Spatial Statistics for Distributional Data in Bayes Spaces: From Object-Oriented Kriging to the Analysis of Warping Functions

In the presence of increasingly massive and heterogeneous spatial data, geostatistical modeling of distributional observations plays a key role. Choosing the “right” embedding space for these data is of paramount importance for their statistical processing, to account for their nature and inherent constraints. The Bayes space theory is a natural embedding space for (spatial) distributional data and was successfully applied in varied settings. The aim of this work is to review the state-of-the-art methods for spatial dependence modeling and prediction of distributional data, while shedding light on the strong links between Compositional Data Analysis, Functional Data Analysis, and, more generally, Object-Oriented Data Analysis, in the context of spatial statistics. We propose extensions of these methods to the multivariate setting, and discuss the applicability of the Bayes space approach to the spatial modeling of phase variability in Functional Data Analysis.