High-Dimensional Adaptive Function-on-Scalar Regression

TL;DR

This paper introduces AFSL, a novel adaptive regularization method for high-dimensional function-on-scalar regression, enabling effective variable selection and estimation in genetic studies with many predictors.

Contribution

It develops a functional version of the oracle property for AFSL, applicable even when predictors vastly outnumber observations, advancing statistical methods for genetic data analysis.

Findings

AFSL successfully identifies important genetic mutations affecting lung growth.

The method demonstrates strong theoretical properties in high-dimensional settings.

Simulation and real data analyses validate AFSL's effectiveness.

Abstract

Applications of functional data with large numbers of predictors have grown precipitously in recent years, driven, in part, by rapid advances in genotyping technologies. Given the large numbers of genetic mutations encountered in genetic association studies, statistical methods which more fully exploit the underlying structure of the data are imperative for maximizing statistical power. However, there is currently very limited work in functional data with large numbers of predictors. Tools are presented for simultaneous variable selection and parameter estimation in a functional linear model with a functional outcome and a large number of scalar predictors; the technique is called AFSL for $\textit{Adaptive Function-on-Scalar Lasso}.$ It is demonstrated how techniques from convex analysis over Hilbert spaces can be used to establish a functional version of the oracle property for AFSL over any real separable Hilbert space, even when the number of predictors, $I$, is exponentially large compared to the sample size, $N$. AFSL is illustrated via a simulation study and data from the Childhood Asthma Management Program, CAMP, selecting those genetic mutations which are important for lung growth.