  We propose a novel measure of statistical depth, the metric spatial depth,
for data residing in an arbitrary metric space. The measure assigns high (low)
values for points located near (far away from) the bulk of the data
distribution, allowing quantifying their centrality/outlyingness. This depth
measure is shown to have highly interpretable geometric properties, making it
appealing in object data analysis where standard descriptive statistics are
difficult to compute. The proposed measure reduces to the classical spatial
depth in a Euclidean space. In addition to studying its theoretical properties,
to provide intuition on the concept, we explicitly compute metric spatial
depths in several different metric spaces. Finally, we showcase the practical
usefulness of the metric spatial depth in outlier detection, non-convex depth
region estimation and classification.
