| Graphical models is an important tool to study the relationship among random variables,where the nodes represent random variables and the edges represent conditional dependency of two random variables.In addition to the node variables,there often exists additional information,which we call covariates.The covariates may further influence the dependency relationships.However,most of the existing work on graphical models only considers the node variables.In this paper,we study the problem of structure learning of graphical models from the data with covariates.Specifically,the contributions can be summarized into two aspects as follows:Firstly,in the framework of sparse regularization,we propose a sparse Gaussian graphical models with the covariates by assuming that the conditional dependency between two variables is a linear function of the covariates.Our model is interpretable and easy to sovle.We then employ the coordinate descent algorithm to solve the model.Our experiments show that the covariates dependent Gaussian graphical model performs better than the counterpart without covariates,which shows the effectiveness and efficiency of the proposed model.Secondly,we utilize the graphical model tool to study the PM2.5 of 31 provincial capital cities and municipalities in China.We reconstruct the network which represent the conditional depedency relations among 31 cities in terms of PM2.5.Based on this network,we further employ the spectral clustering method to detect the community structure of these 31 cities.The results can help to deal with the haze in some sense. |
