  We seek to model a collection of time series arising from multiple entities
interacting over the same time period. Recent work focused on modeling
individual time series is inadequate for our intended applications, where
collective system-level behavior influences the trajectories of individual
entities. To address such problems, we present a new hierarchical
switching-state model that can be trained in an unsupervised fashion to
simultaneously explain both system-level and individual-level dynamics. We
employ a latent system-level discrete state Markov chain that drives latent
entity-level chains which in turn govern the dynamics of each observed time
series. Feedback from the observations to the chains at both the entity and
system levels improves flexibility via context-dependent state transitions. Our
hierarchical switching recurrent dynamical models can be learned via
closed-form variational coordinate ascent updates to all latent chains that
scale linearly in the number of individual time series. This is asymptotically
no more costly than fitting separate models for each entity. Experiments on
synthetic and real datasets show that our model can produce better forecasts of
future entity behavior than existing methods. Moreover, the availability of
latent state chains at both the entity and system level enables interpretation
of group dynamics.
