*YANG Yuanxi*

Xi’an Institute of Surveying and Mapping

Yuanxi@pub.xaonline.com

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I. ROBUST ESTIMATION

Within the last four years, much attention has been paid to parameter estimation for correlated observations and Lp estimation, in the robust estimation field. A set of self-contained theory system on robust estimation is researched based on equivalent variance-covariance

^{[11]}. A new robust estimator for correlated observations (RECO) by bi-factor equivalent weight elements was developed, which keeps the symmetry and the correlation of the observations unchanged^{[35,37]}.The robust estimators of parameters and variance factor with minimum mean square error are derived, all parameters of which are determined according to statistics

^{[26]}.The parameter estimation problem when outliers and ill-conditioning exist simultaneously is researched. A class of new estimators, shrunken type robust estimators, are proposed by grafting the biased estimation techniques philosophy into the robust estimator, and their properties are discussed

^{[3]}. The un-biasness for the robust estimation of the location parameters is discussed, based on the hypothesis that the weight function is an anti-symmetrical function^{[40]}. The unbiased estimation of posterior variance and the optimal robust estimate have been studied. It is demonstrated that for stochastic error model and mean shift error model, they have the same estimate formula^{[17]}.

Robust Kalman filtering model and its application in GPS monitoring networks arestudied.The Robust Kalman filtering model based on the rules of the influences of the outliers on the state vectors is derived^{[39]}. The theory and the method as well as the robustness of the parameter estimation based on the principle of information spread are researched^{[24]}. Givens-Gentleman orthogonal transformation is applied to robust estimation. The formulas of posterior root mean square error and the covariance matrix of the parameter estimates have been derived. The calculation of equivalent weights is discussed. The numerical stability is analyzed for extended Givens-Gentleman orthogonal transformation^{[29]}.

Lestimator is an influential class of robust estimation which has been widely studied in China. The_{p}p-norm distribution is a distributive class which includes the most frequently used distributions such as Laplace, normal and rectangular ones. Thep-norm distribution can be represented by the linear combination of Laplace distribution and normal distribution or by the linear combination of normal distribution and rectangular distribution approximately. The approximate distribution has the same first four-order moments as the originalp-norm distribution. Because every density function used in the approximate formulas has a simple form, using the approximate density function to replace thep-norm ones will simplify the problems ofp-norm distributed data processing obviously^{[19]}. Based on the theory of theM-estimation for the location parameters, the influence functions of theLestimates of the parameters (one dimension and multi-dimension) are derived, the robustness of_{p}Lestimation when the observational errors obey the contaminated distribution is analyzed, and the asymptotic variance and efficiency of the_{p}Lestimation have been determined too_{p}^{[46]}.The optimal

Lestimation based on the global minimum of the asymptotic variance, and the optimal value of parameter_{P}phas been determined under the contaminated distribution. The results show that theLestimation with the value of parameter_{p}pbetween 1.2 and 1.5 is the optimal estimation if the data contain different outliers^{[46]}.The mathematical model of

Lestimation is given. The general method of solving_{∞}Lestimation with linear programme is advanced, and according to the dual principle of linear programme, the dual model of_{∞}Lestimation is given_{∞}^{[48]}. Based on the formula of probability density and distribution function of monistic Laplace distribution, the probability density of the median estimator is derived, when the sum of observations is even. The authors proved that theL-estimates of parameters are unbiased_{1}^{[27]}.Chinese geodesist suggested instant, or through a few steps, approaches to the optimal solutions, based on the existing experiences, and using simplex method

^{[42]}.In the applications of robust estimation, the special displacement of strain models is determined by means of robust estimation technique

^{[21]}. The robust estimation was applied in the parameter estimation for a dynamic model of the sea surface^{[31]}and geodetic datum transformation^{[32]}as well as the determination of systematic errors of satellite laser range^{[33]}.

II. OUTLIER DETECTING AND THEORY OF RELIABILITYA new idea and a distinctive method for outlier detecting and estimating are proposed by Ou

^{[12]}. The determinate and analytic representation relationship between real errors and observation values is applied to fitting the real errors. The unique solution on real errors in this equation is obtained by using the idea of “Quasi-Stable Adjustment”, the new concept on “Quasi-Accurate Observation” is presented. Then the rank-deficiency equations on real errors are resolved by adding the conditions in which the minimum of the norm of the real errors related to quasi-accurate observations is restrained. The new method called as “Quasi-Accurate Detection of gross errors (QUAD)” is derived^{[13]}. The key of QUAD is how to select the Quasi-accurate Observations (QAO) reasonably. An effective scheme, which is divided into two steps, is summarized by Institute of Geodesy and Geophysics of Chinese Academy of Sciences^{ [15]}. In the first step called “Preliminary Selection”, the observations are classified into 4 sorts among which only the “sort 2” and the parts of “sort 3” are able to be chosen as QAOs. In the second step, called “Fine Selection”， the observations corresponding to small absolute values of the estimates of errors, which are calculated at the previous step, are chosen as QAOs.On the basis of expectation drift model, the relationship between outlier correcting and deleting is researched

^{[18]}. It is demonstrated that on the basis of the same expectation drift model, outlier correcting and deleting are equivalent in the estimates of parameter and the square sum of residuals.

In the area of the quality control of the surveying data processing, a new and united index of reliability is introduced

^{[16]}. This index is determined completely by the construction of a system (such as a surveying network), and it is not necessary to include the probability element (non-central parameter_{ }) which is effected on by some factors chosen subjectively. It is shown that the hypothesis test may not detect all gross errors but the quasi-accurate detection may do it easily.The reliability for the situation of correlated observations is also discussed^{[14]}. In the correlated observations, the redundancy measures are over ［0,1］, at times even some of them are negative, so that the redundancy measures may not reflect the characteristics of the reliability. A new measurable indication of reliability is suggested: i.e.γ=(_{i})PR/_{ii}p, here_{ii}is the weight matrix of observations,Pthe adjustment factor matrix, the subscriptRirepresents the place of the observationi. γis in ［0，1］. The smaller_{i}γis, the poorer the reliability corresponding to the observation_{i}iis. The new indicationγis united one. It is suitable not only for independent observations, but also for correlated observations._{i}

III. ADAPTIVE KALMAN FILTERINGThe reliability of the linear Kalman filtering results usually degrades when the kinematic model noise is not accurately modeled in filtering or the measurement noises at any measurement epoch are not normally distributed. A new adaptively robust filtering is proposed based on the robust M (Maximum likelihood type) estimation

^{[34]}. It consists in weighting the influence of the updated parameters in accordance with the magnitude of discrepancy between the updated parameters and the robust estimates obtained from the kinematic measurements and in weighting individual measurement at each discrete epoch. A general estimator for adaptively robust filter is developed, which includes the estimators of classical Kalman filter, adaptive Kalman filter, robust Kalman filter, sequential least squares (LS) adjustment and robust sequential adjustment. In addition to the robustizing properties, feasibility in implementation of the new filter is achieved through the equivalent weights of the measurements and the predicted state parameters. The relations of analytical expressions and covariance matrices between the basic random vectors, such as residual vectors, innovation vectors and correction vectors of predicted state, are derived and discussed. The shortcomings of covariance matrices by the averages of the residual vectors, innovation vectors and correction vectors of the kinematic states timing their transposes within a fixed window, are analyzed. The robust filtering, the Sage adaptive filtering and the adaptively robust filtering are compared. It is shown, by derivations, that the new adaptively robust filtering is not only simple in calculation but also robust in controlling the measurement outliers and kinematic state disturbing^{[36]}. An improved adaptive Kalman filter combining the Sage adaptive filtering and a new adaptively robust Kalman filter with an adaptive factor is proposed^{[30]}. If the state is smooth, then the Sage filter is applied, otherwise the new adaptive Kalman filter works.An improved method of adaptive Kalman Filtering for GPS high kinematic positioning is also proposed based on Sage filtering

^{[6]}.

IV. PARAMETER ESTIMATION FOR NONLINEAR MODELSIn the parameter estimation of n

onlinear model，a direct calculation method is proposed.The influences of quadric terms and cubic terms have been taken into consideration. The precision of estimators can be evaluated by classical method^{[23]}. The estimators obtained by the new method have fine statistic characters. The estimation of the unit weight variance is studied^{[25]}. Two objective functions of nonlinear model are given under two kinds of parameter estimation guides^{[7]}. The iterative methods of nonlinear functions are discussed. Based on that, two sorts of best solutions─circulating search method which does not depend on functional derivatives and iterative method that is based on differential theory—are presented.Nonlinear curvature measures of strength for the model of adjustment of free-networks with general rank deficiency are studied

^{[9]}. On the basis of concerning the nonlinear curvature measures of strength for the model of adjustment, nonlinear curvature measures of strength for the model of adjustment of free-networks with general rank deficiency is put forward^{[9]}. Two new computational methods to nonlinear adjustment of free-networks in consideration of the second-order derivatives, are put forward from the precise orthogonality condition equation to nonlinear least squares^{[10]}.In addition to the three existing nonlinear squares algorithms—Gauss-Newton method, damped least squares method and quasi-Newton method on least squares, a better algorithm—SQPM (Sequential Quadratic Programming Method) as one of the most powerful algorithms of nonlinear programming is applied. And the step-length policy of SQPM is improved in order to advance the iterative convergence

^{[2]}. An improved Marquardt method applied into nonlinear adjustment is proposed^{[8]}.Tao described the quality requirement of the inversion problem of parameter estimation. The resolution of the inversion theory and the precision of the parameter estimation are compared and analyzed. In order to get favorable results, he puts forward several schemes for the nonlinear adjustment. The principles of the parameter choice and corresponding statistical test methods are presented

^{[22]}.The generalized normal distribution is defined by the generating function of stochastic vectors, and the variance matrix of normal stochastic vectors is spread from the full rank matrix to the singular matrix. According to the property of generating function, the digital feature of the quadratic function of normal stochastic vectors is inferred, and the result is applied to error propagation. The expression of the error propagation formula with regard to the quadratic for non-linear function is obtained

^{[47]}.

V. OTHER DEVELOPMENTS IN DATA PROCESSINGA new biased estimator called partial root root estimator is proposed through analyzing the corresponding properties of root root estimator, Stein shrunken estimator, and Sclove partial shrunken estimator. Its properties are discussed

^{[4,5]}. A direct solution to generalized ridge estimate is studied^{[38]}.T

he rule of fitting is researched.In many problems of data processing, by appropriate use of the method of fitting, indefinite rank-deficiency problems can be converted into relatively definite ones, giving valuable results^{[43]}. A few remarks on collocation is made^{[44]}. The current solution of collocation is least squares one under the bi-fitting constraint. The stochastic parameters and the observational errors are treated equally, as errors. It results in that the estimate of the systematic parameters is the best one, but the estimate of the random parameters is not the best estimate, which regard to the realization to be sought, it is biased. The covariance of the estimate is not suitable to be a measure of accuracy of the estimates of the random parameters. The bi-fitting rule is grounded on maximal probability density, but at the same time it leads the estimate of the random parameters toward its expectation, which results in a self-deviation^{[45]}. As one of applications of collocation, a new approximation model with random variables is proposed^{[44]}.A combined adjustment of the nationwide astro-geodetic and GPS geodetic networks was accomplished early in 1998. The principles adopted and the models used in the adjustment and present results are reported

^{[28]}. As a result of adjustment, the systematic scale deviation of the astro-geodetic network has been removed, its local distortion mitigated and its integral precision improved. And more importantly a geocentric geodetic reference system as materialized by the coordinates of nearly 50000 geodetic points has been set up with better than one half meter accuracy for horizontal components^{[28]}.Some strategy on solving high order normal equation for combined adjustment of astro-geodetic and space networks by using conjugate gradient method was researched.A scheme of adjusting the coefficients and a strategy of separation as well as mergence between the adjustment and inversion are proposed^{[1]}.The Helmert method for Variance Component Estimation (VCE) can be successfully applied to Precise Orbit Determination (POD). An improved Helmert method for variance component estimation is developed by using the priori information of the estimated parameters to avoid the occurrence of negative variance

^{[41]}.A semiparametric regression and model refining is researched

^{[20]}. It is hopefully to be a new research direction in the coming years.

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