Depth measures quantify central tendency in the analysis of statistical and geometric data. Selecting a depth measure that is simple and efficiently computable is often important, e.g., when calculating depth for multiple query points or when applied to large sets of data. In this work, we introduce \emph{Hyperplane Distance Depth (HDD)}, which measures the centrality of a query point $q$ relative to a given set $P$ of $n$ points in $\mathbb{R}^d$, defined as the sum of the distances from $q$ to all $\binom{n}{d}$ hyperplanes determined by points in $P$. We present algorithms for calculating the HDD of an arbitrary query point $q$ relative to $P$ in $O(d \log n)$ time after preprocessing $P$, and for finding a median point of $P$ in $O(d n^{d^2} \log n)$ time. We study various properties of hyperplane distance depth and show that it is convex, symmetric, and vanishing at infinity.

翻译：深度测度在统计与几何数据分析中用于量化中心趋势。选择一种计算简单且高效的深度测度通常至关重要，例如在计算多个查询点的深度或应用于大规模数据集时。本文提出一种新的深度测度——\emph{超平面距离深度（HDD）}，用于衡量查询点 $q$ 相对于 $\mathbb{R}^d$ 空间中给定点集 $P$（包含 $n$ 个点）的中心性，其定义为 $q$ 到由 $P$ 中所有 $\binom{n}{d}$ 个点确定的超平面距离之和。我们提出了在预处理 $P$ 后，以 $O(d \log n)$ 时间复杂度计算任意查询点 $q$ 相对于 $P$ 的 HDD 的算法，以及以 $O(d n^{d^2} \log n)$ 时间复杂度寻找 $P$ 的中位点的算法。我们研究了超平面距离深度的多种性质，证明其具有凸性、对称性以及在无穷远处趋于零的特性。