On Exact Computation of Tukey Depth Central Regions

The Tukey (or halfspace) depth extends nonparametric methods toward multivariate data. The multivariate analogues of the quantiles are the central regions of the Tukey depth, defined as sets of points in the d-dimensional space whose Tukey depth exceeds given thresholds k. We address the problem of fast and exact computation of those central regions. First, we analyze an efficient Algorithm (A) from Liu, Mosler, and Mozharovskyi, and prove that it yields exact results in dimension d = 2, or for a low threshold k in arbitrary dimension. We provide examples where Algorithm (A) fails to recover the exact Tukey depth region for d > 2, and propose a modification that is guaranteed to be exact. We express the problem of computing the exact central region in its dual formulation, and use that viewpoint to demonstrate that further substantial improvements to our algorithm are unlikely. An efficient C++ implementation of our exact algorithm is freely available in the R package TukeyRegion.图基（Tukey）深度（亦称半空间深度）将非参数统计方法推广至多变量数据场景。分位数的多变量对应形式为图基深度中心区域，其被定义为d维空间中Tukey深度超出给定阈值k的点集。本文针对上述中心区域的快速精确计算问题展开研究。首先，本文分析了Liu、Mosler与Mozharovskyi提出的高效算法（算法A），并证明该算法在维度d=2时，或在任意维度下针对低阈值k的场景中，均可得到精确计算结果。本文给出了算法A在维度d>2时无法还原精确图基深度区域的示例，并提出了一种可保证计算精确性的改进方案。本文将精确中心区域的计算问题转化为对偶形式，并基于该视角证明，对本文算法进行进一步的实质性改进存在较大难度。本文精确算法的高效C++实现可通过R包TukeyRegion免费获取。