  This article provides an overview on the statistical modeling of complex data
as increasingly encountered in modern data analysis. It is argued that such
data can often be described as elements of a metric space that satisfies
certain structural conditions and features a probability measure. We refer to
the random elements of such spaces as random objects and to the emerging field
that deals with their statistical analysis as metric statistics. Metric
statistics provides methodology, theory and visualization tools for the
statistical description, quantification of variation, centrality and quantiles,
regression and inference for populations of random objects, inferring these
quantities from available data and samples. In addition to a brief review of
current concepts, we focus on distance profiles as a major tool for object data
in conjunction with the pairwise Wasserstein transports of the underlying
one-dimensional distance distributions. These pairwise transports lead to the
definition of intuitive and interpretable notions of transport ranks and
transport quantiles as well as two-sample inference. An associated profile
metric complements the original metric of the object space and may reveal
important features of the object data in data analysis. We demonstrate these
tools for the analysis of complex data through various examples and
visualizations.
