Variable screening and spatial smoothing in Fréchet regression with application to diffusion tensor imaging
Lei Yan, Xin Zhang, Zhou Lan, Dipankar Bandyopadhyay, Yichao Wu, the Alzheimer’s Disease Neuroimaging Initiative
The Annals of Applied Statistics · 2025 · Volume 19 · Issue 1
Abstract
Modern applications in medical imaging often include high-dimensional predictors and spatially dependent responses in the non-Euclidean space. For example, in imaging-genetics studies, our objective is to study the relationship between single-nucleotide polymorphisms (SNPs), a high-dimensional predictor vector, and diffusion tensor imaging (DTI) responses, which are thousands to millions of voxelwise 3×3 symmetric positive definite (SPD) matrices. In this paper we develop a fast and pragmatic method of regressing spatially associated random responses on a high-dimensional predictor set. Specifically, we focus on two related problems: fast variable screening of high-dimensional predictors and smoothing techniques for non-Euclidean spatially associated responses. Under a Fréchet regression framework (which handles regression of SPD matrix-variate responses on covariates in Euclidean space), we propose a two-stage approach, where a screening method (using distance covariances in metric spaces) is employed to mitigate high-dimensionality (Stage 1), followed by deriving a closed-form solution that powers elegant smoothing of the spatially associated SPD responses (Stage 2). We investigate the finite-sample properties of our method using synthetic data generated under various settings and present illustration via analysis of an imaging-genetics (DTI responses with genetic and demographic predictors) dataset, derived from the Alzheimer’s Disease Neuroimaging Initiative 2. Code for implementing our proposed method is available in the GitHub link: https://github.com/leiyan-ly/Frechet-regression.
Keywords
Diffusion tensor imaging, distance covariance, Fréchet regression, smoothing, sure screening property
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