Statistical Analyses And Applications For Missing Data Based On EM Algorithm

EM algorithm is a powerful iterative algorithm in estimating parameters of the Maximum Likelihood Estimation, which is proposed by Dempster, Laird and Rubin, it presents a standard and universal framework in solving the MLE with missing data. There are two kinds of missing situations: on one hand, the missing data evidently exists in our models, on the other hand, in order to solve the complicated computation of the MLE of the observed data, adding some extra data will turn the difficult process into a series of simple problems of optimization, in that way, the original data naturally becomes the incomplete data. Since EM algorithm owns so many good qualities, it is widely used in many analyses with missing data, there is no doubt that some scholars regard EM algorithm and missing data as a twin sister. Meanwhile, mixture model is not only an efficient tool in analyzing complicated social phenomenon with simple structure but also provides a model for data’s homogeneity and heterogeneity, so mixture model gains great interests of many people nowadays.The contents of our research can be summarized as following aspects:Firstly, we mainly outline the worth and background of our paper, as well as the current situation of domestic and foreign scholars’ research.Secondly, we introduce the reasons and the mechanism of missing data; then, we give a simple introduction of related theory of the EM algorithm. In order to have a deep knowledge of the EM algorithm, several explanations of the algorithm are proposed.Thirdly, we mainly make an introduction of the application of the questionnaire with missing data.In order to present the practical values of the EM algorithm, we set the Origin-Destination questionnaire as an example and use the algorithm to replenish the missing data, as well as estimate the parameter.Finally, we investigate the parameter estimate of the Gaussian mixture model and the Binomial-Poisson hierarchy as the represent of the multiple hierarchy model, when we analyze the two-step Gaussian mixture model, we separate the observed value into two types: in the first situation, regarding the data as two part, we obtain the iterative formula under the complete-data and the incomplete-data situations; in the second situation, the data is regarded as an integrated part, since the process of parameter estimation under incomplete-data is the same as the first situation, here we just estimate the parameter of complete-data; at last, we perform a random simulation with Matlab software to illustrate the convergence and other good qualities of EM algorithm.From the process of parameter estimation of Binomial-Poisson multiple hierarchy model, the EM algorithm shows excellent qualities in solving the parameter estimation of the complicated mixture models.