# Measure Selection for Functional Linear Model

## TL;DR

This paper introduces a flexible functional linear model that adaptively chooses the measure defining the function space, improving predictive accuracy over traditional models especially for complex data.

## Contribution

It proposes a novel data-adaptive measure selection method for functional linear models, extending the framework beyond the standard Lebesgue measure.

## Key findings

- Improved predictive performance with the adaptive measure approach.
- Consistent outperformance over traditional models in simulations.
- Effective application to COVID-19 and health survey data.

## Abstract

Advancements in modern science have led to an increased prevalence of functional data, which are usually viewed as elements of the space of square-integrable functions $L^2$. Core methods in functional data analysis, such as functional principal component analysis, are typically grounded in the Hilbert structure of $L^2$ and rely on inner products based on integrals with respect to the Lebesgue measure over a fixed domain. A more flexible framework is proposed, where the measure can be arbitrary, allowing natural extensions to unbounded domains and prompting the question of optimal measure choice. Specifically, a novel functional linear model is introduced that incorporates a data-adaptive choice of the measure that defines the space, alongside an enhanced function principal component analysis. Selecting a good measure can improve the model's predictive performance, especially when the underlying processes are not well-represented when adopting the default Lebesgue measure. Simulations, as well as applications to COVID-19 data and the National Health and Nutrition Examination Survey data, show that the proposed approach consistently outperforms the conventional functional linear model.