  Many time series can be modeled as a sequence of segments representing
high-level discrete states, such as running and walking in a human activity
application. Flexible models should describe the system state and observations
in stationary "pure-state" periods as well as transition periods between
adjacent segments, such as a gradual slowdown between running and walking.
However, most prior work assumes instantaneous transitions between pure
discrete states. We propose a dynamical Wasserstein barycentric (DWB) model
that estimates the system state over time as well as the data-generating
distributions of pure states in an unsupervised manner. Our model assumes each
pure state generates data from a multivariate normal distribution, and
characterizes transitions between states via displacement-interpolation
specified by the Wasserstein barycenter. The system state is represented by a
barycentric weight vector which evolves over time via a random walk on the
simplex. Parameter learning leverages the natural Riemannian geometry of
Gaussian distributions under the Wasserstein distance, which leads to improved
convergence speeds. Experiments on several human activity datasets show that
our proposed DWB model accurately learns the generating distribution of pure
states while improving state estimation for transition periods compared to the
commonly used linear interpolation mixture models.
