Gaussian Process Forecast with multidimensional distributional entries - Francois Bachoc; Alexandra Suvorikova; Jean-Michel Loubes; Vladimir Spokoiny (Humboldt-Universität zu Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series", 2018)
In this work, we propose to define Gaussian Processes indexed by multidimensional distributions. In the framework where the distributions can be modeled as i.i.d realizations of a measure on the set of distributions, we prove that the kernel defined as the quadratic distance between the transportation maps, that transport each distribution to the barycenter of the distributions, provides a valid covariance function. In this framework, we study the asymptotic properties of this process, proving micro ergodicity of the parameters.
Keywords: C00; 330; Gaussian Process; Kernel methods; Wasserstein Distance