  We consider probabilistic models for sequential observations which exhibit
gradual transitions among a finite number of states. We are particularly
motivated by applications such as human activity analysis where observed
accelerometer time series contains segments representing distinct activities,
which we call pure states, as well as periods characterized by continuous
transition among these pure states. To capture this transitory behavior, the
dynamical Wasserstein barycenter (DWB) model of Cheng et al. in 2021 [1]
associates with each pure state a data-generating distribution and models the
continuous transitions among these states as a Wasserstein barycenter of these
distributions with dynamically evolving weights. Focusing on the univariate
case where Wasserstein distances and barycenters can be computed in closed
form, we extend [1] specifically relaxing the parameterization of the pure
states as Gaussian distributions. We highlight issues related to the uniqueness
in identifying the model parameters as well as uncertainties induced when
estimating a dynamically evolving distribution from a limited number of
samples. To ameliorate non-uniqueness, we introduce regularization that imposes
temporal smoothness on the dynamics of the barycentric weights. A
quantile-based approximation of the pure state distributions yields a finite
dimensional estimation problem which we numerically solve using cyclic descent
alternating between updates to the pure-state quantile functions and the
barycentric weights. We demonstrate the utility of the proposed algorithm in
segmenting both simulated and real world human activity time series.
