Iteratively reweighted total least squares: a robust estimation in errors-in-variables models

Transcription

1 : a robust estimation in errors-in-variables models V Mahboub*, A R Amiri-Simkooei,3 and M A Sharifi 4 In this contribution, the iteratively reweighted total least squares (IRTLS) method is introduced as a robust estimation in errors-in-variables (EIV) models The method is a follow-up to the iteratively reweighted least squares (IRLS) that is applied to the Gauss Markov and/or Gauss Helmert models, when the observations are corrupted by gross errors (outliers) In a relatively new class of models, namely EIV models, IRLS or other known robust estimation methods introduced in geodetic literature cannot be directly applied This is because the vector of observations or the coefficient matrix of the EIV model may be falsified by gross errors The IRTLS can then be a good alternative as a robust estimation method in the EIV models This method is based on the algorithm of weighted total least squares problem according to the traditional Lagrange approach to optimise the target function of this problem Also a new weight function is introduced for IRTLS approach in order to obtain better results A simulation study and an empirical example give insight into the robustness and the efficiency of the procedure proposed Keywords: Errors-in-variables model, Weighted total least squares, Robust estimation, Iteratively Reweighted total least squares Introduction If the observations are distorted by gross errors in addition to random ones, one can use robust estimation techniques The concept of robustness has been already introduced in the last years The term robust was coined in statistics by [4] Various definitions of greater or lesser mathematical rigor are possible for the term, but in general, referring to a statistical estimator, it means insensitive to small departures from the idealised assumptions for which the estimator is optimized [9], [3] In the last two decades, several publications concerning robust estimation and outlier detection methods have been published in geodetic literature We may for instance refer to [8], [], [], [7], and [3] None of them, however, has been applied to the relatively new class of models named errors-in-variables (EIV) models, which can be solved using the total least squares (TLS) method developed by [7], although [9] and [0] have Department of Surveying and Geomatics Engineering, Geodesy Division, Faculty of Engineering, University of Tehran, North Kargar Ave, Amir- Abad, Tehran, Iran Department of Surveying Engineering, Faculty of Engineering, University of Isfahan, , Isfahan, Iran 3 Acoustic Remote Sensing Group, Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg, 69 HS, Delft, The Netherlands 4 Department of Surveying and Geomatics Engineering, Geodesy Division, Faculty of Engineering, University of Tehran, North Kargar Ave, Amir- Abad, Tehran, Iran *Corresponding author, vahid_mahboobk@yahoocom proposed a method for outlier detection in EIV model based on the traditional statistical test The method is, however, applicable when only one outlier appears, either in the observation vector or in the coefficient matrix In the last decade, there has been a growing demand to use the TLS method, generally in engineering sciences and particularly in geodetic surveying applications Therefore, a robust estimation method for EIV models needs to be established Although some researchers such as [8] and [5] have investigated this problem in the statistical literature, the methods proposed can only be applied to a linear regression model In addition, the methods are difficult to use We aim to adapt the methods for many engineering applications This came out of the real and simulated examples that show, in order to achieve more accurate results, one has to further develop the method and adopt it for engineering applications In the present contribution, one approximate approach is introduced for robust estimation in EIV models which is coming from the fields of optimal control and filtering rather than the field of mathematical statistics It is the iteratively reweighted total least squares (IRTLS) which is a followup to the iteratively reweighted least squares (IRLS) that was originally introduced by [] into the geodetic applications To develop the IRTLS algorithm, we select one algorithm among the several existing algorithms that solve the TLS problem This algorithm was developed by [5] for the weighted total least squares (WTLS) problem It is based on the traditional Lagrange approach of which ß 03 Survey Review Ltd Received 9 November 0; accepted 8 May 0 9 Survey Review 03 VOL 45 NO 39 DOI 079/75706Y

2 Mahboub et al its target function was proposed by [3] The algorithm has a general structure and can be applied to many geodetic applications such as linear regression and affine transformation The algorithm has also the capability to be reweighted This paper is organised as follows In the next section, the EIV model and the WTLS algorithm are introduced We then introduce the IRTLS problem An algorithm to solve an IRTLS problem is proposed In a later section, a simulation study and an empirical example give insight into the robustness of the IRTLS and the efficiency of the algorithm proposed Finally, we conclude the paper Weighted total least squares In many geodetic surveying applications, one usually assumes that only the observables are contaminated by random noise There are, however, some cases that the model itself has also random noise In this case, the Gauss Markov model is replaced by an EIV model, which is expressed as follows y~ ða{ E A Þxz e y () where underlines indicates random variables, and one has e y e A : ~ e y vec( E A ) & 0 0, s 0 Q y 0 0 Q A where y is the m6 observation vector, e y is the m6 vector of observational noise, A is the m6ncoefficient matrix of input variables (observed), E A is the corresponding m6n matrix of random noise, x is the n6 parameters vector (unknown), D E y ~s 0 Q y and DðE A Þ~s 0 Q A are the corresponding (partly known) dispersion matrices of size m6m and mn6mn, with s 0 the unknown variance component The WTLS estimator is used to solve the EIV model defined in equations () and () To achieve this goal, [5] introduced the generalised total least squares (GTLS)[3] argued that GTLS does not solve the WTLS problem, because the so called weights of GTLS does not actually refer to a covariance matrix of the matrix observations To avoid confusion, the GTLS is now more commonly known as equilibrated TLS What we call weighted TLS follows, in fact, the geodetic tradition and is based on the inverse of the dispersion matrix (variance covariance) In case of a diagonal covariance matrix, element-wise WTLS was first introduced by [7], who claim that in general the problem has no closed-form solution and its computation involves solving a non-convex optimization problem [3] and [4] have similarly introduced an iterative method according to the traditional Lagrange approach to solve WTLS They are, however, restricted to the condition P A ~Q { A ~(P 06P x ), where : denotes the Kronecker product and P 0 is of size m6m and P x of size n6n Based on the traditional Lagrange approach, another algorithm has recently been developed for the WTLS problem [5] It has a general structure and can be applied to many geodetic applications such as linear regression model and an affine transformation Also in [5] it was proved that the WTLS approach preserves the structure of the coefficient matrix in an EIV model when based on the prefect description of the dispersion matrix () Furthermore, as it becomes clear in the present contribution, this algorithm is appropriate to be reweighted The target (Lagrange) function proposed by [3] is W : ~e T y Q{ y e y ze T A Q{ A e Azl T y{ax{e y z(x T 6I m )e A (3) where l5m6 (unknown) Lagrange multiplier vector and I m 5 m6m identity matrix From the first derivatives of equation (3) with respect to the variables e y, e A, l and x, we get the following four equations LW Le T y LW Le T A ~Q { y ~ e y {^l~0 (4) ~Q { A ~ e A z(^x6i m )^l~0 (5) LW Ll T ~y{ax{~ e y z(^x T 6I m ) ~ e A ~0 (6) LW Lx T ~{(AT^l{ ~ E T A^l)~0 (7) where (, ) and ( ˆ ) represent predicted and estimated ones respectively We readily obtain the residual vectors from equations (4) and (5) as follows ~ ey ~Q y^l (8) ~ ea ~vec ~ E A ~{Q A (^x6i m )^l (9) As a starting point one may use ^x(0) ~ A T Q { {A A T Q { y and apply the following iterative algorithm [5]: Step : h ~ Q yz ^x i { (i{) T6I m Q ^x(i{) A 6I m (0) ^l (i) ~ () y{a^x (i{) ~ I n6^l (i) T Q A ^x(i{) 6I m () { ^x(i) ~ A T Az A A T yz y (3) for i5,, Step : Repeat step until one has ^x(i) {^x (i{) ƒd (4) where d is a chosen relative error (threshold value) for the convergence As it was shown by [5], the WTLS algorithms including the above algorithm will deliver the optimal Structured TLS solution as soon as the proper Q A matrix is introduced, in agreement with the following theorem Theorem? The prefect description of the dispersion matrix Q A of the coefficient matrix A according to the Survey Review 03 VOL 45 NO 39 93

3 Mahboub et al following five rules guarantees the complete structured approach for the WTLS problem: If an element of A is repeated, one must use the 00% correlation between the two repeated elements Consequently, these two elements have a covariance equal to their unique variance If an element of A is repeated with a negative sign, one must use a 00% correlation between these two repeated elements Consequently, these two elements have a negative covariance equal to their (minus) unique variance 3 If an element of A is fixed, one must put zero in the corresponding row and column of the dispersion matrix of the coefficient matrix 4 If there are two different elements, it is evident that one puts the corresponding covariance if they are correlated, otherwise it is zero 5 Even in the homoscedastic case the above rules can be applied, one simply uses the number if an element is random otherwise 0 for the fixed element As mentioned, IRTLS is a follow-up to the IRLS that was originally introduced by [] into geodetic applications Although iterative reweighting methods are sometimes not reliable as they can provide approximate solutions in some cases, these methods can easily be applied, because they are based on the L norm or, in this research, the TLS estimator Moreover, their degree of robustness is comparable to the robust estimators, such as M-estimates, S-estimates and L-estimates, see eg [8], [9], [3] Before proposing the IRTLS algorithm, it is necessary to introduce a theorem of advanced linear algebra as follows ([4]) Theorem 3? (The special case of Schur decomposition): Let A be an n6n real symmetric matrix Then there is an orthogonal n6n matrix S and a diagonal matrix L whose diagonal elements are the eigenvalues of A, such that S T AS~L (5) S T S~I n It is to be noted that the special case of Schur decomposition which is a powerful decomposition is not the well known eigenvalue decomposition We prefer to use the special case of Schur decomposition, since the columns of orthogonal matrix S provide a basis with much better numerical properties than a set of eigenvectors which appear in the eigenvalue decomposition The following algorithm is proposed for IRTLS: Step : Perform Schur decomposition to Q A and Cholesky decomposition to Q y as follows Q A ~SLS T Q y ~G T y G (6) y where G is an upper triangular matrix from the diagonal and upper triangle of matrix Q y, satisfying the equation Q y ~G T y G y Step : Use initial values of the residuals e A and e y, and form the diagonal matrixes W A and W y (see the direction to provide such weights in Note ) Step 3: ~ { G T y W{ y G y z(^x (i{) T6I m )SW {0:5 A LW {0:5 A S T i{ ( ^x ð Þ 6I m ) (7) ^l (i) ~ y{a^x (i{) ~ I n6^l (i) T SW {0: 5 A LW{0: 5 A ^x (i) ~ A T Az { A A T yzr i (8) S T ^x(i{) 6I m (9) ðþ y (0) i~,, Step 4: The procedure should iterate from step until no further improvement of the results is possible The weight matrixes W A and W y in equations (7) (9) should be accommodated according to the residuals of each iteration given in equations (8) and (9) Therefore, the weight matrixes W A and W y in equations (7) and (9) should be replaced with W A (i) and W y (i), meaning that they are modified in each iteration Remarks Note : The weight matrixes W are of diagonal type and their diagonal elements are functions of the residuals Here, we use the Huber s M-estimator weights, which are given as [9] and [8] 8 e j >< ƒs0 k ½WŠ jj ~ s 0 k e j j~,, () >: s0 k e j (7) Typically, k is chosen as k5?5 or and s 0 is the square root of the variance component It should be noted that the variable ks ej = e j, in which sej is the standard deviation of the residual e j, should be more reasonable than ks 0 = e j in the heterogeneous case A possible estimation of s 0, based on the proposed WTLS of the previous section, is given by [5] ~ e T s 0 (i)~ y P ~ y e y z ~ e T A P ~ A e A ~ ^l (i)t R (i){ ^l (i) () m{n m{n In a forthcoming publication, a more general approach is introduced to estimate s 0 Note : In the first step, the Schur decomposition is performed to Q A instead of the Cholesky decomposition It is because the coefficient matrix A, in an EIV model, can usually have fixed or deterministic elements and consequently Q A can have columns or rows with elements all equal to zero In this case, Q A is not positive definite and hence one cannot apply the Cholesky decomposition to such a matrix Therefore, a more generalised decomposition algorithm, namely the Schur decomposition, is used Note 3: There are a few technical limitations to employ the TLS techniques to the TLS problems One should in particular be careful in the selection of an appropriate TLS technique for solving a special TLS problem In the IRTLS algorithm, the observations can be correlated when the dispersion matrix of the coefficient matrix is generated by the assumptions of Theorem? See also [6] 94 Survey Review 03 VOL 45 NO 39

4 Mahboub et al Note 4: It should be noted that in contrast to the WTLS [6] and other existing TLS approaches in the field of engineering applications, in the IRTLS approach, the dispersion matrixes Q A and Q y are updated in each iteration according to the algebraic decompositions and the weight functions in order to make the TLS approach insensitive to the grass errors similar to the traditional IRLS approach for the GM model There is no guarantee that the IRTLS approach gives the correct result in all cases because of the empirical constant k and the square root of the variance component s 0 which appear in equation () or even the general form of the weight function in equation () The constant k may take a smaller value in the TLS adjustment compared to the LS adjustment since more variables exist in the EIV model in contrast to the GM model This results in the smaller residuals in the EIV models To obtain an optimum empirical constant for the IRTLS algorithm, further research is needed Similar reasoning can be given to the general form of the weight function A possible alternative weight function is proposed as follows 8 e j ƒs0 k >< ½WŠ jj ~ s 0 k j~,, : (3)! e j >: s0 k e j Both weight functions are examined in the next section The estimation method of variance component of the IRTLS algorithm is not robust One may use a median based variance scale (see [6]) as an approximate robust estimation of the variance component In a forthcoming publication, we introduce a proper approach to achieve this goal Numerical results and discussion For verifying the IRTLS algorithm presented, two examples are given The first example is a simulated twodimensional affine transformation for which weighted data have been generated The second one is a linear regression model in which a real data set is used (homoscedastic case) Both examples contain gross errors and are used frequently in engineering surveying and geodesy Also the results on IRTLS based on the two weight functions in equations () and (3) are compared Example : two-dimensional affine transformation The issue of reference frame transformation has a very long history in the geodetic sciences including geodesy, photogrammetry, mapping, and engineering surveying Whenever a set of points is given for which coordinate estimates are available in two systems (along with their covariance matrix), the transformation parameters can be estimated on the basis of a suitable model [4] In this simulated example, the planar linear affine transformation, also known as the six-parameter transformation, will be employed as follows " # " #" #" # " x t cos b {sin b 0 sx x o ~ z c # ~ y t sin b cos b m s y y o c " #" # " a b x0 z c # a b y 0 c (4) This transformation employs six physical parameters: s x and s y for scales along the x and y axes respectively, b for rotation, c and c for the shifts along the x and y axes respectively, and a non-perpendicularity (or affinity) m between the two axes of the system to be rotated The physical parameters s x, s y, m and b can be replaced by the mathematical parameters a, a, b and b x t and y t are the transformed coordinates in the target system, while x o and y o are the original coordinates being observed in the start system In terms of the observations, equations, equation (4) will have the following structure 3 a b x t ~ x o y o c y t x o y o a (5) The multivariate model of equation (5) is defined as follows (see eg []) 3 3 x t () y t () x o () y o () 3 a a ~ b b 5(6) x t (n) y t (n) x o (n) y o (n) c c where n is the number of the identical points Unfortunately, a few authors still use the classical TLS approach to adjust such a problem where the coefficient matrix is structured or patterned Contributions that neglect this very structure such as the one by [] and several followers must hence be dismissed [] For more details see [6] Suppose that the coordinates of 3 points in the start system are transformed by the parameters a 50?9, b 50?8, c 5, a 50?6, b 50?7 and c 55 into the coordinates in the target system These coordinates are given in Table Then the coordinates of the points in the start and target systems are corrupted by white Gaussian noise but the points differ in precision Q start ~I 6Q S (7) Q target ~I 6Q T (8) where Q S and Q T are given as Q S ~0 : 005Diagð½,,3,,5,4,,7,,,8,3,6ŠÞ b c Table Point coordinates without any error in start (x o and y o ) and target system (x t and y t ) Point x o y o x t y t Survey Review 03 VOL 45 NO 39 95

5 Mahboub et al Table Estimated parameters using the WTLS and IRTLS for 0 series of simulated data without any outlier Parameters a b c a b c Average s Table 3 Estimated parameters using the WTLS for 0 series of simulated data when there exists an outlier in observations Parameters a b c a b c Average s Q T ~0 : 005Diagð½,3,6,,,8,4,3,6,5,4,5,ŠÞ In fact, we assume the same standard deviation (precision) for each coordinate of a point Finally, a bias with the magnitude of m is added to the coordinates of point 4 in the start system The results of post-adjustment shows that the bias is reduced to 5 cm for IRTLS based on the Huber weight function and to cm based on the proposed weight function (equation (0)), which shows that the IRTLS approach is less sensitive to an outlier than other existing techniques of estimation in EIV models This process has been repeated 0 times, because we are interested in obtaining reliable results Therefore, in each simulation, the white Gaussian noise generator produces different noise In each simulation, the transformation parameters have been estimated by the WTLS and IRTLS algorithms based on the two weight functions when an outlier exists and when there is no outlier The average (over 0 runs) estimated parameters by the two algorithms along with their standard deviations, with and without an outlier, are given in Tables 5 When there is no outlier, the results of IRTLS and WTLS are not significantly different from each other, and hence they are only shown once in Table The results on WTLS, IRTLS (Huber weight function) and IRTLS (proposed weight function), when there exists an outlier, are given in Tables 3 5 respectively The IRTLS approach based on the proposed weight function can make the best improvement in three latter transformation parameters, namely a, b and c To make a better validation for the accuracy of each method, in the rest of this section, we use check points showing the goodness of fit for each method in all parts of the model To examine the results of the WTLS and IRTLS algorithms based on the two weight functions, with and without an outlier, 0 check points were generated in the simulation process The vectors of residuals in each check point for all methods with the position of the point in the target system are shown in Fig Also the magnitude of these vectors (in meters) in each check point shows the goodness of fit for each method (Fig ) The IRTLS approach is more robust than the classical approach and improves the accuracy of transformation parameters well The IRTLS approach based on the proposed weight function in equation (3) made the maximum improvement of 0 cm, while the method based on the Huber s weight function in equation () made only the maximum improvement of 4 cm Figures and can interpret the results of Tables 5 and also show the efficiency of the IRTLS approach particularly when one employs the proposed weight function; however, we use another indicator to substantiate the improvement of the results which has been made by IRTLS approach particularly that based on the proposed weight function This indicator is the estimated variance component using each method as follows (see Table 6) As it is seen, the IRTLS approach particularly based on the proposed weight function has the minimum variance component among other methods We should note that the IRTLS method depends on several experimental conditions which were discussed above Therefore, although the IRTLS can mitigate the effect of an outlier (bias) to a large extent, it can still leave part of the bias to the estimated solution compared to the correct unbiased solution We also note that we could further improve the results using the proposed weight function When there is no outlier, the IRTLS and WTLS approaches give correct result Example : application to linear regression Orthogonal regression is employed whenever both X and Y coordinates are noisy In such a case, the Table 4 Estimated parameters using the IRTLS for 0 series of simulated data when there exists an outlier in observations based on Huber s weight function Parameters a b c a b c Average s Table 5 Estimated parameters using the IRTLS for 0 series of simulated data when there exists an outlier in observations based on proposed weight function Parameters a b c a b c Average s Survey Review 03 VOL 45 NO 39

6 Mahboub et al The highlighted residuals vectors in each check point using all methods with the position of control points denoted by asterisk (*) in the target system predicted residuals of TLS approach are orthogonal to the fitted line; consequently, linear regression by TLS method is also named orthogonal regression In the last two decades, scientists in the field of mathematical statistics carried out research on making this method more robust, see eg [8], [6] and [5] They have proposed robust estimators of orthogonal regression in the field of mathematical statistics such as S-estimates of orthogonal regression and M-estimates of orthogonal regression, though they are not yet applicable to the engineering sciences Therefore, to contrast our work with proper criteria, a data set is selected that has already been used by different groups of statisticians such as [8], [0] and [6] They have all employed their rigorous powerful estimators to this data set They also show that there is more than one outlier in this data set Therefore, the recent method of [9] and [0] cannot be applied to this data set Robust orthogonal regression The magnitude of residual vector (in m) for each check point using different methods Survey Review 03 VOL 45 NO 39 97

7 Mahboub et al 3 Line fitting by different estimators methods can also be used to identify multidimensional outliers in situations when classical methods are not very reliable, as, for example, when outliers occur in bunches and mask each other In fact, robust orthogonal regression can help to find projections which are interesting from the outlier detection point of view [8] [0] presented pairs of empirical measurements The measurements were obtained by two different methods Table 6 Estimated variance component using each method for transformation Method WTLS IRTLS with Huber weight function IRTLS with the proposed weight function s Table 7 Twenty pairs of empirical data presented in [0] Point No X Y Point No X Y for the x coordinate and y coordinates The 0 pairs of this data set are presented in Table 7 The assumption that both measurements are subject to random errors with equal variances seems reasonable [8] The M-estimates and S-estimates of orthogonal regression and classical orthogonal regression were reported by [6] The results of the M and S estimates of orthogonal regression, the results of classical orthogonal regression, the results of WTLS solution and the results of simple IRTLS algorithm based on the two weight functions are given in Table 8 and shown in Fig 3 As it is seen in Fig 3 and Table 8, the IRTLS approach improved the results of classical orthogonal regression and WTLS approach very well, since the results of the proposed ways particularly based on the proposed weight function are close to the results of M and S estimates of orthogonal regression which show that the proposed methods, particularly that based on the proposed weight function, are close to the maximum likelihood estimation In addition, it is more applicable than the statistics robust estimators, since it is based on the L norm minimisation principle There is, however, still a bias in the IRTLS solution, which is equal to? in the intercept and 0? in the slope which can create a significant bias by increasing the value of the coordinates One of the reasons of the bias of the IRTLS approach is because of the experimental weight functions used in the IRTLS algorithm (the section on Iterativly reweighted total least squares ) This leads us to reduce this bias to 0?94 in the intercept and 0?05 in the slope based on the proposed weight function in equation (3) Table 8 Estimated lines parameters by different estimation methods Estimation TLS WTLS IRTLS (based on Huber s weight function) IRTLS (based on proposed weight function) S estimates of orthogonal regression M estimates of orthogonal regression Intercept Slope Survey Review 03 VOL 45 NO 39

8 Mahboub et al Table 9 Estimated variance component using each method for transformation Method WTLS IRTLS with Huber s weight function IRTLS with the proposed weight function s Another statistical indicator is the estimated variance component using each method as follows (see Table 9) As it is seen, the IRTLS approach particularly based on the proposed weight function has the minimum variance component among other methods Conclusions In this research, IRTLS has been introduced as a robust estimation method in the EIV models This method is based on the algorithm of WTLS according to the traditional Lagrange approach to optimise the target function of this problem It can be easily implemented since it is based on the L norm minimisation principle, and consequently no linear programming scheme is required Moreover, as shown in the numerical example, in some cases the IRTLS considerably improves the accuracy of the classical TLS and WTLS approach when a few outliers exist in the data This IRTLS algorithm can be applied to several applications There is, however, still a bias in the IRTLS solution As it was discussed in the section on, the nature and source of this bias is likely due to the empirical constant k and the square root of the variance component s 0 appeared in equation () or even the general form of the weight function in equation () This led us to improve the results using a different weight function as an alternative for the Huber s weight function Further work is likely required to come up with a weight function that gives the most accurate solution Our final remark regards the necessity of introducing the statistics robust estimators of the EIV models into the engineering applications, which is currently in progress for future work References Akyilmaz, O, 007 Total least-squares solution of coordinate transformation Survey Review, 39: Amiri-Simkooei, A R, 003 Formulation of L norm minimization in Gauss Markov models Journal of Surveying Engineering, 9(): Baselga, B, 007 Global optimization solution of robust estimation Journal of Surveying Engineering, 33(3): Box, G E P, 953 Non-normality and tests on variances Biometrika, 40: Brown, M, 98 Robust line estimation with errors in both variables Journal of the American Statistical Association, 77: Fekri, M, Ruiz-Gazen, A, 004 Robust weighted orthogonal regression in the errors-in-variables model Journal of Multivariate Analysis, 88: Golub, G and van Loan, C, (980) An analysis of the total least squares problem SIAM Journal on Numerical Analysis, 7: Hekimoglu, S and Berber, M, 003 Effectiveness of robust methods in heterogeneous linear models Journal of Geodesy, 76: Huber, P J, 98 Robust statistics Wiley, New York 0 Kelly, G, 984 The influence function in the errors in variables problem The Annals of Statistics, : Koch, K R, 999 Parameter estimation and hypothesis testing in linear models Springer, Berlin Krarup, T, Juhl, J and Kubik, K, 980 Götterdämmerung over least squares adjustment Proceedings of 4th Congress of the International Society of Photogrammetry, B3: Launer, R L and Wilkinson, G N (eds), 979 Robustness in statistics Academic Press, New York 4 Magnus, J R, 988 Linear structures London School of Economics and Political Science, Charles Griffin and Company LTD, Oxford University Press, London 5 Mahboub, V, 0 On weighted total least-squares for geodetic transformations Journal of Geodesy, 86: Mahboub, V, 0 Discussion of An improved weighted total least squares method with applications in linear fitting and coordinate transformation by Xiaohua Tong; 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