  Graphical models have been used extensively for modeling brain connectivity
networks. However, unmeasured confounders and correlations among measurements
are often overlooked during model fitting, which may lead to spurious
scientific discoveries. Motivated by functional magnetic resonance imaging
(fMRI) studies, we propose a novel method for constructing brain connectivity
networks with correlated replicates and latent effects. In a typical fMRI
study, each participant is scanned and fMRI measurements are collected across a
period of time. In many cases, subjects may have different states of mind that
cannot be measured during the brain scan: for instance, some subjects may be
awake during the first half of the brain scan, and may fall asleep during the
second half of the brain scan. To model the correlation among replicates and
latent effects induced by the different states of mind, we assume that the
correlated replicates within each independent subject follow a one-lag vector
autoregressive model, and that the latent effects induced by the unmeasured
confounders are piecewise constant. The proposed method results in a convex
optimization problem which we solve using a block coordinate descent algorithm.
Theoretical guarantees are established for parameter estimation. We demonstrate
via extensive numerical studies that our method is able to estimate latent
variable graphical models with correlated replicates more accurately than
existing methods.
