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When the total least squares (TLS) solution is used to solve the parameters in the errors-in-variables (EIV) model, the obtained parameter estimations will be unreliable in the observations containing systematic errors. To solve this problem, we propose to add the nonparametric part (systematic errors) to the partial EIV model, and build the partial EIV model to weaken the influence of systematic errors. Then, having rewritten the model as a nonlinear model, we derive the formula of parameter estimations based on the penalized total least squares criterion. Furthermore, based on the second-order approximation method of precision estimation, we derive the second-order bias and covariance of parameter estimations and calculate the mean square error (MSE). Aiming at the selection of the smoothing factor, we propose to use the U curve method. The experiments show that the proposed method can mitigate the influence of systematic errors to a certain extent compared with the traditional method and get more reliable parameter estimations and its precision information, which validates the feasibility and effectiveness of the proposed method.

The observation errors of independent variables in model prediction are typically neglected in traditional regression models, which leads to the decreased prediction accuracy. In this paper, a new regression prediction method that is based on the partial errors-in-variables (partial EIV) model is proposed, which takes into account the observation errors of all variables. It observes that each element of the coefficient matrix is an expression or a function. The partial EIV model is transformed into a linear model and constructed in the form of indirect adjustment for iterative solution. By considering the errors in the independent variables when predicting the corresponding dependent variables with the partial EIV model, the proposed method can achieve higher prediction accuracy. [ABSTRACT FROM AUTHOR]

The partial errors-in-variables (PEIV) model is an structured errors-in-variables (SEIV) model reformulated by collecting all the random elements of the coefficient matrix into an auxiliary random error vector to better treat the case where the coefficient matrix (or 'data matrix') A and observation vector y share common elements. In the usual PEIV model, the case of A and y sharing common elements has not been treated explicitly. However, in some cases, A and y do have common elements, taking the autoregressive problem as an example. To apply the PEIV model to SEIV problems with correlated matrix A and vector y, we propose an extended model named general PEIV (GPEIV). We also provide a corresponding algorithm for the total least-squares solution within the GPEIV model. A weight-matrix construction method, which is suitable for PEIV and GPEIV models, is also presented. The GPEIV model, the algorithm and the weight-matrix construction method are tested using two groups of numerical examples. The results show that the proposed model and corresponding algorithm can solve both uncorrelated and correlated SEIV problems, and the weight-matrix construction method is simple and efficient. [ABSTRACT FROM AUTHOR]

Due to great computation burden of the original computational formula for partial errors-in-variables (EIV), an alternative formula for parameter estimation in the partial EIV model was proposed to promote computational efficiency. Unfortunately, more iteration is required for obtaining the weighted total least-squares (WTLS) solution of the partial EIV model with above two formulae. Consequently, those methods are not proper for larger data problem in terms of computational efficiency. To circumvent this difficulty, a new formula and the fast algorithm are proposed based on the Lagrange extreme value method. Through some numerical examples with coordinate transformation, it is illustrated that the new developed method is an effective strategy to conduct the WTLS adjustment for the partial EIV model. The computational efficiency is significantly improved at least 80%. [ABSTRACT FROM AUTHOR]

The computation of weighted total least squares (WTLS) estimate of partial EIV model requires more iterations and more computation burden. Therefore, the study has proposed a new algorithm for computing the WTLS estimate of partial EIV model by improving objective function based on the weighted LS principle and applying the differential and inversion transformation of matrix. The results of numerical examples show that the new algorithm requires less iterations and more superior in the sense of computational efficiency. [ABSTRACT FROM AUTHOR]

The purpose of this note is to derive alternative formulae for parameter estimation in partial errors-in-variables models to those given in Xu et al. (J Geod 86:661-675, ). We show that these new formulae are more compact and more straightforward to interpret the corrections of the random elements of the design matrix. Nevertheless, each formula has its own computational advantage, depending on the relationship between the number of measurements y and that of the independent random elements of the design matrix. The original formula is computationally more efficient, if the number of measurements y is significantly larger than that of the independent random elements of the design matrix; otherwise, the new alternative formula is much more efficient. [ABSTRACT FROM AUTHOR]