Research On The Total Least Squares Based On Gray Theory And Its Applications In Subgrade Settlement Prediction

With the development of economy,society and science and technology in our country,the state has placed higher demands on the transportation.Subgrade settlement observation is an important part of deformation observation of traffic engineering.With the popularization of nonballasted-track technology,the national requirements for subsidence prediction accuracy become higher and higher.Effected by the observation conditions,engineering status and unexpected environment factors,subsidence deformation observation data are usually small samples and non-equidistant time series.Gray theory is used to reveal the evolution of things under the background of few information,and has unique advantages for data processing.Subgrade settlement followed the early sinking rate is fast,then gradually slow down,and finally tends to the stability of the law of change.The linear change of gray albino differential equation obeys the law of subsidence,therefore,gray theory is widely used in subsidence deformation prediction.The gray theory albino differential equation solves the classical Gauss-Markov model in the condition of Least Squares(LS).However,when the observed data contains errors,the coefficient matrix formed by adding the observed data also includes the error.If LS solution is used again,the unknown parameter to be solved is biased.The total least squares(TLS:Total Least Squares)algorithm takes account of the error of the observation matrix and the coefficient matrix.Considering the overall error,TLS solution is proved to be asymptotically unbiased.In this paper,we mainly study the Least Squares solution method,Total Least Squers solution method and its extension algorithm of gray theory,and apply the related research results to the subgrade settlement prediction.By comparing the common subgrade prediction models,the advantages and disadvantages of each algorithm are systematically analyzed.The main contents are as follows: 1)The GM(1,1)and Gray Verhulst least square method are deduced in detail.According to the fact that the survey data may contain gross errors,the gray theory is combined with the resistance theory to form a gray-scale robust estimation.This paper also deduce the least squares and total least squares estimates of Multi-points grey theory.2)The estimation method of gray theory of total least square is deduced in detail,including singular value decomposition solution and Lagrange approximation solution.Aiming at the condition that the coefficient matrix of gray albino differential equation contains constants columns and the observation weights may be different,GM(1,1)weighted total least square method is proposed.Based on the existence of pathological problems in solving equations,the pathological solutions of the GM(1,1)albino differential equations are researched in this paper.3)Gray theory and total least squares theory are extended correspondingly in this paper,the gray theory is combined with the resistance theory,and the total least square method is used to estimate,and then,a gray by using robust total least squares method is proposed.In view of the trend of saturated settlement of high-speed railway subgrade settlement,gray Verhulst model using total least squares estimation is proposed.Because the total least squares estimation will increase the number of model parameters greatly,a Gray Verhulst model solution which takes into account the structure of the coefficient matrix is proposed.The computational efficiency and prediction accuracy are also researched on this method.4)According to the actual subgrade settlement monitoring project,observation data is a small sample,non-equidistant situation,thus,this paper proposed a non-equidistant equidistant treatment method by using GM(1,1)move least squares method considering the law of settlement variation.The gray robust total least squares estimation and the gray Verhualst model least squares method which takes the coefficient matrix structure into account are applied to the subgrade settlement test.Through the analysis of settlement monitoring of high-speed railway subsidence,the effectiveness and feasibility of the proposed algorithms in the settlement prediction are verified.