Regularized regression on compositional trees with application to MRI analysis

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 analysing a brain imaging data set from studies of Alzheimer's disease, our method identifies meaningful associations between memory decline and volume of brain regions that are consistent with current understanding.