scholarly journals Weighted total least squares formulated by standard least squares theory

2012 ◽  
Vol 2 (2) ◽  
pp. 113-124 ◽  
Author(s):  
A. Amiri-Simkooei ◽  
S. Jazaeri
Keyword(s):  
Least Squares ◽  
Covariance Matrix ◽  
General Structure ◽  
Simulated Data ◽  
Coefficient Matrix ◽  
Total Least Squares ◽  
Unit Weight ◽  
Unknown Parameters ◽  
Weighted Total Least Squares ◽  
Orthogonal Projectors

Weighted total least squares formulated by standard least squares theoryThis 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 asQA=Qn⊗Qm, while the second example, a 2-D affine transformation, takes a general structure of the covariance matrixQA.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 thenonlinearGauss-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.

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

2016 ◽  
Vol 6 (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.

scholarly journals A universal and fast method to solve linear systems with correlated coefficients using weighted total least squares

2021 ◽  
Author(s):  
Matthias Wurm
Keyword(s):  
Numerical Solution ◽  
Least Squares ◽  
Linear Systems ◽  
Covariance Matrix ◽  
Total Least Squares ◽  
Fast Method ◽  
Universal Method ◽  
Computational Costs ◽  
Weighted Total Least Squares ◽  
Special Cases

Abstract Especially in metrology and geodesy, but also in many other disciplines, the solution of overdetermined linear systems of the form Ax≈b with individual uncertainties not only in b but also in A is an important task. The problem is known in literature as weighted total least squares. In the most general case, correlations between the elements of [A,b] exist as well. The problem becomes more complicated and can—except for special cases—only be solved numerically. While the formulation of this problem and even its solution is straightforward, its implementation—when the focus is on reliability and computational costs—is not. In this paper, a robust, fast, and universal method for computing the solution of such linear systems as well as their covariance matrix is presented. The results were confirmed by applying the method to several special cases for which an analytical or numerical solution is available. If individual coefficients can be considered to be free of errors, this can be taken into account in a simple way. An implementation of the code in MATLAB is provided.

Weighted total least squares problems with inequality constraints solved by standard least squares theory

2020 ◽  
Author(s):  
Xie Jian ◽  
Long Sichun
Keyword(s):  
Quadratic Programming ◽  
Least Squares ◽  
Target Function ◽  
Inequality Constraints ◽  
Point Clouds ◽  
Coefficient Matrix ◽  
Total Least Squares ◽  
Programming Algorithm ◽  
Model Parameters ◽  
Weighted Total Least Squares

<p>The errors-in-variables (EIV) model is applied to surveying and mapping fields such as empirical coordinate transformation, line/plane fitting and rigorous modelling of point clouds and so on as it takes the errors both in coefficient matrix and observation vector into account. In many cases, not all of the elements in coefficient matrix are random or some of the elements are functionally dependent. The partial EIV (PEIV) model is more suitable in dealing with such structured coefficient matrix. Furthermore, when some reliable prior information expressed by inequality constraints is considered, the adjustment result of inequality constrained PEIV (ICPEIV) model is expected to be improved. There are two kinds of algorithms to solve the ICPEIV model under the weighted total least squares (WTLS) criterion currently. On the one hand, one can linearize the PEIV model and transform it into a sequence of quadratic programming (QP) sub-problems. On the other hand, one can directly solve the nonlinear target function by common used programming algorithms.All the QP algorithms and nonlinear programming methods are complicated and not familiar to the geodesists, so the ICPEIV model is not widely used in geodesy.   </p><p>In this contribution, an algorithm based on standard least squares is proposed. First, the estimation of model parameters and random variables in coefficient matrix are separated according to the Karush-Kuhn-Tucker (KKT) conditions of the minimization problem. The model parameters are obtained by solving the QP sub-problems while the variables are determined by the functional relationship between them. Then the QP problem is transformed to a system of linear equations with nonnegative Lagrange multipliers which is solved by an improved Jacobi iterative algorithm. It is similar to the equality-constrained least squares problem. The algorithm is simple because the linearization process is not required and it has the same form of classical least squares adjustment. Finally, two empirical examples are presented. The linear approximation algorithm, the sequential quadratic programming algorithm and the standard least squares algorithm are used. The examples show that the new method is efficient in computation and easy to implement, so it is a beneficial extension of classical least squares theory.</p>

Fitting Weighted Total Least-Squares Planes and Parallel Planes to Support Tolerancing Standards

2012 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Point Contact ◽  
Coordinate Measuring Machine ◽  
Total Least Squares ◽  
Point Sampling ◽  
Least Squares Fitting ◽  
Weighted Total Least Squares ◽  
Measuring Machine ◽  
Fitting In ◽  
Value Decomposition

We present elegant algorithms for fitting a plane, two parallel planes (corresponding to a slot or a slab) or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3×3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact Coordinate Measuring Machine). However, we demonstrate that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

scholarly journals Generalized Nonlinear Least Squares Method for the Calibration of Complex Computer Code Using a Gaussian Process Surrogate

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 985
Author(s):  
Youngsaeng Lee ◽  
Jeong-Soo Park
Keyword(s):  
Gaussian Process ◽  
Least Squares ◽  
Covariance Matrix ◽  
Process Model ◽  
Least Squares Method ◽  
Nonlinear Least Squares ◽  
Computer Code ◽  
Test Functions ◽  
Unknown Parameters ◽  
Complex Computer

The approximated nonlinear least squares (ALS) method has been used for the estimation of unknown parameters in the complex computer code which is very time-consuming to execute. The ALS calibrates or tunes the computer code by minimizing the squared difference between real observations and computer output using a surrogate such as a Gaussian process model. When the differences (residuals) are correlated or heteroscedastic, the ALS may result in a distorted code tuning with a large variance of estimation. Another potential drawback of the ALS is that it does not take into account the uncertainty in the approximation of the computer model by a surrogate. To address these problems, we propose a generalized ALS (GALS) by constructing the covariance matrix of residuals. The inverse of the covariance matrix is multiplied to the residuals, and it is minimized with respect to the tuning parameters. In addition, we consider an iterative version for the GALS, which is called as the max-minG algorithm. In this algorithm, the parameters are re-estimated and updated by the maximum likelihood estimation and the GALS, by using both computer and experimental data repeatedly until convergence. Moreover, the iteratively re-weighted ALS method (IRWALS) was considered for a comparison purpose. Five test functions in different conditions are examined for a comparative analysis of the four methods. Based on the test function study, we find that both the bias and variance of estimates obtained from the proposed methods (the GALS and the max-minG) are smaller than those from the ALS and the IRWALS methods. Especially, the max-minG works better than others including the GALS for the relatively complex test functions. Lastly, an application to a nuclear fusion simulator is illustrated and it is shown that the abnormal pattern of residuals in the ALS can be resolved by the proposed methods.

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