Measure selection for functional linear model

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 L2. Core methods in functional data analysis, such as functional principal component analysis, are typically grounded in the Hilbert structure of L2 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.