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

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.

💡 Research Summary

The paper addresses prediction in functional linear regression when the observational errors are heteroscedastic, i.e., the variance of the error term depends on the predictor function. The conventional approach reduces dimensionality by projecting the functional covariates onto the leading eigenfunctions of the ordinary (unweighted) covariance operator—essentially a functional principal component analysis (FPCA). While FPCA is optimal under homoscedastic errors, it can be inefficient when variances differ across observations because it ignores the error structure and may allocate too much importance to directions that are noisy.

To remedy this, the authors propose a weighted least‑squares (WLS) framework that incorporates an estimate of the error variance into the basis construction. Specifically, each observation receives a weight (w_i = 1/\sigma_i^2), where (\sigma_i^2) is the (possibly unknown) error variance for that observation. The weighted covariance operator (\Gamma_w = \mathbb{E}