Gaussian Graphical Model Selection By The E-MS Algorithm

Graphical model,which is a powerful tool to deal with the complex high-dimensional problems,has been widely used in various fields,such as bioinformatics and machine learning.As an important and challenging problem in the graphical model study,the graphical model selection has attracted numerous scholars' attention.In the presence of complete data,Meinshausen and Bühlmann(2006),Yuan and Lin(2007)and Friedmam,Hastie and Tibshirani(2008)proposed a series of model section methods for penalized likelihood maximization.In the presence of missing data,St?dler and Bühlman(2012)used the EM algorithm to maximize the penalized observation likelihood;Thai,Hunter and Akametalu(2014)proposed the m-CCCP which converges faster than the EM algorithm.Jiang,Nguyen and Rao(2015)indicated that the traditional model selection method based on EM has some limitations is dealing with model selection problem in the presence of missing data,and proposed the E-MS algorithm,which is more effective than the model selection method based on EM.In this paper,we use the E-MS algorithm to deal with Gaussian graphical model selection problem in the presence of missing data.We introduce the E-MS algorithm,and then present the concrete steps of Gaussian graphical model selection by the E-MS algorithm in the case of missing data and the corresponding theoretical derivation.Via simulation study in the situation when the number of vertex is 3,4 and 5,we compare the E-MS algorithm and the traditional method of EM with BIC,and verify that the E-MS algorithm has a better performance in dealing with the Gaussian graphical model selection problem.With the increase of the vertex number,the number of candidate models increases exponentially.The optimal model is difficult to be selected in the MS step of each iteration.For the high-dimensional Gaussian graphical model selection problem,we use the simulated annealing(SA)to optimize the MS step,and find the optimal model in the current iteration via the random search method.The specific optimization method and procedure is presented in this paper,and the feasibility of E-MS with SA method is verified through numerical simulation.