| In modern society,the technology which develops rapidly brings a lot of data with huge scale.Analyzing these data and fetching useful information presents new chal-lenges to statistics.Particularly,using the graphical model to encode the conditional dependency structure among multiple variables based on a large number of data is not only challenge but also practical.The precision matrix,which is the inverse of the co-variance matrix,is closely related to the graphical model.When the dimensionality is high,that is,the sample size is much smaller than the sample dimensions,it is difficult to estimate the precision matrix by firstly calculating the sample covariance matrix and then inversing it due to the singularity of the sample covariance matrix.Although there are many ways to estimate the precision matrix by optimizing the maximum likelihood function or transforming it into linear model estimation,when these methods are in the face of the ultra-high dimensions,computational complexity is still be concerned.Thus,under ultra-high dimensional settings,a screening procedure is generally sug-gested before variable selection to reduce computational costs.However,most existing screening methods examine the marginal correlations,thus not suitable to discover the conditional dependence in graphical models.To overcome this issue,we propose a new procedure called graphical uniform joint screening(GUS)for edge identification in graphical models.Instead of screening out edges nodewisely,GUS utilizes a uniform threshold for all statistics indicating the significance of different edges to adapt to vari-ous kinds of graphical structures.We demonstrate that GUS enjoys the sure screening property and even the screening consistency by preserving the rankings of the signifi-cant edges.Furthermore,a scalable implementation of GUS is developed for big data applications.Simulation and real data studies are provided to illustrate the effectiveness of the proposed method. |
