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.

翻译：我们提出了一种新的统计深度度量——度量空间深度，适用于任意度量空间中的数据。该度量对位于数据分布主体附近（远离）的点赋予高（低）值，从而能够量化其中心性/异常性。研究表明，这一深度度量具有高度可解释的几何性质，使其在标准描述性统计难以计算的对象数据分析中具有吸引力。所提出的度量在欧几里得空间中可退化为经典的空间深度。除了研究其理论性质外，为提供直观理解，我们显式计算了多个不同度量空间中的度量空间深度。最后，我们展示了度量空间深度在异常检测、非凸深度区域估计和分类中的实际应用价值。