013; Jazaeri et al. 2014; Xu and Liu 2014). These algorithms were individually developed using different approaches, but all of themwere based on the first derivative of theWTLSobjective function. F”
- Data-snooping procedure applied to errors-in-variables models (2013, methodology): “(4) results in the following equations (Amiri-Simkooei and Jazaeri, 2012) 1 ˆ y y ye Q Q y Ax (5) and 1ˆ ˆvec A A A m ye E Q x I Q y Ax , (6) where Stud.”
- A structured and constrained Total Least-Squares solution with cross-covariances (2013, background): “Amiri-Simkooei and Jazaeri (2012) and Xu et al. (2012) formulated WTLS based on the well-known nonlinear least squares theory allowing one to directly use the existing body of statistical knowledge of the least squares theory.”
- Application of least squares variance component estimation to errors-in-variables models (2013, methodology): “The weighted total least squares estimate of the unknown parameters x is (Amiri-Simkooei and Jazaeri 2012) x̂ = (( A − Ẽ A )t Q−1ỹ ( A − Ẽ A ))−1 × ( A − Ẽ A )t Q−1ỹ ( y − Ẽ A x̂ ) (8) which resembles the standard least squares estimate in which à = A − Ẽ A plays the role of the design…”
- 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”
- A total least squares solution for geodetic datum transformations (2014, methodology): “This adjustment has been named TLS in the field of mathematics, although it is also known as the standard LS, as it is applied in the EIV model (see Neitzel 2010; Amiri-Simkooei and Jazaeri 2012).”