Data-snooping procedure applied to errors-in-variables models

Objectives To solve the problem that the coefficient matrix and the observation vector have structured characteristics for many practical applications in the field of surveying and mapping, that is, the coefficient matrix and the observation vector contain fixed quantities (or even fixed columns) and random quantities, and the random quantities at different positions are linearly related. Methods Starting from the errors-in-variables (EIV) function model, the augmented matrix composed of the coefficient matrix and the observation vector is expressed as an affine function form, and the function model is reconstructed by the variable projection method. Then, a structured total least squares (STLS) estimation algorithm is derived by the Lagrange method. Results The example results show that the proposed algorithm is consistent with the existing weighted or structured total least squares estimation algorithms that can solve structured problem in the coefficient matrix and the observation vector. Compared with the weighted total least squares (WTLS) estimation algorithms, the proposed algorithm only needs to establish a positive definite weight matrix of independent random variables and reduces the number of estimates. Compared with other STLS estimation algorithms, the proposed algorithm takes into account the overall structure of the coefficient matrix and the observation vector. Conclusions This shows the effectiveness of the proposed algorithm. The STLS estimation and the WTLS estimation ensure the statistical optimality from the perspective of the function model and the stochastic model, respectively.

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.

Although the analytical solutions for total least-squares with multiple linear and single quadratic constraints were developed quite recently in different geodetic publications, these methods are restricted in number and type of constraints, and currently their computational efficiency and applications are mostly unknown. In this contribution, it is shown how the weighted total least-squares (WTLS) problem with arbitrary applicable constraints can be solved based on a Newton type methodology. This iterative process with quadratic convergence is expanded upon to become a compact solution for the WTLS with or without constraints. This compact solution is then further interpreted as a universal formula for the symmetrical adjustment of the errors-in-variables model which represents affine, similarity and rigid transformations in two- and three-dimensional space. Furthermore, statistical analysis of the constrained WTLS including the first-order approximation of precision and the bias was investigated. In order to substantiate our proposed method’s applicability, it was used to solve the affine, similarity and rigid transformation problem in two- and three-dimensional cases, where the structure of the coefficient matrix and multiple constraints were taken into account simultaneously.

A novel interference alignment (IA) scheme based on the errors-in-variables (EIV) mathematic model has been proposed to overcome the channel state information (CSI) estimation error for the MIMO interference channels. By solving an equivalently unconstrained optimization problem, the proposed IA scheme employing a weighted total least squares (WTLS) algorithm can obtain the solution to a constrained optimization problem for transmit precoding (TPC) matrices and minimizes the distortion caused by imperfect CSI according to the EIV model. It is shown that the design of TPC matrices can be realized through an efficient iterative algorithm. The convergence of the proposed scheme is presented as well. Simulation results show that the proposed IA scheme is robust over MIMO interference channels with imperfect CSI, which yields significantly better sum rate performance than the existing IA schemes such as distributed iterative IA, maximum signal-to-interference-plus-noise ratio (Max SINR), and minimum mean square error (MMSE) schemes.

The weighted total least-squares (WTLS) estimate for the partial EIV (PEIV) model is very susceptible to outliers. In view of outliers in observations and coefficient matrix, a robust estimate of WTLS which makes using of iteratively reweighted technique (IRTLS) for the PEIV model is developed by combining a new two-step iterated algorithm of the WTLS estimate with the sensitivity-analysis based on the robust M-estimation. The uniformly most powerful test statistics are constructed to determine the down-weighting factor and the variance of unit weight is estimated by least median squares (LMS) method possessing high break-down point. Depended on the PEIV model, two different down-weighting schemes are presented. In the first scheme down-weighting is only implemented for the coefficient matrix and not for observations when the elements of the coefficient matrix are estimated, and the second scheme is contrary. The most attractive aspect is that the suggested computational formulae are the same with the traditional robust least-squares (LS) methods. The two-dimensional affine transformation and liner fitting example are analyzed, and some comparisons are performed with different available weight functions. The proposed approach (Scheme 1) proves to be a powerful tool in detecting outliers for the PEIV model.

The registration of 3-D point clouds is an important procedure during the terrestrial laser scanning data processing. Recently, due to their high flexibility and the powerful mathematical model, a large amount of least-squares-based (LSs-based) methods are proposed and widely applied to estimate the transformation parameters of 3-D point clouds registration. In these LSs-based methods some based on the generalized Gauss–Markov model do not correct the influence of random errors on source 3-D point clouds. Although there are other methods based on the errors-in-variables (EIV) model, they are inapplicable for transformation problems with large rotation angles and arbitrary scale ratio. In addition, the gross errors are usually ignored in previous studies on 3-D point clouds registration, which, however, exists commonly and could distort the registration severely. Aiming to avoid the influence of gross errors and extend its application, an advanced outlier detected total least-squares (OD-TLS) method is proposed in this paper. Based on the generalized EIV model OD-TLS performs a seven-parameter 3-D similarity transformation with large rotation angles and arbitrary scale ratio. The random errors of both source and target 3-D point clouds are considered. Furthermore, outliers are detected and removed automatically by combining the data snooping method with total least-squares (TLS) estimation. In order to indicate the benefits of OD-TLS, comparative experiments with the LS3D and weighted total least squares (WTLS) on synthetic and real-world scanned 3-D point clouds were performed. The experimental results show OD-TLS not only enhances the registration accuracy but also increases its robustness.

An improved mixed total least squares method for strain inversion from distance changes

Targeting the adjustment of the errors-in-variables (EIV) model with equality and inequality constraints, a general solution that is similar to the classical least square adjustment is proposed based on the penalty function and the weight in measurement. Firstly, we take the equality constraints as inequality constraints that do not satisfy the constraint conditions and construct the penalty functions of equality and inequality constraints, respectively. Thus, the inequality constrained optimization problem is transformed into an unconstrained optimization problem. Then the detailed calculation formula and approximate accuracy evaluation formula of the general solution are deduced. The iteration formula of the general solution is easy regarding comprehension and applicable in implementation. It can not only solve the EIV model with equality and inequality constraints respectively, but also address the EIV model with equality and inequality constraints simultaneously. In addition, it can promote the Gauss–Markov (G-M) model with equality and inequality constraints. Finally, three examples (i.e., equality constraints, inequality constraints and those with equality and inequality constraints) are validated, indicating that the general solution is effective and feasible. The results show that the general solution is effective and feasible.

In this study, the observation model of Kalman Filter (KF) is extended to an errors-in-variables (EIV) model because the observations may exist in the design matrix of the observation model. Then, a robust total least squares method (RTLS) is introduced into the KF, and a robust total Kalman filter (RTKF) algorithm is derived. The RTKF is a simple, flexible and effective algorithm. It is simple because its computational formulae are similar to the computational formulae of a standard KF; it is flexible because it can be used in a wide range of applications; it is effective because the influence of outliers on estimated results is weakened. Finally, the simulated example of the indoor location and the empirical example of pseudorange differential positioning are used to demonstrate the performance of the RTKF algorithm. The results prove the validity, robustness, and reliability of the RTKF in dealing with the outliers that exist in both observation vector and design matrix of the EIV model. Furthermore, the results of the empirical example show that the RTKF improves the precision of a pseudorange differential positioning compared with KF and robust Kalman filter (RKF) algorithms regardless the observation model has outliers or not in this empirical example.

This paper addresses the identification of multiple-input single-output (MISO) errors-in-variables (EIV) models, in which all input and output measurements are corrupted by white noise with unknown variances. To achieve consistent parameter estimation while reducing the need to estimate all unknown noise variances, a bias compensation principle based method called the bias compensated weighted least squares (BCWLS) method was previously proposed for singleinput single-output (SISO) EIV models. The present paper extends the BCWLS method, originally developed for the SISO case, to the MISO case. Simulation results demonstrate the effectiveness of the proposed recursive algorithm.

Total least squares (TLS) is a technique that solves the traditional least squares (LS) problem for an errors-in-variables (EIV) model, in which both the observation vector and the design matrix are contaminated by random errors. Four- and seven-parameter models of coordinate transformation are typical EIV model. To determine which one of TLS and LS is more effective, taking the four- and seven-parameter models of Global Navigation Satellite System (GNSS) coordinate transformation with different coincidence pointsas examples, the relative effectiveness of the two methods was compared through simulation experiments. The results showed that in the EIV model, the errors-in-variables-only (EIVO) model and the errors-in-observations-only (EIOO) model, TLS is slightly inferior to LS in the four-parameter model coordinate transformation, and TLS is equivalent to LS in the seven-parameter model coordinate transformation. Consequently, in the four- and seven-parameter model coordinate transformation, TLS has no obvious advantage over LS.

In linear regression model, regressors are assumed fixed in repeated sampling. This assumption is not always satisfied especially in business, economics and social sciences. Consequently in this paper, effort is made to compare the performances of some estimators of linear model with autocorrelated error terms when normally distributed regressors are fixed (non – stochastic) with when they are stochastic. The estimators are the ordinary least square (OLS) estimator and four feasible generalized least estimators which are Cochrane Orcutt (CORC), Hidreth – Lu (HILU), Maximum Likelihood (ML), Maximum Likelihood Grid (MLGD) estimator. These estimators are compared using the finite properties of estimators' criteria namely; sum of biases, sum of variances and sum of the mean squared error of the estimated parameter of the model at different levels of autocorrelation and sample size through Monte – Carlo studies. Results show that at each level of autocorrelation the estimated value of the criteria with stochastic regressor are much lesser than that of the fixed regressor for all the estimators except CORC when the sample size is small (n=20) and the level of autocorrelation is very high . More comparatively, it is observed that the same estimator(s) that is more efficient with fixed regressors is also more efficient with stochastic regressors except when the sample size is large (n = 80) and the level of autocorrelation is either low or high . At these instances, the CORC / HILU estimator is more efficient with fixed regressors while the ML / MLGD estimator is more efficient with stochastic regressors Keywords: Fixed Regressors, Stochastic Regressors, Linear Model, Autocorrelated error, OLS estimator, Feasible GLS estimatorsGlobal Journal of Pure and Applied Sciences Vol. 14 (3) 2008: pp. 363-370

Recognition that measurement error in the independent (predictor) variable of data used for regression modeling necessitates special methods of model fitting dates back to the late 1800s. In the ensuing 100 years attention was devoted almost entirely to measurement error in linear regression models, at which time advances in computing opened the door to the nonlinear statistical modeling generally, and to the study of measurement error in nonlinear regression modeling in particular. This article contains an introduction to the problem of measurement error in regression using linear regression measurement error models to illustrate key ideas. It includes a discussion of the types of measurement error commonly studied (classical and Berkson, differential and nondifferential) and how those types arise in practice. A summary of the consequences of ignoring measurement error when analyzing data measured with error is presented, with detailed results given for simple and multiple linear measurement error models. Pointers to the literature on both linear and nonlinear measurement error models are provided.

Analyses of variables measured with error are often flawed when measurement error is ignored. An extensive literature on linear measurement error models is available by [1], and more recently, for non-linear measurement error models by [2] and [3]. We investigate problems encountered in the analysis of longitudinal change, where the difference in some outcome variable is modeled on a set of regressor variables in a linear regression model. This type of analysis has become quite popular in biomedical research, and the results are often biased when the outcome variable is measured with error and its observed baseline value is included as a covariate in the fitted model. [4] demonstrated the effects of measurement error in the analysis of change in wall thickness of the common carotid artery. They showed the naive analysis led to erroneous findings due to the measurement error bias, even when the regressor variables were assumed to be measured precisely. In this paper, we present a method-of-moments correction for measurement error bias, provided the measurement error variance is known or can be estimated. This work extends the work of [4] to include regressor variables that are measured with error.

This paper considers structural errors-in-variable regression models with replicated observations. Some classical estimators, for example, least squares estimators are not consistent in general under such models. Modifying moment estimation and rank estimation, some estimators in linear structural errors-in-variable models have been constructed. For these estimators, strong convergence properties and asymptotical normality have been obtained.