In this paper, we study and analyze the regularized least squares for functional linear regression model. The approach is to use the reproducing kernel Hilbert space framework and the integral operators. We show with a more general and realistic assumption on the reproducing kernel and input data statistics that the rate of excess prediction risk by the regularized least squares is minimax optimal.

Introduction

There are increasing cases in practice where the data are collected in the form of random functions or curves. This type of data is becoming more prevalent throughout science, engineering and financial market, as automated on-line data collection facilities are becoming more ubiquitous. Many classical statistical tools and models for multivariate analysis, such as principal components analysis, canonical correlation analysis and linear model are then extended to the infinite-dimensional functional domain. In this paper, we consider the functional linear model where Y is a scalar response, X : I → R is a square integrable functional predictor defined over compact domain I ⊂ R, α۰ is the intercept, β۰ : I → R is the slope function. and  is the random noise with mean 0 and finite variance σ ۲ . Functional linear model was introduced by J.O. Ramsay and C.J. Dalzell [11] and first written in its commonly encountered form (1) by T. Hastie and C. Mallows [7]. Some recent research on the statistical analysis of (1) includes [2, 3, 6, 15, 16]. In this paper, we focus on the random design where X is a path of a square integrable stochastic process defined over I and is independent of . Without loss of much generality, throughout the paper we assume E(X) = 0 and the intercept α۰ = ۰, since the intercept can be easily estimated. Let L 2 be the Hilbert space of square integrable functions on I (with respect to Lebesgue measure) with standard inner production < u, v >= R I u(s)v(s)ds and norm kuk = R I u 2 (s)ds
۱/.