  We develop a novel exploratory tool for non-Euclidean object data based on
data depth, extending the celebrated Tukey's depth for Euclidean data. The
proposed metric halfspace depth, applicable to data objects in a general metric
space, assigns to data points depth values that characterize the centrality of
these points with respect to the distribution and provides an interpretable
center-outward ranking. Desirable theoretical properties that generalize
standard depth properties postulated for Euclidean data are established for the
metric halfspace depth. The depth median, defined as the deepest point, is
shown to have high robustness as a location descriptor both in theory and in
simulation. We propose an efficient algorithm to approximate the metric
halfspace depth and illustrate its ability to adapt to the intrinsic data
geometry. The metric halfspace depth was applied to an Alzheimer's disease
study, revealing group differences in the brain connectivity, modeled as
covariance matrices, for subjects in different stages of dementia. Based on
phylogenetic trees of 7 pathogenic parasites, our proposed metric halfspace
depth was also used to construct a meaningful consensus estimate of the
evolutionary history and to identify potential outlier trees.
