A tree approach for variable selection and its random forest

The Sure Independence Screening (SIS) provides a fast and efficient ranking for the importance of variables for ultra-high dimensional regressions. However, classical SIS cannot eliminate false importance in the ranking, which is exacerbated in nonparametric settings. To address this problem, a novel screening approach is proposed by partitioning the sample into subsets sequentially and creating a tree-like structure of sub-samples called SIS-tree. SIS-tree is straightforward to implement and can be integrated with various measures of dependence. Theoretical results are established to support this approach, including its "sure screening property". Additionally, SIS-tree is extended to a forest with improved performance. Through simulations, the proposed methods are demonstrated to have great improvement comparing with existing SIS methods. The selection of a cutoff for the screening is also investigated through theoretical justification and experimental study. As a direct application, classifications of high-dimensional data are considered, and it is found that the screening and cutoff can substantially improve the performance of existing classifiers.