Book cover

Total least squares r pdf

Total least squares r pdf. i)2: 75 80 85 90 95 6 8 10 12 14 16 18 X = % HS grad MLR is just like SLR. 7. For ill-posed problems regularization is necessary to May 1, 1992 · It is shown that the solution to a total linear least squares problem satisfies a quadratic matrix equation, which turns into an algebraic Riccati equation when the matrix of unknowns is square. However, care must be paid to the proper Oct 20, 2023 · The uncertain total least squares estimation can fully consider errors in the given data and the uncertain robust total least squares linear regression method can effectively eliminate outliers in Jan 1, 2008 · The total least squares (TLS) method is a successful approach for linear problems if both the system matrix and the right hand side are contaminated by some noise. It is a generalization of Deming regression and also of orthogonal regress Mar 19, 2010 · Recent advances in total least squares approaches for solving various errors-in-variables modeling problems are reviewed, with emphasis on the following generalizations: 1. ∑. An almost entirely separate discussion of the same issue is found under the label \total least squares" (TLS), which is where one will nd most of the modern literature on this topic. $$\begin {aligned} \mathbf {A}\mathbf {x}\approx \mathbf {b Sep 17, 2010 · In this contribution it is shown that the so-called “total least-squares estimate” (TLS) within an errors-in-variables (EIV) model can be identified as a special case of the method of least-squares within the nonlinear Gauss–Helmert model. Semantic Scholar extracted view of "Tikhonov regularization for weighted total least squares problems" by 0; 1) is the intercept and slope of the straight line with the minimum sum of squared vertical distance to the data points X. SSE = Σ (y – ŷ)². Such systems may be solved using Tikhonov-like regularized total least squares (R-TLS) methods. Recall that dist ( v , w )= A v − w A is the distance between the vectors v and w . Let A be an m × n matrix and let b be a vector in Rn. For ill-posed TLS problems regularization is necessary to stabilize the computed solution. Efficient and robust computational algorithms were developed and properties of the resulting estimators were established in the errors-in-variables setting. Form the augmented matrix for the matrix equation A T Ax = A T b , and row reduce. LEAST SQUARES 1. ˆ ′= Y. In comparison to the standard Gauss-Markov Model (GMM), the EIV-Model is less informative whereas the Feb 6, 2024 · Total least squares. These of the corresponding weighted total-least squares solution (WTLSS) is derived along with an algorithm for its com-putation. 3 - Least Squares: The Theory. Researchers appreciate Apr 8, 2018 · 11. The least squares approximation Xbls is obtained as a solution of the optimization problem {Xbls , ∆Bls } := arg min k∆BkF subject to AX = B + ∆B. Technique used for the modeling and analysis of numerical data. 2 Geometry The vectors x 1;:::;x p span some subspace X of Rn. The total least-squares method was introduced by Golub and Van Loan [1], [2] as a solution technique for an overdetermined system of equations AX ≈ B, where A ∈ R m × n and B ∈ R m × d are the given data and X ∈ R n × d is unknown. In contrast to the EIV-model, the nonlinear GH-model does not impose any restrictions on the form of functional relationship between the quantities Jul 1, 2013 · The achievements and new progresses in the estimation properties, algorithms, and applications of total least squares in the recent decade are summarized in this paper. The R-TLS solution x to (7), with the inequality constraint re-placed by equality, is a solution to the problem The IRLS (iterative reweighted least squares) algorithm allows an iterative algorithm to be built from the analytical solutions of the weighted least squares with an iterative reweighting to converge to the optimal l p approximation [7], [37]. R2 = 1 − Sres Stot R 2 = 1 − S r e s S t o t. In comparison to the standard Gauss-Markov Model (GMM), the EIV-Model is less informative whereas the a Least Squares (LS) technique is used for the transforma-tion procedure. The R-TLS solution x to (7), with the inequality constraint re-placed by equality, is a solution to the problem Mar 1, 2024 · In this section we improve the Gauss-Newton iteration via \ (x_ {k+1}=x_k+\alpha h_k\) by replacing previous unit step length with a more reasonable step length. The least squares estimate ( Y = % in poverty b. E, r. Today I learned about total least squares and that princomp () function (principal component analysis, PCA) can be used to perform it. It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non Oct 1, 2007 · The total least-squares method is a natural generalization of the least-squares approximation method when the data in both A and B is perturbed. We also get an equivalent interpretation of the descent direction \ (h_k\). In this paper we summarize two iterative methods which are based on a sequence of eigenproblems. Jan 1, 1999 · Total least squares, although more computationally demanding than ordinary least squares, can be used in any situation where the latter is applicable. 0 b. 67-79 (January 2011) FITTING COMPOSITE HORIZONTAL CURVES USING THE TOTAL LEAST-SQUARES METHOD Said M. 36. Form the augmented matrix for the matrix equation ATAx = ATb, and row reduce. The algorithm is based on quadratic constraint total least-squares (QC-TLS) method and gives an explicit solution. The residual r is uniquely determined and the solution x is unique if and only if rank(A) == n. Easa1 and Fujian Wang2 1 Professor, Department of Civil Engineering, Ryerson University, Toronto, Ontario, Canada 2 Associate Professor, Department of Civil Engineering, Zhejiang University, Hangzhou 310027, China ABSTRACT The parameters of horizontal curves are A tutorial on the total least squares method for fitting a straight line and a plane Leonardo Romero Muñoz, Moisés García Villanueva and Cuauhtémoc Gómez Suárez, Facultad de Ingeniería Eléctrica, UMSNH, México analysis, statistics or linear algebra, or serve as a transition Abstract—The classic least squares regression fits a line to Mar 7, 2009 · The Total Least-Squares (TLS) approach as applied to the Errors-In-Variables (EIV) model is the proper method to treat problems where all the data are affected by random errors. As the geometry underlying the problem setting greatly contributes to Jan 10, 2023 · I am trying to fit a model that linearly relates two variables using R. This chapter provides an introduction to ordinary least squares (OLS) regression analysis in R. Mar 1, 2014 · least mean square (LMS) and the recursive least-squares (RLS) algorithms [1], assume that the input signal is obser ved accurately and only the filter output is corrupted by noise. The total least squares method was introduced by Golub and Van Loan [25, 27] as a solution technique for an overde-termined system of equations AX ≈ B, where A ∈ Rm×n and B ∈ Rm×d are the given data and X ∈ Rn×d is unknown. But if there is not a clear assignment of “dependent” and “independent” variables, then it makes more sense to TIKHONOV REGULARIZATION AND TOTAL LEAST SQUARES 187 less than kLxTLSk2. Another approach that could be introduced as an alternative methodology is the Total Least Squares (TLS) that is considerably a new approach in geodetic applications. g. Published in Numerische Mathematik 1 March 2002. It is shown how Tikhonov's regularization method can be recast in a total least squares formulation suited for problems in which both the coefficient matrix and the right-hand side are Sep 1, 2010 · The class of total least squares methods has been growing since the basic total least squares method was proposed by Golub and Van Loan in the 70's. r, where A = U LV T, V is p and orthogonal, L is and lower triangular, U has the same shape as A with orthonormal columns, and where L has the form L = C 0 E F where C (r)encapsulates the “large” singular values of A, (E ; F) (p) approximately encapsulate the p r smallest singular values of A, and the last p r columns of V encapsulate the A common problem in a Computer Laboratory is that of finding linear least squares solutions. . Recent advances in total least squares approaches for solving various errors‐in‐variables modeling problems are reviewed, with emphasis on the following generalizations: 1. This is the null model and the residual sum of squares for this model will actually be called the total sum of squares: TSS. ) The core step in the singular spectrum analysis as well Survey Review, 43, 319 pp. When the coefficient matrix is contaminated by some noise, the LS solution is biased, while the TLS solution is unbiased. May 18, 1998 · The method of ``Total Least Squares'' is proposed as a more natural way (than ordinary least squares) to approximate the data if both the matrix and and the right-hand side are contaminated by ``errors''. Published in SIAM Journal on Matrix 1 October 1999. C. In general, the Least-Squares (LS) estimation is used in geodetic problems. Here is a method for computing a least-squares solution of Ax = b: Compute the matrix ATA and the vector ATb. For the least-squares (LS) problem, we describe a way to 5 days ago · In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. EYY. Also, let r= rank(A) be the number of linearly independent rows or columns of A. Jun 10, 2008 · A novel algorithm for source location by utilizing the time difference of arrival (TDOA) measurements of a signal received at spatially separated sensors is proposed. (That is, the set of all vectors of the form x 1b 1 + :::x pb p is a vector subspace of R n). Mathematics. 4 CHAPTER 1. We learned that in order to find the least squares regression line, we need to minimize the sum of the squared prediction errors, that is: Q = ∑ i = 1 n ( y i − y For the special case of regularization with a squared Euclidean norm, it is shown that ${\\mathcal{G}}$ is unimodal and provided an alternative algorithm, which requires only one spectral decomposition. The solutions to robust optimization problems are sometimes too conservative because of the focus on worst-case performance. Related algorithms and Apr 23, 2022 · To identify the least squares line from summary statistics: Estimate the slope parameter, b1 b 1, using Equation 7. Σ represents a sum. The minimimizing vector in X is often denoted by by. 895288 * Color + 3. Hansen, D. The applicability is shown using two numerical examples: (i) a straight-line ﬁt where the coordinates involved have been observed with different precision, and (ii) multiple regression applied to a data set which appea- Dec 31, 2015 · Abstract. Describes a powerful optimization algorithm which iteratively solves a weighted least squares approximation problem in order to solve an L_p approximation problem. ii ′ while the residual sum of squares for the linear fit will be written ( ) 2. The solution of total least squares problems, i. G. ∑∑. Nov 1, 2012 · The total least squares estimation is the linear transformation of the least squares estimation, and the expection of the TLS solution is also the linear transformation of the expection of the LS solution. ( 2 ), has been incorporated through ( 8 ), ( 17 ) cannot be an unconstrained minimum problem because there are inherent constraints on R , i. 4 7. 1 R(A), where r == b-Ax is the residual and R(A) the range space of A. PDF. i b. ˆ 2. 5 Practical: Least-Squares Solution De nition 3. Lem- TIKHONOV REGULARIZATION AND TOTAL LEAST SQUARES 187 less than kLxTLSk2. Statisticians refer to squared residuals as squared errors and their total as the sum of squared errors (SSE), shown below mathematically. [24,25] have employed an RQ-based RTLS method for the Sep 1, 2010 · sion AX ≈ B, B ∈ R m × d, of the total least squares problem allows data ﬁtting by a space of a general dimension r , n ≥ r ≥ 1. Exploits the relationship between two or more variables so that we can gain information about one of them through knowing values of the other. Oct 1, 2022 · We review estimators/algebraic solutions of type MINOLESS, BLIMBE, BLUMBE, BLUUE, BIQUE, BLE, BIQUE, and total least squares. Here an example data: Recipe 1: Compute a least-squares solution. Feb 1, 2005 · TLDR. Oct 31, 2015 · the standard R 2, the raw moment r 2, Buse's R 2 for weighted least squares, the R 2 measures introduced by Wooldridge (1991) for models that include trends and/or sea sonal dummies , as well as Consistency of the structured total least squares estimator in a multivariate model. For ill-posed TLS problems Apr 11, 2013 · Mathematical method known as total least squares or orthogonal regression or error-in-variables. minimize. Note that though the constraint on the errors, i. That function returns a coefficient β such that y = βx. Cambridge A least-squares solution of the matrix equation Ax = b is a vector K x in R n such that. Jun 18, 2021 · It is shown that the regularized total least squares (RTLS) problem can be reformulated as a nonlinear least squares problem and can be solved by the Gauss–Newton method. However, its exact solution is rather complicated, and the accuracies of estimated. The package REGULARIZATION TOOLS consists of 54 Matlab routines for analysis and solution of discrete ill-posed problems, i. the use of weighted norms as a measure of the data perturbation size, capturing prior knowledge about uncertainty in the data; 2. The term “least squares” comes from the fact that dist ( b , Ax )= A b − A K x A is the Oct 1, 1987 · Given {pi} and { p’i }, we present an algorithm for finding the least-squares solution of R and T, which is based on the singular value decomposition (SVD) of a 3 × 3 matrix. It should be good for me (more accurate). , solving $\min_{E,r} \| [E, r]\|_F$ subject to $(A+E)x=b+r$, arises in numerous applications. Paige, Z. We investigate the total least square problem (TLS) with Chebyshev norm instead of the traditionally used Frobenius norm. 1,622. November 13, 2018. It is sometimes called the tted Dec 1, 2006 · One of these mathematical algorithms is called the Total Least Squares (TLS). dist ( b , A K x ) ≤ dist ( b , Ax ) for all other vectors x in R n . Regression can be used for prediction, estimation, hypothesis testing, and modeling causal relationships. The methods in the deming package are concerned with the problem of comparing two assays, both of which are measured with error. Journal of Statistical Planning and Inference, to appear. If rank(A) < n, we seek the unique least squares solution x 1-N(A), which is called the pseudo-inversesolution. 2 == −. Now that we have the idea of least squares behind us, let's make the method more practical by finding a formula for the intercept a 1 and slope b. Jun 4, 2021 · Abstract. These problems arise in a variety of areas and in a variety of contexts. The LS estimation is based on the probability theory, adjusting a set of observations to make the sum of the squares of the residuals minimum. The least squares method minimizes ky xkas x runs through all vectors in X. Terry Therneau Mayo Clinic. Nov 3, 2021 · Partial least squares structural equation modeling (PLS-SEM) has become a standard approach for analyzing complex inter-relationships between observed and latent variables. We consider the following classical definition of TLS problem, see e. 751401. 758415 * Quality + 1. 731548 * Color + 4. Here is a method for computing a least-squares solution of Ax = b : Compute the matrix A T A and the vector A T b . The LS estimation is based on the Oct 1, 1999 · Tikhonov Regularization and Total Least Squares. TLDR. 1 Introduction. in the sense of least squares. A set of efficient algorithms has been developed previously to solve the TLS problem, and a few procedures have been proposed to treat TLS problems with linear A least squares solution x is characterized by r. The two solutions x and x to the two regularized problems in (5) and (7) have a surprising relationship, explained by the following theorem. 1 Excerpt. In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression and also of orthogonal regression, and can be applied Error-contaminated systems A x ≈ b, for which A is ill-conditioned, are considered. and Van Huffel S. Here we present a parameter independent algorithm for the approximate R-TLS solution Apr 1, 2021 · The concise review systematically summarises the state-of-the-art variants of Moving Least Squares (MLS) method. The pls package implements principal component regression (PCR) and partial least squares regression (PLSR) in R (R Development Core Team 2006b), and is freely available from the can express weighted-sum objective as ordinary least-squares objective: kAx−yk2 +µkFx−gk2 = A √ µF x− y √ µg 2 = Ax˜ −y˜ 2 where A˜ = A √ µF , y˜= y √ µg hence solution is (assuming A˜ full rank) x = A˜TA˜ −1 A˜Ty˜ = ATA+µFTF −1 ATy +µFTg Regularized least-squares and Gauss-Newton method 7–7 Regression. (y. I always use lm () in R to perform linear regression of y on x. Strakoš. The total least squares (TLS) method is a well-known technique for solving an overdetermined linear system of equations Ax ≈ b, that is appropriate when both the coefficient matrix A and the right-hand side vector b are Introduction to total least squares. Consider the overdetermined system of equations. Let A be an m × n matrix and let b be a vector in R n . Mar 1, 2010 · Total least squares methods. Total least squares (TLS) is a method for treating an overdetermined system of linear equations ${\\bf A} {\\bf x} \\approx {\\bf b}$, where both the matrix ${\\bf A}$ and the vector ${\\bf b The Total Least Squares (TLS) problem is a well known tech- nique for solving overdetermined linear systems of equations Ax ≈ b, A ∈ Rm×n , b ∈ Rm (m > n), in which both the matrix A and the right hand side b are affected by errors. 5. So, provided you use the model given by lm, (Pearson)-squared and R-squared are equivalent. The theory reveals the necessary and sufficient condition for preserving the smallest singular value of a matrix while appending (or deleting) a column, which represents a basic matrix theory result for updating May 1, 2010 · The usual least-squares adjustment within an Errors-in-Variables (EIV) model is often described as Total Least-Squares Solution (TLSS), just as the usual least-squares adjustment within a Random Effects Model (REM) has become popular under the name of Least-Squares Collocation (without trend). Consistency of elementwiseweighted total least squares estimator in a multivariate errors-in-variables model AX=B. In this case, it’s the sum of all residuals squared. 1 The Overdetermined System with more Equations than Unknowns If one poses the l Jan 1, 2008 · The use of techniques such as approximate maximum likelihood (AML) algorithm [9], total least square (TLS) [11] and quadratic constraint solution (QCS) approaches [10] have been suggested. Linear least squares problems are particularly difficult to solve because they frequently involve large quantities of data, and they are ill-conditioned by their very nature. With m > n, typically there is no exact solution for X, so that an approximate one is Jul 1, 1997 · Regularization by Truncated Total Least Squares. This yields a relaxed Gauss-Newton iteration (RGN-MTLS) for MTLS problems. We present a Matlab toolbox which can solve basic problems related to the Total Least Squares (TLS) method in the modeling. The total least squares (TLS) method is an appropriate approach for linear systems when not only the right-hand side but also the system matrix is contaminated by some noise. Iterative Reweighted Least Squares ∗. However, if you use the model from the total sum of squares regression, and try to use the latter equation, you will get a slightly different result. 1 The Solutions of a Linear System Let Ax = b be an m nsystem (mcan be less than, equal to, or greater than n). Least squares estimation means minimize () ( )2 SA Y A ′=−. 04619. Total Least Squares: Deming, Theil-Sen, and Passing-Bablock Regression. Noting that the point ( x¯,y¯ x ¯, y ¯) is on the least squares line, use x0 = x¯ x 0 = x ¯ and y0 = y¯ y 0 = y ¯ along with the slope b1 b 1 in the point-slope equation: Sep 1, 2015 · As an alternative choice, the total least square (TLS) method can effectively deal with the measurement noises from sensors [23]. 5. , deviation between • what we actually observed (y), and • what we would observe if x = ˆx, and there were no noise (v = 0) least-squares estimate is just xˆ = (ATA)−1ATy Least-squares 5–12 Oct 1, 2007 · Introduction. In least squares (LS) we have only a perturbation of the right hand side whereas Total Least Squares (TLS) considers perturbations of the vector of observations c and of the m × n data matrix A. MLS method is a mathematical tool which could render cogent support in data Jul 31, 2011 · Possible Duplicate: How to calculate Total least squares in R? (Orthogonal regression) I have to implement Total Least Squares model in R instead of lm() (linear regression) Who don't underst Jan 1, 2002 · The total least squares (TLS) method is a well-known technique for solving an overdetermined linear system of equations Ax ≈ b, that is appropriate when both the coefficient matrix A and the Least-squares applications • least-squares data ﬁtting • growing sets of regressors least-squares ﬁt: choose x to minimize total square ﬁtting error: Xm i=1 Aug 6, 2015 · So it is concluded that the Wahba's problem is in fact a total least-square problem rather than the least-square problem as is widely known. The Cramer-Rao bound PRIT problem has been formulated [9]-[ 121. , [4, 13]. In this tutorial note, we give a elementary unified view of ordinary and total least squares problems and their solution. Motivated by the recently popular probabilistic algorithms for low-rank approximations, randomized algorithms for the Tls and the truncated total least squares solutions of large-scale discrete ill-posed problems are developed, which can greatly reduce the computational time and still keep good accuracy. The regression equation defined by ordinary least squares is Price = 4. The resulting regression equation is Price = 5. The least-squares approximation X ^ ls is obtained as a solution of the optimization problem (LS) { X ^ ls, Δ B ls } ≔ arg min X, Δ B ∥ Δ B ∥ F subject to AX = B + Δ B. 1x. So I'm trying to use the odregress() function of the pracma package which performs an Orthogonal Linear Regression via PCA. Wei et al. (2004). the 3. The application to a uniform linear array cently, a total least squares (TLS) approach to the ES- is treated in some detail, and a generalization of ESPRIT to include row weighting is discussed. This paper proposes and test an iterative algorithm based on Lanczos bidiagonalization for computing truncated TLS solutions and expresses the results in terms of the singular value decomposition of the coefficient matrix rather than the augmented matrix, which leads to insight into the filtering Mar 1, 2002 · Scaled total least squares fundamentals. This new algorithm 2 Total Least Squares (Orthogonal) Regression In classical least-squares regression, errors are deﬁned as the squared distance from the data points to the ﬁtted function, as measured along a particular axis direction. 1. The use of Chebyshev norm is motivated by the need for robust Jan 1, 2007 · Ron Wehrens. Jul 29, 2011 · Total Least Squares; Deming regression; Your multiple questions on CrossValidated, here and R-Help imply that you need to do a bit more work to describe exactly what you want to do, as the terms "Total least squares" and "orthogonal regression" carry some degree of ambiguity about the actual technique wanted. n i=1. [PDF] May 30, 2023 · Download PDF Abstract: With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. By illustrative examples we show how to use the TLS method for solution of: - linear regression model - nonlinear regression Aug 6, 2011 · The usual least-squares adjustment within an Errors-in-Variables (EIV) model is often described as Total Least-Squares Solution (TLSS), just as the usual least-squares adjustment within a Random Effects Model (REM) has become popular under the name of Least-Squares Collocation (without trend). This is a technique used to explore whether one or multiple variables (the independent variable or X) can predict or explain the variation in another variable (the dependent variable or Y ). Theorem 2. 1 The Total Least Squares Problem. Van Huﬀel and P . Two approaches for regularizing large scale TLS problems are discussed, one which is based on adding a quadratic constraint and a Tikhonov type regularization concept. Expand. Apr 9, 2020 · In this study, we establish a general total least squares (TLS) theory which allows the errors-in-variables model with different constraints to formulate all transformation models, including 2005. You’ll see a lot of sums in the least squares line formula section! The total least squares method is a natural generalization of the least squares approximation method when the data in both A and B is perturbed. The total least squares method is in principle a method for treating overdetermined linear systems of equations where both the coefficient matrix and the right-hand side are subject to errors. 0. e. Golub, P. , systems of linear equations whose coefficient matrix has the properties. 1 Approximation Methods of approximating one function by another or of approximating measured data by the output of a mathematical or computer Jan 1, 2004 · Guo and Renaut [A r egularized total le ast squares algorithm, in T otal Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications, S. 3. The highlight is the simultaneous determination of the first moment and the second central moment of a probability distribution in an inhomogeneous multilinear estimationby the so-called E-D correspondence as well as its May 21, 2018 · Objective matching allows the least-squares problem to consistently match or outperform other state-of-the-art techniques, including both ridge and LASSO regression, on simulations and real-world data sets. 2 Jun 18, 1998 · Abstract and Figures. Dec 29, 2018 · Total least squares (aka TLS) is one of regression analysis methods to minimize the sum of squared errors between a response variable (or, an observation) and a predicated value (we often say a fitted value). I need to fit a Orthogonal Linear Regression (total least squares). The method of ``Total Least Squares'' is proposed as a more natural way (than ordinary least squares) to approximate the data if both the matrix and and the right-hand side least-squares estimation: choose as estimate xˆ that minimizes kAxˆ−yk i. The most popular and standard method of this is Ordinary least squares (aka OLS), and TLS is one of other methods that take different Sep 17, 2022 · Recipe 1: Compute a Least-Squares Solution. Here, cells A15, B15, and C15 contain the means for the Color, Quality, and Price sample data. O’Leary. Markovsky and Van Hu el [4] present a good overview of the area, which has a rich literature of algorithms and extensions. In this study, in order to determine point displacements, 3-D Jan 6, 2010 · The total least squares (TLS) method is a successful approach for linear problems if both the system matrix and the right hand side are contaminated by some noise. 4. i which leads to A. 409061 * Quality – 6. Google Scholar Kukush A. OLS regression belongs to a family of techniques One of these mathematical algorithms is called the Total Least Squares (TLS). The focus is on efficient implementation Jan 1, 2014 · An iterative solution of weighted total least-squares adjustment. Then,1 b 62range(A) ) no solutions b 2range(A) ) 1n r solutions with the convention that 10 = 1. Total least-squares (TLS) adjustment is used to estimate the parameters in the errors-in-variables (EIV) model. Given the matrix equation Ax = b a least-squares solution is a solution ^xsatisfying jjA^x bjj jjA x bjjfor all x Such an ^xwill also satisfy both A^x = Pr Col(A) b and AT Ax^ = AT b This latter equation is typically the one used in practice. Golub et al, 1999, presented a direct algorithm for the solution of the Lagrange multiplier formulation for the R-TLS problem. In response to a growing interest in Total Least Squares (TLS) and Errors-In-Variables (EIV) modeling by researchers and practitioners, well-known experts from several disciplines were invited to prepare an overview paper and present it at the third international workshop on TLS and EIV modeling held in Leuven, Belgium, August 27-29, 2001. kn wh ox xt eb ch lm ql pt gn