Weighted least squares methods for prediction in the functional data linear model

TL;DR

This paper proposes weighted least squares methods for functional linear regression prediction, demonstrating their effectiveness over traditional PCA-based approaches, especially with heteroscedastic errors, supported by theoretical and numerical analysis.

Contribution

It introduces an alternative basis selection via weighted least squares for functional linear regression, improving prediction accuracy under heteroscedastic errors.

Findings

Weighted least squares basis can outperform PCA basis in heteroscedastic settings.

The method remains effective even with inaccurate variance models.

Advantages are consistent across all dimensions of the problem.

Abstract

The problem of prediction in functional linear regression is conventionally addressed by reducing dimension via the standard principal component basis. In this paper we show that an alternative basis chosen through weighted least-squares, or weighted least-squares itself, can be more effective when the experimental errors are heteroscedastic. We give a concise theoretical result which demonstrates the effectiveness of this approach, even when the model for the variance is inaccurate, and we explore the numerical properties of the method. We show too that the advantages of the suggested adaptive techniques are not found only in low-dimensional aspects of the problem; rather, they accrue almost equally among all dimensions.