Functional Partial Least-Squares: Optimal Rates and Adaptation

Abstract

We consider the functional linear regression model with a scalar response and a Hilbert space-valued predictor, a well-known ill-posed inverse problem. We propose a new formulation of the functional partial least-squares (PLS) estimator related to the conjugate gradient method. We provide the first optimality result for functional PLS showing that the estimator achieves the (nearly) optimal convergence rate on a class of ellipsoids and propose a data-driven early stopping rule that adapts to the unknown degree of ill-posedness. We find in simulations that our estimator performs favorably compared to the principal component regression estimator and requires a smaller number of functional components. Using our estimator, we study the nonlinear temperature effect on corn and soybean yields and find some evidence of the adaptation of US agriculture to climate change.