[Paper Review] Interpretable Principal Components Analysis for Multilevel Multivariate Functional Data, with Application to EEG Experiments

Jun Zhang, Greg J. Siegle+2 more|arXiv (Cornell University)|Sep 17, 2019

Spectroscopy and Chemometric Analyses36 references4 citations

TL;DR

This paper proposes a novel interpretable principal component analysis method for multilevel multivariate functional data, combining sparsity across frequency bands and temporal localization via convex optimization with block Frobenius and matrix L1-norm penalties. Applied to EEG data, it reveals that elevated beta/gamma power in the occipital cortex is significantly linked to reduced dissociation in trauma-exposed individuals, offering new neurophysiological insights into emotional dysregulation.

ABSTRACT

Many studies collect functional data from multiple subjects that have both multilevel and multivariate structures. An example of such data comes from popular neuroscience experiments where participants' brain activity is recorded using modalities such as EEG and summarized as power within multiple time-varying frequency bands within multiple electrodes, or brain regions. Summarizing the joint variation across multiple frequency bands for both whole-brain variability between subjects, as well as location-variation within subjects, can help to explain neural reactions to stimuli. This article introduces a novel approach to conducting interpretable principal components analysis on multilevel multivariate functional data that decomposes total variation into subject-level and replicate-within-subject-level (i.e. electrode-level) variation, and provides interpretable components that can be both sparse among variates (e.g. frequency bands) and have localized support over time within each frequency band. The sparsity and localization of components is achieved by solving an innovative rank-one based convex optimization problem with block Frobenius and matrix $L_1$-norm based penalties. The method is used to analyze data from a study to better understand reactions to emotional information in individuals with histories of trauma and the symptom of dissociation, revealing new neurophysiological insights into how subject- and electrode-level brain activity are associated with these phenomena.

Motivation & Objective

• To address the lack of methods for analyzing multilevel multivariate functional data in neuroscience, particularly EEG data with hierarchical (subject and electrode-level) and multivariate (frequency bands) structures.
• To develop a functional principal component analysis method that simultaneously achieves sparsity across variates (frequency bands) and localization in time for improved interpretability.
• To decompose total variation into subject-level and replicate-within-subject-level (electrode-level) components to disentangle whole-brain and regional brain activity patterns.
• To enable detection of neurophysiological phenomena such as blunted or sustained responses to emotional stimuli across different brain regions and individuals.
• To provide a flexible, generalizable framework that reduces to standard FPCA in special cases and is implemented in an R package for broader adoption.

Proposed method

• Proposes a localized sparse-variate FPCA (LVPCA) framework that models multilevel multivariate functional data through separable subject-specific and replicate-within-subject latent processes.
• Uses a convex optimization problem based on rank-one approximation to estimate principal components with both sparsity (via matrix L1-norm penalties) and temporal localization (via block Frobenius norm penalties).
• Employs method of moments (MoM) to estimate subject-level and within-subject covariance operators under a separable covariance structure, enabling hierarchical decomposition of variation.
• Incorporates regularization through block Frobenius and matrix L1-norm penalties to induce sparsity across frequency bands and localized support within each band’s time domain.
• Adapts the framework for unbalanced designs by modifying covariance estimators to use available cross-products, and allows pre-smoothing for sparse or irregularly spaced time grids.
• The method is implemented in an R package 'LVPCA' and generalizes to multi-way, nested, or crossed designs, with special cases recoverable by setting regularization parameters to zero.

Experimental results

Research questions

• RQ1 How can we jointly model multilevel (subject and electrode) and multivariate (frequency bands) functional data in EEG studies to uncover interpretable neural patterns?
• RQ2 What is the role of beta and gamma band power in the occipital cortex in relation to dissociation symptoms among individuals with a history of trauma?
• RQ3 Are there distinct patterns of alpha band activity in the frontal cortex that differentially predict dissociation in trauma-exposed versus non-exposed individuals?
• RQ4 Can a unified FPCA framework simultaneously achieve sparsity across frequency bands and temporal localization within bands to improve neurophysiological interpretability?
• RQ5 How do subject-level and electrode-level components interact to explain neural responses to emotional stimuli in trauma-related disorders?

Key findings

• Elevated beta and gamma power in the occipital cortex relative to other brain regions is significantly associated with reduced dissociation symptoms (adjusted p-value = 0.036) in individuals with a history of trauma.
• A trend is observed where increased alpha power in the right frontal cortex (F8 and FC6) is positively associated with higher dissociation in trauma-exposed individuals, though not significant after multiple testing correction (adjusted p-value = 0.082).
• The method successfully isolates subject-level components that capture whole-brain variability and replicate-within-subject components that reflect regional (electrode-level) variation, enabling disentangled interpretation.
• The LVPCA method identifies localized, sparse components that highlight specific time-frequency patterns linked to neurophysiological mechanisms of emotional processing.
• The approach reveals that neural responses to emotional stimuli differ between trauma-exposed and non-exposed individuals, particularly in the distribution of power across frequency bands and brain regions.
• The method is generalizable and reduces to standard FPCA, sparse FPCA, or localized FPCA when regularization parameters are set to zero, demonstrating flexibility and robustness.