# Covariate Adjusted Functional Mixed Membership Models

## TL;DR

This paper introduces a new statistical model that adjusts for covariates in functional data, offering insights into brain development differences in children with autism.

## Contribution

The paper introduces a covariate-adjusted functional mixed membership model with identifiability guarantees and applies it to EEG data in autism.

## Key findings

- The model allows for covariate-dependent structures in functional mixed membership frameworks.
- Individuals with ASD show smaller developmental changes in alpha oscillations compared to typically developing children.
- The method provides novel insights into heterogeneity in developmental brain changes.

## Abstract

Mixed membership models are a flexible class of models used for unsupervised learning that allow each observation to partially belong to multiple clusters or features. In this article, we extend the framework of functional mixed membership models to allow for covariate-dependent modeling structures. The framework uses a multivariate Karhunen-Loève decomposition, which allows for a scalable and flexible model. Within this framework, we establish a set of sufficient conditions to ensure the identifiability of the mean, covariance, and allocation structure up to a permutation of the labels. This article is primarily motivated by studies on functional brain imaging through electroencephalography (EEG) of children with autism spectrum disorder (ASD). Using the proposed framework, we provide novel insight into the heterogeneity of developmental changes in alpha oscillations and show that individuals with ASD have smaller developmental changes compared to their typically developing counterparts.