  Multi-task learning leverages shared information among data sets to improve
the learning performance of individual tasks. The paper applies this framework
for data where each task is a phase-shifted periodic time series. In
particular, we develop a novel Bayesian nonparametric model capturing a mixture
of Gaussian processes where each task is a sum of a group-specific function and
a component capturing individual variation, in addition to each task being
phase shifted. We develop an efficient \textsc{em} algorithm to learn the
parameters of the model. As a special case we obtain the Gaussian mixture model
and \textsc{em} algorithm for phased-shifted periodic time series. Furthermore,
we extend the proposed model by using a Dirichlet Process prior and thereby
leading to an infinite mixture model that is capable of doing automatic model
selection. A Variational Bayesian approach is developed for inference in this
model. Experiments in regression, classification and class discovery
demonstrate the performance of the proposed models using both synthetic data
and real-world time series data from astrophysics. Our methods are particularly
useful when the time series are sparsely and non-synchronously sampled.
