The Principle of Total Least Squares and Applications in Surveying Data Processing

• Main Authors: CHEN,YING-JIE, 陳映潔
• Other Authors: LEE,LIANG-HWEI
• Format: Others
• Language: zh-TW
• Published: 2017
• Online Access: http://ndltd.ncl.edu.tw/handle/ser39a

碩士 === 國立高雄應用科技大學 === 土木工程與防災科技研究所 === 106 === The Least-Squares (LS) method is one of the most basic and widely method for data processing. But the application has a premise that the random error exists only in the observation vector, without considering the coefficient matrix interfered by error (the Gauss-Markov model, G-M model). However, most measurement problems are called variable containing the error model {error-in-variables, EIV), because some of the elements of the coefficient matrix interfered by the personnel model, the environment, equipment, and other factors, so that the coefficient matrix is not completely accurate, the coefficient matrix and observation vector also contains the error. From statistical view, it is not reasonable to solve the EIV model with LS, and can not get the optimal solution. Therefore, there is an urgent need for a more reasonable method to solve such problems. Golub and Van Loan advanced the Total Least-Squares (TLS) to solve the ETV model in the eighties of last century. TLS minimizes all variables which need to be fixed, so we can use this method to establish a more reasonable data processing model and get more accurate results. In the past three decades, the TLS as a new data processing method has become a hot topic. At present, this method has been successfully applied to spectral analysis, parameter estimation, automatic control, image processing, system identification, signal processing, and other related fields. At the same time, the TLS has made a big advance. The improved models have been proposed. Based on the advantages of the TLS, it is theoretical and worth to introduce this method into the measurement field. Someone did some exploration research in Utilization of TLS in the problem of measurement data processing, but there were few systematic studies. This paper systematically expounded the basic ideas and principles of TLS, compared to the LS, a dissection of the advantages and disadvantages of the TLS was made in the basic ideas and principles level. Then we deduced in detail the most commonly TLS model and compared each model, and derived a new solution method-improved WTLS in connection with the characteristics of the EIV model of measurement data processing. On the basis of the original model and the calculation method, the weighted whole least squares method is deduced. Finally, we use different methods to solve and compare the one-dimensional straight line fitting, two-dimensional Cartesian coordinate transformation and three-dimensional arbitrary rotation angle reference transformation problem in the measurement data processing, and use the advantages of the whole least squares method Which is close to the real situation and has important theoretical significance, use value and validity in the application of engineering measurement data.