Xi’an Institute of Surveying and Mapping
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. 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.
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. 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. 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.
Robust Kalman filtering model and its application in GPS monitoring networks are studied. The Robust Kalman filtering model based on the rules of the influences of the outliers on the state vectors is derived. The theory and the method as well as the robustness of the parameter estimation based on the principle of information spread are researched. 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.
Lp estimator is an influential class of robust estimation which has been widely studied in China. The p-norm distribution is a distributive class which includes the most frequently used distributions such as Laplace, normal and rectangular ones. The p-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 original p-norm distribution. Because every density function used in the approximate formulas has a simple form, using the approximate density function to replace the p-norm ones will simplify the problems of p-norm distributed data processing obviously. Based on the theory of the M-estimation for the location parameters, the influence functions of the Lp estimates of the parameters (one dimension and multi-dimension) are derived, the robustness of Lp estimation when the observational errors obey the contaminated distribution is analyzed, and the asymptotic variance and efficiency of the Lp estimation have been determined too.
The optimal LP estimation based on the global minimum of the asymptotic variance, and the optimal value of parameter p has been determined under the contaminated distribution. The results show that the Lp estimation with the value of parameter p between 1.2 and 1.5 is the optimal estimation if the data contain different outliers.
The mathematical model of L∞ estimation is given. The general method of solving L∞ estimation with linear programme is advanced, and according to the dual principle of linear programme, the dual model of L∞ estimation is given. 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 the L1-estimates of parameters are unbiased.
Chinese geodesist suggested instant, or through a few steps, approaches to the optimal solutions, based on the existing experiences, and using simplex method .
In the applications of robust estimation, the special displacement of strain models is determined by means of robust estimation technique. The robust estimation was applied in the parameter estimation for a dynamic model of the sea surface and geodetic datum transformation as well as the determination of systematic errors of satellite laser range.
II. OUTLIER DETECTING AND THEORY OF RELIABILITY
A new idea and a distinctive method for outlier detecting and estimating are proposed by Ou. 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. 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 . 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. 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. 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. 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. γi=(PR)ii/pii, here P is the weight matrix of observations, R the adjustment factor matrix, the subscript i represents the place of the observation i. γi is in ［0，1］. The smaller γi is, the poorer the reliability corresponding to the observation i is. The new indication γi is united one. It is suitable not only for independent observations, but also for correlated observations.
III. ADAPTIVE KALMAN FILTERING
The 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. 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. An improved adaptive Kalman filter combining the Sage adaptive filtering and a new adaptively robust Kalman filter with an adaptive factor is proposed. 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.
IV. PARAMETER ESTIMATION FOR NONLINEAR MODELS
In the parameter estimation of nonlinear 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. The estimators obtained by the new method have fine statistic characters. The estimation of the unit weight variance is studied. Two objective functions of nonlinear model are given under two kinds of parameter estimation guides. 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. 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. 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.
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. An improved Marquardt method applied into nonlinear adjustment is proposed.
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.
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.
V. OTHER DEVELOPMENTS IN DATA PROCESSING
A 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.
The 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. A few remarks on collocation is made. 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. As one of applications of collocation, a new approximation model with random variables is proposed.
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. 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. 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.
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.
A semiparametric regression and model refining is researched. It is hopefully to be a new research direction in the coming years.
 Duan W and Yuan X. 1999, Some strategy on solving high order normal equation for combined adjustment of astro-geodetic and aspace networks by using conjugate gradient method. Acta Geodaetica et Cartographica Sinica, 28(3): 205-208. Fan D. 2001, Study on iterative algorithm of nonlinear least squares adjustment by parameters. Journal of Institute of Surveying and Mapping, 18(3): 173-175. Gui Q, Zhang J and Guo J. 2000, Shrunken type-robust estimation. Acta Geodaetica et Cartographica Sinica, 29(3): 224-228. Gui Q, Guo J, Yang Y and Sui L. 2001a, Biased estimation in Gauss-Markov model not of full rank. AVN, 11-12: 390-393. Gui Q, Yang Y and Guo J. 2001b, Biased estimation in Gauss-Markov model with constraints. AVN, 1: 28-30. Hu G and Ou J. 1999, The improved method of adaptive Kalman filtering for GPS high kinematic positioning. Acta Geodaetica et Cartographica Sinica, 28(4): 290-294. Li C, Xu W, Zeng Z and Huang L. 2001, Solutions and traits of adjustment equations in space of nonlinear functions. Journal of Institute of Surveying and Mapping, 18(1): 8-11. Li G, Wan J and Tao H. 2001, Nonlinear data processing based on improved Marquardt method. Journal of Institute of Surveying and Mapping, 18(3): 167-169. Liu G and Zhang C. 1999, Nonlinear curvature measures of strength for the model of adjustment of free-networks with general rank deficiency. Journal of Institute of Surveying and Mapping, 16(2): 101-105. Liu G, Zhang C and Sun Q. 1999, Nonlinear adjustment of free-networks with general rank deficiency. Journal of Institute of Surveying and Mapping, 16(3): 101-105. Liu J, Yao Y and Shi C. 2001, Theoretic research on robustified least squares estimator based on equivalent variance-covariance. Geo-spatial Information Science, 4(4): 1-8. Ou J. 1999, Quasi-accurate detection of gross errors (QUAD). Acta Geodaetica et Cartographica Sinica, 28(1): 15-20. Ou J. 1999, On the property of the estimators of real errors determined by the method of the quasi-accurate detection of gross errors. Acta Geodaetica et Cartographica Sinica, 28(2): 144-147. Ou J. 1999, On the reliability for the situation of correlated observations. Acta Geodaetica et Cartographica Sinica, 28(3): 189-194. Ou J. 2000, Selection of quasi-accurate observations and “hive off” phenomena about the estimators of real errors. Acta Geodaetica et Cartographica Sinica, 29(1): 5-10. Ou J. 2001, On the theory about quality control over surveying data. Engineering of Surveying and Mapping, 10(2): 6-10. Peng J. 2000, The unbiased estimation of posterior variance of gross error and the optimal robust estimate. Journal of Institute of Surveying and Mapping, 17(2): 82-85. Song L and Yang Y. 1999, On outlier correcting and deleting. Bulletin of Surveying and Mapping, No.6: 5-6. Sun H. 2001, Approximate representation of the p-norm distribution. Journal of Wuhan Technical University of Surveying and Mapping, 26(3): 222-225. Sun H and Wu Y. 2002, Semiparametric regression and model refining. Geomatics and Information Science of Wuhan University, 27(2): 172-174. Tao B. 2000, Robust homogeneous analysis of displacement-strain models. Acta Geodaetica et Cartographica Sinica, 29(4): 293-296. Tao B. 2001, Non-linear adjustment of deformation inversion model. Journal of Wuhan Technical University of Surveying and Mapping, 26(6): 504-508. Wang X. 1999, A direct solution method of parameter estimation of nonlinear model. Journal of Wuhan Technical University of Surveying and Mapping, 24(1): 64-67. Wang X. 1999, The theory, method and robustness of the parameter estimation based on the principle of information spread. Journal of Wuhan Technical University of Surveying and Mapping, 24(3): 240-244. Wang X. 2000, Estimation of the unit weight variance in nonlinear model adjustment. Journal of Wuhan Technical University of Surveying and Mapping, 25(4): 358-361. Wang Z and Zhu J. 1999, Optimal estimation under contaminated error model. Acta Geodaetica et Cartographica Sinica, 28(1): 51-56. Wang Z, Ou J and Chen X. 2001, The un-biasness of L1-estimation. Journal of Institute of Surveying and Mapping, 18(2): 105-106. Wei Z, Huang W, Yang J, Yang D, Wang Z and Tao W. 2000, Combined adjustment of nationwide astro-geodetic and space-geodetic network combined adjustment of nationwide astro-geodetic and space-geodetic network. Acta Geodaetica et Cartographica Sinica, 29(4): 283-288. Wen Y, Xi X, Wang W and Yang Y. 2000, Givens-Gentleman orthogonal transformation algorithm of robust estimation. Acta Geodaetica et Cartographica Sinica, 29(3): 198-203. Xu T and Yang Y. 2000, The improved method of Sage adaptive filtering. Science of Surveying and Mapping, 25(3): 22-24. Yang Y, Wen Y, Xiong J and Yang J. 1999, Robust estimation for a dynamic model of the sea surface. Survey Review, 35: 2-10. Yang Y. 1999, Robust estimation of geodetic datum transformation. Journal of Geodesy, 73: 268-274. Yang Y, Cheng MK, Shum, CK and Taplay BD. 1999, Robust estimation of systematic errors of satellite laser range. Journal of Geodesy, 73: 345-349. Yang Y, He H and Xu G. 2001, Adaptively robust filtering for kinematic geodetic positioning. Journal of Geodesy, 75(2/3): 109-116. Yang Y, Song L and Xu T. 2001, New robust estimator for the adjustment of correlated GPS networks, First Inter. Symp. on Robust Statistics and Fuzzy Techniques in Geodesy and GIS, edited by Carosi,A and Kutterer, H. IGP-Bericht Nr. 295, pp. 199-208. Yang Y, He H and Xu T. 2001, Adaptive robust filtering for kinematic GPS positioning. Acta Geodaetica et Cartographica Sinica, 30(4): 293-298. Yang Y, Song L and Xu T. 2002, Robust parameter estimation for geodetic correlated observations. Acta Geodaetica et Cartographica Sinica, 31(2): 95-99. You Y and Wang X. 2002, Direct solution to generalized ridge estimation. Geomatics and Information Science of Wuhan University, 27(2): 175-178. Yu X and Lu W. 2001, Robust Kalman filtering model and its application in GPS monitoring networks. Acta Geodaetica et Cartographica Sinica, 30(1): 27-31. Zang D, Zhang L and Zhou S. 2000, The un-biasness for the robust estimation of the location parameters. Journal of Institute of Surveying and Mapping, 17(1): 1-2. Zhang F, Feng C and Huang C. 2000, Application of the improved Helmert method for variance component estimation in precise orbit determination. Acta Geodaetica et Cartographica Sinica, 29(3): 216-223. Zhou J. 1999a, On the rule of fitting. Acta Geodaetica et Cartographica Sinica, 28(4): 283-284. Zhou J. 1999b, Solution based on experience of the extreme problems in surveying, and some remarks on LAS. Acta Geodaetica et Cartographica Sinica, 28(1): 11-14. Zhou J. 2001a, A new approximation model by collocation. Acta Geodaetica et Cartographica Sinica, 30(4): 286-287. Zhou J. 2001b, Further on collocation. Acta Geodaetica et Cartographica Sinica, 30(4): 283-285. Zhou S, Deng K and Chen Y. 1999, The optimal Lp estimation under prescribed distribution. Acta Geodaetica et Cartographica Sinica, 28(1): 45-50. Zhou Q. 2000, Research of method of L∞ estimation. Science of Surveying and Mapping, 25(2): 55-57. Zhou Q. 1999, Generalized normal distribution and property of its quadratic function. Engineering of Surveying and Mapping, 8(1): 4-11. Zhu J and Zeng Z. 1999, The theory of surveying adjustment under contaminated error model. Acta Geodaetica et Cartographica Sinica, 28(3): 215-220.