Research On Jackknife Method For Total Least Square Adjustment

In the field of geodetic data processing,how to improve the theory of total least squares(TLS)adjustment is a valuable issue which is worthy of study.The total least squares adjustment mainly includes two parts: parameter estimation and precision estimation.In order to obtain more accurate parameter estimation and more reasonable precision information,this paper introduces the Jackknife method,which aims to further develop and extend the parameter estimation and precision estimation of total least squares theory by sampling the observation data.The detailed researches are as follows:The Jackknife resampling parameter estimation method for weighted total least squares(WTLS)is researched.In order to make the evaluation results of weighted total least squares parameters more accurate,this paper introduces the Jackknife method to sample the observation data and make full use of the sample for multiple calculations.Based on the algorithm of error-in-variables(EIV)model,we combine it with Jackknife-1 and Jackknife-d.The calculation methods of Jackknife-1-WTLS and Jackknife-d-WTLS are proposed,and the value of d is further studied.Meanwhile,the two methods are applied to the linear regression model and the planar coordinate transformation model.The results show that the proposed methods of Jackknife of weighted total least squares are more effective than least squares(LS),weighted total least squares and least squares resampling methods in improving the quality of parameter estimation,which verifies the validity and feasibility of the methods.The Jackknife method for precision estimation of weighted total least squares is researched.Few studies have been conducted on the precision estimation of weighted total least squares by using the approximate function probability distribution method.And the existing Monte Carlo(MC)method needs to simulate a lot and need to know the priori information of observation.In order to further improve the total least squares precision estimation theory,this paper bases on the partial error-in-variables(Partial EIV)model,the delete-1 Jackknife method and delete-d Jackknife method are proposed.The biases and standard deviations or covariance of parameter estimations are calculated by these proposed methods.And the specific steps of the precision estimation of these two methods are given.Applying these methods to the linear regression model and the coordinate transformation model,and comparing with other methods,we can see that the Jackknife methods for precision estimation can obtain more stable and reasonable precision results and are very adaptive.When the amount of the observed data is large.it can reduce the amount of calculation and improve the computational efficiency.The method in this paper could provide an idea for further study on the precision estimation for total least squares.The Jackknife estimation method for the variance component of Partial EIV model is researched.In order to further improve the quality of parameter estimation based on the variance component estimation,this paper introduces the Jackknife method into the variance component estimation based on the Partial EIV model.On the basis of this,the bias of the variance component parameter estimation is considered.The bias of the parameter estimation is corrected which is calculated by the Jackknife method.Two schemes for estimating parameters are given.The proposed methods are applied to three examples.The results all show that both methods can obtain more accurate parameter estimation results than the variance component estimation.The method of bias correction can obtain the optimal parameter estimation,which can further effectively improve the quality of parameter estimation.A new Partial EIV model algorithm for big rotation angle's 3D coordinate transformation model is proposed and the Jackknife parameter estimation method is applied to the example of big rotation angle's 3D coordinate transformation.For the case where the observation vector and the coefficient matrix of the coordinate transformation model contain random errors,the Partial EIV model of the big rotation angle's 3D coordinate transformation is derived after the Taylor series expansion is performed on the seven-parameter approximation.A total least squares calculation method based on Partial EIV model for big rotation angle's 3D coordinate transformation is proposed,which solves the problem that it is difficult to design the weight matrix when the complex coefficient matrix contains random errors.Meanwhile,the Jackknife parameter estimation is applied to the more complex big rotation angle's 3D coordinate transformation,and the steps for calculating the seven parameters are given.The feasibility and applicability of the total least squares algorithm based on the Partial EIV model of big rotation angle's 3D coordinate transformation are verified by the example.The seven-parameter result obtained by the Jackknife method is slightly more accurate than the result of the Partial EIV model algorithm.The result of the error of unit weight is smaller than the total least squares method and is significantly closer to the prior error of unit weight,and the precision is higher.