Research On Non-linear M-Estimates And Its Application

The classical data processing and data analysis theory is built on the basis that the errors are normally distributed and the function model is linear. It has a set of complete methods and relative subjects based on the Least squares principle. In the science of surveying and mapping, it has formed the following chief tools used for data processing and data analysis: (1) The parameter estimation formula; (2) The unit weight variance estimation formula and its Chi-square statistic and F-statistic; (3) the Baarda data snooping formula; (4) Chi-square statistic and F-statistic used for the linear hypothesis test; (5) The Helmert variance component estimation formula. Except the parameter estimation (1), the other four items (2)⁵ all require that the variance-covariance matrix of the basic vector including the observations, the estimated parameters, the residuals and the adjustments of the observations are explicit and computable. Because the Least Squares estimation of the linear model is a linear estimation, that is, the basic vector can be represented as the linear function of the observations. So it is not difficult to compute the variance-covariance matrix of the basic vector. To solve the non-robustness of the Least Squares method, Huber defined the M-estimation of the unknown parameters, but it can be also used for data processing and data analysis without gross errors. The cause is that the observation errors are not normally distributed possibly, and the variance and variance component estimation based the Least Squares residuals are not globally optimal. For M-estimation, except the parameters estimation (1) has gotten deeply studied and formed a complete set of algorithm, the other four items all has no useable formula based on M-method and M-residuals. The cause is that the M-estimation of the linear model is non-linear. That is, the basic vector can't be represented as the linear function of the observations, which makes that the basic vector has no computable variance-covariance matrix. This thesis aims at solving the above-mentioned problems and building a set of perfect data processing and data analysis system based on M-estimationIn Chapter Two, the data processing and data analysis system based on the Least Squares principle gets in detail discussed, especially the variance component estimation. The relation of Helmert variance component estimation and the quadric unbiased estimation of the variance component is point out. For the Gauss-Markov model with unequal precision, Helmert method and the quadric unbiased estimation method have the equal estimation value; If the Least Squares residuals are used, the quadric unbiased estimation formula is the Helmert formula; If the other residuals are used, the quadric unbiased estimation formula has the same coefficient matrix as the Helmert formula in the solution equation of the variance component, but its constant vector is different from the Helmert method. Because the Least Squares method can't accurately locate the gross errors, so the Helmert method and the quadric unbiased method can't find and classify gross errors efficientlyBecause the Least Squares method can't accurately locate the gross errors, so the Helmert method and the quadric unbiased method can't find and classify gross errors efficiently and are not robust.In Chapter Three, the definition of M-estimation and its existence conditions get in detail researched. They are extended in surveying practice to be suitable for the rank-deficient network. The general solution formula of the M-estimation in rank-deficient network is derived, its transformation formula of solutions under different reference basis is proved to be the same as the Least Squares; The maximum sum likelihood estimation are studied when the errors are normally distributed. The appropriate variance factor, considering the efficiency and robustness, is determined. The least infinite norm estimation is studied, and the algorithm based on the modification accumulation function and its robustified algorithm is constructed.Chapter Four derives the linear representation of the basic vector of M-estimate including the rank-deficient network, and defines the third nuisance parameter, and gets the variance-covariance matrix of the basic vector. Under the condition that the errors are normally distributed, the mathematical form of the nuisance parameters based on Lp-estimation and the maximum sum likelihood estimation is derived Chapter Five makes research on the variance estimation of the linear function model, and derives the uniform form of the variance estimation independent to the estimation criteria and the error distribution. The example shows that the form can be used to assess the precision of M-estimation including the Least Squares estimation; The variance estimation formula based on the estimation criteria and the error distribution is derived, it can be used to assess the precision of M-estimation under the condition that the error distribution is known.Chapter Six makes research on the variance component estimation based on M-residuals. For the normal errors, the variance component estimation formula and its algorithm based on Lp-estimation and the maximum sum likelihood estimation is derived. Its precision formula of the variance component estimation is also derived in this chapter. Because the M-estimation with robustness can accurately locate gross errors, the variance component estimation formula derived in this thesis differs from the Helmert method; it can accurately classify the gross errors and the normal errors and. The application shows that the formula derived in this paper is robust.Chapter Seven makes research on the relation of the observation errors to the M-residuals; M-residual is the linear function of the influence function. The formula can explain why the M-estimation with robustness is robust. In the formula, because the influence function of M-estimation with robustness is bounded, the gross error almost completely casts on the corresponding residual. The Baarda statistic based on the M-residual with the known variance is derived, it can be used to snoop gross errors under the given confidence level.Chapter Eight makes research on the detecting technology of gross errors and the precision assessing under the contamination model. For the error mean shift model, gross errors are located first by the Baarda statistic based on M-residual, secondly the design matrix of the linear model is right determined. Then the variance estimation formula independent to the error distribution and estimation criteria is used to compute the unit weight variance for assessing the precision; For the variance increasing model, after locating the gross errors, the variance component corresponding to the gross error is considered as being different from its prior value. The variance component estimation formula based on M-residuals in Chapter Six is used to iterate and find the unit weight variance. The computation example shows that the result found by the two methods is reliable and can be used to assess the precision of the contamination model Chapter Nine makes research on the error distribution and its derivation method. Under the hypothesis that the solution equation of the maximum sum likelihood estimation is equal to that of the Lp-estimation, the p-norm distribution is got; Under the hypothesis that the solution equation of the maximum product likelihood estimation is equal to that of Lp-estimation with the density weight, the found error distribution as p=2 is the Cauch distribution; Under the hypothesis that the maximum sum likelihood estimation and Lp-estimation has the same solution equation, the derived error distribution is the power function. It is a bounded distribution. The generality variance and the variance of the maximum likelihood estimation based on the power distribution are got in this chapter The obtained contents and conclusions in this paper are the supplement and perfection for the theory of data processing and data analysis based on M-method. It can be not only used for the surveying data processing and analysis, but also used in the other fields, such as economy, communication, control and so on.