  Intensive longitudinal data (ILD) collected in mobile health (mHealth)
studies contain rich information on multiple outcomes measured frequently over
time that have the potential to capture short-term and long-term dynamics.
Motivated by an mHealth study of smoking cessation in which participants
self-report the intensity of many emotions multiple times per day, we describe
a dynamic factor model that summarizes the ILD as a low-dimensional,
interpretable latent process. This model consists of two submodels: (i) a
measurement submodel--a factor model--that summarizes the multivariate
longitudinal outcome as lower-dimensional latent variables and (ii) a
structural submodel--an Ornstein-Uhlenbeck (OU) stochastic process--that
captures the temporal dynamics of the multivariate latent process in continuous
time. We derive a closed-form likelihood for the marginal distribution of the
outcome and the computationally-simpler sparse precision matrix for the OU
process. We propose a block coordinate descent algorithm for estimation.
Finally, we apply our method to the mHealth data to summarize the dynamics of
18 different emotions as two latent processes. These latent processes are
interpreted by behavioral scientists as the psychological constructs of
positive and negative affect and are key in understanding vulnerability to
lapsing back to tobacco use among smokers attempting to quit.
