Two Algorithms with High Breakdown Points Applied in Linear Regression EIV Model

Objectives: Linear regression model is a basic model in the field of geodesy. To consider the structure of the coefficient matrix with the fixed column, the mixed least squares and total least squares method (mixed LS-TLS) is implemented. However, it is easily contaminated by outliers. The M-estimator results depend on the initial value and are extremely prone to convergence badly. To increase the robustness, we propose two algorithms with high breakdown points for linear regression EIV models, namely, the weighted total least median of squares (WTLMS) method and the weighted total least trimmed squares (WTLTS) method. Methods: The two algorithms are extensions of traditional algorithms and use a more general stochastic model. Their breakdown points are near 50% and the two algorithms have two equivariant properties:scale equivariance and affine equivariance. The estimation formula of variance components is given. Since their objective functions are not differentiable, WTLMS and WTLTS get the solutions by the resampling algorithm and the feasible set algorithm in the EIV model respectively. Results: The results show that:(1) The result of the M-estimator is biased heavily from the real line, while the two proposed algorithms in this paper can obtain results close to the true value. Their performances are significantly better than M-estimator in terms of root mean squares errors (RMSE) and standard deviation (STD). The efficiency of the two algorithms is not high, which can be further improved when the results of the two algorithms are used as the initial value of the M-estimator. The breakdown points of the two algorithms are close to 50% in the real data, which is extremely robust. (2) In the experiment of the lidar data, the performance of the proposed methods is better than the M-estimator. Conclusions: The two proposed algorithms have outstanding robustness, but their complexities are high and their efficiency is not ideal. We will focus to find an easy solution with higher efficiency.