本文選題：線性回歸　切入點(diǎn)：穩(wěn)健總體最小二乘法　出處：《太原理工大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：在生產(chǎn)實踐和科學(xué)實驗中,由于觀測程序和校核條件的不完善,測量數(shù)據(jù)采集過程中會不可避免地出現(xiàn)粗差。因此,如何消除或減弱粗差對參數(shù)估計的影響成為測繪學(xué)科中又一研究課題。隨著穩(wěn)健估計理念的問世,國內(nèi)外學(xué)者們提出了穩(wěn)健最小二乘(RLS)法,但是RLS法僅能顧及觀測向量受粗差影響的情況而忽略了系數(shù)矩陣,故而在此基礎(chǔ)上提出了能夠同時兼顧觀測向量和系數(shù)矩陣中含粗差情況的穩(wěn)健總體最小二乘(RTLS)法。線性回歸是測量數(shù)據(jù)處理中最常用的函數(shù)模型,針對線性回歸模型中自變量和因變量包含粗差的情況,有學(xué)者利用選權(quán)迭代的思想推導(dǎo)出基于線性回歸模型的穩(wěn)健總體最小二乘迭代公式和解算步驟。與此同時,一些學(xué)者通過個別算例中RTLS法得到比RLS法更小的單位權(quán)中誤差,就得出RTLS法優(yōu)于RLS法的結(jié)論。然而,就目前而言,并沒有明確的理論研究說明線性回歸中RLS法和RTLS法的優(yōu)劣性,僅憑個別算例就說明兩種參數(shù)估計方法的有效性太過片面,且僅以單位權(quán)中誤差的變化難以說明哪種參數(shù)估計方法更可靠,因此有必要對穩(wěn)健總體最小二乘法在線性回歸中的相對有效性進(jìn)行研究。本文針對不同誤差影響模型下穩(wěn)健總體最小二乘法在線性回歸中的應(yīng)用加以研究。按照誤差的不同分布可分為三種誤差影響模型:(1)僅觀測值含有隨機(jī)誤差和粗差;(2)系數(shù)矩陣含隨機(jī)誤差和粗差,觀測值僅含有隨機(jī)誤差;(3)觀測值含隨機(jī)誤差和粗差,系數(shù)矩陣僅含有隨機(jī)誤差。通過一元~五元線性回歸算例,對RLS法和RTLS法在多元線性回歸中的相對有效性進(jìn)行了初步比較,并在此基礎(chǔ)上運(yùn)用仿真實驗的方法,針對一元~五元線性回歸模型,分別討論在不同誤差影響模型、不同穩(wěn)健估計方法、不同觀測值個數(shù)以及不同斜率或不同粗差大小等情形下RLS法和RTLS法在多元線性回歸中的相對有效性。無論哪種誤差影響模型,當(dāng)一元線性回歸模型的斜率較小時(約為tan15°),很難說明RLS法和RTLS法哪個更有效;當(dāng)一元線性回歸模型的斜率較大時(約為tan45°或tan75°),第一和第三種誤差影響模型下,RLS法優(yōu)于RTLS法;第二種誤差影響模型下,RTLS法優(yōu)于RLS法。對于二元~五元線性回歸,第一種誤差影響模型下,RLS法優(yōu)于RTLS法;第二種誤差影響模型下,RTLS法優(yōu)于RLS法;第三種誤差影響模型下,很難說RLS法與RTLS法哪個更有效。
[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 擄), 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 擄or tan75 擄), 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.
【學(xué)位授予單位】：太原理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：P207.1

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