A Robust Weighted Total Least Squares Method

For many applications in geomatics,such as linear regression,coordinate transformation and GPS elevation fitting,parameter estimation is an essential means.Conventionally,least-squares(LS)method is the solution,which only considers the errors in the observation vector.In order to take into consideration the possible errors in the coefficient matrix,total least-squares(TLS)method has been developed in the recent years.Weighted total least-squares(WTLS)methods have also been developed with unequal precision observations.In particular,WTLS method based on Gauss-Helmert model,WTLS method based on Newton-Gauss approach and WTLS method with Lagrange multipliers were proposed.However,WTLS method does not take into consideration the possible gross errors in observations,which may lead to a reduction in the robustness and reliability of parameter estimations.In order to solve this problem,in this dissertation,Lagrange multipliers(LM)are employed to make WTLS method rigorous and the IGG(Institute of Geodesy and Geophysics)weight function is employed to make WTLS method more robust and reliable,resulting in a robust weighted total least-squares with Lagrange multipliers and the IGG weight function(RWTLS-LM-IGG)method.Two measures(i.e.the variance component and the mean square error)are adopted to quantitatively evaluate the performance of the proposed method.A comparative experiment with WTLS and existing robust WTLS(i.e.robust weighted total least-squares with Gauss-Helmert model and Huber weight function)methods is conducted with simulation data sets(with different numbers and magnitudes of gross errors)and three sets of real-life data(i.e.linear regression,coordinate transformation and GPS height fitting).The results of simulation datasets experiments show that the variance component and the mean square error obtained by using WTLS and existing robust WTLS methods increase almost linearly with an increase in the numbers and magnitudes of gross errors,but these values obtained by using the proposed method are almost stable,which means an effective reduction of the influence of the gross errors by the proposed method as compared with WTLS and existing robust WTLS methods.It is also found that the larger the numbers and magnitudes of gross errors,the more obvious such a reduction.Furthermore,the experiment results with three sets of real-life data are consistent with the results of simulation datasets experiments.The autoregressive model for time series prediction is a common method in settlement prediction.In the traditional parameter estimation of autoregressive model,LS method is the solution,which only considers the errors in the observation vector.In order to consider the errors in the coefficient matrix and the possible gross errors in observations,in this dissertation,RWTLS(i.e.RWTLS-LM-IGG)method is proposed to estimate parameters of autoregressive model for bridge pier settlement prediction in high-speed railway.A comparison with LS,robust LS(RLS)and WTLS methods is conducted and two sets of observed data are used in this evaluation.The results of experiments show that the variance components and the mean absolute values of predictive residuals obtained by WTLS and RWTLS methods are smaller than those by using LS and RLS methods in the case of modeling data without gross errors,and the variance component and the mean absolute value of predictive residuals obtained by RWTLS method is the smallest in the case of modeling data with gross errors.It shows that autoregressive model settlement prediction for bridge pier by using RWTLS method is more reliable and accurate than LS,RLS and WTLS methods in high-speed railway.The linear regression classification is a fairly simply and but efficient method in image recognition.In the general parameter estimation of linear regression classification model,LS is adopted.However,remote sensing images may contain the gaussian and salt and pepper noise,and the grey values of corresponding position of different remote images are not the same absolutely in the same scene,which may lead to a reduction in the classification accuracy of scene classification for remote sensing images.In order to solve this problem,in this dissertation,RWTLS(i.e.RWTLS-LM-IGG)method is proposed to estimate parameters of linear regression classification model.A comparison with LS,RLS and WTLS methods is conducted and three sets of scene data of remote sensing images are used in this evaluation.It shows that the RLS and RWTLS methods is able to better detect and resist the bigger Gaussian noise and the salt and pepper noise so that the higher classification accuracies can be acquired than those of the LS and WTLS methods.At the same time,RLS and RWTLS methods can also reduce the influence of the bigger random errors or gross errors because of the difference of remote sensing images in the same scene.More importantly,the RWTLS method can handle the errors in observation vector and coefficient matrix,and effectively reduce the influence of gross errors so that the higher classification accuracy becomes feasible.