  Modeling the temporal behavior of data is of primordial importance in many
scientific and engineering fields. Baseline methods assume that both the
dynamic and observation equations follow linear-Gaussian models. However, there
are many real-world processes that cannot be characterized by a single linear
behavior. Alternatively, it is possible to consider a piecewise-linear model
which, combined with a switching mechanism, is well suited when several modes
of behavior are needed. Nevertheless, switching dynamical systems are
intractable because of their computational complexity increases exponentially
with time. In this paper, we propose a variational approximation of piecewise
linear dynamical systems. We provide full details of the derivation of two
variational expectation-maximization algorithms, a filter and a smoother. We
show that the model parameters can be split into two sets, static and dynamic
parameters, and that the former parameters can be estimated off-line together
with the number of linear modes, or the number of states of the switching
variable. We apply the proposed method to a visual tracking problem, namely
head-pose tracking, and we thoroughly compare our algorithm with several state
of the art trackers.
