本文選題：EM算法　切入點：穩(wěn)健估計　出處：《中國地質(zhì)大學(北京)》2017年碩士論文　論文類型：學位論文

【摘要】：傳統(tǒng)測量手段單一,數(shù)據(jù)質(zhì)量可控性較強,故在經(jīng)典的測量數(shù)據(jù)處理理論中,通常是以高斯-馬爾科夫模型作為線性函數(shù)模型,并運用加權(quán)最小二乘方法得到待求參數(shù)的線性最優(yōu)無偏估值,并根據(jù)誤差傳播律進行精度評定。但是,隨著測量手段的豐富、數(shù)據(jù)來源的多樣性,觀測數(shù)據(jù)結(jié)構(gòu)更為復雜,不同類觀測值先驗權(quán)信息有可能不準確,且同時可能含有粗差以及系統(tǒng)誤差等,顯然,在此種情況下經(jīng)典的最小二乘參數(shù)估計的方法不再適用。對于異方差結(jié)構(gòu)數(shù)據(jù),當先驗信息不準確的時候,需要重新定權(quán),此即方差分量估計研究的范疇。方差分量估計的研究較為成熟,針對觀測數(shù)據(jù)自相關(guān)、互相關(guān)、負方差等問題都有大量的研究;針對實際應(yīng)用中的效率問題,有學者對常用的方法提出了相應(yīng)的簡化算法。當觀測值中存在異常值時,也有大量的國內(nèi)外學者進行了研究,最為常見的有穩(wěn)健M估計。但是,對于含有粗差以及異方差的觀測數(shù)據(jù),穩(wěn)健M估計通常只給出未知參數(shù)的估值,并無法給出異常值的具體數(shù)值。運用期望最大化(EM)算法,將隨機誤差以及粗差值作為缺失觀測,不僅可以計算得到未知參數(shù)估值,而且可以得到方差分量估值以及異常值估值。極大似然估計在求解方差分量時得到的為有偏估值,而限制性極大似然估計,消除了未知參數(shù)帶來的自由度的損失,是一種無偏估計方法�；贓M迭代方法的限制性極大似然估計,不僅繼承了基于EM迭代的極大似然估計的優(yōu)點,同時也可以得到理論上無偏的方差分量估值。根據(jù)上述分析,本文主要研究內(nèi)容如下:1.系統(tǒng)研究高斯-馬爾科夫模型下的最小二乘參數(shù)估計的方法及其特點,假設(shè)檢驗方法,以及驗后方差分量估計的方法,并介紹了一種改進的方差分量估計的方法LSMINQUE。2.研究線性混合模型、EM算法及其性質(zhì),研究基于EM算法的極大似然以及限制性極大似然方差分量估計方法的優(yōu)缺點,并通過模擬算例進行了分析驗證。3.結(jié)合限制性極大似然估計以及EM算法的優(yōu)點,推導出改進的混合模型下穩(wěn)健方差分量估計的方法,并用模擬GPS控制網(wǎng)數(shù)據(jù)算例證明該方法的有效性。
[Abstract]:The traditional measurement method is single, the data quality is controllable, so in the classical measurement data processing theory, Gao Si-Markov model is usually used as the linear function model. The weighted least square method is used to obtain the linear optimal unbiased estimation of the parameters to be solved, and the accuracy is evaluated according to the error propagation law. However, with the abundance of measurement means and the diversity of data sources, the structure of observation data is more complex. The prior weight information of different observational values may be inaccurate, and it may also contain gross error and system error. Obviously, the classical least square parameter estimation method is no longer suitable for heteroscedasticity structure data in this case. When the prior information is not accurate, we need to redefine the weight, that is, the category of variance component estimation. The research of variance component estimation is more mature, and there are a lot of research on autocorrelation, cross-correlation and negative square difference of observation data. In view of the efficiency problem in practical application, some scholars have put forward the corresponding simplified algorithm for the common methods. When there are outliers in the observed values, a large number of domestic and foreign scholars have also studied, the most common one is robust M estimation. For observational data with gross error and heteroscedasticity, robust M estimators usually only give estimates of unknown parameters, and cannot give specific values of outliers. The random errors and gross errors are considered as missing observations by using the expectation maximization (EMV) algorithm. The estimations of variance components and outliers can be obtained not only by calculating the unknown parameters, but also by using the maximum likelihood estimator to solve the variance component, while the restricted maximum likelihood estimator can be used to solve the variance component. The loss of degree of freedom caused by unknown parameters is eliminated, and it is an unbiased estimation method. The limited maximum likelihood estimation based on EM iteration not only inherits the advantage of maximum likelihood estimation based on EM iteration. According to the above analysis, the main contents of this paper are as follows: 1. The methods and characteristics of least squares parameter estimation under Gao Si-Markov model are studied systematically. This paper also introduces an improved method of variance component estimation, LSMINQUE.2.Study on the EM algorithm of linear mixed model and its properties. The advantages and disadvantages of maximum likelihood and restricted likelihood variance component estimation methods based on EM algorithm are studied, and the simulation examples are given to verify that .3. combining the advantages of restrictive maximum likelihood estimation and EM algorithm, The robust variance component estimation method under the improved hybrid model is derived, and the effectiveness of the method is proved by a numerical example of simulated GPS control network data.
【學位授予單位】：中國地質(zhì)大學(北京)
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：P207

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