Title: On Wavelet-Based Methods for Scalar-on-Function Regression
Description: This thesis consists of work done on three projects which extend and employ wavelet-based functional linear regression. In the first project, we propose a wavelet-based approach to functional mixture regression. In our approach, the functional predictor and the unknown component-specific coefficient functions are projected onto an appropriate wavelet basis and simultaneous regularization and estimation are achieved via an l1-penalized fitting procedure that is carried out using an expectation-maximization algorithm. We provide an efficient fitting algorithm, propose a technique for constructing non-parametric confidence bands, demonstrate the performance of our methods through extensive simulations, and apply them to real data in order to investigate the relationship between fractional anisotropy profiles and cognitive function in subjects with multiple sclerosis. In the second project, we propose a new wavelet-based estimator for estimating the coefficient function in a functional linear model.

Our estimator attempts to take account of the structured sparsity of the wavelet coefficients used to represent the coefficient function in the fitting procedure. We propose a characterization of the neighborhood structure of wavelet coefficients and exploit this structure in our estimation procedure. We discuss the motivation for our penalized estimator, describe the fitting procedure which can be carried out with existing software, and examine properties of the estimator through simulation. The third and final project explores three novel approaches to using functional data derived from optical coherence tomography devices for diagnosing glaucoma. The first approach uses wavelet-based functional logistic regression to develop predictive models based on measures of retinal nerve fiber layer (RNFL) thickness. The estimates are obtained via an elastic net penalized fitting procedure. The second and third approaches consist of using novel measures of RNFL characteristics to discriminate between healthy and glaucomatous eyes.

The three new approaches are compared with commonly used predictive models using data from a case-control study of African American subjects recruited by ophthalmologists at the Harkness Eye Center of Columbia University.