Functional Linear Regression Via Canonical Analysis1

We study regression models for the situation where both dependent and independent variables are square integrable stochastic processes. Questions concerning definition and existence of the corresponding functional linear regression models and some basic properties are explored. We derive a representation of the regression parameter function in terms of the canonical components of the processes involved. This representation establishes a connection between functional regression and functional canonical analysis and leads to new approaches for functional linear regression analysis. Several specific procedures for the estimation of the regression parameter function using canonical expansions are explored. As an application example, we present an analysis of mortality data for cohorts of medflies, obtained in experimental studies of aging and longevity.