Most-predictive design points for functional data predictors

We suggest a way of reducing the very high dimension of a functional predictor, X, to a low number of dimensions chosen so as to give the best predictive performance. Specifically, if X is observed on a fine grid of design points t(1),..., t(r), we propose a method for choosing a small subset of these, say t(i1),..., t(ik), to optimize the prediction of a response variable, Y. The values t(ij) are referred to as the most predictive design points, or covariates, for a given value of k, and are computed using information contained in a set of independent observations (X-i, Y-i) of (X, Y). The algorithm is based on local linear regression, and calculations can be accelerated using linear regression to preselect the design points. Boosting can be employed to further improve the predictive performance. We illustrate the usefulness of our ideas through simulations and examples drawn from chemometrics, and we develop theoretical arguments showing that the methodology can be applied successfully in a range of settings.