3 research outputs found
Wasserstein Regression
The analysis of samples of random objects that do not lie in a vector space
is gaining increasing attention in statistics. An important class of such
object data is univariate probability measures defined on the real line.
Adopting the Wasserstein metric, we develop a class of regression models for
such data, where random distributions serve as predictors and the responses are
either also distributions or scalars. To define this regression model, we
utilize the geometry of tangent bundles of the space of random measures endowed
with the Wasserstein metric for mapping distributions to tangent spaces. The
proposed distribution-to-distribution regression model provides an extension of
multivariate linear regression for Euclidean data and function-to-function
regression for Hilbert space valued data in functional data analysis. In
simulations, it performs better than an alternative transformation approach
where one maps distributions to a Hilbert space through the log quantile
density transformation and then applies traditional functional regression. We
derive asymptotic rates of convergence for the estimator of the regression
operator and for predicted distributions and also study an extension to
autoregressive models for distribution-valued time series. The proposed methods
are illustrated with data on human mortality and distributional time series of
house prices