On a general definition of the functional linear model

TL;DR

This paper proposes a comprehensive framework for the functional linear model that encompasses various existing models, including those based on inner products, marginals, and projections, supported by theoretical consistency results and practical experiments.

Contribution

It introduces a unified general formulation of the functional linear model, covering multiple existing models and providing theoretical and experimental validation.

Findings

The general model includes standard and marginal-based functional linear models.

Consistency results are established for the proposed framework.

Experimental results demonstrate practical advantages of the unified approach.

Abstract

A general formulation of the linear model with functional (random) explanatory variable $X = X(t), t \in T$ , and scalar response Y is proposed. It includes the standard functional linear model, based on the inner product in the space $L^2[0,1]$, as a particular case. It also includes all models in which Y is assumed to be (up to an additive noise) a linear combination of a finite or countable collections of marginal variables X(t_j), with $t_j\in T$ or a linear combination of a finite number of linear projections of X. This general formulation can be interpreted in terms of the RKHS space generated by the covariance function of the process X(t). Some consistency results are proved. A few experimental results are given in order to show the practical interest of considering, in a unified framework, linear models based on a finite number of marginals $X(t_j)$ of the process $X(t)$.