Double screening via Gram-Schmidt orthogonalization for linear regression with interactions

The linear model containing interaction effects naturally expands the linear model by considering the interactions between the variables. When selecting the variables, it is very important to maintain the genetic conditions between the main effect and the interaction effect. Existing regularization methods usually achieve this goal by solving complex optimization problems, which usually requires high calculation costs, so it is not feasible for high-dimensional data. In this article, we have proposed a new two-stage variable selection method, called double screening via Gram-Schmidt orthogonalization (DSGS), for linear model with interaction effects. In the first stage, we combine the main effects selected by orthogonalization with the main effects corresponding to the interaction terms selected by Pearson correlation to obtain an important set of main effects. In the second stage, the SCAD method is used to make the final selection of main effects and interaction effects. In terms of model, this method maintains genetic conditions; in terms of calculation, the DSGS method is simple and fast. And we have theoretically proven that the screening results of the first stage have the sure independence screening property. Numerical results also prove the performance of this method.