Mixture of functional linear models and its application to CO2-GDP functional data

Functional linear models are important tools for studying the relationship between functional response and covariates. However, if subjects come from an inhomogeneous population that demonstrates different linear relationship between the response and covariates among different subpopulations/clusters, a single functional linear model is no longer adequate for the data. A new class of mixtures of functional linear models for the analysis of heterogeneous functional data is introduced. Identifiability is established for the proposed class of mixture models under mild conditions. The proposed estimation procedures combine the ideas of local kernel regression, functional principal component analysis and EM algorithm. A generalized likelihood ratio test based on a conditional bootstrap is given as to whether the regression coefficient functions are constant. A Monte Carlo simulation study is conducted to examine the finite sample performance of the new methodology. Finally, the analysis of CO2-GDP data reveals the dynamic patterns of relationship between CO2 and GDP among different countries.