Measure Selection for Functional Linear Model

2025-08-30

AI Summary

Traditional functional data analysis relies on the Lebesgue measure over a fixed domain, limiting its adaptability to unbounded domains or non-uniformly distributed data—thereby constraining model expressivity and predictive accuracy. To address this, we propose a data-driven framework for adaptive measure selection, constructing linear functional models within a Hilbert space endowed with an arbitrarily defined measure. By optimizing the inner-product structure, our approach enhances model flexibility. The method integrates measure-adaptive functional principal component analysis with generalized functional regression, enabling principled modeling on unbounded domains and non-standard data distributions. Extensive experiments on synthetic data, as well as real-world COVID-19 epidemiological and NHANES health survey datasets, demonstrate that our measure-adaptive models significantly outperform conventional Lebesgue-based approaches in prediction accuracy. These results empirically validate that judicious choice of measure—not merely the functional form—is critical for improving statistical performance in functional data analysis.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Web Mining and Content Analysis: Web measurements
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success

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.

Problem

• Optimizing measure selection for functional linear models
• Enhancing predictive performance with data-adaptive measures
• Extending functional analysis beyond Lebesgue measure constraints

Innovation

• Data-adaptive measure selection for functional space
• Enhanced functional principal component analysis framework
• Flexible measure framework extending to unbounded domains