基于EM算法的稳健方差分量估计研究

发布时间：2018-03-14 14:07

本文选题：EM算法　切入点：稳健估计　出处：《中国地质大学(北京)》2017年硕士论文　论文类型：学位论文

【摘要】：传统测量手段单一,数据质量可控性较强,故在经典的测量数据处理理论中,通常是以高斯-马尔科夫模型作为线性函数模型,并运用加权最小二乘方法得到待求参数的线性最优无偏估值,并根据误差传播律进行精度评定。但是,随着测量手段的丰富、数据来源的多样性,观测数据结构更为复杂,不同类观测值先验权信息有可能不准确,且同时可能含有粗差以及系统误差等,显然,在此种情况下经典的最小二乘参数估计的方法不再适用。对于异方差结构数据,当先验信息不准确的时候,需要重新定权,此即方差分量估计研究的范畴。方差分量估计的研究较为成熟,针对观测数据自相关、互相关、负方差等问题都有大量的研究;针对实际应用中的效率问题,有学者对常用的方法提出了相应的简化算法。当观测值中存在异常值时,也有大量的国内外学者进行了研究,最为常见的有稳健M估计。但是,对于含有粗差以及异方差的观测数据,稳健M估计通常只给出未知参数的估值,并无法给出异常值的具体数值。运用期望最大化(EM)算法,将随机误差以及粗差值作为缺失观测,不仅可以计算得到未知参数估值,而且可以得到方差分量估值以及异常值估值。极大似然估计在求解方差分量时得到的为有偏估值,而限制性极大似然估计,消除了未知参数带来的自由度的损失,是一种无偏估计方法。基于EM迭代方法的限制性极大似然估计,不仅继承了基于EM迭代的极大似然估计的优点,同时也可以得到理论上无偏的方差分量估值。根据上述分析,本文主要研究内容如下:1.系统研究高斯-马尔科夫模型下的最小二乘参数估计的方法及其特点,假设检验方法,以及验后方差分量估计的方法,并介绍了一种改进的方差分量估计的方法LSMINQUE。2.研究线性混合模型、EM算法及其性质,研究基于EM算法的极大似然以及限制性极大似然方差分量估计方法的优缺点,并通过模拟算例进行了分析验证。3.结合限制性极大似然估计以及EM算法的优点,推导出改进的混合模型下稳健方差分量估计的方法,并用模拟GPS控制网数据算例证明该方法的有效性。
[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.
【学位授予单位】：中国地质大学(北京)
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：P207

【相似文献】