Fast function-on-scalar regression with penalized basis expansions.

Area of Science:

• Statistics
• Functional Data Analysis
• Computational Statistics

Background:

• Functional regression models analyze data where responses or predictors are functions.
• Basis function expansions with roughness penalties are common for fitting these models, preventing overfitting.
• Existing penalized ordinary least squares (P-OLS) methods, while effective, can be computationally intensive.

Purpose of the Study:

• To develop and present a computationally efficient penalized generalized least squares (P-GLS) alternative to P-OLS.
• To introduce algorithms for efficient implementation of both P-OLS and P-GLS with automatic smoothing parameter selection.
• To extend statistical inference capabilities, including pointwise confidence intervals, permutation tests, and novel model selection criteria.

Main Methods:

• Recasting penalized ordinary least squares (P-OLS) as a generalized ridge regression estimator.
• Developing a penalized generalized least squares (P-GLS) estimator as an alternative.
• Implementing efficient algorithms for both estimators with automatic smoothing parameter selection.
• Utilizing simulation studies and real-world data (fMRI, weather) for validation.

Main Results:

• Demonstrated computational efficiency improvements for both P-OLS and P-GLS.
• Successfully implemented automatic selection of optimal smoothing parameters.
• Developed and illustrated methods for pointwise confidence intervals, simultaneous inference via permutation tests, and pointwise model selection.
• Validated the methods through simulation and application to functional magnetic resonance imaging (fMRI) and Canadian weather data.

Conclusions:

• The proposed P-GLS and enhanced P-OLS methods offer significant computational advantages for functional regression.
• Automatic smoothing parameter selection and advanced inference techniques enhance the practical utility of these models.
• The developed R package provides accessible tools for researchers to apply these advanced functional data analysis techniques.