Projected Statistical Methods for Distributional Data on the Real Line with the Wasserstein Metric

We present a novel class of projected methods, to perform statistical\nanalysis on a data set of probability distributions on the real line, with the\n2-Wasserstein metric. We focus in particular on Principal Component Analysis\n(PCA) and regression. To define these models, we exploit a representation of\nthe Wasserstein space closely related to its weak Riemannian structure, by\nmapping the data to a suitable linear space and using a metric projection\noperator to constrain the results in the Wasserstein space. By carefully\nchoosing the tangent point, we are able to derive fast empirical methods,\nexploiting a constrained B-spline approximation. As a byproduct of our\napproach, we are also able to derive faster routines for previous work on PCA\nfor distributions. By means of simulation studies, we compare our approaches to\npreviously proposed methods, showing that our projected PCA has similar\nperformance for a fraction of the computational cost and that the projected\nregression is extremely flexible even under misspecification. Several\ntheoretical properties of the models are investigated and asymptotic\nconsistency is proven. Two real world applications to Covid-19 mortality in the\nUS and wind speed forecasting are discussed.\n