  In this article we discuss some of the consequences of the mixed membership
perspective on time series analysis. In its most abstract form, a mixed
membership model aims to associate an individual entity with some set of
attributes based on a collection of observed data. Although much of the
literature on mixed membership models considers the setting in which
exchangeable collections of data are associated with each member of a set of
entities, it is equally natural to consider problems in which an entire time
series is viewed as an entity and the goal is to characterize the time series
in terms of a set of underlying dynamic attributes or "dynamic regimes".
Indeed, this perspective is already present in the classical hidden Markov
model, where the dynamic regimes are referred to as "states", and the
collection of states realized in a sample path of the underlying process can be
viewed as a mixed membership characterization of the observed time series. Our
goal here is to review some of the richer modeling possibilities for time
series that are provided by recent developments in the mixed membership
framework.
