Approximate inference in switching linear dynamical systems using Gaussian mixtures

Abstract

© Cambridge University Press 2011. The linear dynamical system (LDS) (see Section 1.3.2) is a standard time series model in which a latent linear process generates the observations. Complex time series which are not well described globally by a single LDS may be divided into segments, each modelled by a potentially different LDS. Such models can handle situations in which the underlying model ‘jumps’ from one parameter setting to another. For example a single LDS might well represent the normal flows in a chemical plant. However, if there is a break in a pipeline, the dynamics of the system changes from one set of linear flow equations to another. This scenario could be modelled by two sets of linear systems, each with different parameters, with a discrete latent variable at each time s t ∈ {normal, pipe broken} indicating which of the LDSs is most appropriate at the current time. This is called a switching LDS (SLDS) and used in many disciplines, from econometrics to machine learning [2, 9, 15, 13, 12, 6, 5, 19, 21, 16]. The switching linear dynamical system At each time t, a switch variable s t ∈ {1, …, S} describes which of a set of LDSs is to be used. The observation (or ‘visible’) variable v t ∈ R V is linearly related to the hidden state h t ∈ R H by Here s t describes which of the set of emission matrices B(1), …, B(S) is active at time t.