On The Research And Discussion Of Nonlinear Estimation Method

The classical adjustment theory, which has a set of complete methods and relative methods based on the Least squares principle, is built on the basis that the errors are normally distributed and the function model is linear, For a long time, the least squares method is continuously at the prominent position in various adjustments, because it was mainly the limitation of calculate tools that restricted other methods to further development at that time.Along with the arrival of the Computer Age, the matrix algebra, the functional analysis, the optimized theory and the probability statistics are widely applied one after another to the survey data processing and the analysis, which causes the appearance of the new adjustment method to be possible. Besides, society's continuous progress, also need highly accurate survey data to serve the economic development. In order to continue to enrich and consummate the adjustment theory system, this thesis starts with the rise of the nonlinear estimation theory, the basic thought and its the present situation of the research, analyzes and summarizes the research present situation of the domestic and foreign nonlinear estimation method in detail, based on the review of the adjustment basic theory, and obtains the main reason presently which restricted the further development of the nonlinear estimation theory lies in:(1) In mathematics, nonlinear theory has not formed the integrative theory system yet;(2) After taking into consideration of high step Taylor, the parameter estimation launched by the weak nonlinear model is extremely complex, it is not certainly suitable for the concrete project;(3) The strong nonlinear model parameter estimation and the error propagation theory is still a blank, which causes the nonlinear estimation theory research to be unchangeable.In view of these questions, this thesis discusses the general theory of nonlinear M estimator in detail, summarizes its main achievement and the applied aspect, including the iteration algorithm, the pollution distribution, the influence function and the gradually advanced variance;This thesis discusses the definition of M estimation and its existence conditions, relaxes the M estimation condition required by the statistician, and permits the objective function to have many extreme points as well as the rank-deficient of the design matrix, which makes "the Least Squares which have probability density weight" and "the maximum sum likelihood estimation" form the unification relations with the M estimation. The solution formula of the M-estimation in the rank-deficient network is proved to be the same as the Least Squares estimation, the M method is studied which with errors density are normally distributed,the algorithm based on the modification accumulation function and its robustified algorithm is studied.Parameter estimation based on the general M estimation, the residual errors as well as the estimation of observations all is a complex nonlinear function of observations.It is difficult to derive their variance-covariance matrix.And the linear expression of residual error and observations as well as the unnecessary parameter of the non-linear M estimation which is normally distributed are discussed, and the variance-covariance matrix of the basic vector is obtained. The paper deduces the linear representation of the rank-deficient network M estimation and the basic vector, as well asthe corresponding variance-covariance matrix. The paper presents the shapely characteristicηof a control net and that, whenη≤η₀ thedistribution of the residual errors is V_i～g(λ,0,λσ₀²), here V_i is a residual error of Least Squares andλ= r/n is general reliability of a control net.The validity is explained through simulate computation of virtual vertical control nets. The distribution shows that whenλ→1, the residual errors approach the true errors. With the increase ofλ, degree of residual errors approaching to true errors increases. Whenλ= 0 , V_i = 0, residual errors donot stand for any true errors.The thesis analyze and discuss the method of nonlinear estimation, which is not only benfit for the fundamental research, but also can be used in economy, communications, control and the data processing and analysis of other scientific domains.