Bayesian temporal biclustering with applications to multi-subject neuroscience studies

TL;DR

A Bayesian model for temporal biclustering featuring nested partitions, where a time-invariant partition of subjects induces a time-varying partition of measurements, which indicates that the proposed model can combine information from potentially many subjects to discover a set of interpretable, dynamic patterns.

Abstract

We consider the problem of analyzing multivariate time series collected on multiple subjects, with the goal of identifying groups of subjects exhibiting similar trends in their recorded measurements over time as well as time-varying groups of associated measurements. To this end, we propose a Bayesian model for temporal biclustering featuring nested partitions, where a time-invariant partition of subjects induces a time-varying partition of measurements. Our approach allows for data-driven determination of the number of subject and measurement clusters as well as estimation of the number and location of changepoints in measurement partitions. To efficiently perform model fitting and posterior estimation with Markov Chain Monte Carlo, we derive a blocked update of measurements' cluster-assignment sequences. We illustrate the performance of our model in two applications to functional magnetic resonance imaging data and to an electroencephalogram dataset. The results indicate that the proposed model can combine information from potentially many subjects to discover a set of interpretable, dynamic patterns. Experiments on simulated data compare the estimation performance of the proposed model against ground-truth values and other statistical methods, showing that it performs well at identifying ground-truth subject and measurement clusters even when no subject or time dependence is present.

Figures (12)

• Figure 1: Illustration of proposed biclustering model for an idealized scenario with 11 subjects, 5 measurements and 6 time steps. (Left) Data displayed as a three-dimensional array. A cell represents an observation for a specific subject, measurement and time step. (Right) The proposed model clusters subjects into profiles and measurements into states. A profile is a specific configuration of state sequences over time shared by a group of subjects. The state assigned to an observation determines its likelihood parameters. In this example, there are 3 profiles and 6 states. As suggested by the figure, multiple subjects may be associated with the same profile, states can be shared across profiles, states are likely to persist over time and the probability of state changes is shared across measurements, within a profile.
• Figure 2: Directed graphical representation of our MMTB model. The likelihood of observations $Y_{i,r,t}$ at time $t = 1, \dots, T$ depends on the profile $s_i$ of subject $i = 1, \dots, N$. When $s_i = z$ and the state $c_{r,t}^{(z)}$ of measurement $r$ equals $k$, then $Y_{i,r,t}$ has a distribution $F_{\boldsymbol{\theta}_k}$, for $r \in 1, \dots, R$ and $k \in \{1, \dots, K\}$. $\boldsymbol{\theta}_0$ parameterize the hyperprior on $\boldsymbol{\theta}_k$. Profiles $s_i$ are sampled from a categorical distribution with probabilities $\boldsymbol{\pi} = (\pi_1, \dots, \pi_Z)$. For subjects with profile $s_i = z$, the state of measurement $r$ at time $t$ is the same as at $t-1$ when the persistence indicator $\gamma_{r,t}^{(z)}$ equals $1$, which has probability $a_t^{(z)}$. Otherwise, the state is re-sampled from a categorical distribution with probabilities $\boldsymbol{\omega}^{(z)}= (\omega^{(z)}_{1}, \dots, \omega^{(z)}_{K})$. Yellow, diamond-shaped nodes denote hyperparameters with pre-specified values.
• Figure 3: Performance of MCMC with blocked versus marginal updates of state and persistence indicator sequences. Subplots show average and 90% credible intervals across 30 simulated datasets featuring both time and subject dependence. Simulation details are detailed in Sec. \ref{['sec:simulations']} (third simulation scenario). Lower values of binder loss (BL) and mean absolute error (MAE) are preferred; higher values of log-likelihood and f-measure are preferred.
• Figure 4: fMRI data after pre-processing for two subjects and all selected brain regions of interest (ROIs). The vertical axis plots BOLD signal averaged across voxels in a ROI over a 10-second window. Vertical dashed lines demark five activation blocks where subjects squeeze a ball.
• Figure 5: fMRI data: Estimated probability that any pair of subjects be assigned to the same profile. Subjects are sorted by estimated profile memberships, as marked with orange blocks.