We consider the 2-Wasserstein space of probability measures supported on the
unit-circle, and propose a framework for Principal Component Analysis (PCA) for
data living in such a space. We build on a detailed investigation of the
optimal transportation problem for measures on the unit-circle which might be
of independent interest. In particular, we derive an expression for optimal
transport maps in (almost) closed form and propose an alternative definition of
the tangent space at an absolutely continuous probability measure, together
with the associated exponential and logarithmic maps. PCA is performed by
mapping data on the tangent space at the Wasserstein barycentre, which we
approximate via an iterative scheme, and for which we establish a sufficient a
posteriori condition to assess its convergence. Our methodology is illustrated
on several simulated scenarios and a real data analysis of measurements of
optical nerve thickness