Clusterwise Functional Linear Regression Models

Classical clusterwise linear regression is a useful method for investigating the relationship between scalar predictors and scalar responses with heterogeneous variation of regression patterns for different subgroups of subjects. This paper extends the classical clusterwise linear regression to incorporate multiple functional predictors by representing the functional coefficients in terms of a functional principal component basis. We estimate the functional principal component coefficients based on M-estimation and K-means clustering algorithm, which can classify the data into clusters and estimate clusterwise coefficients simultaneously. One advantage of the proposed method is that it is robust and flexible by adopting a general loss function, which can be broadly applied to mean regression, median regression, quantile regression and robust mean regression. A Bayesian information criterion is proposed to select the unknown number of groups and shown to be consistent in model selection. We also obtain the convergence rate of the set of estimators to the set of true coefficients for all clusters. Simulation studies and real data analysis show that the proposed method is easily implemented, and it consequently improves previous works and also requires much less computing burden than existing methods.