Application Study On The Total Least Squares Method In Multiple Linear Regression Under Different Error Influence Models

A large number of function models are regression models in the field of surveying and mapping, and the linear regression model is the most commonly used model. The main task of linear regression problem is to calculate the regression coefficients and the most commonly used method is the classical least squares(LS) method. However, the LS method only takes the errors of observation vector into account, and errors of the error of the coefficient matrix of error equation of linear regression model will not be considered or be ignored because of anthropogenic causes, which will lead to the adjustment results of the LS method inaccurate. The total least squares(TLS) method is proposed to solve the problem that the errors existing in the observation vector and the coefficient matrix can not be considered simultaneously. With the development and deepening of the total least squares theory, it has been widely used in power systems, medicine, biology, statistics, graphics and other fields. At the same time, it also has received more and more scholars’ attention in the surveying and mapping disciplines.In view of the condition of both the observation vector and the coefficient matrix containing random errors, at present many scholars mainly studied the unitary linear regression model using the TLS method, and concluded that TLS method had smaller mean square errors of unit weight and adjustment values closed to the true values than the LS method in some special cases. However, the researches on the TLS method in multiple linear regression are still rarely, and it is unknown whether the similar conclusions can be obtained in the multiple linear regression models. Therefore, it is necessary to make further efforts to study the TLS method in multiple linear regression, and then can provide the basis for expansion of the scope of the application of the TLS method.Based on the the existing research results of the TLS algorithm, this paper studies the application of the TLS method in the multiple linear regressions. The difference of three kinds of error influence models are analyzed: the first model is EIV(Errors-in-variables) model(i.e. the model of the observation vector and the coefficient matrix contain random errors simultaneously); the second model is EIVO(Errors-in-variables-only) model(i.e. the model of the coefficient matrix contains random errors only); the third model is EIOO(Errors-in-observations-only) model(i.e. the model of the observation vector contains the random errors). At the same time, this paper takes numerical examples of the 1~5 linear regressions to illustrate the difference of the three kinds of errors influence models. In addition, this paper takes 1~5 linear regressions as examples which contain different numbers of observations and different error distributions, discuss the relative effectiveness of the TLS method and LS method, and then determine the more relatively effective parameter estimation method in the application of the multiple linear regressions.The study of this paper indicates that the adjustment results of the TLS method are not stable, and there universally exists estimated values drift(EVD) phenomenon in the simulation experiments. However, the adjustment results of the LS method are relatively stable and generally does not exist EVD phenomenon. From the statistical perspective, the numbers of occurrences of the TLS method’ estimated values drift are more than the LS method’s, and the degree of the EVD is greater than the LS method’s. Overall, for the three kinds of error models EIV, EIVO and EIOO, the TLS method’s mean square residual true errors(MSRTE) are greater than the LS method’s. The above conclusions suggest that the TLS method is the relatively less effective parameter estimation method compared with the LS method. Relatively speaking, the LS method’s results are more reliable than the TLS method’s results in 1~5 linear regression. It is necessary to take the EVD of the TLS method into account in application. Otherwise, it may cause unpredictable consequences.