  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.
