Hierarchical Adaptive Regression Kernels for Regression With Functional Predictors

Abstract

We propose a new method for regression using a parsimonious and scientifically interpretable representation of functional predictors. Our approach is designed for data that exhibit features such as spikes, dips, and plateaus whose frequency, location, size, and shape varies stochastically across subjects. We propose Bayesian inference of the joint functional and exposure models, and give a method for efficient computation. We contrast our approach with existing state-of-the-art methods for regression with functional predictors, and show that our method is more effective and efficient for data that include features occurring at varying locations. We apply our methodology to a large and complex dataset from the Sleep Heart Health Study, to quantify the association between sleep characteristics and health outcomes. Software and technical appendices are provided in the online supplementary materials.

Key Words:

• Electroencephalogram
• Functional data analysis
• Functional linear model
• Kernel mixture
• Lévy adaptive regression kernels
• Nonparametric Bayes

ACKNOWLEDGMENTS

The authors thank the referees and Associate Editor for their thorough reading and excellent recommendations, and Jeff Goldsmith for providing software and assistance in using PFR. Drs. Crainiceanu and Ruppert are supported by the grant R01NS060910 from the National Institute of Neurological Disorders and Stroke, and Dr Woodard is supported by the National Science Foundation grant CMMI-0926814. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of Neurological Disorders and Stroke or the National Institutes of Health.