Wavelet-Based Scalar-on-Function Finite Mixture Regression Models

Classical finite mixture regression is useful for modeling the relationship between scalar predictors and scalar responses arising from subpopulations defined by the differing associations between those predictors and responses. Here we extend the classical finite mixture regression model to incorporate functional predictors by taking a wavelet-based approach in which we represent both the functional predictors and the component-specific coefficient functions in terms of an appropriate wavelet basis. In the wavelet representation of the model, the coefficients corresponding to the functional covariates become the predictors. In this setting, we typically have many more predictors than observations. Hence we use a lasso-type penalization to perform variable selection and estimation. We also consider an adaptive version of our wavelet-based model. We discuss the specification of the model, provide a fitting algorithm, and apply and evaluate our method using both simulations and a real data set from a study of the relationship between cognitive ability and diffusion tensor imaging measures in subjects with multiple sclerosis.