Robust Estimation Of Variance Components Based On Expectation-Maximization Algorithm

In classical measurement data processing theory,the Gaussian-Markov model is usually used as a linear function model,and the weighted least squares method is used to obtain the bestunbiasedlinearestimator of unknown parameters.Thenthe information of estimators isobtained by using the error propagation law.However,with the abundance of measurement methods and the diversity of data sources,the observations structure is complex,the priori information may be inaccurate,and there may be gross errors and systematic errors,etc..In such conditions,the classical least squares parameter estimation method is not exactly valuable.For heteroscedasticity data,when the priori information is inaccurate,weight needs to be re-calculated,which is the category of variance component estimation.There are many literature on variance components estimation,and a lot of researches have been taken on the autocorrelation cross correlation and negative variance of the observation,at the same time,the corresponding simplified method is put forward for the commonly used method.A great number of researchers have studied the outliers in the observation and tried dozens of methods,among which the most popular one is robust M-estimates.However,for observations with gross errors and heterogeneity,the robust M estimate only calculates the estimated value of the modified unknown parameter and does not give a value of the outliers.UsingEM algorithm,the random error and the gross error aretreated as missing data,not only the posteriorvariance-covariance matrix of the observation can be well obtained,the estimation of the outliers can also be calculated.A property of ML estimation is that in estimating variance components it takes no account of the degrees of freedom that are involved in estimating unknown parameters,that is,the estimate of variance components is not unbiased.The combination of EM algorithm and REML estimation not only inherits the advantages of EM algorithm based on maximum likelihood estimation,but also can obtains more accurate unknown parameters.Based on the above analysis,this paper mainly studies the following points:1.Researching the Gauss-Markov Model and Least Square methodsystematically.The hypothesis tests for parameters estimation and system errors and the methods variance component estimation are introduced.An improved method of variance component estimation LSMINQUE is introduced.2.Analysing the linear mixed model,EM algorithm and its properties,at the same time,pointing out the advantages and disadvantages of the maximum likelihood and restricted constrained maximum likelihood variance component estimation methods based on EM algorithm.The numberical result shows the properties of EM algorithm.3.Combining the advantages of restricted maximum likelihood estimation method and EM algorithm,the method of estimating the robust variance component in the mixed model is deduced,and by simulating the data of GPS control network,the validity of the method is proved as well.