  This paper introduces a linear state-space model with time-varying dynamics.
The time dependency is obtained by forming the state dynamics matrix as a
time-varying linear combination of a set of matrices. The time dependency of
the weights in the linear combination is modelled by another linear Gaussian
dynamical model allowing the model to learn how the dynamics of the process
changes. Previous approaches have used switching models which have a small set
of possible state dynamics matrices and the model selects one of those matrices
at each time, thus jumping between them. Our model forms the dynamics as a
linear combination and the changes can be smooth and more continuous. The model
is motivated by physical processes which are described by linear partial
differential equations whose parameters vary in time. An example of such a
process could be a temperature field whose evolution is driven by a varying
wind direction. The posterior inference is performed using variational Bayesian
approximation. The experiments on stochastic advection-diffusion processes and
real-world weather processes show that the model with time-varying dynamics can
outperform previously introduced approaches.

Machine Learning