Switching State-Space Models

We introduce a statistical model for times series data with nonlinear dynamics which iteratively segments the data into regimes with approximately linear dynamics and learns the parameters of each of those regimes. This model combines and generalizes two of the most widely used stochastic time series models|the hidden Markov model and the linear dynamical system|and is related to models that are widely used in the control and econometrics literatures. It can also be derived by extending the mixture of experts neural network model (Jacobs et al., 1991) to its fully dynamical version, in which both expert and gating networks are recurrent. Inferring the posterior probabilities of the hidden states of this model is computationally intractable, and therefore the exact Expectation Maximization (EM) alogithm cannot be applied. However, we present a variational approximation which maximizes a lower bound on the log likelihood and makes use of both the forward{backward recursions for hidden Markov models and the Kalman lter recursions for linear dynamical systems.