Michael I. Jordan Computer Science Division and Department of Statistics, University of California, Berkeley, CA 94720, USA

In this chapter 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. For example, a person (entity) can be associated with various defining characteristics (attributes) based on observed pairwise interactions with other people (data). Likewise, one can describe a document (entity) as comprised of a set of topics (attributes) based on the observed words in the document (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 (Rabiner, 1989) and switching state-space model (Kim, 1994), 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 model-

Membership Models and

ing possibilities for time series that are provided by recent developments in the mixed membership framework.