  A model involving Gaussian processes (GPs) is introduced to simultaneously
handle multi-task learning, clustering, and prediction for multiple functional
data. This procedure acts as a model-based clustering method for functional
data as well as a learning step for subsequent predictions for new tasks. The
model is instantiated as a mixture of multi-task GPs with common mean
processes. A variational EM algorithm is derived for dealing with the
optimisation of the hyper-parameters along with the hyper-posteriors'
estimation of latent variables and processes. We establish explicit formulas
for integrating the mean processes and the latent clustering variables within a
predictive distribution, accounting for uncertainty on both aspects. This
distribution is defined as a mixture of cluster-specific GP predictions, which
enhances the performances when dealing with group-structured data. The model
handles irregular grid of observations and offers different hypotheses on the
covariance structure for sharing additional information across tasks. The
performances on both clustering and prediction tasks are assessed through
various simulated scenarios and real datasets. The overall algorithm, called
MagmaClust, is publicly available as an R package.
