不同误差影响模型下稳健总体最小二乘法在线性回归中的应用研究

发布时间：2018-03-10 05:03

本文选题：线性回归　切入点：稳健总体最小二乘法　出处：《太原理工大学》2017年硕士论文　论文类型：学位论文

【摘要】：在生产实践和科学实验中,由于观测程序和校核条件的不完善,测量数据采集过程中会不可避免地出现粗差。因此,如何消除或减弱粗差对参数估计的影响成为测绘学科中又一研究课题。随着稳健估计理念的问世,国内外学者们提出了稳健最小二乘(RLS)法,但是RLS法仅能顾及观测向量受粗差影响的情况而忽略了系数矩阵,故而在此基础上提出了能够同时兼顾观测向量和系数矩阵中含粗差情况的稳健总体最小二乘(RTLS)法。线性回归是测量数据处理中最常用的函数模型,针对线性回归模型中自变量和因变量包含粗差的情况,有学者利用选权迭代的思想推导出基于线性回归模型的稳健总体最小二乘迭代公式和解算步骤。与此同时,一些学者通过个别算例中RTLS法得到比RLS法更小的单位权中误差,就得出RTLS法优于RLS法的结论。然而,就目前而言,并没有明确的理论研究说明线性回归中RLS法和RTLS法的优劣性,仅凭个别算例就说明两种参数估计方法的有效性太过片面,且仅以单位权中误差的变化难以说明哪种参数估计方法更可靠,因此有必要对稳健总体最小二乘法在线性回归中的相对有效性进行研究。本文针对不同误差影响模型下稳健总体最小二乘法在线性回归中的应用加以研究。按照误差的不同分布可分为三种误差影响模型:(1)仅观测值含有随机误差和粗差;(2)系数矩阵含随机误差和粗差,观测值仅含有随机误差;(3)观测值含随机误差和粗差,系数矩阵仅含有随机误差。通过一元~五元线性回归算例,对RLS法和RTLS法在多元线性回归中的相对有效性进行了初步比较,并在此基础上运用仿真实验的方法,针对一元~五元线性回归模型,分别讨论在不同误差影响模型、不同稳健估计方法、不同观测值个数以及不同斜率或不同粗差大小等情形下RLS法和RTLS法在多元线性回归中的相对有效性。无论哪种误差影响模型,当一元线性回归模型的斜率较小时(约为tan15°),很难说明RLS法和RTLS法哪个更有效;当一元线性回归模型的斜率较大时(约为tan45°或tan75°),第一和第三种误差影响模型下,RLS法优于RTLS法;第二种误差影响模型下,RTLS法优于RLS法。对于二元~五元线性回归,第一种误差影响模型下,RLS法优于RTLS法;第二种误差影响模型下,RTLS法优于RLS法;第三种误差影响模型下,很难说RLS法与RTLS法哪个更有效。
[Abstract]:In production practice and scientific experiment, due to the imperfection of observation procedure and check condition, gross error will inevitably occur in the process of measuring data acquisition. How to eliminate or reduce the influence of gross error on parameter estimation has become another research topic in surveying and mapping. With the advent of robust estimation concept, the robust least squares (RLS) method has been proposed by scholars at home and abroad. However, the RLS method can only take into account the effect of gross error on the observation vector and ignore the coefficient matrix. Therefore, a robust total least squares (RTLS) method, which can take into account both the observation vector and the gross error in the coefficient matrix, is proposed. Linear regression is the most commonly used function model in the measurement data processing. For the case that independent variables and dependent variables contain gross errors in linear regression models, some scholars use the idea of weight selection iteration to deduce the robust global least square iterative formula and calculation steps based on linear regression model, and at the same time, Some scholars have obtained smaller unit weight mean error by RTLS method than RLS method in individual examples, and have concluded that RTLS method is superior to RLS method. However, at present, there is no clear theoretical study on the advantages and disadvantages of RLS method and RTLS method in linear regression. A few examples show that the validity of the two parameter estimation methods is too one-sided, and it is difficult to explain which parameter estimation method is more reliable only by the variation of the error in the unit weight. Therefore, it is necessary to study the relative effectiveness of robust population least squares method in linear regression. In this paper, the application of robust population least squares method in linear regression under different error influence models is studied. The different distribution of errors can be divided into three kinds of error influence model: 1) the observed values only contain random error and gross error.) the coefficient matrix contains random error and gross error. The observed values only contain random errors and gross errors, and the coefficient matrix contains only random errors. The relative effectiveness of RLS method and RTLS method in multivariate linear regression is preliminarily compared by the example of linear regression between one and five variables. On the basis of this, the simulation experiment method is used to discuss the different error influence models and the different robust estimation methods for the linear regression models with one or five variables. The relative validity of RLS method and RTLS method in multivariate linear regression with different number of observed values and different slope or gross error. When the slope of univariate linear regression model is small (about tan15 掳), it is difficult to explain which RLS method or RTLS method is more effective, and when the slope of univariate linear regression model is larger (about tan45 掳or tan75 掳), the first and third error influence model is better than RTLS method. The second error influence model is superior to the RLS method, the first error influence model is superior to the RTLS method, the second error influence model is superior to the RLS method, and the third is the error influence model. It is difficult to say which RLS method is more effective than RTLS method.
【学位授予单位】：太原理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：P207.1

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