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>

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 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.

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.

Gravity interpretation with the aid of quadratic programming

Geophysics ◽  
1980 ◽  
Vol 45 (3) ◽  
pp. 403-419 ◽  
Author(s):  
N. J. Fisher ◽  
L. E. Howard
Keyword(s):  
Quadratic Programming ◽  
Least Squares ◽  
Theoretical Data ◽  
Linear Constraints ◽  
Upper And Lower Bounds ◽  
Programming Algorithm ◽  
Least Squares Problem ◽  
The Matrix ◽  
Gravity Interpretation ◽  
Value Decomposition

The inverse gravity problem is posed as a linear least‐squares problem with the variables being densities of two‐dimensional prisms. Upper and lower bounds on the densities are prescribed so that the problem becomes a linearly constrained least‐squares problem, which is solved using a quadratic programming algorithm designed for upper and lower bound‐type constraints. The solution to any problem is smoothed by damping, using the singular value decomposition of the matrix of gravitational attractions. If the solution is required to be monotonically increasing with depth, then this feature can be incorporated. The method is applied to both field and theoretical data. The results are plotted for (1) undamped, nonmonotonic, (2) damped, nonmonotonic, and (3) damped, monotonic solutions; these conditions illustrate the composite approach of interpretation where both damping techniques and linear constraints are used in refining a solution which at first is unacceptable on geologic grounds while fitting the observed data well.

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