  The study of dependence between random variables is the core of theoretical
and applied statistics. Static and dynamic copula models are useful for
describing the dependence structure, which is fully encrypted in the copula
probability density function. However, these models are not always able to
describe the temporal change of the dependence patterns, which is a key
characteristic of financial data. We propose a novel nonparametric framework
for modelling a time series of copula probability density functions, which
allows to forecast the entire function without the need of post-processing
procedures to grant positiveness and unit integral. We exploit a suitable
isometry that allows to transfer the analysis in a subset of the space of
square integrable functions, where we build on nonparametric functional data
analysis techniques to perform the analysis. The framework does not assume the
densities to belong to any parametric family and it can be successfully applied
also to general multivariate probability density functions with bounded or
unbounded support. Finally, a noteworthy field of application pertains the
study of time varying networks represented through vine copula models. We apply
the proposed methodology for estimating and forecasting the time varying
dependence structure between the S\&P500 and NASDAQ indices.
