Structured Local Sparsity For Partially Functional Quantile Regression Model With Interactions

With the advancement in data science,much attention has been received on the problem of functional data.In practice,people also have access to scalar variables besides functional covariates.Despite works focus on partially functional models,none of them put efforts on developing interaction models for both functional covariates and scalar covariates.A large number of literature studies showed that considering the interactions between variables will effectively improve the accuracy of estimation and prediction.In additional,response variable usually possesses heterogeneity or thick tail.Inspired by above problems,we proposed partially functional interaction quantile regression model.Based on the proposed model,we developed a penalized quantile regression with sparse group functional MCP penalty to accommodate the hierarchical constraints.The proposed approach has three distinctive features:(1)It enables us to explore the entire conditional distribution of the response variable,and more robust than least squares regression;(2)It gives us a more predictive model;(3)It delivers us a more explanatory estimator by imposing hierarchical conditions.We provided efficient algorithms for proposed methods based on Local Quadratic Approximation(LQA)and MM methods.We also provide some theoretical support,showing that the proposed estimators enjoy hierarchical interaction selection and estimation consistency under certain conditions.Furthermore,simulation studies and a real data analysis on Tecator datasets showed the superiority of the proposed method.