Interpretable discriminant analysis for functional data supported on random nonlinear domains with an application to Alzheimer's disease

TL;DR

This paper presents a new interpretable classification framework for functional data on nonlinear, random manifolds, applied to Alzheimer's disease diagnosis using cortical surface features, avoiding covariance estimation.

Contribution

It introduces a regularized multivariate functional linear regression model for nonlinear manifold data, with theoretical analysis and real-world application to neuroimaging data.

Findings

Identified cortical features predictive of Alzheimer's disease

Achieved accurate out-of-sample prediction error bounds

Discovered neurobiologically consistent discriminant directions

Abstract

We introduce a novel framework for the classification of functional data supported on nonlinear, and possibly random, manifold domains. The motivating application is the identification of subjects with Alzheimer's disease from their cortical surface geometry and associated cortical thickness map. The proposed model is based upon a reformulation of the classification problem as a regularized multivariate functional linear regression model. This allows us to adopt a direct approach to the estimation of the most discriminant direction while controlling for its complexity with appropriate differential regularization. Our approach does not require prior estimation of the covariance structure of the functional predictors, which is computationally prohibitive in our application setting. We provide a theoretical analysis of the out-of-sample prediction error of the proposed model and explore the finite sample performance in a simulation setting. We apply the proposed method to a pooled dataset from the Alzheimer's Disease Neuroimaging Initiative and the Parkinson's Progression Markers Initiative. Through this application, we identify discriminant directions that capture both cortical geometric and thickness predictive features of Alzheimer's disease that are consistent with the existing neuroscience literature.