Functional Mixed Membership Models

Abstract

Mixed membership models, or partial membership models, are a flexible unsupervised learning method that allows each observation to belong to multiple clusters. In this paper, we propose a Bayesian mixed membership model for functional data. By using the multivariate Karhunen-Loève theorem, we are able to derive a scalable representation of Gaussian processes that maintains data-driven learning of the covariance structure. Within this framework, we establish conditional posterior consistency given a known feature allocation matrix. Compared to previous work on mixed membership models, our proposal allows for increased modeling flexibility, with the benefit of a directly interpretable mean and covariance structure. Our work is motivated by studies in functional brain imaging through electroencephalography (EEG) of children with autism spectrum disorder (ASD). In this context, our work formalizes the clinical notion of "spectrum" in terms of feature membership proportions. Supplementary materials, including proofs, are available online. The R package BayesFMMM is available to fit functional mixed membership models.

Keywords: Bayesian Methods; EEG; Functional Data; Mixed Membership Models; Neuroimaging.

Fig. 1

Generative model illustration. (Left panel) Data generated under the functional clustering framework. (Right panel) Data generated under a mixed membership framework.

Fig. 2

R-MISE values for the latent feature means and cross-covariances, as well as RMSE values for the allocation parameters, evaluated as sample size increases.

Fig. 3

AIC, BIC, DIC, and the average log-likelihood evaluated for each of the 10 simulated data sets.

Fig. 4

(Left Panel) Recovered means of model-based functional clustering with 4 clusters. (Right Panel) Alpha frequency patterns for a sample of EEG recordings from the T8 electrode of children (TD and ASD), styled by estimated cluster membership.

Fig. 5

(Top Panel) Posterior median and 95% credible intervals (pointwise CI in dark gray and simultaneous CI in light gray) of the mean function for each functional feature. (Bottom Panel) Posterior median of each individual’s (feature-1)-membership, stratified by clinical cohort (cohort average (feature-1)-membership denoted by triangles).