  We present a novel class of projected methods, to perform statistical
analysis on a data set of probability distributions on the real line, with the
2-Wasserstein metric. We focus in particular on Principal Component Analysis
(PCA) and regression. To define these models, we exploit a representation of
the Wasserstein space closely related to its weak Riemannian structure, by
mapping the data to a suitable linear space and using a metric projection
operator to constrain the results in the Wasserstein space. By carefully
choosing the tangent point, we are able to derive fast empirical methods,
exploiting a constrained B-spline approximation. As a byproduct of our
approach, we are also able to derive faster routines for previous work on PCA
for distributions. By means of simulation studies, we compare our approaches to
previously proposed methods, showing that our projected PCA has similar
performance for a fraction of the computational cost and that the projected
regression is extremely flexible even under misspecification. Several
theoretical properties of the models are investigated and asymptotic
consistency is proven. Two real world applications to Covid-19 mortality in the
US and wind speed forecasting are discussed.
