Halfspace Depth

The halfspace depth (HD, also called Tukey depth) is a nonparametric tool of multivariate statistics that has found many applications in exploratory analysis, estimation, and statistical testing. For a dataset, it can be defined as the minimum portion of data points in the ‐space that can be cut off by a hyperplane passing through a given point; in a probabilistic setting, it is defined as the smallest probability of a closed halfspace containing a given point. The HD can be seen as a generalization of quantiles to multivariate data based on a projection pursuit principle. It naturally induces a robust multivariate median, and the contours of the HD function give information about the geometry of the underlying dataset/probability distribution. We provide a high‐level overview of the most important theoretical properties and applications of the HD for multivariate data. We also outline several extensions of the HD to additional settings, such as halfspace‐like depths for directional, object, or functional data. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Nonparametric Methods Statistical and Graphical Methods of Data Analysis > Robust Methods