Regularized regression on compositional trees with application to MRI analysis

TL;DR

This paper introduces a novel regularized regression method tailored for compositional tree-structured data, enabling effective component selection and yielding consistent estimators, with applications demonstrated in brain imaging for Alzheimer's disease.

Contribution

It presents a transformation-free, tree-structured regularization approach for sparse regression on compositional trees, with proven consistency and superior simulation performance.

Findings

Higher accuracy than competing methods in simulations

Identifies meaningful brain region associations in Alzheimer's data

Method is both consistent and model selection consistent

Abstract

A compositional tree refers to a tree structure on a set of random variables where each random variable is a node and composition occurs at each non-leaf node of the tree. As a generalization of compositional data, compositional trees handle more complex relationships among random variables and appear in many disciplines, such as brain imaging, genomics and finance. We consider the problem of sparse regression on data that are associated with a compositional tree and propose a transformation-free tree-based regularized regression method for component selection. The regularization penalty is designed based on the tree structure and encourages a sparse tree representation. We prove that our proposed estimator for regression coefficients is both consistent and model selection consistent. In the simulation study, our method shows higher accuracy than competing methods under different scenarios. By analyzing a brain imaging data set from studies of Alzheimer's disease, our method identifies meaningful associations between memory declination and volume of brain regions that are consistent with current understanding.