Supervised basis functions applied to functional regression and classification

In fitting functional linear models, including scalar-on-function regression (SoFR) and function-on-function regression (FoFR), the intrinsically infinite dimension of the problem often demands an limitation to a subspace spanned by a finite number of basis functions. In this sense, the choice and construction of basis functions matters. We discuss herein certain supervised choices of basis functions for regression/classification with densely/sparsely observed curves, and give both numerical and theoretical perspectives. For SoFR, the functional principal component (FPC) regression may fail to provide good estimation or prediction if the response is highly correlated with some excluded FPCs. This is not rare since the construction of FPCs never involves the response. We hence develop regression on functional continuum (FC) basis functions whose framework includes, as special cases, both FPCs and functional partial least squares (FPLS) basis functions. Aiming at the binary classification of functional data, we then propose the continuum centroid classifier (CCC) built upon projections of functional data onto the direction parallel to FC regression coefficient. One of the two subtypes of CCC (asymptotically) enjoys no misclassification. Implementation of FPLS traditionally demands that each predictor curve be recorded as densely as possible over the entire time span. This prerequisite is sometimes violated by, e.g., longitudinal studies and missing data problems. We accommodate FPLS for SoFR to scenarios where curves are sparsely observed. We establish the consistency of proposed estimators and give confidence intervals for responses. FPLS is widely used to fit FoFR. Its implementation is far from unique but typically involves iterative eigen decomposition. We introduce an new route for FoFR based upon Krylov subspaces. The method can be expressed in two equivalent forms: one of them is non-iterative with explicit forms of estimators and predictions, facilitating the theoretical derivation; the other one stabilizes numerical outputs. Our route turns out to be less time-consuming than other methods with competitive accuracy.