  In this paper we propose a flexible class of multivariate nonlinear
non-Gaussian state space models, based on copulas. More precisely, we assume
that the observation equation and the state equation are defined by copula
families that are not necessarily equal. For each time point, the resulting
model can be described by a C-vine copula truncated after the first tree, where
the root node is represented by the latent state. Inference is performed within
the Bayesian framework, using the Hamiltonian Monte Carlo method, where a
further D-vine truncated after the first tree is used as prior distribution to
capture the temporal dependence in the latent states. Simulation studies show
that the proposed copula-based approach is extremely flexible, since it is able
to describe a wide range of dependence structures and, at the same time, allows
us to deal with missing data. The application to atmospheric pollutant
measurement data shows that our approach is suitable for accurate modeling and
prediction of data dynamics in the presence of missing values. Comparison to a
Gaussian linear state space model and to Bayesian additive regression trees
shows the superior performance of the proposed model with respect to predictive
accuracy.
