Statistical Analysis of Functions on Surfaces, With an Application to Medical Imaging

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

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 Alzheimer's Disease Neuroimaging Initiative and 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.

Anatomically compliant modes of variations: New tools for brain connectivity

Anatomical complexity and data dimensionality present major issues when analysing brain connectivity data. The functional and anatomical aspects of the connections taking place in the brain are in fact equally relevant and strongly intertwined. However, due to theoretical challenges and computational issues, their relationship is often overlooked in neuroscience and clinical research. In this work, we propose to tackle this problem through Smooth Functional Principal Component Analysis, which enables to perform dimensional reduction and exploration of the variability in functional connectivity maps, complying with the formidably complicated anatomy of the grey matter volume. In particular, we analyse a population that includes controls and subjects affected by schizophrenia, starting from fMRI data acquired at rest and during a task-switching paradigm. For both sessions, we first identify the common modes of variation in the entire population. We hence explore whether the subjects' expressions along these common modes of variation differ between controls and pathological subjects. In each session, we find principal components that are significantly differently expressed in the healthy vs pathological subjects (with p-values < 0.001), highlighting clearly interpretable differences in the connectivity in the two subpopulations. For instance, the second and third principal components for the rest session capture the imbalance between the Default Mode and Executive Networks characterizing schizophrenia patients.