Dimensionality Reduction Method For Interaction Model Based On Two-stage Sliced Inverse Regression

The processing of high-dimensional data has been increasingly stressed in modern statistics.We usually need to take into account interactions when dealing with the statistical inference in practices,which reflect the combined action on a single dependent variable by two independent ones.It doubled the difficulty when dealing with high dimensional data.Taking the second order or even high order interaction into account makes it even more difficult.When we deal with the interaction effects model or high dimensional interactions,almost all existing methods are less effective because of the large dimensionality.Therefore the statistical analysis and modeling of high-dimensional data are the main focuses in recent statistic researches.The two popular research areas in statistics are the model selection and the dimension reduction.Here we pay more attention to the dimension reduction.In this paper,we present a dimension reduction method for interaction model based on two-stage sliced inverse regression.In the first step,we consider a main effect model and use the sliced inverse regression(SIR)to select the main effects that affect the response variable.In the second step,we consider an interaction model that involves only the main effects selected from the first step as well as all possible second order interactions.We use the partial inverse regression(PIRE)to do the dimension reduction.We conduct simulation studies and real data analysis to compare our two-stage sliced inverse regression with SIR and partial least square method(PLS).When the dimension is high and the data comes from linear model,our proposed method is very similar to PLS,both of which are better than the PIRE.When the data comes from non-linear model,our proposed method outperforms PIRE and PLS clearly.