Generalized Shrunken Type GM Estimation And Its Application In GPS Data Processing

With the development of surveying theory and observation technologies, there has been an increasing complexity in surveying targets, thus posing the frequently occurring phenomenon that there are not only ill-conditions but also outliers in the observation data. The problems that ill-conditions in design matrix and outliers in observation space exist simultaneously are considered and addressed in present robust-biased estimations, yet the situation in which ill-conditions in design matrix, outliers in design space and outliers in observation space co-exist still has not been considered. The author, just taking this as his point of departure, presents an effective measure of resolving such a problem. Therefore, outliers in design space, viz. high-leverage outliers, are analyzed systematically with a combined application of the modern probability and statistics theory as well as the matrix method, generalized M (GM) estimations are proposed, and generalized shrunken type GM (GSGM) estimations are constructed and presented by incorporating GM estimations into biased estimations. The main conclusions are as follows:(1) From the perspective of surveying adjustment, such concepts as high-leverage value and high-leverage observation value in independent observation are defined; in terms of the properties of high-leverage value outliers are re-classified; the role of hat covering of high-leverage outliers is analyzed, which designates a wise working direction as to how to make M estimations resist high-leverage outliers, namely how to construct GM estimations.(2) M estimations can't overcome the distribution of outliers in design space in independent observation. To circumvent this problem, the formula of M estimation are improved, GM estimations have been proposed and discussed, and the leverage value are used in the design of equivalent weight. The generalized equivalent weight is constructed. The algorithms of GM estimations in independent observation are studied and proposed.(3) Some concepts such as the high-leverage value, high-leverage observation value are defined by reliability theory based on the particularity of the adjustment factor matrix. So the outliers in correlate observation are re-classified. The GM estimation are proposed, corresponding generalized equivalent weight are also designed and constructed. The algorithms about GM estimation in correlative observation are researched and proposed.(4) To tackle the complex situation of the co-existence of ill-conditioning in design matrix, outliers in design space and outliers in observation space, GSGM estimations are prevented by grafting the biased estimation techniques philosophy into the GM estimations, and a number of GSGM estimations such as ordinary ridge type GM (ORGM) estimation, principal component type GM (PCGM) estimation, combined ridge with principal component type GM (CRPCGM) estimation, combined ridge with shrunken estimation type GM (CRSGM) estimation are constructed to meet the demands of the practical problems of surveying adjustment. Some important problems in GSGM estimations such as generalized equivalent weight matrix construction, chosen of biased parameter and the iterative algorithms are discussed and solved.(5) Under certain conditions, it has been proved that GSGM estimations are superior to GM estimation and biased estimations under the criterion of MSE (the mean squared error) and the condition numbers, that is to say, GSGM estimations are more capable to resist the outliers in design space and observation space and the ill-condition in design matrix than GM estimation or biased estimation.(6) The characteristics of the Double-Difference (DD) model of Global Positioning System (GPS) rapid positioning are analyzed, and the new computing scheme are design and propose based on the DD model. It is carried out in three steps. Firstly, calculating float solution based on GSGM estimation. Secondly, searching for the integer ambiguities by the LAMBDA methods based on cofactor matrices of GSGM estimation. Finally, computing the fixed solution by GM estimation.A number of numerical example and practical applications prove that GSGM estimators proposed in this paper are highly efficient. They can not only resist the influence of ill-conditions in design matrix but also overcome difficulty caused by outliers in design space and outliers in observation space exist simultaneously.