On the functional regression model and its finite-dimensional approximations

The problem of linearly predicting a scalar response Y from a functional (random) explanatory variable \(X=X(t),\ t\in I\) is considered. It is argued that the term “linearly” can be interpreted in several meaningful ways. Thus, one could interpret that (up to a random noise) Y could be expressed as a linear combination of a finite family of marginals \(X(t_i)\) of the process X, or a limit of a sequence of such linear combinations. This simple point of view (which has some precedents in the literature) leads to a formulation of the linear model in terms of the RKHS space generated by the covariance function of the process X(t). It turns out that such RKHS-based formulation includes the standard functional linear model, based on the inner product in the space \(L^2[0,1]\), as a particular case. It includes as well all models in which Y is assumed to be (up to an additive noise) a linear combination of a finite number of linear projections of X. Some consistency results are proved which, in particular, lead to an asymptotic approximation of the predictions derived from the general (functional) linear model in terms of finite-dimensional models based on a finite family of marginals \(X(t_i)\), for an increasing grid of points \(t_j\) in I. We also include a discussion on the crucial notion of coefficient of determination (aimed at assessing the fit of the model) in this setting. A few experimental results are given.